September 2001

NASA/TM-2001-211046

On Multifunctional Collaborative Methods

in Engineering Science

Jonathan B. Ransom

Langley Research Center, Hampton, Virginia
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September 2001

NASA/TM-2001-211046

On Multifunctional Collaborative Methods

in Engineering Science

Jonathan B. Ransom

Langley Research Center, Hampton, Virginia
Available from:

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)

7121 Standard Drive 5285 Port Royal Road

Hanover, MD 21076-1320 Springfield, VA 22161-2171

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iii

ABSTRACT

Engineers are challenged to produce better designs in less time and for less cost.

Hence, to investigate novel and revolutionary design concepts, accurate, high-fidelity

data must be assimilated rapidly into the design, analysis and simulation process. This

data assimilation should consider diverse mathematical modeling and multi-discipline

interactions necessitated by concepts exploiting advanced materials and structures.

Integrated high-fidelity methods with diverse engineering applications provide the

enabling technologies to assimilate these high-fidelity, multi-disciplinary data rapidly at

an early stage in the design. These integrated methods must be multifunctional,

collaborative and applicable to the general field of engineering science and mechanics.

Multifunctional methodologies and analysis procedures are formulated for

interfacing diverse domain idealizations including multi-fidelity modeling methods and

multi-discipline analysis methods. These methods, based on the method of weighted

residuals, ensure accurate compatibility of primary and secondary variables across the

domain interfaces. Methods are developed for scalar-field and vector-field problems in

engineering science with extensions to multidisciplinary problems. Results are presented

for the scalar- and vector-field developments using example patch test problems. In

addition, results for torsion, thermal, and potential flow problems are presented to

demonstrate further the effectiveness of the scalar-field development. Results for plane

stress and plane flow problems are presented for the vector-field development. Results

for all problems presented are in overall good agreement with the exact analytical

solution or the reference numerical solution.
iv

The multifunctional methodology presented provides an effective mechanism by

which domains with diverse idealizations are interfaced. This capability rapidly provides

the high-fidelity data needed in the early design phase. Moreover, the capability is

applicable to the general field of engineering science and mechanics. Hence, it provides

a collaborative capability that accounts for interactions among engineering analysis

methods.
v

TABLE OF CONTENTS

Page

LIST OF TABLES ............................................................................................................vii

LIST OF FIGURES .........................................................................................................viii

Chapter

I. INTRODUCTION ............................................................................................... 1

1.1. MOTIVATION .................................................................................... 1

1.2. CONTINUUM MECHANICS FOUNDATIONS ............................... 7

1.3. LITERATURE REVIEW FOR COLLABORATIVE METHODS... 15

1.4. OBJECTIVES AND SCOPE............................................................. 18

II. MULTIFUNCTIONAL APPROACH FOR SCALAR-FIELD PROBLEMS . 21

2.1. GENERAL......................................................................................... 21

1.2. DISCIPLINE SPECIFICS.................................................................. 22

1.3. SINGLE-DOMAIN FORMULATION.............................................. 27

1.4. MULTIPLE-DOMAIN FORMULATION ........................................ 32

1.5. SPATIAL MODELING FOR MULTIPLE DOMAINS.................... 43

1.6. COMPUTATIONAL IMPLICATIONS ............................................ 67

1.7. VERIFICATION TEST CASE.......................................................... 71

III. MULTIFUNCTIONAL APPROACH FOR VECTOR-FIELD PROBLEMS 76

3.1. GENERAL......................................................................................... 76

1.2. CONTINUUM MECHANICS FOUNDATIONS ............................. 77

1.3. DISCIPLINE SPECIFICS.................................................................. 84

1.4. SINGLE-DOMAIN FORMULATION.............................................. 93

1.5. MULTIPLE-DOMAIN FORMULATION ........................................ 99

1.6. SPATIAL MODELING FOR MULTIPLE DOMAINS.................. 106

1.7. COMPUTATIONAL IMPLICATIONS .......................................... 138

1.8. VERIFICATION TEST CASE........................................................ 142

IV. REPRESENTATIVE SCALAR-FIELD APPLICATIONS ......................... 148

4.1. GENERAL....................................................................................... 148

4.2. TORSION OF PRISMATIC BAR................................................... 148

1.3. HEAT CONDUCTION PROBLEM................................................ 158

1.4. POTENTIAL FLOW PROBLEM ................................................... 165

1.5. SUMMARY..................................................................................... 172
vi

Page

Chapter

V. REPRESENTATIVE VECTOR-FIELD APPLICATIONS AND

EXTENSIONS .............................................................................................. 174

5.1. GENERAL....................................................................................... 174

5.2. PLANE STRESS PROBLEM.......................................................... 174

1.3. PLANE FLOW PROBLEM............................................................. 186

1.4. EXTENSIONS TO MULTIPLE DISCIPLINES............................. 195

1.5. SUMMARY..................................................................................... 201

VI. CONCLUSIONS AND RECOMMENDATIONS ....................................... 202

6.1. GENERAL....................................................................................... 202

6.2. CONCLUSIONS.............................................................................. 202

6.3. RECOMMENDATIONS FOR FUTURE WORK .......................... 207

APPENDICES

A. OVERVIEW OF STEPS IN ANALYSIS AND SIMULATION ................. 209

B. CUBIC SPLINE INTERPOLATION MATRICES...................................... 216

C. DERIVATION OF INTERFACE GEOMETRY.......................................... 222

C.1. GENERAL ..................................................................................... 222

C.2. GEOMETRY REPRESENTATION.............................................. 223

REFERENCES................................................................................................................ 227
vii

LIST OF TABLES

Table Page

Table 2.1. Shape Functions for a Nine-Node Quadrilateral Finite Element. .................. 55

Table 2.2. Results of the Multifunctional Approach for the Patch Test Problems. ........ 75

Table 3.1. Results of the Multifunctional Approach for the Cantilevered Plate........... 147

Table 4.1. Normalized Twisting Moment for the Prismatic Bar................................... 155

Table 4.2. Normalized Maximum Shear for the Prismatic Bar..................................... 158
viii

LIST OF FIGURES

Figure Page

Figure 2.1. Geometric Representation of Two-Dimensional Domain. ............................ 22

Figure 2.2. Geometric Configuration of Prismatic Bar.................................................... 23

Figure 2.3. Boundary Definitions for Two-Dimensional Subdomains............................ 34

Figure 2.4. One-Dimensional Finite Difference Element Configuration......................... 50

Figure 2.5. Two-Dimensional Finite Difference Element Configuration........................ 52

Figure 2.6. Interface Definition........................................................................................ 60

Figure 2.7. Two-Dimensional Rectangular Domain........................................................ 72

Figure 2.8. Spatial Discretization for Two-Dimensional Rectangular Domain............... 74

Figure 3.1. Analysis Domain and Boundary Conditions of Cantilevered Plate............. 144

Figure 3.2. Multiple-Domain Discretization of Cantilevered Plate. ............................... 144

Figure 3.3. Central Difference Template Applied at a Corner........................................ 145

Figure 4.1. Prismatic Bar with Rectangular Cross-Section. .......................................... 149

Figure 4.2. Analysis Domain and Boundary Conditions for Prismatic Bar with

Rectangular Cross-Section........................................................................... 150

Figure 4.3. Analysis Domain, Boundary Conditions and Typical Mesh for One Quadrant

of Prismatic Bar with Rectangular Cross-Section. ...................................... 152

Figure 4.4. Multiple-Domain (11 × 11)/(21 × 21) Idealization...................................... 153

Figure 4.5. Analysis Domain and Boundary Conditions for the Steady-State Heat

Conduction in a Square Plate....................................................................... 160

Figure 4.6. Homogeneous (3 × 5)/(2 × 3) Idealization. ................................................. 161

Figure 4.7. Temperature Distribution Along Insulated Edge of Square Plate. .............. 164

Figure 4.8. Spatial Discretization for Inclined Interface for Square Plate..................... 165
ix

Figure Page

Figure 4.9. Temperature Distribution Along Insulated Edge of Square Plate with Inclined

Interface. ...................................................................................................... 165

Figure 4.10. Domain of Flow Around Cylinder.............................................................. 166

Figure 4.11. Analysis Domain of Flow Around Cylinder............................................... 167

Figure 4.12. Spatial Discretization for One Quadrant of Domain of Flow Around

Cylinder....................................................................................................... 168

Figure 4.13. Contour Plot of Velocity Potential for Flow Around Cylinder. ................. 169

Figure 4.14. Contour Plot of Horizontal Velocity Component for Flow Around

Cylinder....................................................................................................... 169

Figure 4.15. Contour Plot of Transverse Velocity Component for Flow Around

Cylinder....................................................................................................... 170

Figure 4.16. Tangential Velocity for Flow Around Cylinder. ........................................ 172

Figure 5.1. Domain of Plate with Central Circular Cutout. ............................................ 176

Figure 5.2. Geometric Configuration for One Quadrant of Plate with Central Circular

Cutout. .......................................................................................................... 177

Figure 5.3. Finite Element Models for One Quadrant of Infinite Plate with Central

Circular Cutout. ............................................................................................ 177

Figure 5.4. Longitudinal Stress Distribution along Midwidth and Midlength for Infinite

Plate with Central Circular Cutout. .............................................................. 179

Figure 5.5. Displacement Magnitude Distribution for Infinite Plate with Central

Cutout. .......................................................................................................... 180

Figure 5.6. Longitudinal Stress Resultant Distribution for Infinite Plate with Central

Cutout. .......................................................................................................... 180

Figure 5.7. Finite Element Models for One Quadrant of Finite-Width Plate with Central

Cutout. .......................................................................................................... 182

Figure 5.8. Convergence of Longitudinal Stress Distribution along Midlength for FiniteWidth Plate with Central Circular Cutout. ................................................... 185
x

Figure Page

Figure 5.9. Longitudinal Stress Distribution along Midwidth and Midlength for FiniteWidth Plate with Central Circular Cutout................................................... 185

Figure 5.10. Displacement Magnitude Distribution for Finite-Width Plate with Central

Circular Cutout............................................................................................ 186

Figure 5.11. Longitudinal Stress Resultant Distribution for Finite-Width Plate with

Central Circular Cutout............................................................................... 186

Figure 5.12. Geometric Configuration for Fluid Squeezed Between Parallel Plates...... 188

Figure 5.13. Finite Element Models for Fluid Squeezed Between Two Parallel Plates.188

Figure 5.14. Horizontal Velocity for the Flow Between Two Parallel Plates................. 192

Figure 5.15. Pressure Distribution Near Centerline for the Flow Between Two Parallel

Plates. .......................................................................................................... 193

Figure 5.16. Shear Stress Distribution Near Plate Boundary for the Flow Between Two

Parallel Plates.............................................................................................. 193

Figure 5.17. Longitudinal Stress Distribution Near Plate Boundary for the Flow Between

Two Parallel Plates...................................................................................... 194

Figure 5.18. Transverse Stress Distribution Near Centerline for the Flow Between Two

Parallel Plates.............................................................................................. 194

Figure 5.19. Beam and Wing-box Structural Models. .................................................... 198
1

CHAPTER I

INTRODUCTION

1.1. MOTIVATION

The analysis of revolutionary aerospace and ground vehicles relies heavily on

accurate, efficient and robust computational methodologies such as the finite element and

finite difference methods. To investigate novel and revolutionary design concepts,

accurate, high-fidelity data must be assimilated rapidly into the design, analysis and

simulation process. This data assimilation should consider mathematical modeling

approximations ranging from simple handbook equations, empirically derived relations,

spreadsheets, and design charts to complex continuous and discrete simulation models.

In addition, the data assimilation needs to consider associated multi-discipline

interactions necessitated by advanced design concepts exploiting multifunctional

materials and leading to multifunctional structures. Rapid discipline-centric modeling

techniques allow high-fidelity design trades between cost and performance, and based on

the insight provided by these simulations, design uncertainties and risk assessment may

be evaluated. Integrated multi-discipline analyses allow the assessment of the effects of

multidisciplinary coupling on the system response. New computing systems and

alternative computing strategies have presented new opportunities for optimal design,

analysis, and simulation of aerospace systems. However, integrated high-fidelity

methods with diverse engineering applications provide the enabling technologies to

assimilate high-fidelity, multi-disciplinary data rapidly at an early stage in the design.

 

The journal model for this dissertation is the AIAA Journal.
2

These integrated methods must be multifunctional, collaborative and applicable to the

general field of engineering science and mechanics.

To understand the impact of these integrated methods, the three concomitant

attributes, namely, multifunctional, collaborative, and engineering science and

mechanics, must be described. In the context of this work, multifunctional

characterization has been adopted from the description of new and innovative materials

and structures with multiple capabilities. These systems, referred to as multifunctional

materials and structures, respectively, have several desirable simultaneous properties and

many diverse disciplinary applications. The systems will adapt, react and evolve in

changing environments, and their use will result in a combined system with enhanced

capabilities at less cost and weight. Likewise, multifunctional methods refer to

computational methodologies that have multiple capabilities such as multiple fidelity

modeling, multiple approximation analysis and multidisciplinary analysis. The methods

are computationally efficient while preserving solution accuracy and are applicable to a

wide range of applications in engineering science. Their use in the combined analysis of

complex configurations promises to provide enhanced computational and engineering

capability at less cost and in less time. With these attributes, a multifunctional method

may address the diverse modeling and analysis needs of evolving systems perhaps using

a hierarchical approach including error analysis and risk assessment.

 The collaborative aspect of the computational methods provides a mechanism by

which two or more physical domains are integrated or interfaced and by which two or

more methods or algorithms are shared or interfaced. It is through this interfacing that

the diverse attributes create a unified framework that far exceeds the capability of an
3

individual method. Collaborative methods may integrate domains of different

discretization fidelity, analysis approximations, or disciplines. An example of a

collaborative method is adaptive dynamic relaxation. Explicit direct time integration

algorithms are well-known for their computational efficient, low-memory requirements,

low computational cost per solution step and direct mapping to massively parallel

processing (MPP) systems. Adaptive dynamic relaxation techniques exploit these

features to determine the quasi-static or steady-state response of a structure without

relying on traditional methods requiring the solution of the large sparse matrices.

Collaborative methods provide a mechanism by which the aggregate cost savings related

to computational and modeling requirements are reduced, and analyses, previously

intractable, may be performed. As in the case of the multifunctional materials or

structures, these methods adapt, react and evolve in the changing environments of

engineering science. Engineering science covers the broad perspective of engineering

and includes the integrated application of engineering principles, science, mathematics,

numerical analysis and non-deterministic methods. Problems in fluid flow, solid

mechanics, thermal analysis, and constitutive modeling are representative of those in

engineering science. Engineering science has a multidisciplinary emphasis, and future

methods applicable to the field should possess multifunctional characteristics and a

collaborative nature to further enhance their analysis capabilities and to advance the stateof-the-art in engineering design.

Multifunctional collaborative methods should address four typical steps of

analysis and design, namely, (1) representation or modeling of the geometry, (2)

knowledge-based selection and development of appropriate mathematical models (i.e.,
4

idealization/discretization), (3) solution of the mathematical model (continuous and/or

discrete), and (4) interrogation/assessment of the results. These steps are briefly outlined

in Appendix A. Methodology and analysis procedures that address these basic steps

provide the foundation for enhanced integrated design and analysis tools within the realm

of engineering science. Such multifunctional methodology should allow interaction

between and collaboration with the analyst and designer, among different mathematical

modeling approximations of the physical phenomena, and among multiple engineering

disciplines. A major feature of the methodology is the transfer of data across the

respective interface, whether the interface is one among diverse mathematical

approximations or among diverse disciplines. Computational issues associated with

individual modeling approaches and disciplines are magnified in number and significance

due to the intricate couplings manifesting themselves as a by-product of their interfacing.

Multi-fidelity modeling approaches provide benefits in all of the major steps of

analysis and simulation. These approaches are often characterized by the use of different

approximations among multiple domains of the same continua and multiple domains

involving different continua (e.g., fluid-structure interaction). Analytical and closedform solutions for specific geometries and configurations are often used to eliminate

constraints placed on the analysis due to geometry considerations. Rapid modeling

approaches facilitate the discretization of geometry by providing a capability to model

regions of interest, independently, increasing the discretization fidelity or enhancing the

mathematical approximation only in the desired domains. Thus, for multi-fidelity finite

element modeling approaches, complex and often unsuitable mesh transitioning,

generated manually or using automatic mesh generators, is limited. In addition, multi-
5

fidelity approaches have been developed that allow for the discretization of parts or

components across geographically dispersed locations with minimal concern for the

discretization of the parts along common boundaries or interfaces. Additional research

has provided for accommodation of slight anomalies in the geometric representation

provided by the independently discretized parts as well as parametric definition of the

interface geometry between parts. Multi-fidelity modeling approaches benefit the

solution of the discretized system in that the system size using a multi-domain approach

for global/local modeling may be smaller for a given level of solution accuracy than the

system obtained by standard practices. In addition, in component modeling, the

associated matrices may be reduced by static condensation, which reduces the size and

subsequent solution time of the overall system of equations. Multi-fidelity modeling

approaches allow for the visualization and interrogation of the results only in regions of

interest. Post-processing of secondary results such as stresses and failure parameters may

be isolated to these regions and dynamically computed as the need arises. By reducing

the modeling, computational and visualization time of simulations of aerospace

structures, multi-fidelity modeling approaches promise to enhance the viability of highfidelity analyses early in the design process.

Multidisciplinary coupling approaches involve the interfacing of different

disciplines to account for their interactions and impact on the overall system response.

There are myriad approaches, for example, any combination of approaches that couple

the fluids, thermal, structures, and acoustic disciplines. The traditional independent

approach for multidisciplinary analysis involves loosely coupling the disciplines through

sequential execution of single discipline analyses. Typically this approach requires
6

several iterations among the different analysis methods and analysts and is relatively

inefficient because the discipline specific models are generally incompatible and require

extensive post-processing after each single discipline analysis to transfer (or interface)

data to the next analysis model. Aeroelastic analysis as an interdisciplinary problem,

requires the coupling of the aerodynamic and structural responses. The use of different

spatial discretization procedures and potentially different mathematical modeling

approximations for the aerodynamic model and the structures model gives rise to the

interfacing problem of transferring computed data between the two grid systems.

Moreover, the same issues are prevalent in fluid-thermal-structural analyses and

structural-acoustic analyses. Suitable methodology for addressing these types of

interfacing problems has been developed by many researchers.

The overarching purpose of this research is to investigate multifunctional

collaborative methods, as described herein, that address the engineering design and

analysis needs of multidisciplinary problems in engineering science. This research

focuses on the fundamental relationships among underlying engineering science and

mechanics principles, computational methods and multi-fidelity models, and methods

using basic problems from continuum mechanics. Given its broad applicability with

respect to the field of engineering science, continuum mechanics forms the foundation for

the multifunctional collaborative methods developed in this work. Hence, for

completeness and to establish notation, basic concepts of continuum mechanics are

presented briefly in the next section.
7

1.2. CONTINUUM MECHANICS FOUNDATIONS

Continuum mechanics is the branch of physical sciences concerned with the

deformations and motions of continuous material media under the influence of external

effects1. The effects that influence the bodies appear in the form of forces,

displacements, and velocities which arise from contact with other bodies, gravitational

forces, thermal changes, chemical interactions, electromagnetic effects, and other

environmental changes. In this work, bodies subject to forces of mechanical origin

and/or thermal changes are of primary concern. General principles in the form of integral

or differential equations govern the deformation and motion of the continuum. Hence,

approximation methods and associated concepts are introduced in addition to the basic

concepts of continuum mechanics.

1.2.1. General Principles of Continuous Media

A medium can be generally categorized as a fluid or a solid. A fluid can be

loosely defined as a continuum that does not require external forces to maintain its

deformed shape. When highly compressible it is called a gas and when essentially

incompressible, it is called a liquid. A solid can be loosely defined as a continuum that

requires external forces to maintain its deformed shape. According to its behavior, a

solid may be called elastic, plastic, viscoelastic, thermoelastic, etc. Usually it is assumed

to have a uniform density2

. When a medium deforms, the small volumetric elements

change position by moving along space curves. Their positions as functions of time can

be specified either by the Lagrangian ( X X ( ) x t i i i = , for i=1, 2, 3) or Eulerian

description ( x x ( ) X t i i i = , ). In the Lagrangian description, each particle is tracked in

terms of its initial position with respect to a fixed reference system, Xi, and time. In the
8

Eulerian description, the motion is expressed in terms of the instantaneous position vector

with respect to a moving reference system, xi, and time.

Classical continuum mechanics rests upon equations expressing the balances of

mass, linear momentum, angular momentum, energy, and entropy in a moving body3.

These balance laws apply to all material bodies, whether fluid or solid in composition,

and each gives rise to a field equation. These balance laws are as follows:

i. Principle of conservation of mass

ii. Principle of conservation of linear momentum

iii. Principle of conservation of angular momentum

iv. Principle of conservation of energy

v. Principle of entropy

The principle of conservation of mass states that when the total mass of the body is

unchanged for an arbitrarily small neighborhood of each material point, the mass is

considered to be conserved locally. The conservation of linear momentum represents

Newton’s second law and governs the motion of the continuum under the influence of the

external effects. The principle of conservation of angular momentum is used to show

symmetry of the stress tensor for many engineering materials, and the stress tensor

describes the state of stress of the continuum. The principle of conservation of energy,

also called the first law of thermodynamics, states that energy is conserved if the time

rate of change of the kinetic and internal energy is equal to the sum of the rate of work of

the external forces and all the other energies entering or leaving the body. The second

law of thermodynamics is automatically satisfied and includes the change in entropy of
9

the continuum. More detailed descriptions of these balance laws are presented in Chapter

III.

In deriving the governing equations, the starting point is a statement of the

conservation principle applied to a “control volume” to develop the integral form of the

equation and extract the differential form by using the divergence theorem. A control

volume has a fixed volume in space; its boundary does not deform but allows mass

transfer through it. In contrast, a material volume contains the same quantity of material

at all times; its boundary can deform, and it does not allow mass transfer.

As the continuum moves, in general, properties change with time and space. The

material derivative (substantial or total) must account for these changes depending on the

method of description used. Consider the scalar property as φ, for the Lagrangian

description, the material derivative is:

( )

dt t

dX

dt t X

d X t i

i

j

∂

∂ = ∂

∂

+

∂

∂ = φ , φ φ φ

For the Eulerian description, the material derivative is:

( )

φ φ + ⋅∇

∂

∂ =

∂

∂

+

∂

∂ =

∂

∂

+

∂

∂

+

∂

∂

+

∂

∂ =

v

,

t

x

v

t

dt

dx

dt x

dx

dt x

dx

dt t x

d x t

i

i

j

φ φ

φ φ φ φ φ 3

3

2

2

1

1

The general conservation equation may be written in integral form or differential

form in conservative or divergence form. However, when considering the differential

form, an equivalent representation is often obtained by working out the divergence

operator and introducing the material derivative. This leads to a non-conservative form
10

of the differential equation. Although the conservative and non-conservative forms of the

differential equations of the conservation principles are equivalent from a mathematical

point of view, they will not necessarily remain so when a numerical discretization is

performed. The general form of the conservation law is said to be written in conservative

or divergence form. The importance of the conservative form in a numerical scheme lies

in the fact that, if not properly taken into account, a discretization of the differential

equations will lead to a numerical scheme in which all the mass fluxes through the mesh -

cell boundaries will not cancel; hence, the numerical scheme will not keep the total mass

constant4.

1.2.2. Mathematical Approximations

Mathematical problems frequently encountered in engineering science may be

classified as boundary-value and initial-value problems based upon the existence of one

or more supplementary conditions. The differential equation describes a boundary-value

problem if the dependent variable and possibly its derivative are required to have

specified values on the domain boundary. The differential equation describes an initialvalue problem if the dependent variable and possibly its derivative are specified initially

(i.e., t=0). Initial-value problems are generally time dependent.

Partial differential equations governing the motion of general continua are often

of the canonical form Auxx + Buxy + Cu yy = 0 where the coefficients A, B, and C are

real constants, u represents a field variable, and the subscripts, x and y, denote partial

differentiation with respect to the independent variables, x and y. The character of this

quasi-linear, second-order, partial differential equation is determined by the sign of the

discriminant, B 4AC 2 − . The partial differential equation is
11

elliptic for 4 0 2 B − AC <

hyperbolic for 4 0 2 B − AC >

parabolic for 4 0 2 B − AC =

The full significance of the classification of quasi-linear, second-order partial differential

equations as elliptic, hyperbolic, or parabolic is beyond the scope of this work. However,

this classification has proved important for an understanding of the kinds of initial and

boundary conditions one must furnish along with the partial differential equation in order

to determine a unique solution. Moreover, solution methods differ markedly from one

classification to another, which is of particular importance in the field of fluid

mechanics6. For example, boundary conditions are generally imposed all the way around

a rectangular domain (the x-y region) of a two-dimensional flow when the equation is

elliptic, and the solution must have no discontinuities in the second derivatives, except

possibly at singular points where the differential equation is not applicable. Hyperbolic

and parabolic equations, by contrast, have at least one open boundary; thus, boundary

conditions are not usually imposed all around the domain under consideration. The

boundary conditions for at least one variable, usually time, are specified at one end, and

the system is integrated indefinitely. Certain kinds of discontinuities in the second

derivatives are admissible across certain curves in such a way that the differential

equation continues to be applicable in those regions.

Approximate solutions of differential equations (e.g., Ritz, Galerkin, leastsquares, collocation or in general weighted-residual methods) satisfy only part of the

conditions of the problem. For example, either the governing equation or the boundary

conditions may be satisfied only at a few positions rather than at each point. The
12

approximate solution is expanded in a set of known functions with arbitrary parameters.

Two ways to determine the parameters are the method of weighted residuals (MWR) and

the variational method. While the MWR and variational methods are only briefly

discussed here, a more complete discussion of the approaches is given in the literature by

Finlayson7

. In MWR, one works directly with the differential equation and boundary

conditions, whereas in the variational method one tries to satisfy the governing

differential equation in an average sense using a functional related to the differential

equations and/or the boundary conditions. MWR encompasses several methods

(collocation, Galerkin, integral, etc.) and provides a framework to compare and contrast

methods. Variational methods are not applicable to all problems, and thus suffer a lack

of generality. MWR is easy to apply whereas variational methods require manipulation

that can be more complex.

Variational methods provide a means for the determination of the governing

equations. In solid mechanics, the principles of virtual work and stationary potential

energy can be used to derive the governing equations and boundary conditions. The

principle of virtual work demands that for the state of equilibrium, the work of the

impressed forces is zero for any infinitesimal variation of the configuration of the system

that is in harmony with the kinematic constraints. Hence, the variational statement

implicitly imposes the natural boundary conditions. All work statements are derived

from classical laws pertaining to the equilibrium of the particle. Moreover, the virtual

work statement is simply the weak form of the equilibrium equations. For monogenic

forces, this statement leads to the condition that for equilibrium, the potential energy shall

be stationary with respect to all kinematically permissible variations.
13

The original differential equation is said to be the strong form of the problem

while the integral form is typically referred to as the weak form. However, in the strict

sense, particularly for approximation methods such as the Galerkin method, the weak

form is obtained by transferring the differentiation from the dependent variable to the test

functions, which includes the identification of the type of boundary conditions that the

weak form can admit. The purpose of the transfer of differentiation is to equalize the

continuity requirements on the dependent variable and the test function. This results in a

weaker continuity requirement on the solution in the weak form than in the original

equation. In the process of transferring the differentiation, boundary terms that determine

the nature of the natural or essential boundary conditions in the solution are obtained.

The classification of boundary conditions as natural and essential boundary

conditions plays a crucial role in the derivation of the approximate functions. From

variational calculus, consider a partial differential equation in the form,

= Ω 















∂

∂

∂

∂ −















∂

∂

∂

∂ − ∂∂ 0 in 

x u y

F

u y

F

u x

F

where ( , , , ) u ux,u y F = F x y , u u x x = ∂ / ∂ and u u y y = ∂ / ∂ . Transferring the

differentiation from the dependent variable, u, to the test function, v, yields the weak

form of the differential equations in the form

dxdy ds = 0 















∂

∂

+

∂

∂ −

















∂

∂

∂

∂

+

∂

∂

∂

∂

+

∂

∂ ∫ ∫

Ω Γ

x

y

x

x y x

n

u

F

n

u

F

v

u

F

y

v

u

F

x

v

u

F

v

It is at this point that the natural and essential boundary conditions are readily identified.

Generally, specifying coefficients of v and its derivative in the boundary integral

constitute the natural boundary condition. That is,
14

= Γ

∂

∂

+

∂

∂

n q on 

u

F

n

u

F

x

y

x

x

ˆ

is the natural boundary condition. Specification of the dependent variable in the same

form as the arbitrary test function constitutes the essential boundary condition. In the

case presented above, only v appears in the boundary integral. Hence, specifying u on Γ

is the essential boundary condition. The variables involved in the essential boundary

conditions of the problem are identified as primary variables and those in the natural

boundary conditions as the secondary variables in the formulation. The primary variables

are required to be continuous, whereas the secondary variables may be discontinuous in a

problem.

The differential equation is said to describe a scalar-field problem if the

dependent variable is a scalar and requires only the specification of magnitude for a

complete description. A vector-field problem is one that requires the specification of

magnitude and direction. The Poisson equation is an example of a differential equation

describing a scalar-field problem that arises in many fields of engineering science such as

elasticity, heat transfer, fluid mechanics, and electrostatics. The equation of motion is an

example of a differential equation describing the vector-field problem that governs the

motion of general continua. Each of these categories of differential equations will be

discussed in more detail and the concomitant formulations presented in Chapters II and

III.

The basic concepts of continuum mechanics and the ancillary fundamental

concepts of mathematical approximation methods outlined in this section form the basis

for the methodologies developed in this work. In subsequent chapters, the concepts are
15

described further as they relate to the development of multifunctional approaches for

scalar-field and vector-field problems in engineering science.

1.3. LITERATURE REVIEW FOR COLLABORATIVE METHODS

This section includes a literature review of topics related to collaborative methods

for multi-fidelity modeling and analysis. Review of approaches for collaborative

modeling of multiple domains is presented. This review is not intended to be an

exhaustive review of the subject matter but rather to provide sufficient background of the

fundamental concepts applicable to collaborative methods for engineering science. For

more detailed discussions on any of the topics reviewed, the reader is directed to the

referenced reports.

Multi-fidelity modeling, as referred to herein, entails the use of diverse

approximations among multiple domains. Numerous approaches for multi-fidelity

modeling have been developed over the last several decades. Many of these approaches

are commonplace in the analysis and design of aerospace structures. Generally, these

methods focus on modeling to obtain accurate stress data, and they have been used

primarily in an analysis framework rather than as an integral part of the design process.

With the development of rapid equation solvers and fast computer systems with

enormous storage capacities, these methods have the potential for impacting the

preliminary design stage. Research directly applicable to multi-fidelity modeling based

upon the finite element method continues to flourish. Developments pertinent to this

research include substructuring, global/local methods, model synthesis methods (i.e.,

multiple method approaches), submodeling, and finite element interface methods. While
16

all of these methods can be used in a global/local analysis, in general, they provide a

diverse capability for modeling multiple subdomains.

Substructuring, submodeling, and general global/local methods have been

highlighted, for example, by Ransom8 and Ransom and Knight9 and have been further

elaborated on by Rose10. One notable application of substructuring related to recent

advances in computational strategies is the use of neural networks to synthesize or

combine substructures11. In reference 11, substructures are modeled individually with

computational neural networks, and the response of the assembled substructure is

predicted by synthesizing the neural networks. Statically determinant substructures and

statically indeterminate substructures were assembled using a superposition approach and

a displacement collocation approach. Typically, substructuring and submodeling

approaches either require that the finite element nodes along the interdomain boundaries

coincide or make use of restrictive interpolations of displacements to the boundaries of

the local models. The global/local method proposed in reference 8 alleviates the

requirement for nodal compatibility along the local model boundary by introducing a

surface spline interpolation of the displacements from an independent global model to the

boundary of a more refined local model. This uncoupled approach was further extended

to provide global/local model interaction in an iterative approach proposed by Whitcomb

et al.12,13 In addition, global/local methodology for two- and three-dimensional stress

analysis of composite structures has been developed within a common framework by

Knight et al.14

In the context of this work, model synthesis refers to collaborative methodology

that couples or synthesizes two or more dissimilar mathematical models of multiple
17

subdomains. Myriad methods fall into this category. Examples of these methods and

representative references include, but are not limited to, synthesis of finite element and

boundary element methods15,16,17, finite element and Rayleigh-Ritz approximations18,

finite element and finite difference methods19,20, finite element and analytical solutions21,

and finite element and equivalent plate solutions22. Furthermore, an extensive review of

coupling the finite element method and boundary solution procedures has been given by

Zienkiewicz23. In reference 23, the finite element method is generalized to encompass

both the finite difference and the finite volume approaches.

 A new era of multi-fidelity modeling was introduced through the development of

an alternative approach for combining finite element models with different levels of

fidelity, which is referred to in the literature as interface technology. The concept of

interface technology is the genesis for the multifunctional capability presented in this

work. As such, a more extensive review of the literature is presented and the notable

contributions are outlined. The basic concept of the interface technology was discussed

by Housner and Aminpour24. In this work, the fundamental approaches were discussed

for mathematically coupling multiple subdomains whose grid points along common

boundaries did not coincide. Subsequent developments performed by Aminpour et al.25

implemented the basic concepts, extended the work to alternative approximations, and

compared the results for representative benchmark applications. Ransom et al.26

advanced further the technology by recasting the interface technology in the form of an

element, thus facilitating the use of the method for more than two subdomains.

Moreover, the implementation of the method as an element facilitated the inclusion of the

technology into standard commercially available finite element software codes27. Davilá
18

et al.28 extended further the capability for coupling not only along finite element edges as

originally implemented but across finite element faces as well. Rose10 extended the

concept of interface technology to include geometric incompatibility as well as nodal

incompatibility. In this work, the geometry of the subdomains is automatically adjusted

to account for an inaccurate geometry description along the common subdomain

boundaries and for gaps in the boundary definition, which allows for enhanced modeling

flexibility. In addition, extensions have been developed to include geometrically

nonlinear analysis29. The technology has been developed to provide dimensionality

reduction for integrating three-dimensional finite element models within two-dimensional

finite element models26. All of the aforementioned interface technology developments

have focused on a one-dimensional interface along a curve or line. Aminpour et al.30 and

Schiermeier et al.31 have extended the work to a two-dimensional surface interface for

coupling three-dimensional finite element models.

1.4. OBJECTIVES AND SCOPE

The overall objective of this research is to formulate multifunctional methodologies and

analysis procedures for interfacing diverse domain idealizations including multi-fidelity

modeling methods and multi-discipline analysis methods. Specific goals of this research

include:

1. To formulate general methodology providing capability for multifunctional

modeling, analysis, and solution.

2. To identify computational aspects and related algorithms for this methodology.

3. To apply the formulation to scalar- and vector-field applications in engineering

science.
19

The scope of the present work includes the multi-fidelity modeling and analysis of

interfaced domains within the same discipline as well as among multiple disciplines. The

analysis capabilities are limited to scalar- and vector-field problems using both single and

multiple approximation methods within a given domain. The capabilities are developed

considering discrete changes in domain characteristics across the interfaced boundaries,

compatibility with general-purpose finite element codes, applicability for a wide range of

discretization methods and engineering disciplines, and cost-effectiveness related to both

modeling and analysis time. To accomplish the objectives of the present work, numerical

studies are performed to gain insight into the interactions among the interfaced domains

and the computational strategies for the modeling and analysis. Prior to applying the

method to vector-field problems, the proposed method is evaluated with regard to

accuracy and computational implications on representative scalar-field problems.

The organization of the remainder of the dissertation is as follows. A

multifunctional approach for scalar-field problems is presented in Chapter II. Single- and

multiple-domain formulations are presented in the chapter along with a discussion of the

spatial modeling and the computational implications, and numerical results for a

verification test case are presented. The multifunctional approach for vector-field

problems is presented in Chapter III. Single- and multiple-domain formulations are

presented in this chapter along with a discussion of the spatial modeling and the

computational implications, and numerical results for a verification test case are

presented. Numerical results for representative scalar-field problems in engineering

science are presented in Chapter IV, while results for vector-field problems are presented

in Chapter V. In addition, a discussion of extensions of the methodology to multiple
20

discipline coupling is given in Chapter V. Conclusions and recommendations are

presented in Chapter VI. An overview of the steps in analysis and simulation is given in

Appendix A. A derivation of the cubic spline interpolation matrices used in the

multifunctional approach is presented in Appendix B. Details of the geometry

representation along the subdomain interface are given in Appendix C.
21

CHAPTER II

MULTIFUNCTIONAL APPROACH FOR SCALAR-FIELD

PROBLEMS

2.1. GENERAL

The motivation for the consideration of multifunctional approaches for scalarfield problems comes from the fact that methods of approximation such as Ritz, Galerkin,

and other weighted residual methods are based on weak statements of the differential

equations governing the system response. The differential equation is said to describe a

scalar-field problem if the dependent variable is a scalar and requires only the

specification of magnitude for a complete description. The scalar-field problem is a basic

form of the governing differential equations and thus lends itself to forming the

mathematical foundation for the general methodology developed herein. Representative

examples of the scalar-field differential equations in two dimensions are considered

herein, and the mathematical statement is formulated. The concepts developed here are

directly applicable to one-dimensional scalar-field problems; however, the development

is not included in the interest of brevity. The general form of the differential equation

GHVFULELQJDVFDODUILHOGSUREOHPIRUGRPDLQ VHHFigure 2.1) is given by the Poisson

equation, which is of the form

− ∇ ⋅( ) k∇u = Q in Ω (2.1)

subject to the natural boundary condition, h( ) u u q dn

du k + − ∞ = RQ s

, and essential

boundary condition, u u

~ = RQ

p

. The normal derivative, x ny y

u

n

x

u

dn

du

∂

∂

+

∂

∂ = , and nx
22

and ny are the components of the outward normal vector, nWRWKHERXQGLQJVXUIDFH RI

GRPDLQ ,Q(T2.1), the variables k and Q are known coefficients, and the primary

variable or dependent variable is u, which is a function of the independent variables x and

y. In the natural boundary condition, the variables, h and u∞, are the convection

coefficient, and the far-field value of the primary variable, respectively. The terms, q,

k

x

u

∂

∂ , and k

y

u

∂

∂ are the secondary variables that may be described on a portion of the

ERXQGDU\ s. The primary variable, uLVVSHFLILHGRQWKHERXQGDU\ p, and its

prescription to the boundary value, u , constitutes the essential boundary condition. The

complete boundary is defined as p s Γ = Γ + Γ .

Ω

Γp

Γ

s

n

x

y

− ∇( ) k∇u = Q

p s Γ = Γ + Γ

Figure 2.1. Geometric Representation of Two-Dimensional Domain.

2.2. DISCIPLINE SPECIFICS

Equations of the type of Eq. (2.1) arise in many fields of engineering science such

as elasticity, heat transfer, fluid mechanics, and electrostatics. Reddy32 has tabulated

several examples. In this work, the Poisson equation is applied to problems in the solid

mechanics and fluid mechanics disciplines.
23

2.2.1. Solid Mechanics

For applicability of the Eq. (2.1) in solid mechanics, consider a prismatic bar of

constant cross section subjected to equal and opposite twisting moments at the ends as

shown in Figure 2.2(a).

Mt

Mt

x,u

y,v

z,w x

y

α

θz

v

-u

r

O

P′

P

α

 (a) Geometry (b) Partial End View

Figure 2.2. Geometric Configuration of Prismatic Bar.

In general, the cross sections normal to the axis of the bar warp. As a fundamental

assumption, the warping deformation is taken to be independent of the axial location and

is given by

w = w( ) x, y

Assuming that that no rotation occurs at the end z=0 and that the angle of rotation, θ, is

small, the displacement components, u and v, in the x and y coordinate directions, of an

arbitrary point, P, P(x, y), in a plane for constant z, are respectively,

u = −(rθ z)sinα = −yθ z

v = (rθ z)cosα = xθ z (2.2)
24

where the angular displacement of a line segment, OP, from the origin, O, to an arbitrary

point, P, is θz and α is the angle between OP and the x axis (See Figure 2.2(b)). By

substituting Eq. (2.2) into the strain-displacement relations, the following are obtained

ε x = γ xy = ε y = ε z = 0

γ θ γ xθ

y

w

y

x

w

zx zy +

∂

∂ − = ∂∂ = (2.3) ; 

The three-dimensional stress-strain relations given in terms of Lame’s constants for a

linear isotropic solid are given by

z z xz xz

y y yz yz

x x xy xy

G e G

G e G

G e G

σ ε λ τ γ

σ ε λ τ γ

σ ε λ τ γ

= + =

= + =

= + =

2 ; 

2 ; 

2 ; 

where x y z e = ε + ε + ε , ( )( ) ν ν

ν λ

1+ 1− 2 = E , and ( ) +ν = 2 1

E G .

The shear modulus, G, and the quantity, λ, are referred to as the Lame’s constants, and

the modulus of elasticity, E, and the Poisson’s ratio, ν, are material properties.

Substituting the strain-displacement relations of Eqs. (2.3) into the stress-strain relations

gives

σ x = τ xy = σ y = σ z = 0















 +

∂

∂  = 

 



 − ∂

∂

τ = θ τ xθ

y

w

y G

x

w

zx G zy ; (2.4)

Then, the three-dimensional equations of equilibrium,
25

0

0

0

+ = ∂

∂

+

∂

∂

+

∂

∂

+ = ∂

∂

+

∂

∂

+

∂

∂

+ = ∂

∂

+

∂

∂

+

∂

∂

z

xz yz z

y

xy y yz

x

x xy xz

F

x y z

F

x y z

F

x y z

τ τ σ

τ σ τ

σ τ τ

(2.5)

with negligible body forces, simplify to the following equations:

0, 0, = 0

∂

∂

+

∂

∂ = ∂

∂ = ∂

∂

z z x y

zx zy zx zy τ τ τ τ

(2.6)

First, note that the stresses in Eq. (2.4) satisfy exactly the first two equilibrium equations

above (see Eq. (2.5)). Next, Eq. (2.4) can be combined into a single equation by

differentiating the expressions for τzx and τzy by y and x, respectively, and subtracting the

resulting equations. These operations yield the compatibility equation given by

θ τ τG

y x

zx zy = −2

∂

∂

− ∂

∂ . (2.7)

The stress in a bar of arbitrary cross section may thus be determined by solving the third

equation of equilibrium given in Eq. (2.6) along with the equations of compatibility given

in Eq. (2.7) and the given boundary conditions.

This torsion problem is commonly solved by introducing a single stress function.

If such a function, φ(x, y), the so-called Prandtl stress function, is assumed to exist, such

that

y x

zx zy ∂

∂ = − ∂∂ = φ τ φ , τ ; 

then, the equations of equilibrium are automatically satisfied. The equation of

compatibility becomes, upon substituting these expressions for the shear stress,
26

θ φ φ G

x x

2 2

2

2

2

= −

∂

∂

+

∂

∂

Therefore, if the compatibility requirement is to be satisfied, the stress function, φ, must

satisfy Poisson’s equation, Eq. (2.1). The primary variable, u, the constant, k, and the

source variable, Q, are represented by the stress function, φ, the inverse of the shear

modulus, G, and twice the angle of twist per unit length, θ, respectively. Moreover, the

stress function, φ=constant on the surface of the bar.

2.2.2. Fluid Mechanics

For a two-dimensional incompressible irrotational flow, expressions are given for

the velocity components, vx and vy, in terms of the x and y coordinate directions,

respectively. The velocity components should satisfy the continuity condition

= 0

∂

∂

+

∂

∂ ∇ ⋅ = y

v

x

vx y v (2.8)

and the irrotational flow condition

= 0

∂

∂ − ∂

∂

∇ × = y

v

x

v y x v . (2.9)

In terms of the stream function, ψ, the components are given by

x

v

y

vx y ∂

∂Ψ = − ∂∂Ψ = and (2.10)

and in terms of the velocity potential, Φ, the components are

y

v

x

vx y ∂

∂Φ = − ∂∂Φ = − and . (2.11)

Substituting the velocity components, vx and vy, from Eq. (2.10) into the irrotational flow

condition Eq. (2.9), one obtains
27

0 2

2

2

2

=

∂

∂ Ψ

+

∂

∂ Ψ

x y

. (2.12)

Note that the velocity components in terms of the stream function given in Eq. (2.10)

satisfy the continuity condition, Eq. (2.8) identically. Hence, Eq. (2.12) governs the flow

in terms of the stream function, ψ, and is in the form of the Poisson Equation, Eq. (2.1)

where the primary variable, u, the constant, k, and the source variable, Q, are represented

by the stream function, ψ, the density, ρ, and the mass production, σ (normally zero),

respectively.

Substituting the velocity components, vx and vy, from Eq. (2.11) into the

continuity equation, one obtains

0 2

2

2

2

=

∂

∂ Φ

+

∂

∂ Φ

x y

. (2.13)

Note that the velocity components in terms of the velocity potential given in Eq. (2.11)

satisfy the irrotational flow condition, Eq. (2.9), identically. Eq. (2.13) governs the flow

in terms of the velocity potential, Φ, and is in the form of the Poisson Equation, Eq. (2.1),

where the primary variable, u, the material constant, k, and the source variable, Q, are

represented by the velocity potential, Φ, the density, ρ, and the mass production, σ

(normally zero), respectively.

2.3. SINGLE-DOMAIN FORMULATION

In this section, multifunctional methodology for a scalar-field problem over a

single domain is presented in terms of weighted residuals. The method of weighted

residuals is used extensively in fluid mechanics and thus the potential problem is

formulated from this perspective. While the intent of this work is to develop general
28

methodology for multiple domains, the salient features of the weighted residual method

formulation may be investigated and discussed using the single domain. Consider the

general Poisson equation for a two-dimensional domain for field variable, u

− k∇ u = Q 2 (2.14)

 in a domain, Ω, bounded by Γ. In general, the boundary, Γ, can have mixed boundary

conditions with the primary variable, u, prescribed on Γ

p

and the secondary variable, the

flux, q, prescribed on the remaining part of the boundary, Γ

s

 (see Figure 2.1).

In the method of weighted residuals, an approximate solution, u

~ , is used in

expressing u

2 ∇ , then the differential equation, Eq. (2.14), will no longer be satisfied,

and this lack of equality is a measure of the departure of u

~ from the exact solution. The

lack of equality is called the residual, R, and is written as

0 2 R = −k∇ u − Q ≠ .

The residual is orthogonalized by a set of weight functions, Φi and averaged over the

domain. This residual may be written as

( ) d 0 2 ∫ − ∇ − Φ Ω =

Ω

Q i k u

~ . (2.15)

The solution for u

~ is sought in the form 0

1

= ∑ Ψ + Ψ

= i

n

i

i u a ~ 7KHIXQFWLRQV i, are usually

called trial functions, and ai are arbitrary constants. The trial functions satisfy the

KRPRJHQHRXVERXQGDU\FRQGLWLRQVZKLOH 0 satisfies the nonhomogeneous boundary

conditions. Posing the problem to be solved in a generalized weighted residual form33,34

and relaxing the requirement for the approximate solution to satisfy all boundary

conditions, the weighted residual statement may be written in the form
29

∫ ( ) dΩ + ∫ ( )dΓ = 0

Ω Γ

u

~ u B ~ A

where the residual in the satisfaction of the boundary conditions is orthogonalized by a

secondary set of weight functions, Φ , and the differential equation set is represented by

( )

( )

A u ( ) = 0





















=

M

A u

A u

2

1

in the domain, Ω, together with the boundary conditions

( )

( )

B u ( ) = 0





















=

M

B u

B u

2

1

on the boundary, Γ, of the domain. As implied by the matrix notation used previously,

the solution sought may represent a scalar quantity or a vector of several variables.

Similarly, the differential equation may be a single equation or a set of simultaneous

equations. For the system at hand, a scalar quantity is sought and the differential

equation is a single equation. Here, ( ) = −∇ u − Q 2 A u =0, and the essential and natural

boundary conditions, respectively, are represented by

( ) p B1 u = u − u = 0 on Γ

and

( ) s

2 − = 0 on Γ

∂

∂ = q

n

u B u k .

Therefore, considering the approximate solution, u

~ , we may write the general integral

form of the differential equation governing the potential flow as

( ) ( ) d 0

d

d d 1 d 2

2  Γ = 

 



 ∫ − ∇ − Ω + ∫ Γ + ∫ −

Ω Γ Γ

p s

p s

q

n

u k u Q u-u k

~ ~ ~ (2.16)
30

Note that the trial function may be selected so as to satisfy the essential and the natural

boundary conditions; thus, the boundary integrals in Eq. (2.16) are identically zero. In

this formulation, only the essential boundary conditions, i.e.,

p u − u = 0 on Γ ~

are assumed to be automatically satisfied by the choice of the trial functions. Therefore,

Eq. (2.16) is rewritten as

( ) d 0

d

d d 2  Γ = 

 



 ∫ − ∇ − Ω + ∫ −

Ω Γ

s q

n

u k u Q k s

~ ~ (2.17)

or

d 0

d

d d 2

2

2

2

 Γ = 

 



 Ω + − 













 −

















∂

∂

+

∂

∂ ∫ − ∫

Ω Γ

s q

n

u Q k

y

u

x

u k s

~ ~ ~

where x ny y

u

n

x

u

n

u

∂

∂

+

∂

∂ =

~ ~ ~

d

d and 2 = .

In general, the method of weighted residuals does not strictly require the

incorporation of natural boundary conditions into the weak formulation, as in the Ritz

method. However, if the operator permits the weak formulation, continuity requirements

on the primary variable and its derivatives may be relaxed. Moreover, if integration by

parts is possible, one may reduce the order of the highest derivative in the integral form

to eliminate the difficulty of selecting the appropriate weight functions. Thus, in the

formulation herein, the order of differentiation on the primary variable in the integral

equation, Eq. (2.17), is reduced to obtain the weak formulation. In addition,

acknowledging that the primary variable, u, is approximated by u

~ , for simplicity, the

subsequent development is presented in terms of u. Application of the divergence

theorem to Eq. (2.17) yields
31

d 0

d

d d

d d

 Γ = 

 



 − Ω + −

Γ 















∂

∂

+

∂

∂ Ω − 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫ ∫

∫ ∫

Ω Γ

Ω Γ

s

x y

s

q

n

u Q k

n

y

u

n

x

u k

y y

u

x x

u k

 (2.18)

Note that the boundary is presumed to consist of boundaries on which the primary

variable is specified and boundaries on which the secondary variable is specified, and

p s Γ = Γ + Γ . Therefore, the boundary integral on Γ, may be expressed as

∫ ∫ ∫

Γ Γ Γ

Γ 















∂

∂

+

∂

∂ Γ + 















∂

∂

+

∂

∂ Γ = 















∂

∂

+

∂

∂

s

s

x y

p

x y x y n

y

u

n

x

u

n k

y

u

n

x

u

n k

y

u

n

x

u k

p

d d d

Noting that, in the method of weighted residuals, the weight function, Φ, satisfies the

homogeneous boundary conditions for the primary variable (i.e., essential boundary

conditions). Thus, Φ=0 on Γ

p

. Therefore, the boundary integral on Γ

p

 is identically zero

and Eq. (2.18) may be rewritten as

d 0

d

d d d

d

d d  Γ = 

 



 Ω − Γ − Ω + − 















∂

∂

∂

∂

+

∂

∂

∂

∂ ∫ ∫ ∫ ∫

Ω Ω Γ Γ

s s

s s

q

n

u Q k

n

u k

y y

u

x x

u k

Since the weight functions, and , are arbitrary, they may be chosen, without loss of

generality, such that, = . Therefore,

dΩ − dΩ − dΓ = 0 















∂

∂

∂

∂

+

∂

∂

∂

∂ ∫ ∫ ∫

Ω Ω Γ

s

s

Q q

y y

u

x x

u k

or

s

s

Q q

y y

u

x x

u k Ω = Ω + Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂ ∫ ∫ ∫

Ω Ω Γ

d d d (2.19)

The integral form of Eq. (2.19) forms the basis of finite element approximations, which is

summarized in a subsequent section.
32

2.4. MULTIPLE-DOMAIN FORMULATION

In the multiple domain method, the domain of the problem is subdivided into a

number of smaller subdomains. The method is quite similar to the subdomain collocation

method, which is another weighted residual method. In the subdomain collocation

approach, the domain is divided into as many subdomains as there are adjustable

parameters. These parameters are then determined by making the residual orthogonal to

a weight function in an integral sense over each subdomain. Here, as in the singledomain formulation, methodology is presented formulating the general method of

weighted residuals for multiple domains by considering the Poisson equation for a twodimensional domain for a field variable, u. Then,

− k∇ u = Q 2 (2.20)

in the entire domain, Ω, bounded by Γ. For simplicity, the multiple-domain formulation

is presented for two subdomains, Ω1 and Ω2 (see Figure 2.3). Independent

approximations and weight functions are assumed in each of the subdomains and

continuity conditions are used to provide for a continuous solution across the subdomain

interfaces. Thus, Eq. (2.20) is satisfied in each subdomain, independently, i.e.,

2 2 2

2

1 1 1 2

2

1 − k ∇ u = Q in Ω and − k ∇ u = Q in Ω

subject to the boundary conditions on the subdomain boundaries, Γ1 and Γ2. Although

Eq. (2.20) is assumed for uniform constant, k, throughout the domain, it is permitted to be

different in each subdomain. That is, constants, k1 and k2, are used for subdomains Ω1

and Ω2, respectively, to allow for the general case of nonhomogeneous material.
33

At this point, differences between the single- and multiple-domain approaches

become evident. First, the domain, Ω, is now represented by the union of ns subdomains,

Ωi, such that

∑

=

Ω = Ω

ns

i

i

1

.

Second, the bounding surface, Γ, of the domain, Ω, is the union of the exterior surfaces,

E Γi , of the ns subdomains, Ωi, such that

∑

=

Γ = Γ

ns

i

i

1

E

In general, these exterior surfaces, E Γi , may involve mixed boundary conditions with the

primary variable, u, prescribed on p

i Γ and the secondary variable, the flux, q, prescribed

on s Γi such that

s

i

p

i i Γ = Γ + Γ E .

Finally, as a result of the subdomain modeling, the collaborative effort to solve the

problem involves an interior surface interface boundary, I Γi , and the information transfer

across the boundary. Hence, the boundary surface for the i

th subdomain is given by

p s I Γi = Γi + Γi + Γi

 The boundary conditions may be written as

p s q

n

u

u u k 1 1

1

1 1 1 1 0 on 

d

d − = 0 on Γ and − = Γ

and
34

p s q

n

u

u u k 2

2

2 2 2 2 0 on 

d

d 0 on and 2 − = Γ − = Γ .

ΓI

1

s

λ2

λ1

v

Ω1

Ω2

2

s

2

p

Γ

1

p

Γ

2

I

Γ

1

I

Figure 2.3. Boundary Definitions for Two-Dimensional Subdomains.

The residual for each domain is orthogonalized by a set of weight functions, Φi and is

written as

( 1 1) 1 d 1 0 2

1

1

∫ − ∇ − Ω =

Ω

k u Q

and

( 2 2 ) 2 d 2 0 2

2

2

∫ − ∇ − Ω =

Ω

k u Q

where the approximate solution is sought in the form i i

n

i u a 1 0

1

~

1 = ∑ 1 Ψ + Ψ and

i i

n

i u a 2 0

1

~

2 = ∑ 2 Ψ + Ψ 7KHIXQFWLRQV 0i 1iDQG 2i, are trial functions, and a1i and a2i
35

are sets of arbitrary constants. Using the general form outlined in the single-domain

formulation (i.e., ∫ ( ) dΩ + ∫ ( )dΓ = 0

Ω Γ

u

~ u B ~ A ), for each subdomain, one may write

( ) d ( ) u d 0 for 1,2 ~ u B ~ ∫ A Ω + ∫ Γ = =

Ω Γ

i i i i i i

i i

where the residual in the satisfaction of the boundary conditions is orthogonalized by a

secondary set of weight functions, Φi , for subdomain i. Therefore, considering the

approximate solution, u1

~ and u2~ , we may write the general integral form of the

differential equation governing the potential flow for subdomain 1 as

( ) ( ) d 0

d

d d d 1

1

1

1 1

1

1

1 1 1 11 1 1 12 1

2

1 1  Γ = 

 



 ∫ − ∇ − Ω + ∫ Γ + ∫ −

Ω Γ Γ

p s

p s

q

n

u k u Q u -u k

~ ~ ~ (2.21)

and for subdomain 2 as

( ) ( ) d 0

d

d d d 2 2

2

2 2 2 21 2 2 2 22 2

2

2 2

2 2 2

 Γ = 

 



 ∫ − ∇ − Ω + ∫ Γ + ∫ −

Ω Γ Γ

p s q

n

u k u Q u -u k

p s

~ ~ ~ (2.22)

Again, we will presume that the essential boundary conditions, i.e.,

p u u 1 1 1 − = 0 on Γ ~

and

p u u 2 2 2 − = 0 on Γ ~

are automatically satisfied by the choice of the functions, u1

~ and u2~ . Therefore, for

subdomain 1, Eq. (2.21) is rewritten as

( ) d 0

d

d d 1

1 1

1

1

1 1 1 12 1

2

1 1  Γ = 

 



 ∫ − ∇ − Ω + ∫ −

Ω Γ

s

s

q

n

u k u Q k

~ ~ (2.23)

or in its expanded form
36

d 0

d

d d

1 1

1 1

1

1 1 1 1 2

1

2

2

1

2

1 1  Γ = 

 



 Ω + − 













 −

















∂

∂

+

∂

∂ ∫ − ∫

Ω Γs

s q

n

u Q k

y

u

x

u k

~ ~ ~

(2.24)

where i y

i

x

i i n

y

u

n

x

u

n

u

i ∂

∂

+

∂

∂ = ∂

∂~ ~ ~

 and 12 = 1 and 22 = 2 . Similarly, the weighted

residual form for subdomain 2,

d 0

d

d d 2 2

2

2 2 2 2 2

2

2

2

2

2

2 2

2 2

 Γ = 

 



 Ω + − 















−

















∂

∂

+

∂

∂ ∫ − ∫

Ω Γ

s

s

q

n

u Q k

y

u

x

u k

~ ~ ~

(2.25)

The order of differentiation on the primary variable in the integral equations, Eq.

(2.24) and (2.25), is reduced to obtain the weak formulation. In addition, acknowledging

that the primary variables, u1 and u2, are approximated by u1

~ and u2~ , for simplicity, the

subsequent development is presented in terms of u1 and u2. Utilizing the divergence

theorem, Eq. (2.24), can be rewritten, for subdomain 1, yields,

d 0

d

d

d d d

1

1 1 1

1 1

1 1 1

1

1

1 1 1 1 1

1 1

1 1

1 1 1 1

1

 Γ = 

 



 + −



 Γ − Ω













∂

∂

+

∂

∂ Ω − 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫

∫ ∫ ∫

Γ

Ω Ω Γ

s

s

x y

q

n

u k

n Q

y

u

n

x

u k

y y

u

x x

u k

 (2.26)

and similarly, for subdomain 2,

d 0

d

d

d d d

2

2 2 2

2 2

2 2 2

2

2

2 2 2 2 2

2 2

2 2

2 2 2 2

2

 Γ = 

 



 + −



 Γ − Ω













∂

∂

+

∂

∂ Ω − 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫

∫ ∫ ∫

Γ

Ω Γ Ω

s

s

x y

q

n

u k

n Q

y

u

n

x

u k

y y

u

x x

u k

(2.27)

Note that the domain boundary is presumed to consist of boundaries on which the

primary variable is specified, boundaries on which the secondary variable is specified,
37

and boundaries at the subdomain interface. Thus, for subdomain i, I

i

s

i

p

i i Γ = Γ + Γ + Γ .

Here, the boundary on the interface is assumed to be conforming (i.e., represents same

geometry) and I I Γ = Γ i . Therefore, the boundary integral on Γi, may be expressed as

I d d

d d

I

Γ 















∂

∂

+

∂

∂ Γ + 















∂

∂

+

∂

∂

+

Γ 















∂

∂

+

∂

∂ Γ = 















∂

∂

+

∂

∂

∫ ∫

∫ ∫

Γ Γ

Γ Γ

y i

i

x

s i

y i

i

x

i

p

y i i

i

xi

i

y i i

i

x

i

i i s

i

i i i

p

i

i

i

i i

n

y

u

n

x

u

n

y

u

n

x

u

n

y

u

n

x

u

n

y

u

n

x

u

Note that i = 0 on Γi

p

. Eq. (2.27) can be rewritten, for subdomain 1, as

d d 0

d

d

d

d

d d

d

d d

1 1

I 1 1

1 1 1 1 1 1

1

1

I

1

1

1 1 1

1

1 1

1 1 1 1

1

 Γ − Ω = 

 



 + −

Ω − Γ − Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫ ∫

∫ ∫ ∫

Γ Ω

Ω Γ Γ

q Q

n

u k

n

u k

n

u k

y y

u

x x

u k

s

s

s

s

Since the weight functions, 1 and 1, are arbitrary, they may be chosen, such that,

1 = 1. Therefore,

d 0

d d

d

d d

1

1

I 1

1 1 1

1 1 1

I

1

1

1 1

1 1 1 1

1

− Ω =

Ω − Γ − Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫

∫ ∫ ∫

Ω

Ω Γ Γ

Q

q

n

u k

y y

u

x x

u k s

s

 (2.28)

Similarly, for subdomain, Ω2,

d 0

d d

d

d d

2

2

I 2

2 2 2

2 2 2

I

2

2

2 2

2 2 2 2

2

− Ω =

Ω − Γ − Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫

∫ ∫ ∫

Ω

Ω Γ Γ

Q

q

n

u k

y y

u

x x

u k s

s (2.29)
38

In general, for the multiple domain case, the approximation for the primary

variable (e.g., the potential field) must satisfy the following conditions:

i. The primary variable must be continuous and single valued in the subdomain.

ii. The primary variable must be continuous across the interdomain boundary.

iii.The primary variable on the subdomain boundary must satisfy the boundary

conditions.

If the requirement to satisfy interdomain continuity is relaxed, an additional boundary

condition is used, namely,

I u1 − u2 = 0 on Γ .

This constraint can be satisfied in the integral sense as

( ) I

1 2 d 0 on 

I

∫ − Γ = Γ

Γ

λ u u (2.30)

ZKHUH LVD/DJUDQJHPXOWLSOLHUDVVRFLDWHGZLWKWKHVHFRQGDU\YDULDEOHDORQJWKH

common subdomain boundary. Therefore, combining Eqs. (2.28) and (2.29) for the

entire domain, and including the continuity integral at the interdomain boundary yields

( )d 0

d d d ˆ d ˆ d

d d d

I

I I

1 2 2

1 2 1

I

1 2

I

2 2

I

2 2 2 1 1 1 2 2 2 1 1

2 1 1 1

2 2 2 2

1 2

1 1 1 1

1

+ − Γ =

− Ω − Γ − Γ − Γ − Γ



 Ω − Ω













∂

∂

∂

∂

+

∂

∂

∂

∂ Ω + 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫

∫ ∫ ∫ ∫ ∫

∫ ∫ ∫

Γ

Ω Γ Γ Γ Γ

Ω Ω Ω

u u

Q q q q q

Q

y y

u

x x

u k

y y

u

x x

u k

s s

s s

λ

(2.31)

where for subdomain, i, qi ˆ , are the secondary variables along the interdomain

boundary, i i y

i

x

i i

i n

y

u

n

x

u

n

u

∂

∂

+

∂

∂ = = d

d

qˆ . Note that the normals on the interdomain
39

boundary are equal and opposite (see Figure 2.3). That is,n1 = −n2 where

n i j i i i = nx + ny , and it follows that, qˆ = −qˆ = qˆ 1 2 . Therefore,

( )

( )d 0

d d d ˆ d

d d d

I

1 2

I

2 2 2 1 1 1 2 2 2 1 2

2 1 1 1

2 2 2 2

1 2

1 1 1 1

1

I

I

1 2 2

1 2 1

+ − Γ =

− Ω − Γ − Γ − − Γ



 Ω − Ω













∂

∂

∂

∂

+

∂

∂

∂

∂ Ω + 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫

∫ ∫ ∫ ∫

∫ ∫ ∫

Γ

Ω Γ Γ Γ

Ω Ω Ω

u u

Q q q q

Q

y y

u

x x

u k

y y

u

x x

u k

s s

s s

λ

or rearranging

( )d 0 .

d ˆ d d d

d ˆ d d d

I

2 2 I 2

1 1 I 1

I

1 2

2 2 2 2 2 2

I

2 2

2 2 2 2

2

1 1 1 1 1 1

I

1 1

1 1 1 1

1

=

















+ − Γ





















Ω + Γ − Ω − Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂

+





















Ω − Γ − Ω − Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

Γ

Ω Γ Ω Γ

Ω Γ Ω Γ

u u

q Q q

y y

u

x x

u k

q Q q

y y

u

x x

u k

s

s

s

s

λ

 (2.32)

 Note that Eq. (2.32) is written as a single equation for convenience and represents the

sum of terms related to the residual in the governing differential equation within each

subdomain and the continuity constraint for the primary variables along the common

subdomain boundary. However, each of the bracketed terms in Eq. (2.32) must equal

zero individually. These bracketed terms are identical to Eqs. (2.28), (2.29), and (2.30)

which must be satisfied independently.

In this formulation, the two primary field variables, u1 and u2 are approximated

independently, and continuity requirements between these two approximations are

satisfied along the subdomain interface boundary. The use of these approximations and
40

the associated continuity requirements gives rise to the classification of the formulation

as a two-approximation approach.

Now consider a configuration that makes use of a third approximation for the

primary variable along the subdomain interface boundary in addition to the

approximations given along the boundary of each of the subdomains. This primary

variable, v, along the interface is assumed to be independent of the primary variables, u1

and u2, of the subdomains to which it is attached. These independent approximations

give rise to continuity requirements along the interface of the form

I v − u1 = 0 on Γ

I v − u2 = 0 on Γ

These constraints can be satisfied in the integral sense as

( ) I I

1 1 d 0 on 

I

∫ − Γ = Γ

Γ

λ v u (2.33)

( ) I I

2 2 d 0 on 

I

∫ − Γ = Γ

Γ

λ v u (2.34)

where λ1 and λ2 are Lagrange multipliers or weight functions in the form of the

secondary variable along the interface. An additional continuity requirement in terms of

the secondary variable along the common subdomain boundary is required. These

secondary variables, 1qˆ and 2 qˆ , are assumed to be independent of each other. These

independent approximations give rise to continuity requirements along the interface of

the form

I qˆ1 + qˆ2 = 0 on Γ

These constraints can be satisfied in the integral sense as
41

( ) I I ˆ1 ˆ2 d 0 on ˆ

I

∫ + Γ = Γ

Γ

λ q q (2.35)

where λ

ˆ is a Lagrange multiplier or weight function of the form of the primary variable

along the interface. Combining Eqs. (2.28) and (2.29) for the entire domain and

including the three continuity integrals along the interdomain boundary, Eqs. (2.33),

(2.34), and (2.35), yields

() ( ) ( )

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫

∫ ∫

Ω Γ Ω Γ

Γ Γ Γ Γ Γ

Ω Ω

= Ω + Γ + Ω + Γ

Γ − Γ + − Γ + − Γ + + Γ

Ω − 















∂

∂

∂

∂

+

∂

∂

∂

∂ Ω + 















∂

∂

∂

∂

+

∂

∂

∂

∂

s s

s s Q q Q q

q q v u v u q q

y y

u

x x

u k

y y

u

x x

u k

2 2 1 1

I I I I I

1 2

1 1 1 1 1 1 2 2 2 2 2 2

I

1 2

I

2 2

I

1 1

I

2 2

I

1 1

2

2 2 2 2

1 2

1 1 1 1

1

d d d d

ˆ ˆ d ˆ ˆ d ˆ d d d

d d

λ λ λ

or rearranging

() ( ) ( ) ˆ ˆ d 0 ˆ d d

d ˆ d d d

d ˆ d d d

I I I

2 2 I 2

1 1 1 I

I

1 2

I

2 2

I

1 1

2 2 2 2 2 2

I

2 2 2

2 2 2 2

2

1 1 1 1 1 1

I

1 1 1

1 1 1 1

1

=

















+ + Γ 















+ − Γ 















+ − Γ





















Ω − Γ − Ω − Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂

+





















Ω − Γ − Ω − Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂

∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

Γ Γ Γ

Ω Γ Ω Γ

Ω Γ Ω Γ

v u v u q q

q Q q

y y

u

x x

u k

q Q q

y y

u

x x

u k

s

s

s

s

λ λ λ

(2.36)

Again, note that Eq. (2.36) is written as a single equation for convenience and represents

the sum of terms related to the residual in the governing differential equation within each

subdomain and the continuity constraints for the primary and secondary variables along

the common subdomain boundary. Each of the bracketed terms in Eq. (2.36) must equal

zero individually. These bracketed terms are identical to Eqs. (2.28), (2.29), (2.33),

(2.34), and (2.35), which must be satisfied independently.
42

The integral form of Eqs. (2.32) and (2.36) forms the basis for the subsequent

spatial modeling approximations. The spatial modeling approximations are discussed in

detail in the next section. Eqs. (2.32) and (2.36) may be generalized for more than two

subdomains and for multiple interfaces by

∑ ∑ ( ) ∫ ∫ ∑ ∫ ∫

∑ ∫

= Ω Γ = = Γ Γ

+

= Ω

















= Ω + Γ









































+ − Γ − Γ



















 Ω













∂

∂

∂

∂

+

∂

∂

∂

∂

ss

s m m

I ss

i i

ss

m

N

m

s m m m m m m

N

i

n i

j

i ij ij i ij ij j

N

m m m m m m m

u q Q q

y y

u

x x

u k

1 1

( )

1

1 I I

1

1 d ˆ d d d

d

I I

λ

(2.37)

and

( )

∑ ∫ ∫

∑ ∑ ∫ ∫ ∫

∑ ∫

= Ω Γ

= = Γ Γ Γ

= Ω

















= Ω + Γ

















+ − Γ − Γ + Γ



















 Ω













∂

∂

∂

∂

+

∂

∂

∂

∂

s

s m m

I ss

i i i

s

m

N

m

s m m m m m m

N

i

n i

j

ij i ij ij ij ij ij i ij ij

N

m m m m m m m

Q q

v u q q

y y

u

x x

u k

1

1

( )

1

I I I

1

d d

ˆ d ˆ d ˆ d

d

I I I

λ λ (2.38)

where Nss is the number of subdomains in which the entire domain is subdivided, NI is

the number of interfaces connecting the Nss subdomains and n ( )i ss are the number of

subdomains attached to interface i. For example, for one interface connecting two

subdomains, Eq. (2.38) yields in its expanded form
43

() ()

∫ ∫ ∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫

Γ Γ Ω Γ Ω Γ

Γ Γ Γ Γ

Ω Ω

Γ + Γ = Ω + Γ + Ω + Γ

+ − Γ − Γ + − Γ − Γ +



















 Ω













∂

∂

∂

∂

+

∂

∂

∂

∂ Ω + 















∂

∂

∂

∂

+

∂

∂

∂

∂

s s

s s q q Q q Q q

v u q v u q

y y

u

x x

u k

y y

u

x x

u k

2 2 1 1 I1

I

1

I

1

I

1

I

1

I

1

1 2

1 1 1 1 1 1 2 2 2 2 2 2

I

1 12 1

I

1 11 1

I

12 12 1

I

12 1 12 1

I

11 11 1

I

11 1 11 1

2

2 2 2 2

1 2

1 1 1 1

1

ˆ d d d d d ˆ ˆ d ˆ

d ˆ d d ˆ d

d d

λ λ

λ λ

which is identical to Eq. (2.36).

2.5. SPATIAL MODELING FOR MULTIPLE DOMAINS

Although this section is focused on spatial modeling of multiple domains using a

multifunctional development, a brief discussion of spatial modeling for a single domain is

given first, followed by a more detailed discussion for multiple domains. Thus far, a

multifunctional approach based on weighted residuals has been formulated. This

approximation technique provides a mechanism for finding approximate solutions to

problems in mathematical physics and engineering science such as those represented by

the Poisson problem. Selection of the approximating and weighting functions for

complex geometrical shapes and boundary conditions poses a major difficulty for

weighted residual methods. In addition, the methods were generally not regarded as

being computationally competitive compared to the traditional finite difference method.

However, weighted residual methods offer a versatile means by which to formulate finite

element equations where no functional is available. Hence, many of the difficulties

associated with this class of methods are alleviated. The derivation of discrete equations

is an essential component of the approximation technique. Thus, several discretization

approaches are outlined in the next section.
44

2.5.1. Overview of Discretization Methods

Various forms of spatial modeling or discretization of the continuum problem

defined by the differential equations can be used. These forms include, but are not

limited to, the finite difference method, the finite volume method, the finite element

method, and the boundary element method. In such spatial modeling, the infinite set of

numbers, representing the unknown function or functions is replaced by a finite number

of unknown parameters. A brief discussion of each of the aforementioned modeling

methods is given here to provide the foundation for discussion of interfacing such diverse

methods, which is presented in subsequent subsections.

The finite difference method

Of the various forms of spatial modeling, one of the simplest is the finite

difference method. The finite difference method gives a pointwise approximation to the

governing equations. In the finite difference approximation of a differential equation, the

derivatives in the equation are replaced by differential quotients that involve the values of

the solution at discrete mesh points of the domain. The resulting discrete equations are

solved for values of the solution at the mesh points, after imposing the boundary

conditions. While finite difference techniques are widely used in fluid dynamics and heat

transfer and can treat fairly difficult problems, they become hard to use when irregular

geometrical shapes or unusual boundary conditions are encountered. In addition, because

it is difficult to vary the size of different cells in particular regions, the method is not

suitable for problems of rapidly changing variables, such as stress concentration

problems. These adverse attributes are particularly significant in structural analysis.
45

The finite volume method

The finite volume method evolved in the early seventies via the finite difference

approximations and has many proponents in the field of fluid mechanics. The method

takes as its starting point the physical conservation laws in integral form written for small

control volumes around every discrete point. Modifying the shape and location of the

control volumes associated with a given discrete point, as well as varying the rules and

accuracy for the evaluation of the fluxes through the control volume, gives the method

considerable flexibility. Unlike the finite difference method, the finite volume method

can readily handle arbitrary mesh orientation thus making it more amenable to problems

of rapidly changing variables. In addition, by direct discretization of the integral form of

the conservation laws, the basic quantities (e.g., mass, momentum, and energy) will be

conserved at the discrete level. Like the finite difference method, the finite volume

method has been shown to be a special case of the finite element method with nonGalerkin weight functions35.

The finite element method

The finite element method consists of representing a given domain by an

assembly of smaller, geometrically simple subdomains or elements over which the

approximation functions are systematically derived. Then, Ritz-Galerkin approximations

of the governing equations are developed over each element. Finally, the equations over

all elements of the collection are connected by continuity of the primary variables. In the

mathematical literature, the names Petrov-Galerkin are often associated with the use of

weighting functions such that ≠ N , and the names Bubnov-Galerkin are often

associated with the use of weighting functions such that = N , where in the finite
46

element method N are the element shape functions. The latter method is often referred to

as the Galerkin method. The resulting system of equations is sparse, banded, symmetric,

and positive definite. The finite element method is especially well suited for handling

arbitrary shapes or domains. To obtain good accuracy in regions of rapidly changing

variables a large number of small elements must be used. Furthermore, the method is

widely used for the analysis of many engineering problems involving static, dynamic, and

thermal stresses of structures.

The boundary element method

 The boundary element method is an alternative to the finite element method.

Like the finite element method, the boundary element method uses nodes and elements to

discretize the boundary of the domain. Thus, compared to the finite element method, the

dimensionality is reduced by one. The governing differential equations are transformed

into integral identities, which are applicable over a surface or boundary. These integrals

are numerically integrated over the boundary, which is divided into small boundary

segments. The method may be used to model accurately the response in the domain

bounded by its mesh. The method can easily accommodate geometrically complex

boundaries. Furthermore, since all the approximations are restricted to the surface, the

method can be used to model regions with rapidly changing variables with better

accuracy than the finite element method. Complex kernel routines are required to

determine the response for the interior of the domain. Hence, the computational expense

increases quickly if the response at several interior locations is needed. In addition, for

nonlinear problems, the interior must be modeled; thus losing the advantage of reduction
47

in dimensionality. Unlike the finite element system matrix, the analogous boundary

element matrix is small, fully populated, and unsymmetric.

Each of the aforementioned discretization approaches has advantages and

disadvantages specific to the domain of the physical problem or the discipline within

which it is applied. To overcome the disadvantages of the individual methods, coupled or

collaborative methods have been developed. Collaborative methods couple two or more

discretization approaches and make use of a given approach when and where it is best

suited. The interaction between the methods is an essential feature related to the

robustness and accuracy of the combined methods and is a subject of discussion herein.

Moreover, this work focuses on the application of the multifunctional method developed

here to the finite difference and finite element methods and their coupling.

Computational methods using finite-differences for fluids experiencing field

discontinuities such as shock-waves and flow separations have been proven to be

efficient solution techniques. The finite element method has proven to be efficient in

solving for the response of complex aerospace structures, which may contain internal

discontinuous members such as spars, ribs, and bulkheads found in fuselage and wing

structures. In addition, coupled finite difference/finite element methods have been

proposed that make use of the strengths of the each of the modeling methods in the

solution of the aeroelastic problem and elasticity problems in references 36 and 19,

respectively. Thus, both spatial modeling approaches and their coupling will be

discussed in turn.
48

2.5.2. Overview of Single-Domain Spatial Modeling

Finite element discretization

For a single domain, the finite element equations may be obtained by rewriting

Eq. (2.19) over an element domain as

∫ ∫ ∫

Ω Ω Γ

Ω = Ω + Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂

es e e

e e e s Q q

y y

u

x x

u k d d d (2.39)

where superscripts on the domain, Ω, and boundary surface, Γ

s

, integrals denote

integration over the element. In later sections, numeric subscripts will be used to denote

element integration within the specified subdomain. The primary variable is

approximated over the element domain by u = Nue , and using the Galerkin method, the

vector of weight functions is given by = N . Substituting approximations into the

integral equation given in Eq. (2.39) yields

∫ ∫ ∫

Ω Ω Γ

= Ω + Γ 















Ω















∂

∂

∂

∂

+

∂

∂

∂

∂

es e e

e e s

e

e Q q

x x y y

k d d d T T

T T

u N N N N N N

or

e e e k u = f

where ke is the element stiffness matrix, ue is the vector containing the generalized

primary variables, and fe is the element force vector containing the generalized secondary

variables. The element field quantities, k, u, and f, are denoted by a subscript, e.

Assembling these element equations over the entire domain and enforcing continuity of

the primary variable at the interelement boundaries yields the system of equations given

by

Ku = F
49

where ∑ ∫ = Ω

Ω















∂

∂

∂

∂

+

∂

∂

∂

∂ =

nelem

e

e

e x x y y

k

1

T T

d N N N N K ; u is the assembly of all of the nodal

degrees of freedom associated with the primary variables; and

∑ ∫ ∫

Ω Γ

= Ω + Γ

nnodes e e s

es e

Q q

1

T T F N d N d

Finite difference discretization

In the finite difference methods, derivatives are approximated by difference

expressions that transform the derivatives appearing in the partial differential equations to

algebraic equations. For an elliptic partial differential equation, usually timeindependent, the methods result in a system of algebraic equations that are solved using a

direct or iterative solution technique. For hyperbolic and parabolic partial differential

equations, a set of algebraic equations is obtained. These equations are solved either

explicitly or implicitly. For the explicit solution, each equation will yield one unknown.

The matrix of unknown variables is a diagonal matrix and the right-hand-side vector of

the system is dependent on the variables at previous times. For the implicit solution, the

equations are coupled and must be solved simultaneously. Since the system equations are

coupled and more than one set of variables is unknown at the same level, the matrix to be

inverted is non-diagonal. In most cases, however, the structure of the matrix will be

rather simple, such as a block pentadiagonal, block tridiagonal, or block bidiagonal. The

truncation errors, stability and consistency of the numerical scheme are aspects that must

be considered in the development of the methods. The difference expressions are

obtained by Taylor series expansion, using forward, backward or central expansions.

Zienkiewicz and Morgan37 have shown that the finite difference method of approximation
50

is a particular case of collocation with locally defined basis functions. In the collocation

method, the unknown weight function parameters are determined by forcing the residual

in the approximation to vanish at N selected points in the domain. Upon substitution of

the approximation function into the differential equation, the equations can be recast in

weighted residual form by selecting ( ) i i = δ x − x . The weighted form of the residual

reduces to the evaluation of the partial differential equations using the approximate

solution evaluated at the N selected mesh points. For a second-order ordinary

differential equation, the approximate solution, u

~ , may be given as a function of the

solution at neighboring points (see Figure 2.4) as

e

i i

e

i i

e

i u ui N u N u N −1 −1 +1 +1 = + + ~

where e Ni are locally defined quadratic basis functions represented by

1 2 2 1 2 2

; ; 

2 ( )

( )

( )

( )( )

( )

( )

e

e

e

e i

e e

e

i e

e

e

i h

x h x N

h

h x h x N

h

x h x N + = − + = − = − − + .

x

i-1 i i+1

element e

he he

Figure 2.4. One-Dimensional Finite Difference Element Configuration.
51

The locally defined quadratic basis functions, e Ni , given here in the Cartesian coordinate

V\VWHPPD\EHZULWWHQLQWKHHOHPHQWQDWXUDOFRRUGLQDWHV\VWHP DV

− = − ξ ( ) ξ = ( ) −ξ + = ξ ( ) 1+ ξ 2

1 1- ; 1 ; 21

1

2

1

e

i

e

i

e Ni N N

where ξ=x/he. Note that -1 ≤ ξ ≤ 1. These basis functions are the standard Lagrangian

shape functions for three-node one-dimensional finite elements. This derivation for onedimensional problems may be extended to two- and three-dimensional problems. The

derivation is given for two-dimensional problems considering the bi-quadratic shape

functions for a nine-node two-dimensional finite element. A schematic of the finite

difference template and the associated finite element are shown in Figure 2.5 where the

open circles represent grid points in the five-point finite difference template used to

represent second-order derivatives.

The shape functions for a nine-node quadrilateral34 are given in Table 2.1. For

example, the shape function at point i,j-1 is given by

( )( ) ( )( ) 2 2 2

1 1 1

2

1 1 1

2

1 Ni, j− = −ξ −η − −ξ −η .

Similarly,

( )( ) ( )( ) 2 2 2

1 1 1

2

1 1 1

2

1 Ni+ , j = + ξ −η − −ξ −η ,

( )( ) ( )( ) 2 2 2

1 1 1

2

1 1 1

2

1 Ni, j+ = −ξ +η − −ξ −η ,

( )( ) ( )( ) 2 2 2

1 1 1

2

1 1 1

2

1 Ni− , j = −ξ −η − −ξ −η ,

and

( )( ) 2 2 Ni, j = 1−ξ 1−η .
52

Then,

( ) ( ) ( ) ( ) 2

1

2

2 0

2

2 2

1

2

2 2

1

2 1 1 1 1

x

N

h h

N

h x

N i j

e e

i j

e

i j

∂

∂ = = 



 



 = −

∂

∂ =

∂

∂ +

=

− , − , ,

η

η

ξ

,

( ) ( ) ( ) ( )2 0

2

2 2

2

2 2

2 2 21 1 1

e e

i j

e

i j

h h

N

h x

N − = 



 



 = − −

∂

∂ =

∂

∂

η= η

ξ

, , ,

( ) ( ) ( ) ( ) 2

1

2

2 0

2

2 2

1

2

2 2

1

2 1 1 1 1

x

N

h h

N

h y

N i j

e e

i j

e

i j

∂

∂ = = 



 



 = −

∂

∂ =

∂

∂ +

=

, − , − ,

η

ξ

η

, (2.40)

and

( ) ( ) ( ) ( )2 0

2

2 2

2

2 2

2 2 21 1 1

e e

i j

e

i j

h h

N

h y

N − =



 



 = − −

∂

∂ =

∂

∂

ξ = ξ

η

, , .

The standard finite difference representation follows by direct substitution. This

specialization of the finite difference method as a form of the generalized method of

weighted residuals forms the basis for its inclusion in this multifunctional derivation.

i,j i-1,j

i,j+1

i+1,j

i,j-1

he

he

i-1,j-1 i+1,j-1

i-1,j+1 i+1,j+1

Figure 2.5. Two-Dimensional Finite Difference Element Configuration.
53

For a single domain, as in the finite element method, the finite difference

equations may be obtained by interrogating the weighted residual equations over an

element domain where the element, e, surrounds node i (see Figure 2.5). The

approximate solution for the primary variable is given by

m e

M

m

u Nmu u Nu ~ = ∑ =

=

 or 

1

where M is the number of shape functions over the element, and the weight function, Φi,

is given by the Dirac delta function, ( )( ) i i i i δ x − x , y − y = δ x , y . Note that the subscript

i on the weight function is used to denote the subdomain, while the subscript i on the

coordinate values, x and y, is used to represent the point in the physical domain at which

the Dirac delta function is evaluated. Therefore, Eq. (2.39) becomes

( ) ( ) ( ) ( ) es

i i

e

e i i

i i i i e

es e e

Q x y q x y

y

x y

x y

x y

x

k = Ω + Γ 

















 Ω













∂

∂

∂

∂

+

∂

∂

∂

∂ ∫ ∫ ∫

Ω Ω Γ

d , d , d , , δ δ δ δu

N N

Using the identities ()( ) ( ) i i ∫ f x x − x dx = f x

∞

−∞

δ and ( ) ( ) ( )

i x x

i

dx

df x dx

dx

d x x f x

=

∞

−∞

= − ∫ δ

(See Bracewell38), the element equation reduces to

( ) ( ) e i i i i

y y

x x

y y

x x

Q x y q x y

y

N

x

N k

i

i

i

i

, , 2

2

2

2

= +





















∂

∂

+

∂

∂ −

=

=

=

=

u

For the second derivative difference approximation, the number of shape functions of an

element, M=3 and = { } i−1 i i+1

T

e u u u u . Therefore, as in the finite element method,

the resulting finite difference equations may be written in matrix form as

e e e k u = f
54

where ke is the finite difference “element stiffness” matrix, ue is the vector of generalized

primary variables and fe is the finite difference generalized force vector. Assembling the

element equations yields

Ku = F

where u contains all of the nodal degrees of freedom associated with the primary

variables,

∑





















∂

∂

+

∂

∂ = −

=

=

=

=

Nelem

y y

x x

y y

x x

i

i

i

i y

N

x

N k

1 2

2

2

2

K ,

and

= ∑[ ] ( )+ ( )

Nnodes

i i i i Q x y q x y

1

F , , .

While a single spatial modeling approach (i.e., the finite element method or the

finite difference method) is used for the single domain formulation, subdomain modeling

permits multiple discretization strategies to be used in a collaborative manner. These

discretization strategies include homogeneous approaches in which the same

discretization method is used in each subdomain and heterogeneous approaches in which

different discretization methods are used amongst the subdomains. Each of these

discretization strategies is discussed in the following sections.
55

Table 2.1. Shape Functions for a Nine-Node Quadrilateral Finite Element.

Primary Terms Secondary Terms of Shape Functions

− − = ( )( ) 1−ξ 1−η 4

1 Ni 1, j 1 1 2

1 − Ni, j − Ni 1, j 2

1 − − Ni, j 4

1 +

+ − = ( )( ) 1+ ξ 1−η 4

1 Ni 1, j 1 1 2

1 − Ni, j − Ni 1, j 2

1 − + Ni, j 4

1

+

+ + = ( )( ) 1+ ξ 1+η 4

1 Ni 1, j 1 Ni 1, j 2

1 − + 1 2

1 − Ni, j+ Ni, j 4

1 +

− + = ( )( ) 1−ξ 1+η 4

1 Ni 1, j 1 1 2

1 − Ni, j+ Ni 1, j 21 − − Ni, j 4

1 +

− = (1−ξ )( ) 1−η 2

2

1 Ni, j 1 Ni, j 2

1 +

( )( ) 2

2

1

1 Ni+ , j = 1+ ξ 1−η Ni, j 2

1 +

= (1−ξ )( ) 1+η 2

2

1 Ni, j Ni, j 2

1 +

( )( ) 2

2

1

1 Ni− , j = 1−ξ 1−η Ni, j 2

1 +

( )( ) 2 2 Ni, j = 1−ξ 1−η

2.5.3. Multiple-Domain Modeling - Homogeneous Discretization

In this context, homogeneous discretization approaches are applicable to multiple

subdomain discretization. These approaches make use of a single discretization method

among all subdomains in which the domain is subdivided. Of the many spatial modeling

approaches, this work will focus on the finite element and the finite difference methods.
56

Two-approximation interface modeling

For homogeneous domain discretization developed herein, Eq. (2.32) is used to

provide the mathematical basis. The two-approximation formulation, for both the finite

element (FE) and finite difference (FD) methods, may be obtained by rewriting Eq. (2.32)

over an element domain as

( ) ()

∫ ∫

∫ ∫ ∫ ∫

∫ ∫

Γ Ω

Γ Ω Γ Γ

Ω Ω

+ Ω + Γ

− − Γ + − Γ = Ω + Γ



 Ω













∂

∂

∂

∂

+

∂

∂

∂

∂ Ω + 















∂

∂

∂

∂

+

∂

∂

∂

∂

e

s e

e

s e e

e

e e

e e s

e e s

e e

Q q

q u u Q q

y y

u

x x

u k

y y

u

x x

u k

2 2

1 1 e

I

e

I

1 2

2 2 2 2 2 2

1 1 1 1 1 1

I

1 2

I

1 2

2

2 2 2 2

1 2

1 1 1 1

1

d d

ˆ d d d d

d d

λ (2.41)

Note that the integration over the common subdomain boundary, ΓI

, is considered only

for element edges along that boundary.

The form of Eq. (2.41) for the two (FE and FD) methods differs by the form of

the element shape functions and the approximation selected for the weight functions, Φ.

For the generalized element expansion of subdomain i, the independent approximations

for the element generalized primary variables, (i.e., displacements or velocities), interface

secondary variables (i.e., tractions or fluxes), and the weight functions associated with the

secondary variables, are, respectively

i i e i i i i

u = N u ; qˆ = R and = R

where is a vector of unknown coefficients associated with secondary variable, qˆ , and

N and R are matrices of interpolation functions for the element primary and secondary

variables, respectively. The interpolation functions in the matrix R are assumed to be

constants for linear finite elements and linear functions for quadratic finite elements.
57

Substituting these approximations into Eq. (2.41) yields an integral equation in terms of

the weight function, , which is given by

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫

Γ Ω Γ Ω

Γ Γ Γ Γ

Ω Ω

= Ω + Γ + Ω + Γ





















− Γ





















+ Γ





















+ Γ





















− Γ





















Ω















∂

∂

∂

∂

+

∂

∂

∂

∂

+





















Ω















∂

∂

∂

∂

+

∂

∂

∂

∂

e

s

e

e e s e

e

e

e

e

e

e

e

e

e e

e s es

e e

e e e e

Q q Q q

x x y y

k

x x y y

k

2 2 1 1

2

I

1

I I I

2

2

1

1

2 2

T

2 2 2

T

1 1 2

T

1 1 1

T

1

I

2

T

2

I

1

T

1

I

2

T

2

I

1

T

1

2

2

T

2 2

T

2

1 2

1

T

1 1

T

1

1

d d d d

d d d d

d d

R R R N u R N u

u

N N

u

N N

where for i=1,2

∫

Ω

Ω















∂

∂

∂

∂

+

∂

∂

∂

∂ = e

i

i

e

i

i i i i

i

x x y y

k d

T T N N ke ,

( ) ∫

Γ

+ = − Γ

e

e

i i i

i

I

1 T I k 1 R N d p ,

( ) ∫

Γ

= − Γ e

e

i i i

i

I

T I k 1 R d s ,

and

∫ ∫

Ω Γ

= Ω + Γ e

s

i

e

e

i

s i i i e i i i i Q d q d T T fe

Assembling the element equations over the entire domain, enforcing continuity of the

primary variable only within each subdomain and assembling the contributions along the

element edges on the common subdomain boundary, and noting that 1 ue and 2 ue and

1ef and 2 ef are completely uncoupled, yields the system of equations given by
58





















=









































0

f

f

u

u

K K 0

0 K K

K 0 K

2

1

2

1

2

1

1 2

2

1

p p

s

s

 (2.42)

The system of equations given in Eq. (2.42) is obtained based on the initial development

of the weighted residual statement, from Eqs. (2.28) and (2.29),

s

i i i i i i i

i

i i

i i i i

i s

i i i

Q q

n

u k

y y

u

x x

u k Ω − Γ = Ω + Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂ ∫ ∫ ∫ ∫

Ω Γ Ω Γ

d d d

d

d d

I

I

subject to the constraint equation, Eq. (2.30),

( ) I I

1 2 d 0 on 

I

∫ − Γ = Γ

Γ

λ u u .

Here, the first two matrix equations in Eq. (2.42) are obtained from the weighted residual

statement for each subdomain, Eqs. (2.28) and (2.29), and the third matrix equation is

obtained from the constraint on the primary variables along the common subdomain

boundary, Eq. (2.30).

For the finite element modeling, the weight functions are taken to be the finite

element shape functions. That is, i = Ni , and thus, for i=1,2

∫

Ω

Ω 















∂

∂

∂

∂

+

∂

∂

∂

∂ = e

i

i

e

i

i i i i

i

x x y y

k d

T T N N N N ke ,

( ) ∫

Γ

+ = − Γ

e

e

i i i

i

I

1 T I k 1 R N d p ,

( ) ∫

Γ

= − Γ e

e

i i i

i

I

T I k 1 N R d s , (2.43)

and
59

∫ ∫

Ω Γ

= Ω + Γ e

s

i

e

e

i

s i i i e i i i i Q d q d T T fe N N

Here, note that at the element level, T

i pi s

k = k , and consequently, at the global system

level, T

i pi s

K = K .

For the finite difference modeling, the weight functions are taken to be the Dirac

delta function. That is, ( )( ) i i i i i i i = δ x − x , y − y = δ x , y , and thus, for i=1,2

( ) ( ) ∫

Ω =

=

=

= 























∂

∂

+

∂

∂

Ω = 















∂

∂

∂

∂

+

∂

∂

∂

∂ = e

i

i

i

i

i

i

y y

x x

i

y y

x x

i

i ei

i i i i i i i i

i

x y

k

y

x y

x y

x y

x

k 2

2

2

2

d

N , N , N N

ke

δ δ ,

( ) ∫

Γ

+ = − Γ e

e

i i i

i

I

1 T I k 1 R N d p ,

() ( ) () ( ) 1 , d 1 ,

I

I

i i i

i

i i i i

i

x y x y e

e

i ks = − ∫ R Γ = − R

Γ

δ , (2.44)

and

() () ( ) ( ) i i i i i i s i i i i i e i i i i i i x y Q x y q Q x y q x y e

s

i

e

e

i

f , , , , e = ∫ Ω + ∫ Γ = +

Ω Γ

δ d δ d .

Three-approximation interface modeling

For the three-approximation formulation, Eq. (2.36) is used to provide the

mathematical basis for the development. In previous work by Aminpour et al.25, a

similar formulation based on the principle of minimum potential energy is implemented

in the form of an element. In that work as is the case in this study, the interdomain

interface boundary is discretized with a mesh of evenly-spaced pseudo-nodes (open
60

circles in Figure 2.6) that need not be coincident with any of the interface nodes (filled

circles in the figure) of any of the subdomains.

Ω1

Ω2

Ω3 Γ2

Γ3

Γ1 ΓΙ

Pseudo-nodes

Interface

Finite element

nodes

Figure 2.6. Interface Definition.

The generalized element equations may be obtained by introducing the continuity

requirements into the weighted residual statement. Eq. (2.36) can be rewritten over an

element domain as

() ( ) ( )

d d d d .

ˆ ˆ d ˆ ˆ d d d

d d ˆ d

1 1 2 2

I I I I

I 2 1

1 1 1 1 1 1 2 2 2 2 2 2

I

1 2

I

2 2

I

1 1

I

2 2

I

2 1 1

2 2 2 2

1 2

1 1 1 1

1

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫

Γ Ω Γ Ω

Γ Γ Γ Γ

Γ Ω Ω

= Ω + Γ + Ω + Γ

− Γ + − Γ + − Γ + + Γ

Ω − Γ 















∂

∂

∂

∂

+

∂

∂

∂

∂ Ω + 















∂

∂

∂

∂

+

∂

∂

∂

∂

e

s

e

e e s e

e

e

e

e

e

e

e

e

e

e

e e

e s e e s

e e

Q q Q q

q v u v u q q

q

y y

u

x x

u k

y y

u

x x

u k

λ λ λ (2.45)

Note that in the potential energy formulation25, the continuity of the secondary variables

was satisfied through the subsidiary conditions obtained through the minimization of the

potential energy. In this weighted residual formulation, the continuity of the secondary
61

variables is satisfied in a weighted residual sense and the Lagrange multipliers, λi and λ

ˆ ,

are represented by weight functions in the form of the secondary and primary variables,

respectively.

The form of the equations for the finite element and finite difference applications

differs by the form of the element shape functions and the approximation selected for the

weight functions, . For the generalized element expansion of subdomain i, the

independent approximations for the element generalized primary variables, (i.e.,

displacements or velocities), interface secondary variables (i.e., tractions or fluxes), the

weight functions associated with the secondary and primary variables, and the interface

variables, are, respectively

I and ˆ ui = Niue ; qˆi = Ri i ; i = Ri ; = T v = Tu i

where is a vector of unknown coefficients associated with the secondary variables, qˆ ,

and N, R, and T are matrices of interpolation functions for the element primary and

secondary variables, and primary variables along the interface, respectively. The

interpolation functions in the matrix R are assumed to be constants for linear elements

and linear functions for quadratic elements. The interpolation functions in the matrix T

are cubic spline functions. The derivation of this interpolation matrix is given in

Appendix B, and the derivation of the geometry representation, ΓI

, is given in Appendix

C. Substituting these approximations into Eq. (2.45) yields an integral equation in terms

of the weight function, Φ, which is given by
62

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫

Γ Ω Γ Ω

Γ Γ Γ Γ

Γ Γ Γ Γ

Ω Ω

= Ω + Γ + Ω + Γ





















− Γ





















+ Γ





















− Γ





















Γ

+





















+ Γ





















+ Γ





















− Γ





















− Γ





















Ω















∂

∂

∂

∂

+

∂

∂

∂

∂

+





















Ω















∂

∂

∂

∂

+

∂

∂

∂

∂

e

s e e s e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e e

e e s e e s

e e

e e e e

Q q Q q

x x y y

k

x x y y

k

2 2 1 1

2

I I

1

I I

I I I I

2

2

1

1

2 2

T

2 2 2

T

1 1 2

T

1 1 1

T

1

I

2

T

I 2

T I

2

I

1

T

I 1

T I

1

2

I

2

T

1

I

1

T

2

I

2

T

1 2

I

1

T

1

2

2

T

2 2

T

2

1 2

1

T

1 1

T

1

1

d d d d

d d d d

d d d d

d d

R T u R N u R T u R N u

R R T R T R

u

N N

u

N N

where, for i=1,2

∫

Ω

Ω















∂

∂

∂

∂

+

∂

∂

∂

∂ = e

i

i

e

i

i i i i

i

x x y y

k d

T T N N ke

∫

Γ

= − Γ

e

e

i i i

I

d T I kp R N ,

e

e i i i

T I d

I

= − ∫ Γ

Γ

ks R ,

∫

Γ

= Γ e

e

i i

I

d T I k I T R ,

and

e

es

i

e

i

s i i i e i i i i = ∫ Q Ω + ∫ q Γ

Ω Γ

d d T T fe

where integration over the common subdomain boundary, ΓI, is considered only for

element edges along that boundary.
63

Note that all of the element submatrices in the three-approximation formulation

except for the i

k I matrix are identical to those obtained in the two-approximation

formulation. The submatrix, i

k I , does not exist in the two-approximation formulation

but is included in the three-field formulation. This submatrix is associated with the

coupling of the primary variables along the subdomain interface boundaries to those

along the interface.

Assembling the element equations over the entire domain, enforcing continuity of

the primary and secondary variables only within each subdomain and assembling the

contributions along the element edges on the common subdomain boundary, and noting

that 1 ue and 2 ue , and 1ef and 2 ef , and 1 and 2 are completely uncoupled, yields the

system of equations given by





































=









































































0

0

0

f

f

u

u

u

0 K K 0 0

K 0 K 0 0

0 0 0 K K

0 K 0 0 K

K 0 0 K 0

2

1

2

1

I

2

1

T

p I

T

p I

I I

2 s

1 s

2 2

1 1

1 2

2

1

(2.46)

or in a symbolic manner





















=









































0

0

f

u

u

K K 0

0 0 K

K 0 K

I

p I

I

s

where K, u, and f are the assembled stiffness matrix, displacement vector and force

vector for the entire structure, and Kp, Ks, KI, uI, and α are the assembled Kpi, Ksi, KI

i

,

uI, and i for all interfaces.
64

While it is convenient to represent the weighted residual form over the domain

using a single equation, the system of equations, Eq. (2.46) is obtained from the

individual weighted residual expressions over each of the subdomains and the constraint

integrals. The first two matrix equations of the system of equations, Eq. (2.46) are

derived from the weighted residual statement for subdomain i. That is,

∫ ∫ ∫ ∫

Ω Γ Ω Γ

Γ = Ω + Γ

∂

∂ Ω − 















∂

∂

∂

∂

+

∂

∂

∂

∂

s

i i i

s

i i i i i i i

i

i

i i i i

i Q q

n

u k

y y

u

x x

u k d d d d

I

I

1

The third matrix equation of the system results from the reciprocity statement of the

secondary variables. That is,

( ) I I ˆ1 ˆ2 d 0 on ˆ

I

∫ + Γ = Γ

Γ

λ q q .

The fourth and fifth matrix equations result from the continuity requirement for the

primary variables, which is given by

( ) I I

1 I 1 d 0 on 

I

∫ − Γ = Γ

Γ

λ u u

( ) I I

2 I 2 d 0 on 

I

∫ − Γ = Γ

Γ

λ u u

For the finite element development, the weight functions are taken to be the finite

element shape functions (i.e., i = Ni ). For the finite difference development, the

weight functions are taken to be the Dirac delta function (i.e.,

( )( ) i i i i i i i = δ x − x , y − y = δ x , y ). Thus, for i=1,2, the finite element and finite

difference stiffness matrices and force vector, i

ke , pik , iks , and i ef , for the threeapproximation formulation are identical to those obtained for the two-approximation

formulation for the respective discretization approaches.
65

Note that, for both of the discretization methods, the form of the coupling element

matrices that are not in terms of the weight functions are independent of the method of

discretization. That is,

∫

Γ

= − Γ e

e

i i i

I

T I k R N d p ,

and

∫

Γ

= Γ e

e

i i

I

T I

I k T R d

are of the same form for the finite element and finite difference discretizations.

However, since the element shape functions, Ni, differ for the two methods, the interface

matrices, pi k , in general, are not identical. Moreover, in the finite element development,

the weight functions, i , are taken to be the finite element shape functions, Ni ; thus, at

the element level T

s pi i k = k , and at the global system level Ts pi i K = K .

The three-approximation derivation is more general as it allows for the coupling

of the primary variables to an independent approximation. This attribute is particularly

important in the heterogeneous discretization approach described in the next section.

2.5.4. Multiple-Domain Modeling - Heterogeneous Discretization

Heterogeneous discretization approaches make use of different discretization

methods for at least two of the subdomains in which the domain is subdivided. There are

many combinations of spatial modeling approaches; however, this work will focus on the

coupling of the finite element and finite difference methods.

Both the two- or three-approximation multifunctional formulations, discussed for

the homogeneous discretization approach, are applicable to heterogeneous discretization.
66

However, as noted earlier, the three-approximation approach provides additional

flexibility for the interface definition. Thus, only the three-approximation approach is

presented. Hence, the multifunctional weighted residual formulation of Eq. (2.46) is

used. Considering the two domains, upon which this discussion is based, one subdomain

is discretized using the finite element method, and the other subdomain is discretized

using the finite difference method. As before, for the finite element development, the

weight functions are taken to be the finite element shape functions (i.e., i = Ni ), and for

the finite difference development, the weight functions are taken to be the Dirac delta

function (i.e., ( )( ) i i i l i i i = δ x − x , y − y = δ x , y ). As expected, the set of element

matrices becomes a hybrid of the matrices from the finite element method and the finite

difference method. For completeness, these matrices are repeated here considering

subdomain 1 as the finite element subdomain and subdomain 2 as the finite difference

subdomain, and

∫

Ω

Ω















∂

∂

∂

∂

+

∂

∂

∂

∂ = e

e

x x y y

k

1

1 1

1

T

1 1

T

1

1 d

N N N N ke and 

























∂

∂

+

∂

∂ =

=

=

=

=

i

i

i

i

y y

x x

y y

x x x y

k 2

T

2

2

2

T

2

2

2 2

N N

ke ,

∫

Γ

= − Γ e

e

I

1

I

1

T

1 k N R d s and ( ) 2 2 i i k R x , y s = − , (2.47)

and

d d

1 1

1 1

T

1 1 1

T

1 ∫ 1 ∫

Ω Γ

= Ω + Γ e

s

e

e

e s f N Q N q e and ( ) ( ) i i i i f Q x , y q x , y e2 = 2 + 2 ,

and for the two domains, i=1,2,
67

∫

Γ

= − Γ e

e

i i i

I

d T I kp R N ,

and

∫

Γ

= Γ e

e

i i

I

T I

I k T R d .

2.6. COMPUTATIONAL IMPLICATIONS

The two- and three-approximation multifunctional modeling approaches have

been generalized such that they are applicable to both homogeneous and heterogeneous

discretization approaches. Computational implications are presented in this section for

the generalized system of equations, Eqs. (2.42) and (2.46). Implications specific to a

discretization approach are highlighted, where appropriate.

The assembled stiffness matrix K is a block diagonal matrix containing the

stiffness matrices Ki of each of the subdomains along its block diagonal. The interface

“stiffness” matrix thus contains coupling terms that augment the stiffness matrices of the

subdomains along the interface. The two- and three-approximation approaches yield

systems of equations (see Eqs. (2.42) and (2.46)) of similar form and with the same

attributes. Due to the use of Lagrange multipliers in the constraint conditions, the

systems are neither banded nor positive definite. Therefore, standard Cholesky solvers

can not be used, unless full pivoting is performed to obtain the solution. In addition, due

to the generalization for the finite difference approximations, the system of equations is

not necessarily symmetric due to different off-diagonal submatrices, Kp and Ks. The

system unknowns in Eq. (2.46) consist of both primary and secondary variables given by
68

the potential function, uDQGWKHVHFRQGDU\YDULDEOHFRHIILFLHQWV UHVSHFWLYHO\

Generally, the coupling matrices, i

Ks , are of the order of the length of the interdomain

boundary, which results in a marked difference in the magnitude of the off-diagonal

terms of the system matrix compared to its diagonal terms. This characteristic produces

an ill-conditioned matrix whose solution can cause difficulties for some general-purpose

solvers. Hence, the coupling matrix should be scaled such that it is of the same order as

the subdomain stiffness. The upper diagonal submatrix blocks contain uncoupled

subdomain stiffness matrices. The symmetry of the subdomain matrix is determined by

the choice of the weight function, Φ. For the finite element discretization, the subdomain

matrices are symmetric. However, due to the elimination of fictitious nodes for the

imposition of boundary conditions and loads in the finite difference discretization, the

subdomain stiffness matrices, Ki, generally are not symmetric, but they are positive

definite and sparse. The coupling is accomplished through the introduction of the

coupling terms in the matrices, iKp and 

i

Ks , for both approaches. The threeapproximation approach requires an additional matrix, KI . For the three-approximation

approach, the number of additional degrees of freedom associated with the interface is

generally small in comparison with the total number of degrees of freedom in the

subdomains. Thus, modeling flexibility is provided at a relatively small computational

expense. The computational expense in this study may be reduced additionally as the

efficiency of new solution algorithms for the system of equations in Eqs. (2.42) and

(2.46) is increased.

The load transfer mechanism for finite element multiple-domain discretizations

presented by Aminpour et al.25 is generalized for the multifunctional approach, herein.
69

This load transfer mechanism may be interrogated for the two- and three-approximation

formulations by considering the first and second rows of Eqs. (2.42) and (2.46),

respectively. For the three-approximation approach, the matrix equations of interest are

given by

2 2 s 2 2

1 1 s 1 1

2

1

K u K f

K u K f

+ =

+ =

These equations can be partitioned such that they correspond only to the primary

variables, ui on the interdomain boundaries. That is, ui represents a subset of ui ;

hence,

K u K 0

K u K 0

+ =

+ =

2 2 s 2

1 1 s 1

2

1 (2.48)

whereKi denotes interdomain boundary stiffness terms related to ui , and there are no

forces on the interdomain boundary. The expressions given by the product term, Kiui ,

represent the internal fluxes at the i

th interdomain boundary, and thus Eq. (2.48) may be

written as

1 s 1 2 s 2 1 2

f = −K and f = −K . (2.49)

For homogeneous discretization using the finite element method, substituting for

i

Ks from Eq. (2.43) into Eq. (2.49) gives

∫ ∫

Γ Γ

= − Γ = − Γ

e

e

e

e

I I

d ˆ d I

1

T

1 1

I

1

T f1 N1 R N q (2.50)

∫ ∫

Γ Γ

= − Γ = − Γ

e

e

e

e

I I

d ˆ d I

2

T

2 2

I

2

T f2 N2 R N q
70

Examining Eq. (2.50) indicates that the evaluation of the internal fluxes is consistent with

the evaluation of equivalent nodal fluxes in the presence of applied fluxes on the

boundary. In addition, Eq. (2.50) substantiates that the secondary variables along the

interface are represented by distributed fluxes for each of the subdomains.

For homogeneous discretization using the finite difference method, substituting

for 

i

Ks from Eq. (2.44) into Eq. (2.49) gives

1 1 1 1 f = −R = −qˆ (2.51)

2 2 2 2 f = −R = −qˆ

Examining Eq. (2.51) indicates that the evaluation of the internal fluxes is consistent with

nodal fluxes evaluated at points in the presence of applied fluxes on the boundary. In

addition, Eq. (2.50) substantiates that the secondary variables along the interface for this

approach are represented by nodal fluxes for each of the subdomains.

For heterogeneous discretization using the combined finite element and finite

difference methods, substituting for i

Ks from Eq. (2.47) into Eq. (2.49) gives

∫ ∫

Γ Γ

= − Γ = − Γ

e

e

e

e

I I

d ˆ d I

1

T

1 1

I

1

T f1 N1 R N q (2.52)

2 2 2 2 f = −R = −qˆ

Examining Eq. (2.52) indicates for subdomain 1 (the finite element subdomain), that the

evaluation of the internal fluxes is consistent with the evaluation of equivalent nodal

fluxes in the presence of applied fluxes on the boundary. Meanwhile, for subdomain 2

(the finite difference subdomain), the evaluation of the internal fluxes is consistent with

nodal fluxes evaluated at points. This reveals that for this multiple-domain approach, the

secondary variables along the interface for subdomain 1 are represented by distributed
71

fluxes, while for subdomain 2 the secondary variables along the interface are represented

by nodal fluxes. Thus, for this heterogeneous modeling approach, it is required to

transform the interface secondary variables into equivalent quantities.

2.7. VERIFICATION TEST CASE

In this section, the multifunctional methodology for the scalar-field problem is

demonstrated on a verification test case. The application is described, and the associated

results and salient features are discussed. This application is considered a patch test for

the formulation and verifies the applicability of the method for a configuration for which

the solutions are known. Finite difference and finite element solutions for single- and

multiple-domain configurations are presented to provide benchmark solutions for the

multifunctional approach using homogeneous and heterogeneous discretization.

Representative applications from the field of engineering science are presented in

Chapter IV.

2.7.1. Patch Test Problems

 The patch test has proven to be a useful discriminator of the convergence

properties of finite elements and other discretization approaches. A patch test refers to

any problem with an exact solution as a constant state for which the approximating

primary variable is capable of reproducing. The fundamental concept of the patch test for

the scalar-field problem herein is to subject a domain to boundary conditions that

engender a linear or quadratic primary variable field and a constant or linear secondary

variable field throughout the domain. For the governing differential equation of the form

of Eq. (2.1), boundary conditions that serve this purpose are:

i. Specified primary variable on p Γ which emanate from a linear field as
72

1 2 a0 u = a x + a y +

or quadratic field as

( ) 2 3 0

2 2

1 u = a x − y + a x + a y + a

where a1, a2, a3,and a0 are arbitrary constants.

ii. Constant or linear secondary variable on s Γ

1 2 b0 q = b x + b y +

Given these boundary conditions, a solution is sought to the Laplace’s equation. This

governing equation is applicable to a variety of problems in engineering science. For

example, consider the solution for the primary variable, u(x,y), in a rectangular domain

(see Figure 2.7) with boundary conditions of the forms indicated which yield the exact

solution.

Figure 2.7. Two-Dimensional Rectangular Domain.

The problem is given by

x a y b

y

u

x

u = < < < <

∂

∂

+

∂

∂ 0, 0 0 2

2

2

2

,

x

y

(a,b)

q=h(x,y)

q=e(x,y)

u = f(x,y) u = g(x,y)
73

which is known as Laplace’s equation for a planar domain.

Results of the analyses performed have been compared to appropriate reference

solutions and are summarized in Table 2.2 using normalized values. A value of unity

implies perfect agreement with the reference solution. Specified boundary conditions

representing linear, bilinear, and quadratic potential functions are applied to the square

domain. For all cases, the reference solution is the exact solution. For the linear case, a

specified boundary condition of the form

u(0, y) = 2, u(a, y) = a + 2, and q (x,0) = q (x,b) = 0 n n

has been imposed. For the bilinear case, a specified boundary condition of the form

u(0, y) = y, u(a, y) = a + y, and q (x,0) = -1 and, q (x,b) = 1 n n

has been imposed. For the quadratic case, a specified boundary condition of the form

u y y u a y a y q x q x b b n n 0 and 0 0 and 2 2 2 2 ( , ) = − , ( , ) = − , ( , ) = , ( , ) = −

has been imposed. Several analyses have been performed namely, (1) two single-domain

analyses with individual finite element and finite difference discretizations, respectively,

(2) two multiple-domain analyses with homogeneous modeling with individual finite

element and finite difference discretizations, respectively, and (3) one multiple-domain

analysis with heterogeneous modeling with combined finite element and finite difference

discretizations. Results from these analyses are summarized in Table 2.2. In this work, a

five-node central difference template and four-node quadrilateral finite elements are used

to form the models. Spatial modeling is used consistent with single-domain modeling

approaches with a (5 × 5) mesh and a (9 × 9) mesh. The coarse and fine models, shown

in Figure 2.8, are used in the finite element homogeneous modeling. For the finite

difference homogeneous modeling and the heterogeneous modeling, a finite difference
74

mesh is used that has the same number of mesh points as the finite element mesh in the

respective domain.

Figure 2.8. Spatial Discretization for Two-Dimensional Rectangular Domain.

For boundary conditions consistent with linear and bilinear potential functions,

the computed potential and flux results are exact for all analysis types. For boundary

conditions consistent with a quadratic potential function, the error in the computed

potential and flux is approximately 3% for the multiple-domain homogeneous finite

element (MDFE) spatial modeling, and the error is approximately 1% for the multipledomain heterogeneous modeling (MD/HM) with finite difference and finite element

discretization. For the given boundary conditions and element configuration (i.e., square

or rectangular elements), the single-domain finite element (SD/FE) model reproduces the

exact solution using the bilinear finite element. However, for a general element

orientation (i.e., quadrilateral elements), the bilinear element used does not reproduce the

exact solution. Moreover, for the multiple-domain analysis, error is introduced when

combining finite element models of different discretization along the boundary. This

error is due to the use of a higher-order interpolation function (i.e., cubic spline) on the

interface than that used to represent the potential on the finite element edges. The error
75

obtained using the heterogeneous model is smaller than that obtained for the

homogeneous finite element model. This attribute is due to the ability of the finite

difference model to represent accurately the potential function on the interface based on

the higher-order shape function used in the generalization of the finite difference method.

Table 2.2. Results of the Multifunctional Approach for the Patch Test Problems.

Normalized Potential Function, u Normalized Flux, qx

Analysis Order of Potential function Order of Potential Function

Type* Linear Bilinear Quadratic Linear Bilinear Quadratic

SD/FE 1.0 1.0 1.0 1.0 1.0 1.0

SD/FD 1.0 1.0 1.0 1.0 1.0 1.0

MD/FE 1.0 1.0 1.03125 1.0 1.0 1.03125

MD/FD 1.0 1.0 1.0 1.0 1.0 1.0

MD/HM 1.0 1.0 .98958 1.0 1.0 .98958 *

 SD/FE: Single-Domain with Finite Element discretization

 SD/FD: Single-Domain with Finite Difference discretization

 MD/FE: Multiple-Domain with Finite Element discretization

 MD/FD: Multiple-Domain with Finite Difference discretization

 MD/HM: Multiple-Domain with Heterogeneous Modeling (combined finite

difference and finite element discretizations)
76

CHAPTER III

MULTIFUNCTIONAL APPROACH FOR VECTOR-FIELD

PROBLEMS

3.1. GENERAL

While a scalar-field problem is one in which the dependent variable is a scalar and

requires only the specification of magnitude for a complete description, a vector-field

problem is one in which the dependent variable is a vector of components and requires

the specification of magnitude and direction. Many of the concepts outlined for the

scalar-field problem in the previous chapter are readily extendable to the vector-field

problem, which allows further generalization of the multifunctional approach developed

herein. A representative example of the vector-field differential equation in two

dimensions is considered, and the mathematical statement is formulated. The concepts

developed here are directly applicable to one-, two-, and three-dimensional vector-field

problems; however, only the two-dimensional development is included in the interest of

brevity. The general form of the differential equation describing the vector-field problem

governing the motion of a continuum is given by the equilibrium equation

( )

dt

d v b T ρ ρ + ∇ ⋅ = (3.1)

where the variables ρ, b, T and v are the material mass density, the body force per unit

volume, the stress tensor and the velocity vector, respectively. Eq. (3.1) is subject to the

natural boundary condition, t = T⋅n = t on Γs

, and essential boundary conditions,

u = u , on ΓpZKHUHWKHQRUPDOYHFWRUWRWKHERXQGDU\ LVJLYHQE\ i j ˆ ˆ x y n = n + n , and
77

nx and ny are direction cosines of the unit normals, i

ˆ and j

ˆ . In addition, t , and u are

applied tractions, and prescribed displacements, respectively, and v is the initial velocity

vector. The equilibrium equations must be satisfied within the domain. Note that instead

of prescribing the tractions on the boundary, boundary conditions may be given in terms

of displacement or velocity components. Furthermore, boundary conditions on Γ may be

mixed (i.e., surface forces, t, may be prescribed on one part of the boundary and

displacements or velocities may be prescribed on another). The equilibrium equation and

other governing equations of continuum mechanics are discussed in more detail in the

following section.

3.2. CONTINUUM MECHANICS FOUNDATIONS

The conservation of mass, linear momentum, angular momentum, energy, and

entropy give rise to field equations that govern the deformation and motion of a

continuum, and these equations are given in the form of integral or differential equations.

In deriving the governing equations, the starting point is a statement of the conservation

principle applied to a “control volume” to develop the integral form of the equation and

extract the differential form by using the divergence theorem.

3.2.1. Principle of Conservation of Mass

The principle of conservation of mass states that when the total mass of the body

is unchanged for an arbitrarily small neighborhood of each material point, the mass is

considered to be conserved locally. Hence, the rate of increase of the mass inside the

control volume is equal to the net inflow of mass through the control surface.

Mathematically, this principle is given by
78

( )dV = 0 

 



 + ∇ ⋅ ∂

∂ ∫

V t

ρv ρ

Since the integral is equal to zero for arbitrary respective volumes, V, the integrand must

be equal to identically zero everywhere in the domain. The resulting equation, known as

the continuity equation, is well known in fluid dynamics and is given in a conservative

form by

( ) ( ) 0 or = 0 ∂

∂

+

∂

∂

+ ∇ ⋅ = ∂

∂

i

i

x

v

t t

ρ ρ ρ ρ v (3.2)

The differential equation takes on a slightly different form when the derivatives of

products are expanded and the definition of the material derivative is considered. The

resulting non-conservative form is given by

0 d

d 0 or 

d

d = ∂

∂

+ ∇ ⋅ = +

i

i

x

v

t t

ρ ρ ρ ρ v

If the material is incompressible so that the density in the neighborhood of each material

particle remains constant as it moves, the continuity equation takes the simpler form

0 or = 0 ∂

∂ ∇ ⋅ =

i

i

x

v

v (3.3)

This is known as the condition of incompressibility, which is important in classical

hydrodynamics and plasticity theories. The continuity equation is an important partial

differential equation in all branches of continuum mechanics and the discipline-specific

aspects are discussed in the next section.

3.2.2. Conservation of Linear Momentum

The equations of motion, valid in all branches of mechanics, are partial

differential equations derived from the momentum principles of a collection of particles.
79

In this case, it is easier to use integrals over a given mass of material (the material

volume, V ′) rather than over a given spatial volume (the control volume, V). The

Reynolds transport theorem is used to replace the material volume with the control

volume. The conservative form of this theorem is given by

V S

t

V

t V V S

d d d

d

d ∫ ∫ ∫ + ⋅ ∂∂ ′ =

′

φρ φρ φρv n

where φ is the continuum property per unit mass and S

S

∫ ρv ⋅n d is recognized as the mass

flux. The conservation of linear momentum represents Newton’s second law and governs

the motion of the continuum under the influence of the external effects. This principle

states that the time rate of change of momentum is equal to the resultant force, F, acting

on the body. Thus, dt

dL F = where F is the resultant of all external forces and is given

acting on a material volume as V S

V S

= ′ + ′ ∫ ∫ ′ ′

F ρb d t d , and L is the linear momentum

vector on a material volume given by V

V

= ′ ∫

′

L vρ d . First, expressing the conservation

of linear momentum over the material volume and then using the Reynolds transport

theorem to express the equation in terms of the control volume yields the integral

conservative momentum equation given by

V S V S

V S V S

d d

t

b d t d v + v v ⋅n

∂

∂ ∫ ρ + ∫ = ∫ ρ ∫ ρ

Using the divergence theorem and Cauchy’s formula, the conservative differential form

may be obtained as

( ) ( ) vv

v b T ρ ρ ρ + ∇ ⋅ ∂

∂

+ ∇ ⋅ = t
80

The non-conservative form of the differential equations is obtained by expanding the

divergence operator, ∇ ⋅( ) ρvv , and making use of the continuity equation, Eq. (3.3),

yielding

( ) ( )

t

v

x

T

b

t

i

j

ij

i d

d or 

d

d ρ ρ ρ ρ = ∂

∂

+ ∇ ⋅ = +

v b T

3.2.3. Conservation of Angular Momentum

The principle of conservation of angular momentum is used to show symmetry of

the stress tensor, which is used to describe the state of stress of the continuum. In a

collection of particles whose interactions are equal, opposite and collinear forces, the

time rate of change of the total moment of momentum for the given collection of particles

is equal to the vector sum of the moments of the external forces acting on the system. In

the absence of distributed couples, the same principle for a continuum is postulated.

Thus,

() ( ) ( ) V

t

S V

S V V

d

d

d ∫ r × t d + ∫ r × ρb d = ∫ r × ρv

where × denotes the vector cross-product operation. Upon expressing the cross products

in indicial notation, transforming the surface integral to a volume integral (using the

divergence theorem), and using the expression for the material derivative of a volume

integral, the moment of momentum equation is reduced to

ekrsTsr = 0

at each point where ekrs is the permutation operator. This yields

For r=1 T32 - T23 = 0
81

For r=2 T31 - T13 = 0

For r=3 T12 - T21 = 0

establishing the symmetry of the stress tensor in general without any assumption of

equilibrium or of uniformity of the stress distribution. However, the balance of the

couple stresses is assumed. In reference 39, a proof is given for symmetry of the stress

tensor involving the condition that the rates of change of the components of stress remain

finite.

3.2.4. Conservation of Energy

The principle of conservation of energy states that energy is conserved if the time

rate of change of the kinetic and internal energy is equal to the sum of the rate of work of

the external forces and all the other energies that enter or leave the body per unit time.

Such energies supplied may include thermal energy, chemical energy, or electromagnetic

energy. Herein, only mechanical and thermal energies are considered, and the energy

principle takes the form of the well-known first law of thermodynamics. Since the

energy equation involves an additional unknown quantity, the internal energy, the

equation is a useful addition to the equations of continuum mechanics only when it is

possible to relate the internal energy to the other state variables; in traditional

thermodynamics an equation of state furnishes the required relation. The first law of

thermodynamics applied to a material volume may be written as

K& +U& = W + Q

where the superscripted dot, ( )& , represents the derivative with respect to time, and K& is

the rate of increase of the kinetic energy of the material volume ,U& is the rate of increase

of the internal energy of the material volume, W is the rate of work done by the external
82

forces on the material volume, and Q is the rate of heat added to the material volume.

The individual variables are defined as follows:

V

dt

d K

V

= ⋅ ′ ∫

′

d

2

1

ρv v &

u V

dt

d U

V

= ′ ∫

′

ρˆ d &

W V S

V S

= ⋅ ′ + ⋅ ′ ∫ ∫ ′ ′

ρb v d Tv n d

Q S r V

S V

= − ⋅ ′ + ′ ∫ ∫ ′ ′

q n d ρ d

where uˆ is the specific internal energy, q is the heat flux vector and r is the radiative heat

transfer per unit mass. Upon using Reynolds transport theorem to convert the material

volume to the control volume and the divergence theorem to convert the surface integrals

to volume integrals, and performing further algebraic manipulation, the energy takes the

form

V V r V V

t

u

V V V V

d d d d

d

d ∫ ∫ + ∫ + ∫ = − ∇ ⋅q T:D ˆ ρ ρ

where the stress power,T:D , is the scalar product of the stress tensor, T, and the rate of

deformation tensor, D. The differential forms are given by

q T:D ˆ = −∇ ⋅ + r +

t

u

ρ ρ d

d

or

ij ij

i

r T D

t x

u

+ +

∂

∂

ρ = − ρ ˆ q

d

d
83

If only mechanical quantities are considered, the principle of conservation of

energy for the continuum may be derived directly from the equation of motion. This

equation, referred to as the conservation of mechanical energy, states that the rate of

increase of the internal energy equals the heat added per unit time plus the stress power

that is not contributing to the kinetic energy. The equation is given by

q T:D uˆ = −∇ ⋅ + r +

t

ρ ρ d

d (3.4)

3.2.5. Second Law of Thermodynamics

The second law of thermodynamics is automatically satisfied and includes the

change in entropy of the continuum. The entropy is regarded as a measure of change of

energy dissipation with respect to temperature. The relationship expressing conversion of

heat and work into kinetic and internal energies during a thermodynamic process is set

forth in the energy equation. The first law, however, leaves unanswered the question of

the extent to which the conversion process is reversible or irreversible. The basic

criterion for irreversibility is given by the second law of thermodynamics through the

statement on the limitations of entropy production. For a general process, the energy

equation and the second law of thermodynamics are combined yielding

T

Q

t

q

t T

s

ρ = +

d

1 d

d

dˆ

where d is the change in the entropy per unit mass, T is the absolute temperature, sˆ t

q

d

d is

the heat transferred per unit time per unit mass, Q is the dissipative function obtained

from ij

D Q = Tij D using the dissipative or deviatoric stress tensor TD, and the notation d

is used to indicate that the quantity is not an exact differential. The deviatoric stress
84

tensor is defined by ij ij

D Tij = T − pδ where − p is the hydrostatic pressure. For a general

process, Q ≥ 0

t

q

t T

s

d

1 d

d

d ≥ ˆ

and for an adiabatic process,

0

d

d ≥

t

sˆ

where in each of the above equations, the equality condition holds for a reversible

process and the inequality condition holds for an irreversible process.

The general principles of continuum mechanics have been outlined in this section

to provide a foundation for the basic equations governing the motion of general continua.

In the derivation of the balance laws, no differentiation has been made between various

types of substances. The character of the material is brought into the formulation through

appropriate constitutive equations for each material with the constitutive variables being

restricted in their regions of definitions. These and other discipline-specific attributes are

outlined in the following section.

3.3. DISCIPLINE SPECIFICS

The constitutive equations characterize the individual material and its reactions to

applied loads. Hence, in the following section, the discipline-specific attributes of solid

and fluid continua and their impact on the general principles of continuum mechanics are

reviewed. In addition, other salient characteristics of the governing equations for solids

and fluids are discussed.

All constitutive equations must be consistent with the general principles of

continuum mechanics. While impact of the constitution of the continua is discussed for
85

all of the balance laws, emphasis is placed on the principle of conservation of linear

momentum. This principle is the basis for the governing equations of the multifunctional

approach presented herein. This law states that the sum of the body forces together with

the sum of the contact forces is equal to the change of the linear momentum of the

material. The law is used as the basis for describing the motion in both solid mechanics

and fluid mechanics.

3.3.1. Solid Mechanics

The field of solid mechanics has traditionally been characterized by wellformulated analysis of mechanical phenomena occurring in engineering systems,

combined with experiments that explore the basic concepts40. Herein, elasticity theory is

the primary field of solid mechanics discussed. In classical linear elasticity theory, it is

assumed that displacements and displacement gradients are sufficiently small such that

no distinction need be made between the Lagrangian and Eulerian descriptions. It is

further assumed that the deformation processes are adiabatic (no heat loss or gain) and

isothermal (constant temperature). The conservation of mass states that the mass of a

deformed piece of material is the same as the mass of the undeformed material. In

elasticity, based on the small strain assumption, the density, ρ, in the deformed state may

be approximated by the density, ρ0

, in the undeformed state, and the conservation of mass

is identically satisfied.

Moreover, it is convenient to identify a material particle of the continuous body

by giving its initial coordinates. The position coordinates, x, y, z appearing in the partial

derivatives and the integrals in the foregoing derivatives are, however, the instantaneous

positions. For an elastic body in equilibrium, they represent the coordinates of a particle
86

in its new position in the deformed body. When the strains and displacements are small,

it may be possible that the equilibrium conditions are satisfied in the undeformed

configuration of the body. The equilibrium differential equations are strictly applicable

and the stress tensor is strictly symmetric for the nonpolar case only when defined in the

instantaneous deformed position. Even in small strain theory of elasticity, it is necessary

to take account of this attribute in applications where the instability may occur, as in the

buckling of a column or a shell. Asymmetry of the stress tensor also occurs when there is

distributed couple stress6

.

In ideal elasticity, heat transfer is considered insignificant, and all of the input

work is assumed to be converted into internal energy in the form of recoverable stored

elastic strain energy, which can be recovered as work when the body is unloaded. In

general, however, the major part of the input work into a deforming material is not

recoverable energy stored, but dissipated by the deformation process, causing an increase

in the body’s temperature and eventually being conducted away as heat. When thermal

effects are neglected, the energy balance equation may be written as

Tij Dij Tij ij t

u

ε

ρ ρ

1 1

d

d = = ˆ

The internal energy, uˆ , in this case is purely mechanical and is called the strain energy

density (per unit mass)

u Tij ij ε

ρ

d 1 d ˆ =

A material body is said to be ideally elastic when the body recovers (under

isothermal conditions) its original form completely upon removal of the forces causing

deformation, and there is a one-to-one relationship between the state of stress and state of
87

strain. The generalized Hooke’s law relates the nine components of stress to the nine

components of strain

ij Cijkl kl σ = ε

Symmetry of stress and strain reduces the number of material constants in the fourthorder tensor, Cijkl , from 81 to 36. The existence of the strain energy density functional

further reduces the number of constants to 21. The existence of three mutually

orthogonal planes of symmetry reduces the number of constants to nine. Isotropy reduces

the number of constants to two.

For this special case, Hooke’s law reduces to

[ ( )] ij ij kl ik jl il jk kl σ = λδ δ + μ δ δ + δ δ ε (3.5)

where

( ) ( )( ) ν ν

ν λ

ν

μ 1 1 2 ; 2 1 + − = + = = E E G

For i=j=1, the second and third terms of Eq. (3.5) are nonzero if k=1 and l=1. Thus,

σ 11 = λ + 2με11 e

where 11 22 33 e = ε + ε + ε . For i=1 and j=2, the second term of Eq. (3.5) is nonzero if

k=1 and l=2 and the third term is nonzero if k=2 and l=1. Thus,

σ 12 = με12 + με 21 = 2με12 .

Similarly, other components of stress may be defined.

Noting that the linear strain-displacement relationship is given by

( ) ij i j j i u u = , + , 2

1

ε
88

One method of solution of the problems of elasticity is to eliminate the stress components

in the equilibrium equations given in indicial notation as

σ ij, j + ρ( ) bi − v&i = 0 ,

and using Hooke’s law to express the strain components in terms of the displacements.

Eq. (3.5) may be written, with no loss of generality, as

σ ij = λε llδ ij + 2με ij

Solving the boundary-value problem involving 15 equations for 15 unknowns is a

formidable task. There are several ways of formulating the problem in terms of fewer

unknowns and fewer equations. The most straightforward method is to obtain the

stresses in terms of displacement gradients, and then substitute into the equilibrium

equations to obtain three second-order partial differential equations for the three

displacement components. Therefore, in terms of displacements,

( ) σ ij = λul,lδ ij + μ ui, j + u j,i

and

2

2

t

u

v u i

i i

∂

∂ & = && =

Substituting these expressions into the equilibrium equation yields

λul,ljδ ij + μ(ui, jj + u j,ij)+ ρ( ) bi − u&&

i = 0

or

λul,li + μ(ui, jj + u j,ij)+ ρ( ) bi − u&&

i = 0

Noting that l is a dummy index in the term ul,li. The equation may be written as

λu j, ji + μ(ui, jj + u j,ij)+ ρ( ) bi − u&&

i = 0
89

This leads to the field equations of Navier

() ( ) λ + μ u j, ji + μui, jj + ρ bi − u&&

i = 0

or

() ( ) 0 2 μ∇ ui + λ + μ u j, ji + ρ bi − u&&

i = (3.6)

The conditions for the static equilibrium of an elastic body are described by an elliptic

system of nine partial differential equations for the displacements and stresses.

3.3.2. Fluid Mechanics

Fluids whose constitution is described by linear constitutive relations are called

Newtonian fluids. The subject of Newtonian fluids is generally referred to as fluid

mechanics, which encompasses widely diverse topics including, but not limited to,

motion of airplanes and missiles through the atmosphere, the flow of liquids and gases

through ducts, and the transfer of heat and mass by fluid motion. The constitutive

equations for these fluids are given by

σ ij = −Pδ ij + Cijkl Dkl

where P is the thermodynamic pressure and Dkl are the components of the rate of

deformation tensor

( ) kl k l l k D v v = , + , 2

1

For isotropic fluids, the last term in the constitutive equations may be written as

Cijkl Dkl = λDrrδ ij + 2μDij

or

[ ( )] Cijkl Dkl = λ δ ijδ kl + μ δ ikδ jl + δ ilδ jk Dkl

Therefore,
90

[ ( )] σ ij = −Pδ ij + λ δ ijδ kl + μ δ ikδ jl + δ ilδ jk Dkl

and by evaluating the Kronecka delta parameters,

σ ij = −Pδ ij + λDkkδ ij + 2μDij (3.7)

This is the Navier-Poisson law for a Newtonian fluid.

As in linear elasticity, substituting the constitutive equation into the equation of

motion yields

− P, jδ ij + λvk,kjδ ij + μ(vi, jj + v j,ij)+ ρ(bi − v&i ) = 0

expanding gives

− P,i + ( ) λ + μ v j, ji + μvi, jj + ρ(bi − v&i ) = 0

or

( ) i i j ji i jj bi ρv& = −P, + λ + μ v , + μv , + ρ

or in vector form

P ( )( ) v v b vρ = −∇ + λ + μ ∇ ∇ ⋅ + μ∇ + ρ 2

dt

d

Using the Stokes condition, λ μ

3

2 = − , the equations reduce to the Navier-Stokes

equations and are given by

i i j ji i jj i v P v μv ρb μ ρ = − , + , + , +

3 & (3.8)

or

P ( ) v v b vμ ρ μ ρ = −∇ + ∇ ∇ ⋅ + ∇ + 2

dt 3

d (3.9)

In this form, the difference between the Navier equations of solid mechanics, Eq. (3.6)

and the Navier-Stokes equations of fluid mechanics, Eq. (3.8) or (3.9), can be readily
91

considered. In Navier-Stokes equations, there is not only an additional pressure term but

also the equations are nonlinear; this can be seen by examining the acceleration,

i j j

i i

i v v

t

v

dt

dv

v + , ∂

∂ & = = , and from the products of the density, ρ, and the acceleration, v&,

present in the equation. Additional nonlinearities are evident in the continuity equation

given by vk,k=0 (∇⋅v=0). In the linear theory of elasticity, this situation does not occur

since 2

2

t

u

vi

∂

∂ & ≈ and ρ is taken as a constant. The Navier-Stokes equations together with

the continuity equation form a complete set of four equations and four unknowns: the

pressure, P, and the three velocity components, vi.

For steady and low-speed flow of an incompressible fluid (∇⋅v=Dkk=0), for

constant ρ and by making use of the divergence-free condition in Eq. (3.8) or (3.9), the

governing equations take the form

Dkk = 0

− P,i + μvi, jj + ρbi = 0

However, these equations, often referred to as Stokes equations, may be written for twodimensions in the most general form without using the divergence-free condition to

simplify the equations. In so doing, the physical form of the natural boundary conditions

is preserved. The form of these equations is given by

0

2

2

1

1 = ∂

∂

+

∂

∂

x

v

x

v (3.10)
92

2

2 2

2

2

2

1

2

2

1

1

1

1 1

2

2

1

2 2

1

1

2

2

2

b

x

P

x

v

x

v

x

v

x

b

x

P

x

v

x

v

x x

v

μ ρ

μ ρ

= ∂

∂

+

















∂

∂

+















∂

∂

+

∂

∂

∂

∂ −

= ∂

∂

+

































∂

∂

+

∂

∂

∂

∂

+

∂

∂ −

Fluids often behave as though they are inviscid or frictionless. Therefore, it is

useful to investigate the dynamics of an ideal fluid that is incompressible and has zero

viscosity. For frictionless flow of an incompressible fluid, the equations, called Euler’s

equations, may be obtained from the general Navier-Stokes equations. Since in a

frictionless flow, there can be no shear stress present and the normal stress is the negative

of the thermodynamic pressure, the equations of motion are

i i i ρv = −P + ρb , &

or

P b vρ = −∇ + ρ ∂

∂

t

For a general fluid, the character (e.g., elliptic, hyperbolic, or parabolic) of these

equations of motion is determined by the sign of the discriminant. The Navier-Stokes

system of equations, in general, is considered as mixed elliptic, parabolic and hyperbolic

equations. The system of time-dependent Navier-Stokes equations is essentially

parabolic in time and space, although the continuity equation has a hyperbolic structure.

Therefore, they are considered a parabolic hyperbolic system. For the same reason, the

steady-state form of the Navier-Stokes equations leads to elliptic-hyperbolic properties.

In addition, the classification of the differential equation changes with the flow
93

characteristics (i.e., subsonic, supersonic, or transonic), which may create great

difficulties in solution where part of the flow is supersonic and part of it is subsonic.

3.4. SINGLE-DOMAIN FORMULATION

As in the scalar-field problem, methodology for the vector-field problem is

presented formulating the general method of weighted residuals for a single domain.

Consider the equilibrium equation governing the motion, u, of a continuum

∂t

∂

+ ∇ ⋅ = v

ρb T ρ

or in indicial notation

σ ij, j + ρ(bi − v&i ) = 0 in Ω for i, j = 1,2,3 (3.11)

 in a domain, Ω, bounded by Γ. In this work, the equilibrium equations of Eq. (3.11)

describe the motion of a three-dimensional continuum. Hence, the indices, i and j range

from the value of unity to three (i.e., i,j = 1,2,3). This range will apply throughout this

development unless otherwise specified. In general, the boundary, Γ, can have mixed

boundary conditions with the primary variables, u, prescribed on Γ

p

and the secondary

variable, the traction, t, prescribed on the remaining part of the boundary, Γ

s

. In solid

mechanics, the six stress components will be some general functions of the components

of the generalized displacement

[ ] w x y z = u v θ θ θ Tu

where u, v, and w are translational components and x, y and z are rotational

components. In fluid mechanics, the stress components will be functions of the velocity

vector

[ ] 1 2 3

Tu = v v v ,
94

which has similar components to those of the displacement vector. Thus, Eq. (3.11) can

be considered as a general equation of the form of A( ) u = 0 .

The method of weighted residuals is applied to the vector-field problem in this

chapter in the same way as for the scalar-field problem of Chapter II. Hence, an

approximate solution, u

~ , is used in expressing σij,j through the use of stress-strain and

strain-displacement (or stress-rate of strain) relations, then the differential equation, Eq.

(3.11), will no longer be satisfied, and this lack of equality is a measure of the departure

of u

~ from the exact solution. The lack of equality is called the residual, R, and is written

as

Ri = σ ij, j + ρ(bi − v&i) ≠ 0 for i,j = 1,2,3 .

The residual is orthogonalized by a set of weight functions, Φi and may be written as

∫ Φ dΩ = ∫( + − )Φ dΩ = 0

Ω Ω

i i ij j i i i R (b v ) , σ ρ & (3.12)

where the approximate solution is given by m

n

m

= Ψ + ∑amΨ

=1

0 ~

u . As defined before, the

IXQFWLRQV m, are trial functions, and am are arbitrary coefficients. The trial functions

VDWLVI\WKHKRPRJHQHRXVSDUWRIWKHHVVHQWLDOERXQGDU\FRQGLWLRQVZKLOH 0 satisfies the

nonhomogeneous part. Using the general weighted residual form outlined in Chapter II,

∫ ( ) dΩ + ∫ ( )dΓ = 0

Ω Γ

u

~ u B ~ A .

where the residual in the satisfaction of the boundary conditions is orthogonalized by a

secondary weight function, Φ . For the system at hand, a vector quantity is sought and

the differential equation is a simultaneous system of equations. Here,
95

A( ) u = σ ij, j + ρ(bi − v&i ) = 0, and the essential and natural boundary conditions are

represented by

( ) p i i B1 u = u − u = 0 or u − u = 0 on Γ

and

( ) s

i i B2 u = t − t = 0 or t − t = 0 on Γ ,

respectively. Therefore, considering the approximate solution, u

~ , we may write the

general integral form of the differential equation governing the continuum motion as

( ) ( )d ( )d 0 ~ ∫Φ , + ( − ) dΩ + ∫ Φ 1 Γ + ∫ Φ 2 − Γ =

Ω Γ Γ p s

s

i i i p i ij j i i i i i σ ρ b v& u -u t t (3.13)

Note that the approximate solution may be selected to satisfy the essential and the natural

boundary conditions and thus the boundary integral equations in Eq. (3.13) are identically

zero. In this formulation, we will presume that the essential boundary conditions, i.e.,

p i i − = u − u = 0 on Γ ~ or ~

u u 0

are automatically satisfied by the choice of the function, u

~ . Therefore, Eq. (3.13) is

rewritten as

∫Φ ( ) + − dΩ + ∫ Φ ( ) − dΓ = 0

Ω Γ

s

i ij j i i i i i s

(b v ) t t , σ ρ & (3.14)

where Φi2 = Φi .

In the formulation herein, the order of differentiation on the stress term in the

integral equation, Eq. (3.14), is reduced to obtain the weak formulation. Recognizing that

the stress components are functions of the primary variable, u, which is approximated by

u

~ . For simplicity, the subsequent development is presented in terms of u. Application

of the divergence theorem to Eq. (3.14) yields
96

− ∫ Φ dΩ + ∫( )Φ dΓ + ∫ − Φ dΩ + ∫( ) − Φ dΓ = 0

Ω Γ Ω Γ

s

ij i j ij j i i i i i i i s

σ , σ n ρ(b v& ) t t . (3.15)

Note that the domain boundary is presumed to consist of boundaries on which the

primary variable is specified and boundaries on which the secondary variable is specified,

and p s Γ = Γ + Γ . Therefore, the boundary integral on Γ, may be expressed as

( ) () () ∫ ∫ ∫

Γ Γ Γ

Φ Γ = Φ Γ + Φ Γ

p s

s

ij j i p σ ijn j i d σ ijn j i d σ n d .

In the method of weighted residuals, the weight functions, Φ, satisfy the homogeneous

boundary conditions for the primary variable, and thus, Φ=0 on Γ

p

. Therefore, the

boundary integral on Γ

p

 is identically zero and Eq. (3.15) may be rewritten as

− ∫ Φ dΩ + ∫( )Φ dΓ + ∫ − Φ dΩ + ∫( ) − Φ dΓ = 0

Ω Γ Ω Γ

s

i i i i i i

s

ij i j ij j i s s

σ , σ n ρ(b v& ) t t .

Since the weight functions, and , are arbitrary, they may be chosen, without loss of

generality, such that, = − , and using the Cauchy formula, i ijn j t = σ ,

− ∫ Φ dΩ + ∫ − Φ dΩ + ∫ Φ dΓ = 0

Ω Ω Γ

s

ij i j i i i i i s

σ , ρ(b v& ) t (3.16)

or

s

ij i j i i i i i s

∫ Φ Ω = ∫ v − b Φ Ω + ∫ t Φ Γ

Ω Ω Γ

σ , d ρ(& ) d d .

The integral form of Eq. (3.16) is given for a general continuum. If the weight functions,

Φi , are selected to be virtual displacements or velocities,δui , then Eq. (3.16) is given by

− ∫ dΩ + ∫ − dΩ + ∫ dΓ = 0

Ω Ω Γ

s

ij i j i i i i ui u b v u t

s

σ δ , ρ( & )δ δ . (3.17)

The term ui j δ , can be expanded to
97

( ) ij ij j i j δui δ u  = δ ε +ω 

 



 , = ,

ZKHUH ij DQG ij are symmetric and skew-symmetric tensors, respectively. These tensors

are given by

( ) ( ) ij i j j i ij i j j i u u u u = , + , = , − , 2

1 and 2

1

ε ω .

 In solid mechanics, these tensors represent the linear infinitesimal strain-displacement

and linear infinitesimal rotation tensors, respectively. In fluid mechanics, the tensors

represent the linear infinitesimal strain-rate of deformation and vorticity tensors,

UHVSHFWLYHO\1RWLQJWKDW ij is a symmetric tensor and that the product of a symmetric

tensor and a skew-symmetric tensor is zero, Eq. (3.17) may be rewritten as

− ∫ dΩ + ∫ − dΩ + ∫ dΓ = 0

Ω Ω Γ

s

ij ij i i i i ui b v u t

s

σ δε ρ( & )δ δ (3.18)

Eq. (3.18) represents the principle of virtual work where the first integral term represents

the internal virtual work, the second and third terms represents the external virtual work

due to body forces, inertial forces and surface tractions.

In the virtual work development, the term virtual work is loosely used for fluid

mechanics and has been included here to highlight the similarities between solid and fluid

mechanics. Variational techniques for perfect fluids, non-Newtonian fluids and general

Navier-Stokes equations are discussed in Finlayson7. In this work, concentration is given

to the general weighted residual equations, Eq. (3.16), and these equations form the basis

of finite element approximations, which will be presented briefly in a subsequent section.

Thus far, the single domain formulation has been developed for the vector-field

problem focussing on the momentum equation, which is applicable to general continua.
98

However, the motion of a fluid is governed by the conservation laws of mass, momenta,

and energy. In general, these equations consist of a set of coupled nonlinear, partial

differential equations in terms of the velocity components, temperature, and pressure.

When the Reynolds number for the flow is very low, the nonlinear terms due to inertial

effects can be neglected, resulting in a linear boundary value problem. Such a flow is

called Stokes flow41 (see Eq. (3.10)). When temperature effects are not important, the

energy equations are uncoupled from the momentum (i.e., Navier-Stokes) equations.

Thus, for isothermal flows, only the Navier-Stokes, Eq. (3.8), and continuity, Eq.

(3.2), need to be solved. Hence, an additional equation expressing the continuity

condition is included in the weighted residual formulation. In the interest of

completeness, the formulation herein is described using a Newtonian fluid. The laws

governing the flow of Newtonian fluids were reviewed in Section 3.3.2 in which the

equations were specialized to viscous fluids that are subject to the assumption of

incompressibility. Under these conditions, the weighted residual statement of the

equation of continuity, Eq. (3.3), is given by

ˆ d 0 , ∫ Φ Ω =

Ω

j ui (3.19)

where the residual in the continuity condition is orthogonalized by the weight function,

Φˆ , and , j

j

j i

x

v

u

∂

∂ = ∇ ⋅u = . Hence, for fluid mechanics, both Eqs. (3.16) and (3.19) are

the weighted residual statements required to approximate the continuum motion. While

for solid mechanics, since the continuity condition, Eq. (3.3) and likewise Eq. (3.19) are

automatically satisfied, Eq. (3.16) is the only weighted residual statement required to

approximate the continuum motion.
99

3.5. MULTIPLE-DOMAIN FORMULATION

As in the case of the scalar-field problem of Chapter II, the domain of the problem

is subdivided into smaller subdomains. Consider the equilibrium equation governing the

motion, u, of a continuum

σ ij, j + ρ(bi − v&i ) = 0 in Ω for i, j = 1,2,3 (3.20)

in the entire domain, Ω, bounded by Γ. For simplicity, the multiple-domain formulation

is presented for only two subdomains, Ω1 and Ω2 (see Figure 2.3) with a single interface

boundary. Independent approximations and weight functions are assumed in each of the

subdomains and continuity conditions are used to provide for a continuous solution

across the domain. Thus, Eq. (3.20) is satisfied in each subdomain, independently, i.e.,

() () () ( ) ( ) ( ) 2

2 2

1

2

1

1 1

1

1 σ ij, j + ρ (bi − v&

i ) = 0 in Ω and σ ij, j + ρ (bi − v&i ) = 0 in Ω

subject to the boundary conditions on the subdomain boundaries, Γ1 and Γ2, and the

superscripted numbers enclosed by parentheses denote the subdomain. In general, the

boundaries can have mixed boundary conditions with the primary variable, u, prescribed

on Γ

p

and the secondary variable, the traction, t, prescribed on Γ

s

. These boundary

conditions may be written as

( ) ( ) ( ) ( ) s

i i

p

i i u u t t 1

1 1

1 1

1 1

u1 − u1 = 0 or − = 0 on Γ and t − t = 0 or − = 0 on Γ

and

( ) ( ) ( ) ( ) s

i i

p

i i u u t t 2

2 2

2 2

2 2

u2 − u2 = 0 or − = 0 on Γ and t − t = 0 or − = 0 on Γ .
100

For the multiple domain case, the boundary at the interface between the two subdomains

is denoted ΓI. Hence, the subdomain boundaries, Γk, are presumed to include three

boundary types, and these boundaries are given by

 for 1,2 I Γ = Γ + Γ + Γ k = k

s

k

p

k k .

Here, the boundary on the subdomain common boundary is assumed to represent the

same geometry and thus, I I Γ = Γ k . The residual for each domain is orthogonalized by a

set of weight functions, ( ) k Φi and is written as

() () () ( ) ( ) ( ) d 1 0 1 1 1

1

1

,

1

∫ + − Φ Ω =

Ω

ij j i i i σ ρ b v&

and

() () () ( ) ( ) ( ) d 2 0 2 2 2

1

2

,

2

∫ + − Φ Ω =

Ω

ij j i i i σ ρ b v&

where the approximate solution is given by m

n

m

u a m 1

1

1 1 ~ = ∑ Ψ

=

 and m

n

m

u a m 2

1

2 2 ~ = ∑ Ψ

=

. The

IXQFWLRQV 1mDQG 2m , are the trial functions, and a1m and a2m are sets of arbitrary

coefficients. Using the general form outlined previously, (i.e.,

∫ ( ) dΩ + ∫ ( )dΓ = 0

Ω Γ

u

~ u B ~ A ), for each subdomain, one may write

() () ( ) ( ) ( ) ( )d 0 for 1,2 ~ d ~ ∫ Ω + ∫ Γ = =

Ω Γ

k k

k k k k

k k

A u B u

Therefore, considering the approximate solutions, ~( ) 1 u and ~( ) 2 u , the general integral form

of the differential equation governing the motion for subdomain 1 is given by
101

() () () () ( ) () () ( ) ( )

() () ( ) ( )d 0 ~

d ~ ( ) d

1

1

1

1

1

1 1 1

2

1 1 1

1 1

1 1

1

1

, 1

+ Φ Γ =

Φ + − Ω + Φ Γ

∫

∫ ∫

Γ

Ω Γ

s

p

s

i i i

p

i ij j i i i i i

t -t

σ ρ b v& u -u

 (3.21)

and for subdomain 2 as

() () () () ( ) () () ( ) ( )

() () ( ) ( )d 0 ~

d ~ ( ) d

2

2

2

2

2

2 2 2

2

2 2 2

2 1

2 2

2

2

, 2

+ Φ Γ =

Φ + − Ω + Φ Γ

∫

∫ ∫

Γ

Ω Γ

s

p

s

i i i

p

i ij j i i i i i

t -t

σ ρ b v& u -u

 (3.22)

Again, the essential boundary conditions, i.e.,

( ) ( ) p

i i u u 1

1 1

1 1 0 on ~ or ~

u − u = 0 − = Γ

and

( ) ( ) p

i i u u 2

2 2

2 2 0 on ~ or ~

u − u = 0 − = Γ

are identically satisfied by the choice of the functions, 1

~

u and 2

~

u . Therefore, for

subdomain 1, Eq. (3.21) is rewritten as

() () () () ( ) () () ( ) ( ) d ( )d 0

1

1

1

1 1 1

1

1 1

1

1

, 1 ∫Φ + − Ω + ∫ Φ − Γ =

Ω Γs

s

i ij j i i i i i σ ρ b v& t t (3.23)

where ( ) 1 ( ) 1

i2 Φi Φ = . Similarly, for subdomain 2,

() () () () ( ) () () ( ) ( ) d ( )d 0

2

2

2

2 2 2

2

2 2

2

2

, 2 ∫Φ + − Ω + ∫ Φ − Γ =

Ω Γs

s

i ij j i i i i i σ ρ b v& t t (3.24)

where ( ) 2 ( ) 2

i2 Φi Φ = .
102

The order of differentiation on the primary variable in the integral equations, Eq.

(3.23) and (3.24), is reduced to obtain the weak formulation. Using the divergence

theorem Eq. (3.23) can be rewritten, for subdomain 1, as

 

() () () () ( ) () () () ()

() () ( ) ( )d 0

d d ( ) d

1

1

1 1 1

1 1 1

1

1 1 1

1 1

1 1 1

1

1 1 ,

+ Φ − Γ =

− Φ Ω + Φ Γ + − Φ Ω

∫

∫ ∫ ∫

Γ

Ω Ω Γ

s

s

i i i

i j ij ij j i i i i

t t

σ σ n ρ b v&

 (3.25)

and similarly, for subdomain 2,

 

() () () () ( ) () () () ()

() () ( ) ( )d 0

d d ( ) d

2

2

2 2 2

2 2 2

2

2 2 2

2 2

2 2 2

2

2 2 ,

+ Φ − Γ =

− Φ Ω + Φ Γ + − Φ Ω

∫

∫ ∫ ∫

Γ

Ω Ω Γ

s

s

i i i

i j ij ij j i i i i

t t

σ σ n ρ b v&

 (3.26)

Recall that the boundary Γ is presumed to consist of boundaries on which the primary

variable is specified and of boundaries on which the secondary variable is specified, and

boundaries at the subdomain interface, and for subdomain k, I Γ = Γ + Γ + Γ s

k

p

k k .

Therefore, the boundary integral on Γk may be expressed as

()() ( ) () ()() ( ) () ()() ( ) ( )

()() ( ) ( ) ∫

∫ ∫ ∫

Γ

Γ Γ Γ

+ Φ Γ

Φ Γ = Φ Γ + Φ Γ

I

I d

d d d

k

i

k

j

k

ij

s

k

k

i

k

j

k

ij

p

k

k

i

k

j

k

k ij

k

i

k

j

k

ij

n

n n n

s

k

p

k

k

σ

σ σ σ

Noting that, Φ = 0 k

i on p

k Γ . Therefore, the boundary integral on pk Γ is identically zero,

and Eq. (3.25) can be rewritten, for subdomain 1, as
103

() () () () ( ) () () () ( ) ( )

() () ( ) ( ) () () () d ( ) d 0

d d d

1

1 1 1

1

1 1 1

1 1 1 1 1 1 I

1

1 1 ,

1 1

1

1 I

1

1

+ Φ − Γ + − Φ Ω =

− Φ Ω + Φ Γ + Φ Γ

∫ ∫

∫ ∫ ∫

Γ Ω

Ω Γ Γ

i i i

s

i i i

ij j i

s

i j ij ij j i

t t b v

n n

s

s

ρ &

σ σ σ

Since the weight functions, ( ) 1 Φi and ( ) 1 Φi , are arbitrary, they may chosen such that

() () 1 1

i Φi Φ = − , and using the Cauchy formula, ( ) 1 (1) (1)

i ij j t = σ n ,

() () () () ( ) ( ) () ()

() () () ( ) d 0

d d d

1

1 1 1

1

1 1 1 I 1 1

1

1 1 ,

1

1 1

1

I

+ − Φ Ω =

− Φ Ω + Φ Γ + Φ Γ

∫

∫ ∫ ∫

Ω

Ω Γ Γ

i i i

s

i j ij ij j i i i

b v

n t

s

ρ &

σ σ

 (3.27)

Similarly, for subdomain, Ω2,

() () () () ( ) () () ()

() () () ( ) d 0

d d d

2

2 2 2

2

2 2 2 I 2 2

2

2 2 ,

2

2 2

2

I

+ − Φ Ω =

− Φ Ω + Φ Γ + Φ Γ

∫

∫ ∫ ∫

Ω

Ω Γ Γ

i i i

s i j ij ij j i i i

b v

n t

s

ρ &

σ σ

 (3.28)

In the two-approximation formulation for the scalar-field problem, the two

primary field variables, u1 and u2 are approximated independently, and continuity

requirements between these two fields are satisfied at the subdomain interface boundary.

The three-approximation approach, which makes use of a third approximation field for

the primary variables along the subdomain interface boundary in addition to the

approximations given along the boundary of the subdomains, is most general. Hence,

only the three-approximation approach will be discussed for the vector-field problem.

This primary variable, v, along the interface is assumed to be independent of the primary
104

variables, u1 and u2, of the subdomains to which it is attached. These independent

approximations give rise to continuity requirements along the interface of the form

( ) 1 I

1 − = or − = 0 on Γ i i v u 0 v u

( ) 2 I

2 − = or − = 0 on Γ i i v u 0 v u

These constraints can be satisfied in the integral sense as

( ) () () ( ) I 1 1 I I

1 1 d 0 or d 0 on 

I I

∫ − Γ = ∫ − Γ = Γ

Γ Γ

i i i v u λ v u (3.29)

( ) () () ( ) I 2 2 I I

2 2 d 0 or d 0 on 

I I

∫ − Γ = ∫ − Γ = Γ

Γ Γ

i i i v u λ v u (3.30)

where ( ) 1

i λ and ( ) 2i λ are Lagrange multipliers or weight functions in the form of the

secondary variable along the interface. An additional continuity requirement in terms of

the secondary variable along the common subdomain boundary is required. These

secondary variables, ( ) 1 ˆ

i

t and ( ) 2 ˆ

i

t , are assumed to be independent of each other. These

independent approximations give rise to continuity requirements along the interface of

the form

() ( ) 1 2 I ˆ + ˆ = 0 on Γ i i t t

These constraints can be satisfied in the integral sense as

() ( ) ( ) 1 2 I I ˆ ˆ d 0 on ˆ

I

∫ + Γ = Γ

Γ

i i i λ t t (3.31)

where λi

ˆ is a Lagrange multiplier or weight function of the form of the primary variable

along the interface. Combining Eqs. (3.27) and (3.28) for the entire domain, including
105

the three continuity integrals at the interdomain boundary, Eqs. (3.29), (3.30), and (3.31),

and recognizing that () () () ˆ ( ) 1 1 1

i ij j t = σ n and () () () ˆ ( ) 2 2 2i ij j t = σ n yields

() () ( ) () () ( ) () () ( ) ( )

() () ( ) () () ( ) () ( ) ( )

() () () () ()

() () () () () s i i i i i

s i i i i i

i i i i i i i i i

i j ij i j ij i i i i

s

s

b v t

b v t

v u v u t t

t t

2

2 2

2

2 2 2

2

1

1 1

1

1 1 1

1

1 1 I 2 2 I 1 2 I

1 1 I 2 2 I

2

2 2

1 , 1 1 ,

( ) d d

( ) d d

ˆ ˆ d ˆ d d

d d ˆ d ˆ d

2 2

1 1

I I I

I I 1 2

− Φ Ω + Φ Γ

= − Φ Ω + Φ Γ +

+ − Γ + − Γ + + Γ

− Φ Ω − Φ Ω − Φ Γ − Φ Γ

∫ ∫

∫ ∫

∫ ∫ ∫

∫ ∫ ∫ ∫

Ω Γ

Ω Γ

Γ Γ Γ

Ω Ω Γ Γ

&

&

ρ

ρ

λ λ λ

σ σ

 (3.32)

In addition, for fluid mechanics, the continuity equation is given and satisfied

independently over each domain as

() ( ) 2

2

1 , 1

, ui j = 0 in Ω and ui j = 0 in Ω

The weighted residual statements over the domains are given by

() () ( ) ( ) d 0 ˆ d 0 and ˆ 2

2 2

1 , 1 1 ,

1 2

∫ Φ Ω = ∫ Φ Ω =

Ω Ω

i j i i j i u u . (3.33)

Here, note that no integration by parts is used on the continuity equations, and no

relaxation of the differentiability on u can be accomplished since the resulting boundary

conditions would not be physical. Combining Eqs. (3.33) yields

() () ( ) ( ) d 0 ˆ d ˆ 2

2 2

1 , 1 1 ,

1 2

∫ Φ Ω + ∫ Φ Ω =

Ω Ω

i j i i j i u u . (3.34)

The integral form of Eq. (3.32) forms the basis of finite element approximations for solid

mechanics, and both Eqs. (3.32) and (3.34) form the basis for fluid mechanics. These

finite element approximations as well as other approximations will be discussed in more

detail in the next section.
106

3.6. SPATIAL MODELING FOR MULTIPLE DOMAINS

Spatial modeling for multiple domains using the finite element and finite

difference methods for the approximation of the vector-field problem is outlined in this

section. A brief overview of discretization methods is given followed by spatial

modeling for solid and fluid mechanics domains.

3.6.1. Overview of Discretization Methods

Finite element and finite difference discretization methods for the vector-field

problem are outlined in this subsection. For a more detailed discussion the reader should

consult the literature.

The finite element method

 The finite element method for the vector-field problem is developed in the same

manner as for the scalar-field problem. In the vector-field problem, the dependent

variable in the integral equations is a vector of components. In general, the inplane

vector components (e.g., displacements parallel to the x and y axes) are approximated by

the same shape functions. For isoparametric elements, this approximation is the same as

that taken for the shape. For the elasticity problem, the consideration for the straindisplacement relation, the Jacobian transformation, and the displacement gradient

interpolation results in a more complex (the product of three matrices) set of equations

than for the scalar field.

The finite difference method

The finite difference method is ideal for solving the governing partial differential

equations of a continuum. It represents a variety of equations in engineering science;
107

however, the method has not been used in solid mechanics to the same degree as the

finite element method42. The decline in the use of the finite difference method in solid

mechanics is largely due to the limited flexibility of its treatment of boundary conditions.

Most finite difference developments avoid the general problem of boundary conditions in

one of the following ways: (1) a scalar problem, such as those of the previous chapter, is

solved as an example and the boundary conditions are incorporated in the analysis using

arguments based on symmetry of the independent variables in the derivative

approximations or (2) an example is chosen with fixed boundaries to eliminate the

presence of fictitious points. The lack of an intuitive procedure for elimination of the

fictitious or external grid points introduced when a central difference operator is applied

to a boundary point is one cause of the deficiency in the method. For the vector-field

problem discussed herein, a 3×3 central difference template is used to evaluate the

momentum equation, Eq. (3.20). An approach for eliminating the fictitious points based

on physical arguments is presented in reference 43. The fictitious nodes are replaced by

boundary tractions using a set of constitutive equations and the primary variables in the

continuum. These points can then be eliminated, and the boundary tractions are

introduced into the finite difference model. An alternative approach is to construct

special forms of the difference equations for grid points at or near the boundaries44.

These forms make use of forward or backward difference operators to express differential

forms. In general, standard forward or backward difference operators have higher-order

truncation error than the central difference operators used for the differential equation.

Hence, special forms using additional interior grid points are constructed such that the

operators have the same order of truncation error as those operators used for the
108

differential equation. The latter approach is used in this work and will be discussed in

some detail in the discussions of the patch test application given in this chapter.

3.6.2. Overview of Single-Domain Spatial Modeling

For a single domain, the finite element equations may be obtained by rewriting

and manipulating slightly Eq. (3.16) over an element domain as

∫Φ dΩ − ∫Φ − dΩ − ∫Φ dΓ = 0

Ω Ω Γ

e

es e e

s

i i

e

i i i

e

i j ij , σ ρ(b v& ) t (3.35)

ZKHUH ij are the approximate stress fields produced by the stress-strain and straindisplacement (or rate of deformation) relations and approximating the primary variable

over the element domain by u = Nue .

General finite element development

Using the Galerkin method, the weight function is given by = N . Substituting these

approximations into the integral equation given in Eq. (3.35) and writing in matrix form

yields

d d d 0 T T T ∫ Ω − ∫ − Ω − ∫ Γ =

Ω Ω Γ

e

e

s e e

e e s ∂N N ρ(b v&) N t (3.36)

where ∂ is the operator matrix defined, in general, by
109

































































∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ =

z x

z y

y x

z

y

x

0

0

0

0 0

0 0

0 0

,

and the stress vector is given by

[ ]T = σ 11 σ 22 σ 33 σ 12 σ 23 σ 13

General finite difference development

Recall that in the finite difference methods, derivatives are approximated by

difference expressions that transform the derivatives and consequently the partial

differential equation to algebraic expressions and equations, respectively. Upon

substitution of the approximation function into the differential equation, the equations can

be recast in weighted residual form by selecting ( ) i i i = δ x − x , y − y . Note that the

subscript i on the weight function is used to denote the subdomain, while the subscript i

on the coordinate values, x and y, is used to represent the point in the physical domain at

which the Dirac delta function is evaluated. This nomenclature is used throughout the

mathematical formulation presented here. The weighted form of the residual reduces to

the evaluation of the partial differential equations using the approximate solution

evaluated at the N selected mesh points

For a single domain, as in the finite element method, the finite difference

equations may be obtained by interrogating the weighted residual equations over an
110

element domain where the element, e, surrounds grid point i (see Figure 2.5). The

approximate solution for the primary variable is given by

m e

M

m

u Nmu u Nu ~ = ∑ =

=

 or 

1

where M is the number of shape functions over the element. The weight function, Φ, is

given by the Dirac delta function, ( )( ) i i i i δ x − x , y − y = δ x , y . Therefore, Eq. (3.35)

becomes

() ()

( ) d 0

d d

− Γ =

Ω − − Ω

∫

∫ ∫

Γ

Ω Ω

e

es

e e

s

i i i

e

i i i i

e

i i j ij

x y t

x y x y b v

,

, , , ( )

δ

δ σ δ ρ &

and upon making use of properties of the Dirac delta function,

− [ ] ( )( ) − − ( ) = 0

=

= i i i i i i i i i

y y

ij j x xi b x y v x y t x y

i

, , , , σ ρ & . (3.37)

This equation and the equations related to the finite difference formulation that follow are

evaluated at point (xi,yi) where i denotes a point in the physical domain, and no

summation is implied over the xi terms. Eqs. (3.36) and (3.37) are applicable to a general

continuum irrespective of its physical constitution. Discipline-specific constitutive

relations are considered at this point to continue with the finite element and finite

difference developments specific to solid and fluid mechanics. Each of these

developments will be discussed in turn.

Solid mechanics - finite element discretization

For solid mechanics, the constitutive relation relating stress and strain is given by

0 0 = E( − ) +
111

where the strain vector

[ ]T

11 22 33 12 23 13 = ε ε ε 2ε 2ε 2ε ,

E is a matrix of material stiffnesses, and 0

 and 0 are initial stress and strain quantities,

respectively. The strain-displacement relation is given by

= ∂ u = Nue = Bue ∂ .

Implicit in the definition of B is the use of the Jacobian matrix to transform from

Cartesian coordinates to element natural coordinates used in the shape function

development. In addition, in solid mechanics, the acceleration of the continuum is given

by 2

2

∂t

∂ = = u

v& u&& . Moreover, the second time derivative of the primary variable over the

element domain is approximated by u Nue && = && . Substituting the stress-strain, straindisplacement relations and the acceleration into Eq. (3.36) yields

e

e

s e

e e e e

e s

e e

e

e

e

e

+ Ω + Γ

= Ω − Ω 















+ Ω 















Ω

∫ ∫

∫ ∫ ∫ ∫

Ω Γ

Ω Ω Ω Ω

d d

d d d d

T T

0

T

0

T T T

N b N t

B EB u N N u B E B

ρ

ρ &&

 (3.38)

or

e e e e e k u + m u&& = f

where ke is the element stiffness matrix, me is the element mass matrix, ue is the vector

containing the generalized primary variables, ue && is the vector containing the second time

derivative of the generalized primary variables, and fe is the element force vector

containing the generalized secondary variables. Note that the acceleration term can be

considered as an inertial force and included as part of the element force vector.
112

Assembling these element equations over the entire domain and enforcing continuity of

the primary variable at the interelement boundaries yields the system of equations given

by

Mu&& + Ku = F

where ∑ ∫

Ω

= Ω

nelem

e

1 e

T K B EBd ; ∑ ∫

Ω

= Ω

nelem

e

1 e

T M ρN Nd ; u is the assembly of all of the

nodal degrees of freedom associated with the primary variables; u&& is the assembly of all

of the nodal degrees of freedom associated with time derivative of the primary variables,

and = ∑ ∫ Ω − ∫ Ω + ∫ − Ω + ∫ Γ

Ω Ω Ω Γ

nnodes

e e e s e

e

s 1 e e e

T T

0

T

0

T F B E d B d N ρ(b u&&) d N t d

Solid mechanics - finite difference discretization

For solid mechanics, making use of the stress-strain and strain-displacement

relations, and substitution of the primary variable approximations into Eq. (3.37), the

element equation becomes

( ) ( ) i i i i

y y

x x

y y

x x e

y y

x x e

y y

x x x y x y

i

i

i

i

i

i

i

i , , 0

T

0

T T EB u N u = E − + b + t 



 





+ 















=

=

=

=

=

=

=

= ∂ ρ && ∂ ∂ ρ

For the second derivative difference approximation, the number of shape functions, M=3

and = { } i−1 i i+1

T

e u u u u .

Therefore, as in the finite element method the difference equations may be written in the

form

e e e e e m u&& + k u = f

where ke and me are the finite difference “element mass and stiffness” matrices, ue is the

vector of generalized primary variables, ue && is the vector of time derivatives of the
113

generalized primary variables, and fe is the finite difference generalized force vector.

Assembling the element equations yields

Mu&& + Ku = F

where

∑ 















=

=

=

Nelem

y y

x x

i

i 1

T K ∂ EB , ∑ 



 



 = =

=

Nelem

y y

x x

i

i

1

M ρN , u and ue && contain all of the nodal

degrees of freedom associated with the primary variables and its time derivative, and

∑ ( ) ( )



 



 = − + + =

=

=

=

Nnodes

i i i i

y y

x x

y y

x x x y x y

i

i

i

i

1

0

T

0

T F ∂ E ∂ ρb , t , .

Fluid mechanics- finite element discretization

For fluid mechanics, the constitutive relation relating stress and the rate of

deformation, Eq. (3.7), for an incompressible fluid is given by

= − PI

where the viscous stress vector, , is given by

[ ]T

11 22 33 12 23 13 = τ τ τ τ τ τ ,

u denotes the velocity vector, P is the pressure, and I is the identity matrix. The viscous

stress is given by = 2μD where is the shear viscosity of the fluid and D is the rate of

deformation tensor whose components are given by

( ) ij i j j i D u u = , + , 2

1 (3.39)

Hence, the rate of deformation is related to the deformation and may be expressed in the

same form as the strain-displacement relation as
114

= ∂ f u = f Nue = Bf ue D ∂

where f ∂ is a differential operator defined by ∂ = T∂ f and B = T∂ f . The

transformation matrix, T, is used to introduce the scalar multiple of the shear components

of the rate of deformation (see Eq. (3.39)) and is symbolically defined as

















=









































= 0 I

I 0

T

2

1

2

1

2

1

2

1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 0 0

1 0 0 0 0 0

In addition, in fluid mechanics, the acceleration of the continuum is given by

v v

v v

v + ⋅∇

∂

∂ = = dt t

d & . Moreover, the time derivative of the primary variable over the

element domain is approximated by v u Nue & = & = & and NPe P ˆ = . Substituting the

constitutive and rate of deformation relations along with the acceleration into Eq. (3.36)

and rearranging yields

( )

e

e

s e e

e e e

e s

e

e

e

e

e

e

e e

e

= Ω + Γ 















− Ω

















+ Ω 















+ Ω 















Ω

∫ ∫ ∫

∫ ∫ ∫

Ω Ω Γ

Ω Ω Ω

d d d

d d 2 d

T T T

f

T T T

NI P N b N t B ˆ

N N u N Nu N u B B u

ρ

ρ & ρ μ

 (3.40)

or

e e e e e e e e e m u& + c u + k u − q P = f

where the element matrices ke and me and the element force vector, fe, are of similar form

as those obtained in the solid mechanics development, ce, is a nonlinear element matrix
115

resulting from the total derivative of the velocity, ue is the vector containing the

generalized primary variables, Pe is the vector containing the element pressure variables,

and ue & is the vector containing the time derivative of the generalized primary variables.

Hence, in the fluid mechanics development, the rate of change of the velocity - u& is

analogous to the second derivative with respect to time of the displacement (u&& in the

solid mechanics development). Moreover, the first integral term of Eq. (3.40) can be

thought of as an inertial force. Assembling these element equations over the entire

domain and enforcing continuity of the primary variable at the interelement boundaries

yields the system of equations given by

Mu& + C(u)u + Ku − QP = F (3.41)

where ∑ ∫

Ω

= Ω

nelem

e

1 e

f

T K 2μB B d ; ∑ ∫

Ω

= Ω

nelem

e

1 e

T M ρN Nd ;

∑ ( ) = ∫ Ω

Ω

nelem

e

e

1 e

T C ρN Nu N d ; = ∑ ∫ Ω

Ω

nelem

e

1 e

TNI d Q B ˆ ; u is the assembled vector

of all nodal degrees of freedom associated with the primary variables; P is the assembled

vector of all nodal degrees of freedom associated with the pressure, u& is the assembled

vector of all nodal degrees of freedom associated with the time derivative of the primary

variables and = ∑ ∫ Ω + ∫ Γ

Ω Γ

nnodes

e s e

es 1 e

T T F N ρb d N t d .

In addition to the element equations for momentum, Eq. (3.40), the element

equations for continuity must also be developed from Eq. (3.19). Using the Galerkin

method, the weight function corresponding to the continuity equation is given by ˆ Nˆ Φ = .
116

Substituting the approximation for the weight function and the primary variable into Eq.

(3.19) and writing the equation over the element yields

d 0 T = 















∫ Ω

Ω

e

e

e

N u B ˆ

or

0 Tqe ue =

Assembling these element equations yields

0 T − Q u = (3.42)

Equations (3.41) and (3.42) can be combined into one system of equations and written in

matrix form as











 =

















 





−

+ − +

















 





0

F

P

u

Q 0

C(u) K Q

P

u

0 0

M 0

T

& (3.43)

or in a more symbolic form as

MU& + KU = F

where { }T U = u1 u2 u3 P . Hence, the equations for fluid mechanics may be

expressed in the same form as the equations for solid mechanics. Note that the system of

equations, Eq. (3.43), is referred to as the primitive-variable model, the pressure-velocity

model, or the mixed model32. This mixed model results in a system that is nonpositive

definite because of the zeros appearing on the main diagonal. In addition, the

interpolation used for the pressure should be one order less than those that appear for the

velocity field41. Furthermore, the pressure approximation may be discontinuous across

interelement boundaries. In addition, because different orders of approximation are

typically used for the velocity and pressure fields, the pressure may not appear at every
117

node of an element, which can complicate the assembly process. An alternative

formulation, called the penalty function formulation32,41, circumvents this situation by

treating the continuity equation as a constraint among the velocity components. This

formulation is developed here for finite element discretization.

From the weak form in Eq. (3.16), a functional describing the continuum motion

can be obtained. The linear and bilinear forms of the functional over an element when

the two-dimensional velocity field, (v1, v2), satisfies the continuity constraint, Eq. (3.19),

is given by

( )

e

e

s

e

es e e

s

e e s

t

L b v b v t

+ Φ Γ

Φ Φ = − Φ Ω + − Φ Ω + Φ Γ

∫

∫ ∫ ∫

Γ

Ω Ω Γ

d

d d d

2 2

1 2 1 1 1 2 2 2 1 1 , ρ( & ) ρ( & )

( ) ( )( )

e

e

e

e

x

v

x

v

x x

x

v

x x

v

x

B v v

Ω 















∂

∂

+

∂

∂

















∂

∂Φ

+

∂

∂Φ

+

Ω 















∂

∂

∂

∂Φ

+

∂

∂

∂

∂Φ Φ Φ =

∫

∫

Ω

Ω

d

2 d

1

2

2

1

1

2

2

1

2

2

2

2

1

1

1

1

1 2 1 2

μ

, , , μ

.

Note that the pressure does not appear explicitly in the bilinear form. The quadratic

functional is given by

( ) ( )( ) ( )( )

e

e

s

e

e

s e e

e

s

e e s

e

t v

b v v b v v t v

x

v

x

v

x

v

x

v

I v v B v v v v L v v

− Γ

− − Ω − − Ω − Γ

Ω 































∂

∂

+

∂

∂

+ 















∂

∂

+ 















∂

∂ =

= −

∫

∫ ∫ ∫

∫

Γ

Γ Ω Ω

Ω

d

( ) d ( ) d d

d

2

1

, , , , 2

1 ,

2 2

1 1 1 2 2 2 1 1

2

1

2

2

1

2

2

1

2

1

1

1 2 1 2 1 2 1 2

ρ & ρ &

μ

 (3.44)
118

The equations governing the flow of viscous incompressible fluids, Eqs. (3.16), and

(3.19), are equivalent to minimizing of Eq. (3.44) subject to the constraint

( ) 0

1

2

2

1

1 2 = ∂

∂

+

∂

∂ = xv

x

v G v ,v .

In the penalty function method, the constrained problem is reformulated as an

unconstrained problem by minimizing the modified functional

( )( ) [ ] ( ) e m e e

I v v = I v v + ∫ G v v Ω

Ω

d

2

1 2

1 2 1 2 1 2 , , γ ,

where the penalty parameter, γe, can be chosen for each element. The necessary

conditions for the minimum of Im is 0 δIm = or 0 1 δ v Im = and 0 2 δ v Im = .

where δv1 and δv2 denote the first variation with respect to the velocity components, v1

and v2, respectively. Therefore,

( )( )

d 0

( ) d d

2 d

, , d

2

2

1

1

1

1

1 1 1 1 1

1

2

2

1

2

1

1

1

1

1

1 2 1 2 1 1 1

Ω = 















∂

∂

+

∂

∂

∂

∂

+

− − Ω − Γ

 Ω



 





















∂

∂

+

∂

∂

∂

∂

+

∂

∂

∂

∂ =

= + Ω

∫

∫ ∫

∫

∫

Ω

Γ Ω

Ω

Ω

e

e

e

s e

e

e

x

v

x

v

x

v

b v v t v

x

v

x

v

x

v

x

v

x

v

I I G v v G v v

e

e s

e

e

v m v e v

δ

γ

ρ δ δ

δ

μ

δ

μ

δ δ γ δ

&

 (3.45)

and
119

()()

d 0

( ) d d

2 d

, , d

2

2

1

1

2

2

2 2 2 2 2

1

2

2

1

1

2

2

2

2

2

1 2 1 2 2 2 2

Ω = 















∂

∂

+

∂

∂

∂

∂

+

− − Ω − Γ

 Ω



 





















∂

∂

+

∂

∂

∂

∂

+

∂

∂

∂

∂ =

= + Ω

∫

∫ ∫

∫

∫

Ω

Γ Ω

Ω

Ω

e

e

e

s e

e

e

x

v

x

v

x

v

b v v t v

x

v

x

v

x

v

x

v

x

v

I I G v v G v v

e

e s

e

e

v m v e v

δ

γ

ρ δ δ

δ

μ

δ

μ

δ δ γ δ

&

 (3.46)

These two statements, Eqs. (3.45) and (3.46), provide the weak forms for the penalty

finite element model. While, the pressure does not appear in the weak forms explicitly, it

is part of the boundary tractions, t1 and t2. The penalty finite element model is obtained

using Eqs. (3.45) and (3.46), the approximations for the primary variable and the time

rate of change of the primary variable, v = u = Nue and v u Nue & = & = & , respectively, and

by choosing δv1 = δv2 = N . Assembling these element equations over the entire domain

and enforcing continuity of the primary variable at the interelement boundaries yields the

system of equations given by

Mu& + C(u)u + Ku + Su = F (3.47)

where ∑ ∫

Ω

= Ω

nelem

e

1 e

f

T K 2μB B d ; ∑ ∫

Ω

= Ω

nelem

e

1 e

T M ρN Nd ;

∑ ( ) = ∫ Ω

Ω

nelem

e

e

1 e

T C ρN Nu N d ; = ∑ ∫ Ω

Ω

nelem

e

x x

1 e

T d 1 2 S γ N, N, ; u is the assembled

vector of all nodal degrees of freedom associated with the primary variables; u& is the

assembled vector of all nodal degrees of freedom associated with the time derivative of
120

the primary variables, = ∑ ∫ Ω + ∫ Γ

Γ Ω

nnodes

e s e

e

s 1 e

T T F N ρb d N t d , and i x N, denotes

differentiation with respect to the independent variable, xi, i=1, 2.

Eq. (3.47) may be represented in a more symbolic form as

MU& + KU = F

where K = C(u)u + K + S . Note that this penalty finite element method yields a system

of equations in terms of the primary variables, u, and does not include the pressures, P.

The pressures may be obtained from the computed velocity field by

( ) 















∂

∂

∂

∂ − = =

1

2

1

1

1 2

x

v

x

v

G v v

e γ γ

γ γ γ P γ , γ ,

where ( ) 1γ 2γ v ,v is the finite element solution of Eq. (3.47).

Fluid mechanics - finite difference discretization

For fluid mechanics, making use of the stress-rate of deformation constitutive

relation, and substitution of the primary variable approximations along with the

acceleration into Eq. (3.37), the element equation becomes

( )

( ) ( ) i i i i

e

y y

e x x

y y

e x x

y y

x x e e

y y

x x

x y x y

i

i

i

i

i

i

i

i

, ,

T T ˆ

b t

N u X u u I P

= +

















− 















 +



 



  +



 





=

=

=

= =

=

=

=

ρ

ρ & ρ 2μ∂ ∂ ∂

Also, considering continuity,

I u = 0 















=

= e

y y

x x

i

i

T ˆ ∂

The difference equations may be written in the form

e e e e e e e e e m u& + c u + k u − q P = f
121

and

qeue = 0

where ke, ce, qe, and me are finite difference “element” matrices, ue is the vector of

generalized primary variables, ue & is the vector of time derivatives of the generalized

primary variables, and fe is the finite difference generalized force vector. Assembling the

element equations yields

Mu& + C( ) u u + Ku − QP = F (3.48)

− Q u = 0 T (3.49)

where

∑ 















=

=

=

nelem

y y

x x

i

i 1

T K 2μ∂ ∂ , ∑ 



 



 = =

=

nelem

y y

x x

i

i

1

M ρN , ∑ ( ) 



 



 = =

=

nelem

y y

x x e

i

i

1

C ρ X

∑ 















=

=

=

nelem

y y

x x

i

i 1

T ˆ Q ∂ I , u and ue & are vectors that contain all nodal degrees of freedom

associated with the primary variables and its time derivative, and

= ∑[ ] ( ) + ( )

nnodes

i i i i x y x y

1

F ρb , t , . As in the finite element method, Eq. (3.48) and (3.49)

can be combined into one system of equations and written in matrix form as











 =

















 





−

+ − +

















 





0

F

P

u

Q 0

C(u) K Q

P

u

0 0

M 0

T

&

or in a more symbolic form as

MU& + KU = F

where { }T U = u1 u2 u3 P .
122

As in the case for the scalar-field formulation, the shape functions for a nine-node

quadrilateral finite element are used (see Table 2.1). The shape function at point i-1,j-1 is

given by

( )( ) ( )( ) ( )( ) ( )( ) 2 2 2 2

1 1 1 1

4

1 1 1

2

1 1 1

2

1 1 1

4

1 Ni− , j− = −ξ −η − −ξ −η − −ξ −η + −ξ −η .

Similarly,

( )( ) ( )( ) ( )( ) ( )( ) 2 2 2 2

1 1 1 1

4

1 1 1

2

1 1 1

2

1 1 1

4

1 Ni+ , j− = + ξ −η − −ξ −η − + ξ −η + −ξ −η ,

( )( ) ( )( ) ( )( ) ( )( ) 2 2 2 2

1 1 1 1

4

1 1 1

2

1 1 1

2

1 1 1

4

1 Ni+ , j+ = + ξ +η − + ξ −η − −ξ +η + −ξ −η ,

and

( )( ) ( )( ) ( )( ) ( )( ) 2 2 2 2

1 1 1 1

4

1 1 1

2

1 1 1

2

1 1 1

4

1 Ni− , j+ = −ξ +η − −ξ +η − − ξ −η + −ξ −η ,

Then, for a square element

( ) ( ) ( )

2

1 1

2

2

0 0

2

1 1

2

2

1 1

2

4

1

2

1

2

1

4

1 1 1

x

N

h h

N

h x y

N

i j

e e

i j

e

i j

∂

∂ =

=





















 



 = − − +

∂ ∂

∂ = ∂ ∂

∂

+ +

= =

− − − −

,

,

, ,

ξ η

ξ η ξη ξ η

( ) ( ) ( )

2

1 1

2

2

0 0

2

1 1

2

2

1 1

2

4

1

2

1

2

1

4

1 1 1

x

N

h h

N

h x y

N

i j

e e

i j

e

i j

∂

∂ =

= − 



















 



 = − − + +

∂ ∂

∂ = ∂ ∂∂

− +

= =

+ − + −

,

,

, ,

ξ η

ξ η ξη ξ η

(3.50)

The standard finite difference representation follows by direct substitution of Eqs. (3.50)

for the cross-derivative terms of the momentum equation along with Eqs. (2.40) for the
123

second-order terms. As noted previously, a single spatial modeling approach (i.e., the

finite element method or the finite difference method) is used for the single-domain

formulation. While for multiple domains, homogeneous approaches and heterogeneous

approaches are available. That is, the same method in each domain (homogeneous

approach) or different methods in different domains (heterogeneous approach) are

possible combinations of spatial modeling.

3.6.3. Multiple-Domain Modeling - Homogeneous Discretization

These homogeneous approaches make use of a single discretization method

among all subdomains in which the domain is subdivided. The focus of this work is on

the finite element and the finite difference methods as the spatial discretization methods.

For homogeneous domain discretization developed herein, Eq. (3.32) is used to provide

the mathematical basis for the three-approximation formulation. The generalized element

equations, for both the finite element and finite difference methods, may be obtained by

rewriting Eq. (3.32) over an element domain as

() () ( ) () () ( ) () () ( ) ( )

() () ( ) () () ( ) () ( ) ( )

() () () () ()

() () () () () e

e

s e

e

e

s e

e

e

e

e

e

e

e

e

e

e

e e

s i i e i i i

s i i e i i i

i i i i i i i i i

i i i i e i j ij e i j ij

b v t

b v t

v u v u t t

t t

2

2 2

2

2 2 2

2

1

1 1

1

1 1 1

1

1 1 I 2 2 I 1 2 I

1 1 I 2 2 I

2

2 2

1 , 1 1 ,

( ) d d

( ) d d

ˆ ˆ d ˆ d d

d d ˆ d ˆ d

2 2

1 1

I I I

I I 1 2

+ − Φ Ω + Φ Γ

= − Φ Ω + Φ Γ

+ − Γ + − Γ + + Γ

− Φ Ω − Φ Ω − Φ Γ − Φ Γ

∫ ∫

∫ ∫

∫ ∫ ∫

∫ ∫ ∫ ∫

Γ Ω

Γ Ω

Γ Γ Γ

Γ Γ Ω Ω

&

&

ρ

ρ

λ λ λ

σ σ

 (3.51)

Note that in the potential energy formulation25, the continuity of the secondary variables

was satisfied through the subsidiary conditions obtained through the minimization of the
124

potential energy. In this weighted residual formulation, the continuity of the secondary

variables is satisfied in a weighted residual sense and the Lagrange multipliers, ( ) k

i λ and

λi

ˆ , are represented by weight functions in the form of the secondary and primary

variables, respectively.

The form of the equations for the finite element and finite difference applications

differs by the form of the element shape functions and the approximation selected for the

weight functions, Φ. The formulation for solid mechanics and fluid mechanics differs by

the constitutive relations. For the generalized element expansion of subdomain i, the

independent approximations for the element generalized primary variables, (i.e.,

displacements or velocities), interface secondary variables (i.e., tractions or fluxes), the

weight functions associated with the secondary and primary variables, and the interface

variables, are, respectively

 and ˆ ; ; ˆ ; TuI uk = Nkue tk = Rk k k = Rk = T v = k (3.52)

Both the solid and fluid mechanics derivations may be developed from Eq. (3.51), given

the approximations of Eq. (3.52), the appropriate constitutive relation, and the choice of

weight function. Each derivation is presented in turn in the following work.

Solid Mechanics- finite element discretization

Substituting the approximations of Eq. (3.52) into Eq. (3.51) along with the

constitutive equations and using the Galerkin method in which = N , yields
125

e e e e s

e e e e s

e e

e e e e e e e e

es e e e

es e e e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e e e e

2 2

T

2 2 2

T

2 2

(2)

0

T

2 2

(2)

2 0

T

2

1 1

T

1 1 1

T

1 1

(1)

0

T

1 1

(1)

1 0

T

1

I

2

T

I 2

T I

2

I

1

T

I 1

T I

1

2

I

2

T

1

I

1

T

2

I

2

T

1 2

I

1

T

1

2 2

T

1 1 2

T

2 2 2 1

T

1 1 1 2

T

1

d d d d

d d d d

d d d d

d d d d

d d d d

2 2 2 2

1 1 1 1

2

I I

1

I I

I I I I

2

2

1

1

2

2

1

1

+ Ω − Ω + Ω + Γ

= Ω − Ω + Ω + Γ





















− Γ





















+ Γ





















− Γ





















Γ

+





















+ Γ





















+ Γ





















− Γ





















− Γ





















+ Ω





















+ Ω





















+ Ω





















Ω

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

Ω Ω Ω Γ

Γ Ω Ω Ω

Γ Γ Γ Γ

Γ Γ Γ Γ

Ω Ω Ω Ω

B E B N b N t

B E B N b N t

R T u R N u R T u R N u

N R N R T R T R

B E B u B E B u N N u N N u

ρ

ρ

ρ && ρ &&

 (3.53)

where Bk Nk = ∂ for k=1,2 and for the k

th subdomain, the element matrices are

∫

Ω

= Ω e

k

k

e

k k k k d T ke B E B ; ∫

Ω

= Ω e

k

k

e

k k k d T me ρN N

∫

Γ

= − Γ e

e

k k k

I

T I k R N d p ,

e

e k k k

T I d

I

= − ∫ Γ

Γ

ks N R , (3.54)

e

e k k

T I

I d

I

= ∫ Γ

Γ

k T R ,

and

es

k k k e k k i ek

k

k ek

k ek k k es

k

e

k ek ek

= ∫ Ω − ∫ Ω + ∫ Ω + ∫ Γ

Ω Ω Ω Γ

d d d d ( ) T T

0

( ) T

0

T f B E B N ρb N t .
126

Solid Mechanics- finite difference discretization

Substituting the approximations of Eq. (3.52) into Eq. (3.51) along with the

constitutive equations and using the Dirac delta function as the weight function,

( )( ) k i i i i = δ x − x , y − y =δ x , y ,

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

e

e

e

e

e

e

e

e

e

e

e

i

i

i

i

i

i

li

i

i

i

i

i

y y

x x y y

x x

y y

x x

y y

x x

y y

x x y y

x x

y y

x x

y y

x x

e

e

y y

x x

y y

x x

e y y

x x e y y

x x e

y y

e x x

y y

x x

=

= =

=

=

=

=

=

=

= =

=

=

=

=

=

Γ Γ

Γ Γ

Γ Γ

=

=

=

=

=

= =

=

=

=

=

=

+ − + +

= − + +

















− Γ





















+ Γ





















− Γ





















+ Γ





















+ Γ





















 + Γ



 



  −



 



 −





 



  +



 





+ 















+ 















∫ ∫

∫ ∫

∫ ∫

2 2

(2)

0

(2) T

2 0

T

1 1

(1)

0

(1) T

1 0

T

I

2

T

I 2

T I

2

I

1

T

I 1

T I

1

2

I

2

T

1

I

1

T

1 1 2 2

2 2 1 2

T

1 1

T

2

I I

1

I I

I I

1 2 1 2

d d

d d

d d

E b t

E b t

R T u R N u

R T u R N u

R R T R T R

E u E u u u

ρ

ρ

ρ ρ

∂ ∂

∂ ∂

∂ ∂ ∂ ∂ && &&

 (3.55)

where, for k=1,2 and for the k

th subdomain, the element matrices are

∫

Ω =

= = Ω

e

k i

k i

e

k

y y

k i x x d T ke ∂ E ∂ ; ∫

Ω =

= = Ω e

k

i

i k

e

k

y y

x x k m d e ρ

∫

Γ

= − Γ

e

e

k k k

I

T I k R N d p ,

( ) k i i x y k k R , s = − , (3.56)

e

e k k

T I

I d

I

= ∫ Γ

Γ

k T R ,
127

and

i

i

i

i

i

i

i

i y y

x x k

y y

x x k

y y

x x k

y y

x x k ek k =

= =

=

=

=

=

= f = E − + ρb + ρt ( )

0

( ) T

0

T∂ ∂ .

For both the finite element and the finite difference discretization strategies,

assembling the element equations over the entire domain, enforcing continuity of the

primary and secondary variables only within each subdomain and assembling the

contributions along the element edges on the common subdomain boundary, and noting

that 1 ue and 2 ue , 1 ue && and 2 ue && , 1ef and 2 ef , and 1 and 2 are completely uncoupled,

yields the system of equations given by





































=









































































+





































































0

0

0

f

f

u

u

u

0 K K 0 0

K 0 K 0 0

0 0 0 K K

0 K 0 0 K

K 0 0 K 0

u

u

u

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 M 0 0 0

M 0 0 0 0

2

1

2

1

I

2

1

T

p I

T

p I

I I

2 s

1 s

2

1

I

1

1

2

1

2 2

1 1

1 2

2

1

&&

&&

&&

 (3.57)

or





















=









































+









































0

0

f

u

u

K K 0

0 0 K

K 0 K

u

u

0 0 0

0 0 0

M 0 0

T

I

p I

I

s

I &&

&&

where K, M, u, and f are the assembled stiffness matrix, mass matrix displacement

vector, and force vector for the entire structure, and Kp, Ks, KI, uI, and are the

assembled Kpk, Ksk, KI

k

, uI, and k for all interfaces. The assembled stiffness and mass

matrices, K and M, are block diagonal matrices containing the stiffness and mass

matrices, Kk and Mk, of each of the subdomains along its block diagonal. The interface

“stiffness” matrix thus contains coupling terms that augment the stiffness matrices of the
128

subdomains along the interface. All of the interface “stiffness” terms appear in the

stiffness matrix with none in the mass matrix. Similar results may be obtained when

damping is included. As for the scalar-field problem, the three-approximation approach

for vector-field problems yields systems of equations (see Eqs. (3.57)) of similar form

and with the same attributes. Again, due to the generalization for the finite difference

approximations, the system of equations is not necessarily symmetric due to the offdiagonal submatrices, Kp and Ks, nor are they banded or positive definite. Therefore,

standard Cholesky solvers may not be used, unless full pivoting is performed to obtain

the solution. The upper diagonal submatrix blocks contain uncoupled stiffness matrices.

The symmetry of the matrix is determined by the choice of the weight function, Φ. In

general, due to the introduction of fictitious nodes for the imposition of boundary

conditions and loads in the finite difference discretization, the stiffness matrices are not

symmetric but are positive definite and sparse. The coupling is accomplished through the

introduction of the coupling terms in the matrices k Kp and k Ks for both approaches.

The number of additional degrees of freedom associated with the interface element is

generally small in comparison with the total number of degrees of freedom in the

subdomains. Thus, modeling flexibility is provided at a relatively small computational

expense. The computational expense in this study may be reduced additionally as the

efficiency of new solution algorithms for the system of equations in Eq. (3.57) is

increased.

While it is convenient to represent the weighted residual form over the domain

using a single equation, the system of equations, Eq. (3.57) is obtained from the

individual weighted residual expressions over each of the subdomains and the constraint
129

integrals. The first two matrix equations of the system of equations, Eq. (3.57) are

derived from the weighted residual statement for subdomain k. That is,

() () () () ( ) ∫ Φ dΩ − ∫Φ − dΩ − ∫Φ dΓ = 0

Ω Ω Γ

e

e

s e e

k s

i i

k e

i

k

i i

k e

ij

k

i j (b v ) t , σ ρ &

The third matrix equation of the system results from the reciprocity statement of the

secondary variables. That is,

() ( ) ( ) 1 2 I I ˆ ˆ d 0 on ˆ

I

∫ + Γ = Γ

Γ

i i i λ q q .

The fourth and fifth matrix equations result from the continuity requirement for the

primary variables, which is given by

() () ( ) I I d 0 on 

I

∫ − Γ = Γ

Γ

k

i i

k

i λ v u

() () ( ) 2 2 I I d 0 on 

I

∫ − Γ = Γ

Γ

i i i λ v u

Note that the forms of the coupling element matrices that are not in terms of the

weight functions are independent of the method of discretization. That is,

∫

Γ

= − Γ

e

e

k k k

I

T I k R N d p ,

and

∫

Γ

= Γ e

e

k k

I

T I

I k T R d

are of the same form for the finite element and finite difference discretizations.

However, since the element shape functions, Nk, differ for the two methods, the interface

matrices, pk k , in general, are not identical.
130

Fluid Mechanics- finite element discretization

Substituting the approximations of Eq. (3.52) into Eq. (3.51) along with the

constitutive equations and using the Galerkin method in which = N , yields

( )

( )

e e s e e s

e e

e e e e

e e e e e e e

e e e e e e e

es e e s e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e e

e e e

e e e

2 2

T

2 2 2

T

1 1 2

T

1 1 1

T

1

I

2

T

I 2

T I

2

I

1

T

I 1

T I

1

2

I

2

T

1

I

1

T

2

I

2

T

1 2

I

1

T

1

2 2

T

1 1 2

T

1

2 2

T

1 1 2

T

2 2 2 1

T

2

1 1 1

T

2 2 1

T

1 1 2

T

1

d d d d

d d d d

d d d d

d ˆ d ˆ

d 2 d 2 d

d d d

2 2 1 1

2

I I

1

I I

I I I I

2

2

1

1

2

2

1 f

1

2 f

2

2

1

1

2 1

2

1

1

= Ω + Γ + Ω + Γ





















− Γ





















+ Γ





















− Γ





















+ Γ





















+ Γ





















+ Γ





















− Γ





















− Γ





















− Ω





















− Ω





















+ Ω





















+ Ω





















+ Ω





















+ Ω





















+ Ω





















Ω

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫

∫ ∫ ∫

∫ ∫ ∫

Ω Γ Ω Γ

Γ Γ Γ Γ

Γ Γ Γ Γ

Ω Ω

Ω Ω Ω

Ω Ω Ω

N b N t N b N t

R T u R N u R T u R N u

N R N R T R T R

B N I P B N I P

N N u N u B B u B B u

N N u N N u N N u N u

ρ ρ

ρ μ μ

ρ & ρ & ρ

 (3.58)

where Bk Nk = ∂ and Bk f f Nk = ∂ for k=1,2 and the elemental matrices are

∫

Ω

= Ω

e

k

k

e

k k k 2 d f

T ke μB B ; ∫

Ω

= Ω e

k

k

e

k k k d T me ρN N ;

∫ ( )

Ω

= Ω

e

k

k k

e

k k e k k d T ce ρN N u N ; ∫

Ω

= Ω e

k

k

e

k k k d T ˆ q B N I e ;
131

∫

Γ

= − Γ

e

e

k k k

I

T I k R N d p ,

e

e k k k

T I d

I

= − ∫ Γ

Γ

ks N R ,

e

e k k

T I

I d

I

= ∫ Γ

Γ

k T R ,

and

es

k k k

e ek k k k e

s

k

e

k

= ∫ Ω + ∫ Γ

Γ Ω

d d T T f N ρb N t .

In addition to the element equations for momentum, Eq. (3.58), the element

equations for continuity must be considered. Eq. (3.34) is used to provide the

mathematical basis for the continuity equation for multiple domains. Using the Galerkin

method, the weight function corresponding to the continuity equation is given by ˆ Nˆ Φ = .

Substituting the approximation for the weight function and the primary variable into Eq.

(3.34) yields

d d 0 2 2

2

1 1

1

2

T

1 2

T

1 =

























+ Ω

























∫ Ω ∫

Ω Ω

e

e

e

e

e e

N u B ˆ N u B ˆ

Fluid Mechanics- finite difference discretization

Substituting the approximations of Eq. (3.52) into Eq. (3.51) along with the

constitutive equations and using the Dirac delta function as the weight function,

( )( ) k k i i k i i = δ x − x , y − y = δ x , y ,
132

( )

( )

[ ] ( ) [ ] ( )

( ) () () ( ) i i i i i i i i

e e

i i i i

e

y y

e x x

y y

x x

e

y y

e x x

y y

e x x

y y

e x x

e

y y

e e x x

y y

x x e

y y

x x

x y x y x y x y

x y x y

e

e

e

e

e

e

e

e

e

e

e

e

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

, , , ,

d d d d

, , d d

ˆ ˆ

2 2

1 1 2 2

I

2

T

I 2

T I

2

I

1

T

I 1

T I

1

2

I

2

T

1

I

1

T

1 1 2 2

2

T

1

T

2

T

1

T

2 2

1 2 1 1

2

I I

1

I I

I I

1 2

2 2 1 2

1 2 1 1

b t b t

R T u R N u R T u R N u

R R T R T R

IN P IN P

N u N u T N u T N u

N u N u N u N u

= + + +





















− Γ





















+ Γ





















− Γ





















+ Γ





















+ Γ





















− − + Γ

















− ∂ 















− ∂

















+ ∂ ∂ 















+ ∂ ∂

















+

















 +



 



  +



 





∫ ∫ ∫ ∫

∫ ∫

Γ Γ Γ Γ

Γ Γ

=

=

=

=

=

=

=

= =

=

=

= =

=

=

=

ρ ρ

ρ μ μ

ρ & ρ & ρ

(3.59)

where, for k=1,2 and the elemental matrices are

i

k i

y y

k x x

=

= ke = 2 ∂ ∂ T μ ; 

i

i k

y y

x x k =

m = N = e ρ ;

( )

i

i k

y y

x x k e k =

c = u = e ρ ; 

i

k i

y y

k x x

=

= q = I e

T ˆ ∂ ;

∫

Γ

= − Γ

e

e

k k k

I

T I k R N d p ,

( ) k i i x y k k R , s = − ,

e

e k k

T I

I d

I

= ∫ Γ

Γ

k T R ,

and

( ) ( ) ek k i i k i i f = ρb x , y + t x , y .
133

Considering continuity, and using the Dirac delta function as the weight function,

( )( ) k k i i k i i x x , y y x , y ˆ = δ − − = δ , the continuity equation is given by

I u I u = 0 















+ 















=

=

=

= 1 2 2

T

1

T ˆ ˆ e

y y

e x x

y y

x x

i

i

i

i

∂ ∂

For both the finite element and the finite difference discretization strategies,

assembling the element momentum equations, Eqs. (3.58) and (3.59) over the entire

domain, enforcing continuity of the primary variable only within each subdomain, and

noting that 1 ue and 2 ue , 1 ue & and 2 ue & , 1

Pe and 2Pe , 1ef and 2 ef , and 1 and 2 are

completely uncoupled, yields the system of equations given by













































=





























































































+

























































































−

+

























































































+

























































































0

0

0

0

0

f

f

P

P

u

u

u

0 K K 0 0 0 0

K 0 K 0 0 0 0

0 0 0 0 0 K K

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 K 0 0 0 0 K

K 0 0 0 0 K 0

P

P

u

u

u

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 Q 0 0

0 0 0 Q 0 0 0

P

P

u

u

u

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 C 0 0 0 0 0

C 0 0 0 0 0 0

P

P

u

u

u

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 M 0 0 0 0 0

M 0 0 0 0 0 0

2

1

2

1

2

1

I

2

1

T

I p

T

I p

I I

2 s

1 s

2

1

2

1

I

2

1

2

1

2

1

2

1

I

2

1

2

1

2

1

2

1

I

2

1

2

1

2 2

1 1

1 2

2

1

&

&

&

 (3.60)
134

along with











 =

















 





0

0

u

u

0 Q

Q 0

2

1

2

1

or symbolically





























=

























































−

+ −

+





















































0

0

0

f

P

u

u

K K 0 0

Q 0 0 0

0 0 0 K

C(u) K 0 Q K

P

u

u

0 0 0 0

0 0 0 0

0 0 0 0

M 0 0 0

T

T

I

p I

I

s

I &

&

where K, M, C, Q are the assembled coefficient matrices for momentum and continuity,

u and f are the displacement vector and force vector for the entire structure, and Kp, Ks,

KI, uI, and are the assembled Kpk, Ksk, KI

k

, uI, and k for all interfaces.

The first two matrix equations of the system of equations, Eq. (3.60) are derived

from the weighted residual statement for subdomain k. That is,

() () () () ( ) ∫ Φ dΩ − ∫Φ − dΩ − ∫Φ dΓ = 0

Ω Ω Γ

e

e

s e e

k s

i i

k e

i

k

i i

k e

ij

k

i j (b v ) t , σ ρ &

The third matrix equation of the system results from the reciprocity statement of the

secondary variables. That is,

() ( ) ( ) 1 2 I I ˆ ˆ d 0 on ˆ

I

∫ + Γ = Γ

Γ

i i i λ q q .

The fourth and fifth matrix equations result from the continuity requirement for the

primary variables, which is given by

() () ( ) 1 1 I I d 0 on 

I

∫ − Γ = Γ

Γ

i i i λ v u

() () ( ) 2 2 I I d 0 on 

I

∫ − Γ = Γ

Γ

i i i λ v u
135

Note that the forms of the coupling element matrices that are not in terms of the

weight functions are independent of the method of discretization. That is,

∫

Γ

= − Γ

e

e

k k k

I

T I k R N d p ,

and

e

e k k

T I

I d

I

= ∫ Γ

Γ

k T R

are of the same form for the finite element and finite difference discretizations.

However, since the element shape functions, Nk, differ for the two methods, the interface

matrices, pk k , in general, are not identical.

In addition, for the penalty finite element model, the system of equations is of the

same form as given in Eq. (3.60), except that penalty terms are included rather than the

pressure terms. The resulting system of equations is given by





































=





































































+









































































+





































































+





































































0

0

0

f

f

u

u

u

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 S 0 0 0

S 0 0 0 0

u

u

u

0 K K 0 0

K 0 K 0 0

0 0 0 K K

0 K 0 0 K

K 0 0 K 0

u

u

u

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 C 0 0 0

C 0 0 0 0

u

u

u

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 M 0 0 0

M 0 0 0 0

2

1

2

1

I

2

1

2

1

2

1

I

2

1

T

p I

T

p I

I I

2 s

1 s

2

1

I

2

1

2

1

2

1

I

2

1

2

1

2 2

1 1

1 2

2

1

&

&

&

 (3.61)

or symbolically
136





















=







































 + +

+









































0

0

f

u

u

K K 0

0 0 K

C(u) K S 0 K

u

u

0 0 0

0 0 0

M 0 0

T

I

p I

I

s

I &

&

where K, M, C, S are the assembled coefficient, mass, momentum, and penalty matrices,

u and f are the displacement vector and force vector for the entire structure, and Kp, Ks,

KI, uI, and are the assembled Kpk, Ksk, KI

k

, uI, and k for all interfaces. Recall that the

element penalty matrix for the k

th subdomain is given by

( ) ( ) ∫

Ω

= Ω e

k

k

e

k x k x k d

2 , ,

T

1

se N N

3.6.4. Multiple-Domain Modeling - Heterogeneous Discretization

The multifunctional weighted residual formulation of Eqs. (3.57) and (3.60) are

used as the mathematical basis for multiple-domain modeling using heterogeneous

discretization. Considering the two domains upon which this discussion is based, one

subdomain is discretized using the finite element method, and the other subdomain is

discretized using the finite difference method. Again, for the finite element development,

the weight functions for the primary variables, u and P, are taken to be the finite element

shape functions (i.e., k Nk k Nk ˆ ˆ = and = ), and for the finite difference development,

the weight functions are taken to be the Dirac delta function (i.e.,

( )( ) k k i i k i i = δ x − x , y − y = δ x , y ). As expected, the set of element matrices becomes

a hybrid of the matrices from the finite element method and the finite difference method.

For completeness, these matrices are repeated here for the finite element and finite

difference subdomains for solid mechanics as
137

∫

Ω

= Ω e

e

1

1 1 1 1

T

1 k B E B d e and 

i

i

y y

x x

=

= = 2 2

T

2

ke ∂ E ∂ ,

∫

Ω

= Ω e

e

1

1 1 1

T

1 m N N d e ρ and 

i

i

y y

x x

=

= = me 2 ρ 2 ,

and e e e e s e es e e e 1 1

T

1 1 1

T

1 1

(1)

0

T

1 1

(1)

1 0

T

1 1 d d d d

1 1 1 1

= ∫ Ω − ∫ Ω + ∫ Ω + ∫ Γ

Ω Ω Ω Γ

f B E B N ρb N t and

i

i

i

i

y y

x x y y

x x e =

= =

= − + 2 = + 2

(2)

0

(2) T

2 0

T

2 f ∂ E ∂ ρb t .

For fluid mechanics the element matrices are given by

∫

Ω

= Ω e

e

1

1 1f 1

T

1 k 2 B B d e μ and 

i

i

y y

x x

=

= = 2

T

2

ke 2μ∂ ∂ ,

∫

Ω

= Ω e

e

1

1 1 1

T

1 m N N d e ρ and 

i

i

y y

x x

=

= = me 2 ρ 2 ,

( ) ∫

Ω

= Ω e

e e

1

1 1 1 1 1

T

1 c N N u N d e ρ and ( )

i

i

y y e x x=

= 2 2 = 2 2 ce ρ u

∫

Ω

= Ω e

e

1

1 1 1

T

1 N I d B ˆ qe and 

i

i

y y

x x

=

= = 2

T ˆ 2 q I e ∂ ,

and e e s e es e

1 1

T

1 1 1

T

1 d d

1 1

1 = ∫ Ω + ∫ Γ

Ω Γ

f N ρb N t and ( ) ( ) e i i i i x , y x , y 2 2 2 f = ρb + t .

The coupling matrices at the element level are of the same form for both solid and fluid

mechanics and these matrices are given by

∫

Γ

= − Γ e

e

I

1

T I

1 k N R d s and ( ) 2 i i k R x , y s = − ,

and for the two domains, k=1,2,
138

e

e k k k

T I d

I

= − ∫ Γ

Γ

kp R N ,

and

∫

Γ

= Γ e

e

k k

I

T I

I k T R d .

3.7. COMPUTATIONAL IMPLICATIONS

The multifunctional modeling approach for the vector field problem has been

generalized such that it is applicable to solid and fluid mechanics as well as both

homogeneous and heterogeneous discretization approaches. As such the computational

implications are presented in this section for the generalized system of equations, Eqs.

(3.57) and (3.60). Implications specific to a discipline or a discretization approach are

highlighted, where appropriate.

The assembled coefficient matrices, K, M, C, and Q, are block diagonal matrices

containing the matrices, Kk, Mk, Ck, and Qk of each of the subdomains along its block

diagonal. The interface coupling matrix thus contains terms that augment the coefficient

matrices of the subdomains along the interface. All of the interface coupling terms

appear in the coefficient matrix associated with the primary variables with none in the

matrix associated with the time derivative. Again, due to the generalization for the finite

difference approximations, the system of equations is not necessarily symmetric due to

the off-diagonal submatrices, Kp and Ks, nor are they banded or positive definite. Note

that, even for a single domain model, the mixed formulation results in a nonpositive

definite matrix. Therefore, standard Cholesky solvers may not be used, unless full

pivoting is performed to obtain the solution. The upper diagonal blocks contain
139

uncoupled fluid flow coefficient matrices. The symmetry of the matrix is determined by

the choice of the weight function, Φ. For the finite element discretization, the subdomain

matrices are symmetric. In general, due the imposition of boundary conditions and loads

in the finite difference discretization, the coefficient matrices, Kk, are not symmetric but

are positive definite and sparse. The coupling is accomplished through the introduction

of the coupling terms in the matrices k Kp and k Ks for both approaches and each of the

disciplines discussed herein.

In addition, due to the generalization for the finite difference approximations, the

system of equations is not necessarily symmetric due to the off-diagonal submatrices, Kp

and Ks. The system unknowns in Eq. (3.57) and (3.60) consist of both primary and

secondary variables given by the displacements or velocities, u, and the traction

coefficients, , respectively. Generally, the coupling matrices, k Ks , are of the order of

the length of the interdomain boundary, which results in a marked difference in the

magnitude of the off-diagonal terms of the system matrix compared to its diagonal terms.

This characteristic produces an ill-conditioned matrix whose solution can cause

difficulties for some general-purpose solvers. Hence, the coupling matrix should be

scaled such that it is of the same order as the subdomain stiffness.

The load transfer mechanism of the multifunctional approach may be interrogated

for the vector-field problem by considering the first and second rows of Eqs. (3.57) and

(3.60) for solid and fluid mechanics, respectively. In either case the matrix equations of

interest are given for solid mechanics by

2 2 2 2 s 2 2

1 1 1 1 s 1 1

2

1

M u K u K f

M u K u K f

+ + =

+ + =

&&

&&
140

or for fluid mechanics by

2 2 2 2 s 2 2

1 1 1 1 s 1 1

2

1

M u K u K f

M u K u K f

+ + =

+ + =

&

&

These equations can be partitioned such that they correspond only to the primary

variables, uk on the interdomain boundary

K u K 0

K u K 0

+ =

+ =

2 2 s 2

1 1 s 1

2

1 (3.62)

andKk denotes stiffness terms related to uk and there are no forces (including inertial

forces Mkuk && ) on the interdomain boundary and assuming steady fluid flow (i.e.,

Mku& k = 0 ). The expressions given by Kk uk represent the internal fluxes at the

interdomain boundary, and thus Eq. (3.62) may be written as

1 s 1 2 s 2 1 2

f = −K and f = −K . (3.63)

For homogeneous discretization using the finite element method, substituting for

i

Ks from Eq. (3.54) into Eq. (3.63) gives

∫ ∫

Γ Γ

= − Γ = − Γ

e

e

e

e

I I

d ˆ d I

1

T

1 1

I

1

T

1 1 f N R N t (3.64)

∫ ∫

Γ Γ

= Γ = − Γ

e

e

e

e

I I

d ˆ d I

2

T

2 2

I

2

T

2 2 f N R N t (3.64)

Examining Eqs. (3.64) indicate that the evaluation of the internal forces is consistent with

the evaluation of equivalent nodal forces in the presence of applied tractions on the

boundary. In addition, Eq. (3.64) substantiates that the secondary variable along the

interface is represented by distributed forces for each of the subdomains.
141

For homogeneous discretization using the finite difference method, substituting

for 

i

Ks from Eq. (3.56) into Eq. (3.63) gives

1 1 1 1

ˆ f = −R = −t (3.65)

2 2 2 2

ˆ f = −R = −t

Examining Eq. (3.65) indicates that the evaluation of the internal forces is consistent with

nodal forces evaluated at points in the presence of applied tractions on the boundary. In

addition, Eq. (3.65) substantiates that the secondary variable along the interface for this

approach is represented by nodal forces for each of the subdomains.

For heterogeneous discretization using the combined finite element and finite

difference methods, substituting for k Ks from Eq. (3.54) into Eq. (3.56) gives

∫ ∫

Γ Γ

= − Γ = − Γ

e

e

e

e

I I

d ˆ d I

1

T

1 1

I

1

T

1 1 f N R N t (3.66)

2 2 2 2

ˆ f = −R = −t

Examining Eq. (3.66)) indicates, for subdomain 1, that the evaluation of the internal

forces is consistent with the evaluation of equivalent nodal forces in the presence of

applied tractions on the boundary, while for subdomain 2, the evaluation of the internal

forces is consistent with nodal forces evaluated at points. This reveals that for this

multiple domain approach, the secondary variable along the interface for subdomain 1 is

represented by distributed forces, and for subdomain 2, the secondary variable along the

interface is represented by nodal or point forces. Thus for this heterogeneous modeling

approach, it is required to transform the interface secondary variables into equivalent

quantities.
142

3.8. VERIFICATION TEST CASE

In this section, the multifunctional methodology for the vector-field problem is

demonstrated on a verification test case. The application is described and the associated

results and salient features are discussed. This application is considered a patch test for

the formulation and verifies the applicability of the method for a configuration for which

the solutions are known. Finite difference and finite element solutions for single- and

multiple-domain configurations are presented to provide benchmark solutions for the

multifunctional approach using heterogeneous discretization. Representative applications

from the field of engineering science are presented in Chapter V.

3.8.1. Patch Test

As in the scalar-field problem, a patch test is used to determine the effectiveness

of the multifunctional approach applied to a vector-field problem. A cantilevered plate is

subjected to uniform inplane loading at the free end that yields a constant state of strain.

In particular, this loading condition provides verification of the finite difference method

for combinations of displacement and traction boundary conditions, and the method is

validated for both the single- and multiple-domain models.

Problem Statement

The analysis domain and the boundary conditions are shown in Figure 3.1. The

normal and tangential tractions are denoted by Tn and Tt, respectively, in the figure. This

configuration has been used in the combined finite difference and finite element analysis

reported by Dow et al.20, and it is used here to provide a point of comparison. The length

of the plate, L, is 20 in., the width, W, is 8 in., and the thickness, h, is 1 in. The material

system is described by a Young’s modulus of 30,000 psi and a Poisson’s ratio of 0.3. An
143

applied displacement of 0.3 in. is applied at one end, and the opposite end is fixed. The

other sides are free.

For the finite element method, four-node elements are used to discretize the

domain for all applications. Homogeneous discretizations for single- and multipledomain models for the finite element and finite difference methods are presented. For the

finite element discretization of a single domain, a finite element mesh of 20 elements and

4 elements are used in the axial (x-direction) and transverse directions (y-direction) ,

respectively, of the plate. For multiple domains with compatible meshes (i.e., nodal

coincidence is maintained at the interface), two finite element meshes of 10 elements and

4 elements are used in the x- and y-directions, respectively. For the finite difference

discretization of a single domain, a finite difference grid consistent with the finite

element mesh was used. That is, a grid of 21 grid points and 5 grid points are used in the

axial (x-direction) and transverse directions (y-direction), respectively, of the plate.

Similarly, for multiple domains with compatible meshes, two finite difference meshes of

11 grid points and 5 grid points are used in the x-and y-directions, respectively. For

multiple domains with incompatible finite element meshes, one domain is discretized

with 10 elements in the x-direction and 4 elements in the y-direction. While the other

domain is discretized with 20 elements in the x-direction and 8 elements in the ydirection. The multiple-domain discretization is shown in Figure 3.2. The finite

difference discretization is consistent with the finite element mesh discretization.
144

x

L

W

y u=u0

Tn=Tt=0

Tn= Tt = 0

Tt=0

Figure 3.1. Analysis Domain and Boundary Conditions of Cantilevered Plate.

Interface

Figure 3.2. Multiple-Domain Discretization of Cantilevered Plate.

Boundary Conditions for Finite Difference Method

 The finite difference method is extensively tested for the single- and multipledomain configurations to assure that the boundary conditions are being applied correctly.

Generally, for the vector-field problem, a 3×3 or nine-point central difference template is

used to evaluate the momentum equation, Eq. (3.20). On the boundary of the domain, the

template introduces fictitious nodes. In reference 43, the fictitious nodes are eliminated

using traction conditions, Tn and Tt, and the constitutive equations. When the differential

equation is evaluated at the corner of the domain boundary (see point i,j in Figure 3.3), a

fictitious node (point i+1,j+1) is introduced for which there are no additional auxiliary

equations. Thus, to eliminate the degrees of freedom associated with this fictitious node,

non-physical higher-order derivatives of the constitutive equations are introduced that

further complicate the approach. An alternative approach, used herein, is to apply the
145

momentum equation only to the nodes in the interior of the domain, while the differential

equations representing the traction conditions are applied to the boundary nodes. Special

forms44 of the difference equations for grid points at the boundaries are used to avoid the

use of fictitious nodes. These forms make use of higher-order forward or backward

difference operators to express the differential forms in order to maintain the same order

of accuracy as the central difference operator. For multiple-domain spatial modeling, the

momentum equation is applied to nodes on the subdomain interface boundary. The

higher-order backward or forward difference operators are used to introduce the unknown

traction on the interface. This approach yields equations at the interface in terms of the

unknown tractions at that specified interface node only. If a central difference scheme

were used for the traction conditions, the equations on the interface would be in terms of

the unknown tractions at the specified interface node and adjacent interface nodes. In the

latter case, the resulting equations can not be derived from the generalized

multifunctional formulation.

Fictitious Node

Boundary Node

Interior Node

Domain Boundary

i,j

Tn, Tt

Tn, Tt

i+1,j+1

Figure 3.3. Central Difference Template Applied at a Corner.
146

Analysis Results

Several analyses have been performed: (1) two single-domain analyses, one with

finite element discretization and one with finite difference discretization, respectively, (2)

two multiple-domain analyses with homogeneous modeling, one with finite element

discretization in each domain and one with finite difference discretization in each

domain, and (3) one multiple-domain analysis with heterogeneous modeling with

combined finite element and finite difference discretizations. All of the analyses yielded

the exact solution within machine accuracy. Results for the internal forces or stresses

along the interface for the analysis cases are shown in Table 3.1. The results are given at

the locations along the width of the plate normalized by the plate width.

 For the finite element domains, the internal forces, Fx and Fy, obtained from the

multiple-domain analyses are normalized by the value of the force obtained from the

exact solution multiplied by the element length along the edge of the interface. Thus, for

a consistent load and for the finite elements used in this study, a normalized value of

unity represents complete agreement with the exact solution at the interior nodes (i.e.,

1/8≤y/W≤7/8). At the end nodes (i.e., y/W=0 and y/W=1), a normalized value of one half

represents complete agreement with the exact solution.

For the finite difference domains, the stresses, σx and τxy, obtained along the

interface from the multiple-domain analyses are normalized by the value of the normal

stress obtained from the exact solution. Thus, a normalized value of unity represents

complete agreement with the exact solution. Values in Table 3.1 for the normalized

distance along the interface, y/W, annotated with a superscript ‘F’ in parentheses denotes

results obtained from the most refined subdomain (see Figure 3.2).
147

The single-domain analyses with either finite element discretization or finite

difference discretization are in excellent agreement with the exact solution. Moreover,

the interface force and stress results obtained with multiple-domain analyses using

homogeneous modeling with either finite element discretization or finite difference

discretization are in excellent agreement with the exact solution. For the heterogeneous

modeling, the finite difference method was used in the coarsely discretized domain, and

the finite element method was used in the more refined domain. Note that the stresses are

used to compare the accuracy of the solution in the finite difference domain, and the

internal forces are used to compare the accuracy in the finite element domain. The results

obtained from this heterogeneous modeling approach are in overall good agreement with

the exact solution.

Table 3.1. Results of the Multifunctional Approach for the Cantilevered Plate.

Analysis Type*

SD/FE SD/FD MD/FE MD/FD MD/HM

Location

Along

Interface, y/W Fx Fy σx τxy Fx Fy σx τxy σx Fx

0. 0.5 0.00 1.00 0.00 0.5 0.00 1.00 0.00 .999 .499

1/8(F) 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 - .999

1/4 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 .999 .999

3/8(F) 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 - .999

1/2 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 .999 1.00

5/8(F) 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 - 1.00

3/4 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 1.00 1.00

7/8(F) 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 - 1.00

1 0.5 0.00 1.00 0.00 0.5 0.00 1.00 0.00 1.00 .499 *

SD/FE: Single-Domain with Finite Element discretization

 SD/FD: Single-Domain with Finite Difference discretization

 MD/FE: Multiple-Domain with Finite Element discretization

 MD/FD: Multiple-Domain with Finite Difference discretization

 MD/HM: Multiple-Domain with Heterogeneous Modeling (combined finite

difference and finite element discretizations)
148

CHAPTER IV

REPRESENTATIVE SCALAR-FIELD APPLICATIONS

4.1. GENERAL

In this chapter, the multifunctional methodology is demonstrated on several

representative scalar-field applications. The governing partial differential equation for

the scalar-field problem is applicable to a variety of problems in engineering science. A

sampling of these problems include a torsion problem, a heat conduction problem, and a

two-dimensional flow problem. The applications are described, and the associated

multifunctional analysis results and salient features are discussed. Finite difference and

finite element solutions for single- and multiple-domain configurations are presented to

provide benchmark solutions for the multifunctional approach using heterogeneous

spatial discretizations. The finite element models use four-node Lagrange isoparametric

finite elements, and the finite difference model uses a five-point template to approximate

the governing differential equation. Stand-alone finite element software is used to

generate the finite element stiffness matrices. The mathematical computing program

MATLAB is used to generate the finite difference matrices and the interface coupling

matrices and to solve the resulting system of equations.

4.2. TORSION OF PRISMATIC BAR

The torsion of a prismatic bar with a rectangular cross-section is used to

demonstrate the multifunctional capabilities for the Poisson problem. As mentioned in

Section 2.2.1, the torsion problem reduces to the nonhomogeneous partial differential

equation
149

θ φ φ G

x y

2 2

2

2

2

= −

∂

∂

+

∂

∂

in which the stress function, φ , must be constant along the boundary of the cross-section,

LVWKHDQJOHRIWZLVWSHUXQLWOHQJWKRIWKHEDUDQGG is the shear modulus. The

configuration of the bar is shown in Figure 4.1, and the analysis domain and the boundary

conditions, are shown in Figure 4.2.

Mt

Mt

x,u

y,v

z,w

Figure 4.1. Prismatic Bar with Rectangular Cross-Section.

For a solid cross-section, the requirement of a stress-free boundary yields the boundary

condition, φ = 0 , on all four bounding surfaces along the bar length. Because of the

symmetries in the problem, only one quadrant of the rectangular cross-section needs to be

considered. Moreover, this symmetric model is useful in verifying the application of

mixed boundary conditions. That is, the application of boundary conditions in terms of

both primary and secondary variables. The quadrant considered in the symmetric model

is shown in Figure 4.3.

The shear stresses in the cross-section are
150

y x

zx zy ∂

∂ = − ∂∂ = φ τ φ τ , . 

At the ends of the bar, the first moment integrated over the cross-sectional area must

equal the twisting moment. This requirement gives

M = 2∫φ dxdy t

and the twisting moment is related to the angle of twist by

M t = GJθ

where J is the torsional constant.

y

x

2a

2b

φ = 0

φ = 0

φ = 0

φ = 0

Figure 4.2. Analysis Domain and Boundary Conditions for Prismatic Bar with

Rectangular Cross-Section.

The analytical solution46 for the stress function is given by

( )( ) ( )

( ) a

n x

n b a

n y a

n

G a n

n 2

cos

cosh / 2

cosh / 2 1 1 32 1 1 2

1,3,5,... 3 3

2 π

π

π

π

θ φ 



 



 = − − − ∞

=

∑ ,

and by differentiating
151

( )( ) ( )

( ) a

n x

n b a

n y a

n

G a

y

n

n zx 2

cos

cosh / 2

sinh / 2 1 1 16 1 1 2

1,3,5,... 2 2

π

π

π

π

φ θ

τ 



 



 = − − ∂

∂ = − ∞

=

∑

and

( )( ) ( )

( ) a

n x

n b a

n y a

n

G a

x

n

n zy 2

sin

cosh / 2

cosh / 2 1 1 16 1 1 2

1,3,5,... 2 2

π

π

π

π

φ θ

τ 



 



 = − − ∂

∂ = − − ∞

=

∑ .

Assuming that b>a, the maximum shearing stress corresponding to the maximum slope,

is at the middle points (y=0) of the long sides x=±a of the rectangular cross-section.

Substituting x=a, y=0 and recognizing that

5 8

1

3

1 1

2

2 2

π

+ + +L =

yields

( )

G a

n n n b a

θ

π π

τ

















= − ∑

∞

=1,3,5,... max 2 2 cosh / 2

16 1 2 .

In addition, the twisting moment, Mt, is given by

( )( ) 













 = − ∑

∞

=1,3,5,... 5 5

3

t 2

tanh 192 1 2 2 1

3

1

n a

n b

b n

a M G a b π

π

θ .
152

y

x

a

b

φ = 0

0

= 0

∂

∂

x

φ

= 0

∂

∂

y

φ

φ = 0

 

(a) Analysis Domain and Boundary Conditions (b) 6 × 6 Mesh of Grid Points

Figure 4.3. Analysis Domain, Boundary Conditions and Typical Mesh for One Quadrant

of Prismatic Bar with Rectangular Cross-Section.

Spatial Modeling of Prismatic Bar

Analyses are performed for the case of b=2a (i.e., rectangular cross-section),

where a and b are dimensions of the cross-section shown in Figure 4.3(a). Three levels

of grid refinement are used for the spatial modeling, namely meshes of (6 × 6), (11 × 11),

and (21 × 21) grid points, each applied to one quadrant of the domain shown in Figure

4.3(a). A typical idealization for a (6 × 6) mesh of grid points is shown in Figure 4.3(b).

Multiple-domain analyses with the spatial modeling of these three levels of grid

refinement and with coincident nodes along the common subdomain boundary have been

performed for comparison. For the multiple-domain spatial modeling with noncoincident nodes along the common boundary, the mesh discretization of the most

refined domain is consistent with the discretization used in that same region for the

single-domain analysis. The mesh in the less refined domain has half the “element”
153

density of that used in the refined domain. This mesh is referred to by the syntax (11 ×

11)/(21 × 21). The coarse and fine finite element models, shown in Figure 4.4, are used

in the finite element homogeneous spatial modeling. For the finite difference

homogeneous modeling and the heterogeneous modeling, a finite difference mesh is used

that has the same number of grid points as the finite element mesh in the respective

domain.

Figure 4.4. Multiple-Domain (11 × 11)/(21 × 21) Idealization.

Twisting Moment for the Prismatic Bar

Having found the values of the stress function, φ , at the grid points in the solution

domain by the respective spatial discretization approaches, the twisting moment may be

found by repeated application of the trapezoidal rule for numerical integration. The

computed twisting moment is then normalized by the analytical solution. The normalized

twisting moment ( ) analytical Mt Mt obtained using the homogeneous and heterogeneous

spatial modeling approaches are given in Table 4.1. A value of unity indicates perfect

agreement with the analytical solution. Results in Table 4.1 indicate that all analyses are

Insert 11x11

21x21 Mesh

Here
154

in good agreement with the analytical solution. The maximum error in any of the

computed solutions is less than 6%. The maximum error value for the multiple-domain

analyses is less than 3% and is observed for the multiple-domain heterogeneous modeling

analysis (MD/HM) using combined finite difference and finite element discretizations.

Note that some of the error is intrinsic to the coarse approximation of the integral using

the trapezoidal rule. The integration error decreases as the mesh refinement is increased.

A more accurate integration rule such as Simpson’s rule would produce results that are

more accurate. Independent of the integral approximation, the solution accuracy for each

of the modeling methods increases as the mesh refinement increases. For the same

number of nodes or grid points, the finite element discretization yields more accurate

solutions than the finite difference discretization. The results obtained for the singledomain modeling (e.g., SD/FE and SD/FD) and the multiple-domain homogeneous

modeling with coincident nodes along the subdomain boundary are identical or nearly

identical (see the results for (6 × 6), (11 × 11) and (21 × 21) meshes in Table 4.1). These

results validate the multifunctional approach for coincident grid points along the

subdomain boundary. The results obtained for the multiple-domain heterogeneous

modeling approach with coincident grid points along the subdomain boundary are less

accurate than corresponding results obtained using homogeneous modeling but are in

overall good agreement. In addition, with the heterogeneous modeling, the accuracy of

the twisting moment increases as the mesh refinement increases. With multiple-domain

modeling using finite element (MD/FE) discretization and with non-coincident nodes,

the accuracy of the twisting moment is bounded by the accuracy of the less refined (11 ×

11) and more refined (21 × 21) coincident meshes (see the results for the (11 × 11)/(21 ×
155

21) mesh in Table 4.1). For the multiple-domain finite difference (MD/FD)

discretization in both domains with non-coincident nodes, the twisting moment is slightly

less accurate than the results obtained using the (11 × 11) coincident mesh, which is

indicative of the error introduced by the finite difference interface constraints along the

common boundary. For the heterogeneous modeling approach with coincident nodes

along the interface boundary, the twisting moment is less accurate than the homogeneous

approach with either finite element modeling or finite difference modeling. These results

reveal the error introduced in the heterogeneous modeling approach for this problem due

to the interface constraints. However, recall that the twisting moment is a secondary

result, and the errors obtained are larger than those obtained for the primary variable, φ ,

the stress function. For the heterogeneous modeling approach with non-coincident nodes,

the twisting moment is slightly more accurate than the (11 × 11) coincident mesh, which

is indicative of the benefit gained (i.e., more accurate field approximation and interface

constraint) by the combination of the finite element and finite difference discretizations.

Table 4.1. Normalized Twisting Moment for the Prismatic Bar.

Normalized Twisting Moment, ( )analytical M t Mt

Mesh Density Analysis

Type*

(6 × 6) (11 × 11) (21 × 21) (11 × 11)/(21 × 21)

SD/FE 0.9871 0.9944 0.9976 -

SD/FD 0.9743 0.9897 0.9964 -

MD/FE 0.9871 0.9944 0.9976 0.9959

MD/FD 0.9746 0.9898 0.9964 0.9834

MD/HM 0.9498 0.9738 0.9878 0.9749 *

 SD/FE: Single-Domain with Finite Element discretization

 SD/FD: Single-Domain with Finite Difference discretization

 MD/FE: Multiple-Domain with Finite Element discretization

 MD/FD: Multiple-Domain with Finite Difference discretization

 MD/HM: Multiple-Domain with Heterogeneous Modeling (combined finite

difference and finite element discretizations)
156

Maximum Shear Stress for Prismatic Bar

7KHPD[LPXPVKHDUVWUHVV max, occurs at x=a and y=0 and is obtained by

evaluating ∂φ ∂x at that point. For the finite element method, the shear stress may be

obtained from the element shape functions. However, a more general approximation is

used herein to compare the finite element and finite difference computations. Generally,

to determine this partial derivative, ∂φ ∂x , of the stress function, a smooth curve

containing the stress function values at the grid points can be assumed to represent the

function, φ . Newton’s interpolation formula47, used for fitting such a curve, can be used

to define the function that is differentiated and evaluated at x=a to give the value of

maximum shear. However, due to errors introduced in the interpolation for large

amounts of data, a simple backward-difference approximation with the error of the order

of 2 ∆x was used such that

( ) i j i j i j

x a y x x , , ,,

φ φ φ φ 4 3

2

1

2 1

0

− +

∆  = 

 





∂

∂ − −

= =

where the subscripts, i,j, represent the location of the grid point at which the stress

function is sampled (i.e., x=a, y=0 in this case) and ∆x is the distance between the i

th and

the i-1thJULGSRLQW7KHYDOXHVIRUWKHPD[LPXPVKHDUVWUHVV max, obtained using the

multifunctional approach with single-domain (e.g., SD/FE and SD/FD) and multipledomain analyses are normalized by the analytical solution, and these normalized values

are given in Table 4.2. A value of unity indicates perfect agreement with the analytical

solution. The results indicate that all of the analyses are in excellent agreement with the

analytical solution. The maximum error in any of the computed solutions is less than 2%.

This maximum error value is obtained for the multiple-domain heterogeneous modeling
157

analysis (MD/HM). In general, the solution accuracy for each of the modeling methods

increases as the mesh refinement increases. An exception to this characteristic is

observed for the finite element discretization (see the results for the (11 × 11) and the (21

× 21) meshes in Table 4.2). In this case, the results are oscillating about the analytical

solution. For the same number of nodes or grid points, the finite element discretization

yields more accurate solutions than the finite difference discretization. The results

obtained for the single-domain modeling and the homogeneous modeling with coincident

nodes along the subdomain boundary are identical or nearly identical. As in the case for

the twisting moment, this characteristic indicates that the multifunctional approach does

not introduce error for the compatible meshes. The results obtained for the multipledomain heterogeneous modeling approach with coincident grid points along the

subdomain boundary are less accurate than corresponding results obtained using

homogeneous modeling; however, the results are in overall good agreement. In addition,

with the heterogeneous modeling, the accuracy of maximum shear stress increases as the

mesh refinement increases. With multiple-domain modeling using finite element

discretization and with non-coincident nodes, the accuracy of the twisting moment is

bounded by the accuracy of the less refined (11 ×11) and more refined (21 × 21)

coincident meshes (see the results for the (11 × 11)/(21 × 21) mesh in Table 4.2). For the

finite difference discretization in both domains with non-coincident nodes, the twisting

moment is slightly less accurate than the (6 × 6) coincident mesh, which is indicative of

the error introduced by the finite difference interface constraints along the common

boundary. However, the error for all of the finite difference homogeneous analyses is

much less than 1%; thus, the difference in the homogeneous modeling is not appreciable.
158

For the heterogeneous modeling approach with non-coincident nodes, the twisting

moment is slightly less accurate than the (11 × 11) coincident mesh, which, again, is

indicative of the benefit gained (i.e., more accurate field approximation and interface

constraint) by the combination of the finite element and finite difference discretizations.

Table 4.2. Normalized Maximum Shear for the Prismatic Bar.

Normalized Maximum Shear, ( ) max max analytical τ τ

Mesh Density Analysis

Type*

(6 × 6) (11 × 11) (21 × 21) (11 × 11)/(21 × 21)

SD/FE 1.009 0.9997 0.9993 -

SD/FD 0.9940 0.9973 0.9986 -

MD/FE 1.009 0.9997 0.9993 0.9995

MD/FD 0.9942 0.9973 0.9986 0.9940

MD/HM 0.9842 0.9904 0.9948 0.9902 *

 SD/FE: Single-Domain with Finite Element discretization

 SD/FD: Single-Domain with Finite Difference discretization

 MD/FE: Multiple-Domain with Finite Element discretization

 MD/FD: Multiple-Domain with Finite Difference discretization

 MD/HM: Multiple-Domain with Heterogeneous Modeling (combined finite

difference and finite element discretizations)

4.3. HEAT CONDUCTION PROBLEM

In this section, the basic equation of heat conduction is described briefly to

provide a convenient reference for the fundamental concepts and equations governing

conductive heat transfer. The starting point for heat conduction analysis is Fourier’s law

given in Cartesian vector form for an isotropic medium48

q = −k∇T

where q is a vector whose components are the heat flow per unit area in the respective

Cartesian directions, k is the thermal conductivity coefficient that may be a function of

the temperature, T, and
159

i j k

x y ∂z

∂

+

∂

∂

+

∂

∂ ∇ = .

In an isotropic solid with temperature-dependent thermal conductivity, the law of

conservation of energy with Fourier’s law yields the thermal energy equation. The law of

conservation of energy is given by

t

T Q c

z

q

y

q

x

qx y z

∂

∂

+ = 















∂

∂

+

∂

∂

+

∂

∂ − ρ

where Q is the internal heat generation rate per unit volume, is the mass density, c is the

specific heat, and t is time. For constant thermal properties and steady-state heat transfer,

the heat conduction problem reduces to a nonhomogeneous partial differential equation

of the form of Eq. (2.1) and is given by

Q

y

T

x

T k =















∂

∂

+

∂

∂ − 2

2

2

2

In this work, two-dimensional heat conduction in a square plate (see Figure 4.5) is used

to demonstrate the multifunctional capabilities for thermal analysis. For this problem, the

time-independent, heat conduction equation is

 in { } () () x, y 0 1 2

2

2

2

= Ω = < < 















∂

∂

+

∂

∂ − x y kQ

y

T

x

T

: ,

subject to the boundary conditions

{ }

0 on { } lines 0 and 0

0 on lines 1and 1

s

p

= Γ = = = ∂

∂

= Γ = = =

x y

n

T

T x y

.
160

x

y

1.0

T=0

T=0 = 0

∂

∂

x

T

= 0

∂

∂

y

T

1.0

Figure 4.5. Analysis Domain and Boundary Conditions for the Steady-State Heat

Conduction in a Square Plate.

Spatial Modeling of Square Plate

The spatial discretizations in the analyses were selected to be comparable to those

reported by Reddy32 for this problem. Coarse and fine models are used in each of the

subdomains. The coarse model has a (2 × 3) nodal grid, and the fine model has a (3 × 5)

nodal grid. The syntax (m × n) is used to denote spatial modeling with m grid points in

the x-direction and n grid points in the y-direction. The number of grid points, rather than

the number of elements, in the coordinate directions are used to describe the mesh

densities to provide consistency when discussing the finite difference and finite element

discretizations. Combinations of these mesh densities are used for comparative purposes

where the letters C and F are used to denote the coarse and fine models, respectively. A

multiple-domain model with finite element models discretized with a fine (3 × 5) nodal

grid and a coarse (2 × 3) nodal grid is shown in Figure 4.6. Curves labeled C/C or F/F

denote multiple-domain coarse or fine models, respectively, with coincident nodes along

the common subdomain boundary. Multiple-domain analyses with finite element
161

discretization or finite difference discretization are denoted by MD/FE and MD/FD,

respectively. Similarly, multiple-domain analyses using heterogeneous modeling with

the combination of finite difference and finite element discretizations are denoted by

MD/HM.

Figure 4.6. Homogeneous (3 × 5)/(2 × 3) Idealization.

Temperature Distribution for Square Plate

The temperature distribution as a function of the distance along the y=0 line is

shown in Figure 4.7 for the different spatial discretizations and modeling approaches.

The analytical solution for this problem is given by

( ) ( ) () ( ) [ ] [ ] ( )

() () [ ] 















− −

− − − = − + ∑

∞

=1 3 3

2

2 1 cosh 2 1 2

32 1 cos 2 1 2 cosh 2 1 2 1

2

1 ,

n

n

n n

n y n x T x y y

π

π π

π

In addition, a 1-parameter Ritz approximation is given by

( ) ( )( ) 2 2 1 1

16

5 , x y k

Q T x y = − −

x

y
162

Results obtained using the multifunctional approach are compared to the

analytical solution (solid line in the figure) and a Ritz approximation (dashed line in the

figure). Finite element (see Figure 4.7(a)) and finite difference (see Figure 4.7(b))

solutions were obtained using a multiple-domain analysis with homogeneous spatial

discretization and are in excellent agreement with the analytical solution. These results

illustrate that the temperature at x=y=0 obtained with a coarse finite difference mesh is

more accurate than that obtained with a comparable finite element mesh (see curves

labeled MD/FE-C/C and MD/FD-C/C in Figure 4.7(a) and Figure 4.7(b)). This

difference decreases as the meshes are refined, although the finite element model

continues to produce a higher temperature value at x=y=0. The multiple-domain

analyses with non-coincident nodes produce accurate results even at the subdomain

common boundaries. The multiple-domain results for heterogeneous spatial

discretization approaches are shown in Figure 4.7(c) and indicate the effectiveness of the

multifunctional approach. The fine (3 × 5) nodal grid (see Figure 4.6) is discretized with

the finite difference method, and the coarse (2 × 3) nodal grid is discretized with the

finite element method. These results are in overall agreement with the results obtained

with the homogeneous approaches. The homogeneous and heterogeneous results are

compared in Figure 4.7(d) for models with non-coincident nodes with a fine model in the

left domain and a coarse model in the right (see Figure 4.6). These results indicate that

temperatures obtained with the heterogeneous approach are slightly lower than for the

homogeneous approach with either finite element or finite difference discretizations. In

addition, the results, obtained by using the finite difference discretization in one or both

of the domains, illustrate the slight difference in the temperature at the interface from the
163

left and the right domains. However, note that the uniqueness of the solution along the

interface boundary is satisfied only in an integral sense and this slight difference does not

detract from the overall accuracy and effectiveness of the multifunctional approach for

this Poisson problem.

An additional analysis has been performed to demonstrate the multifunctional

capability for an inclined subdomain boundary (boundary not parallel to the y-axis). In

this analysis, multiple-domain modeling with the finite element method is used. The

finite element model used in the analysis has a (3 × 6) mesh of grid points in the left

domain and a (2 × 3) mesh of grid points in the right domain as shown in Figure 4.8. The

results for this multiple-domain finite element analysis are shown in Figure 4.9. These

results (open squares) are compared to the analytical solution (solid line), the Ritz

approximation (dashed line) and the multiple-domain finite element analysis (see Figure

4.6 for the model discretization) with a subdomain boundary parallel to the y-axis (open

circles). The results indicate the effectiveness of the multifunctional approach for the

inclined subdomain boundary.
164

 (a) Multiple-Domain Finite Element (b) Multiple-Domain Finite Difference

 (MD/FE) Modeling (MD/FD) Modeling

 (c) Multiple-Domain Heterogeneous (d) Multiple-Domain Homogeneous and

(MD/HM) Modeling Heterogeneous Modeling

Figure 4.7. Temperature Distribution Along Insulated Edge of Square Plate.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.2 0.4 0.6 0.8 1.0

x

kT(x,0)

Q Analytical Solution

Ritz Approx.

MD/FE- C/C

MD/FE -F/F

MD/FE - C/F

MD/FE - F/C

Interface

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.2 0.4 0.6 0.8 1.0 x

kT(x,0)

Q Analytical Solution

Ritz Approx.

MD/FD- F/F

MD/FD -C/F

MD/FD - F/C

Interface

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.2 0.4 0.6 0.8 1.0

x

kT(x,0)

Q

Analytical Solution

Ritz Approx.

MD/HM- F/F

MD/HM -C/F

MD/HM - F/C

Interface

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.2 0.4 0.6 0.8 1.0

x

kT(x,0) 

Q

Analytical Solution

Ritz Approx.

MD/FE- F/C

MD/FD -F/C

MD/HM - F/C

Interface
165

Figure 4.8. Spatial Discretization for Inclined Interface for Square Plate.

Figure 4.9. Temperature Distribution Along Insulated Edge of Square Plate with Inclined

Interface.

4.4. POTENTIAL FLOW PROBLEM

A two-dimensional fluid flow problem is used to demonstrate the multifunctional

capabilities for a fluid mechanics problem. As shown in Section 2.2.2, the equation

governing irrotational fluid flow reduces to the Laplace equation

x

y Interface

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.2 0.4 0.6 0.8 1.0 x

kT(x,0) 

Q

Analytical Solution

Ritz Approx.

MD/FE- F/C

MD/FE - S/C

Interface
166

0 2

2

2

2

=

∂

∂

+

∂

∂

y

u

x

u

where uFDQEHHLWKHUWKHVWUHDPIXQFWLRQ RUWKHYHORFLW\SRWHQWLDOφ . In this work,

the two-dimensional, steady, inviscid flow between two infinite plates is considered. A

rigid, infinite cylinder or radius, R, with an axis at a right angle to the flow is assumed to

be in the passageway between the plates as shown in Figure 4.10. Far upstream from the

cylinder there is a uniform flow field with a velocity of V0. Because of the symmetries in

this problem, only one quadrant of the domain is considered. The analysis domain and

the boundary conditions on the velocity potential, φ , are shown in Figure 4.11.

2a

2b

V0 R

Figure 4.10. Domain of Flow Around Cylinder.
167

a

b

= 0

∂

∂

y=b y

φ

0

0

V

x x

= ∂

∂

=

φ

= 0

∂

∂

r =R n

φ

x

y

0

0

= ∂

∂

y= y

φ

= 0

x=a φ

r R

θ

Figure 4.11. Analysis Domain of Flow Around Cylinder.

The finite element models used in this problem are shown in Figure 4.12. A reference

solution is obtained using the finite element model shown in Figure 4.12(a). The local

and global finite element models used in the homogeneous and heterogeneous spatial

modeling approaches are shown in Figure 4.12(b). For the heterogeneous modeling, a

finite difference mesh is used in the coarsely refined domain that has the same number of

grid points as the finite element mesh used in the same domain. This discretization

strategy illustrates the use of the finite element method to represent the complex

geometry around the cylinder and the use of the finite difference method away from the

curved boundary where it is most suitable.
168

(a) Reference Model (b) Multiple Domain Model

Figure 4.12. Spatial Discretization for One Quadrant of Domain of Flow Around

Cylinder.

Contour plots for the velocity potential, the horizontal velocity component and the

transverse velocity component are shown in Figure 4.13, Figure 4.14, and Figure 4.15,

respectively. In each of these figures, the results using the multifunctional approach are

compared to results obtained from the single-domain analysis using the reference model

(see Figure 4.12(a)). As shown in the figures, the velocity potential and the velocity

components obtained using the multifunctional approach are in excellent agreement with

the reference solution. In the multiple-domain analyses, the slight discontinuity in the

horizontal and transverse velocity components at the interface (see Figure 4.14(b) and

Figure 4.15(b)) is due in part to the difference in the computation of the velocity across

the interface. Unlike in the single-domain analysis (i.e., reference solution), in the

multiple-domain analyses, the velocities are not averaged across the interface.
169

(a) Single-Domain Model (b) Multiple-Domain Model

Figure 4.13. Contour Plot of Velocity Potential for Flow Around Cylinder.

(a) Single-Domain Model (b) Multiple-Domain Model

Figure 4.14. Contour Plot of Horizontal Velocity Component for Flow Around Cylinder.
170

(a) Single-Domain Model (b) Multiple-Domain Model

Figure 4.15. Contour Plot of Transverse Velocity Component for Flow Around Cylinder.

The analytical potential solution for the tangential velocity around a cylinder in an

infinite domain, valid on the cylinder surface, is given by

1 sinθ 2

2

0 













 = +

r

R

ut V

ZKHUH WKH DQJOH  UDGLDO GLVWDQFHr, and the tangential velocity, ut, can be computed

from the relations

tan , [ ] ( ) , sin cos . 1 2 2 1 2 θ r a x y ut ux θ u y θ

a x

y  = − + = + 

 





− = −

The tangential velocity as a function of the angular distance along the cylinder surface is

shown in Figure 4.16. Results are shown for the tangential velocity around a cylinder in

an infinite domain for which an analytical solution is known and in a finite domain for

which a reference solution is obtained using a refined single-domain finite element

model. For the infinite domain configuration, the plate length to cylinder radius ratio,
171

2a R , and the plate width to cylinder radius ratio, 2b R, are 40 and 20, respectively, and

the domain can be considered as infinite. That is, the cylinder radius, R, is very small

compared to the length, 2a, and the width, 2b. For the finite domain configuration, the

plate length to cylinder radius ratio, 2a R , and the plate width to cylinder radius ratio,

2b R, are 4 and 2, respectively, and the domain is considered to be finite. The tangential

velocity obtained for the multifunctional approach is in overall good agreement with the

analytical solution for the infinite domain and with the reference solution (i.e., singledomain analysis) for the finite domain. Results obtained with homogeneous multipledomain analyses with finite element discretization in each domain are denoted by open

circles in the figure. Results obtained with heterogeneous multiple-domain analyses with

combined finite difference and finite element discretization are denoted by open squares

in the figure. The tangential velocity obtained with the homogeneous modeling approach

is in excellent agreement with the analytical and reference solutions for the infinite and

finite domain configurations. The tangential velocity obtained with the heterogeneous

modeling approach is more accurate for the infinite domain configuration than for the

finite domain configuration. This characteristic is indicative of the performance of the

finite difference approach, for this problem, in a gradient region.
172

Figure 4.16. Tangential Velocity for Flow Around Cylinder.

4.5. SUMMARY

In this chapter, the multifunctional methodology has been described and

demonstrated for a variety of problems in engineering science. These selected problems

included second-order problems of solid mechanics, heat transfer, and fluid mechanics

that can be formulated in terms of one dependent variable. The governing equation in

each case is either the Laplace or the Poisson equation. The analyses performed have

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 10 20 30 40 50 60 70 80 90

Angle, θ

ut(R,θ)

V0

Infinite Domain - Analytical Solution

Infinite Domain - FD/FE

Infinite Domain - FE/FE

Finite Domain - Ref. FE Solution

Finite Domain - FD/FE

Finite Domain - FE/FE

r R

θ
173

demonstrated the effectiveness and accuracy of the solutions obtained for the respective

problems. In all cases, the results obtained using the multifunctional methodology were

in overall good agreement with the reported analytical or reference solution. In the next

chapter, the multifunctional methodology is demonstrated for problems whose motion is

described by coupled partial differential equations expressed in terms of two dependent

variables -- vector-field problems.
174

CHAPTER V

REPRESENTATIVE VECTOR-FIELD APPLICATIONS AND

EXTENSIONS

5.1. GENERAL

In this chapter, the multifunctional methodology is demonstrated on two

representative vector-field applications. The applications are described and the

associated results and salient features are discussed. The applications include a plane

stress problem and a plane flow problem. Finite difference and finite element solutions

for single- and multiple-domain configurations are presented to validate the

multifunctional approach using heterogeneous discretization. The finite element models

use four-node Lagrange isoparametric finite elements, and the finite difference model

uses a nine-point template to approximate the governing differential equation. Standalone finite element software is used to generate the finite element stiffness matrices.

The mathematical computing program MATLAB is used to generate the finite

difference matrices and the interface coupling matrices and to solve the resulting system

of equations. In addition, extensions to multiple discipline analyses are discussed.

5.2. PLANE STRESS PROBLEM

A rectangular plate of uniform thickness subjected to a uniform tensile load and

with a central circular cutout (shown in Figure 5.1) is an ideal example problem with

which to verify the multifunctional approach. The example problem has a variety of

practical applications (i.e., rivet holes, aircraft door and window openings, etc.), and an

exact solution is available46. The plate has been used by many researchers to verify
175

aspects of proposed computational methodologies. For example, the plate problem has

been used by Ransom8 to verify global/local analysis technology, by Aminpour et al.25 to

verify multiple-domain homogeneous modeling using the finite element method, and by

Rose10 to verify an adaptive geometry generator used with a multiple-domain finite

element model. The plate configuration is such that the state of stress is represented by

the condition of plane stress or plane strain. The membrane displacements, u and v, in

the axial (x-direction) and transverse (y-direction) directions, respectively, represent the

plate configuration in plane stress and plane strain.

Two configurations of this problem have been studied: an infinite plate and a

finite-width plate. The infinite plate configuration has a central cutout that is very small

relative to the length and width of the plate, and the exact solution for this problem was

obtained by Timoshenko46. The stress distribution in the neighborhood of the cutout

exhibits a stress concentration, but from Saint-Venant’s principle, the stress distribution

is essentially uniform at distances that are large compared with the radius of the cutout.

The finite-width plate configuration has a larger central cutout relative to the length and

width, and the stress distribution away from the cutout is not uniform. The finite-width

plate with a central circular cutout has been discussed by Howland49 and Peterson50.

For the infinite plate configuration, herein, the length to radius ratio, 2a R , and

the width to radius ratio, 2b R, are 40 and 20, respectively, and the plate can be

considered as infinite. That is, the cutout radius, R, is very small compared to the length,

2a, and the width, 2b. The material system is aluminum with a Young’s modulus of 107

psi, and a Poisson’s ratio of 0.3, and the thickness of the plate, h, is 0.1 in. A uniform

running load, (Nx)0, is applied to each of its ends, and the other sides are free. The plate
176

example problem is used to verify the multifunctional approach for both homogeneous

and heterogeneous spatial modeling. Because of the symmetry that exists, one quadrant

of the domain (see Figure 5.2) is modeled. In addition, boundary conditions are shown in

Figure 5.2 where Tn and Tt denote normal and tangential tractions, respectively. For the

multiple-domain analysis, a refined model is used in the near-field subdomain (i.e., the

local region near the cutout), and a coarse, less-refined model is used in the remainder of

the domain. A single-domain analysis using a finite element model that has the same

number of nodes and elements in the near-field region as the multiple-domain model is

used to obtain a reference solution with which to compare the solution obtained with the

multifunctional approach. The single-domain model and the multiple-domain model

(used in the homogeneous spatial modeling) are shown in Figure 5.3. For the

homogeneous modeling, a finite element (FE) mesh is used in each region. For the

heterogeneous modeling, a finite difference (FD) mesh is used in the far-field region that

has the same number of grid points as the finite element mesh in that region. A finite

element mesh is used in the region near the cutout.

2a

2b ( ) x 0 N () () N h x 0 x 0 = σ

x

y

r

R

Figure 5.1. Domain of Plate with Central Circular Cutout.
177

0

0

=

=

t

n

T

T

a

b

Tn = Tt = 0

u = 0

x

y

v = 0

( )

0

0

=

= −

t

n x

T

T σ

r

A B

D

C

Figure 5.2. Geometric Configuration for One Quadrant of Plate with Central Circular

Cutout.

 (a) Single-Domain Model (b) Multiple-Domain Model

Figure 5.3. Finite Element Models for One Quadrant of Infinite Plate with Central

Circular Cutout.

The exact elasticity solution46 for an infinite plate with a circular cutout loaded in

tension indicates that the stress concentration factor, Kt, is 3.0 at the edge of the cutout

and is given by

( )

0

2

( )

( ) ,

x

x R

t N

N

K

θ π = .

The stress concentration factor is defined as the ratio of the maximum stress resultant,

(Nx)max, to the uniform far-field stress resultant, (Nx)0. Stress concentration factors
178

obtained using the multifunctional approach with homogeneous and heterogeneous

modeling are 3.08 and 3.10, respectively, which is within 2.7% and 3.3% of the elasticity

solution. The stress distributions of the hoop stress resultant (Nx) along the midwidth,

 GHQRWHGDVOLQH$%LQFigure 5.2DQGPLGOHQJWK GHQRWHGDVOLQH&'LQ

Figure 5.2) normalized by the far-field stress resultant (Nx)0, are shown in

Figure 5.4 as a function of the distance from the plate center normalized by cutout radius,

R. The elasticity solution for the stress distribution is given by

( ) ( ) 































− + 













 = + θ θ cos2 3 1 1

2

1

4

4

2

2

0

r

R

r

R Nx Nx

and is shown by the solid lines in the figure. The stress distributions obtained from the

multifunctional analyses using homogeneous modeling are indicated by the open circles

in the figure. The stress distributions obtained from the multifunctional analyses using

heterogeneous modeling are indicated by the open squares in the figure. Excellent

correlation is observed for all analyses.

Contour plots of the magnitude, δ , of the displacement vector (i.e.,

2 2 δ = u + v ) superimposed on the deformed shape and the longitudinal stress

resultant, Nx, are shown in Figure 5.5 and Figure 5.6, respectively. The multiple-domain

analysis results are shown for homogeneous modeling using finite element discretization

in each of the subdomains. To aid visual comparison, the deformation has been

magnified by 10% of the maximum domain dimension. The displacement contour plots

reveal the nearly linear variation along the plate length in the far-field region of the plate

with only local changes near the cutout. The stress resultant contour plots reveal the

uniform stress state away from the cutout and the peak stress in the neighborhood of the
179

cutout. While not shown, the results for the multiple-domain heterogeneous modeling

approach are nearly identical to those shown in Figure 5.5 and Figure 5.6, and thus have

not been included. These contour plots illustrate further the excellent correlation among

the multifunctional approach using homogeneous and heterogeneous modeling and the

single-domain solution.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 5 10 15 20

r/R

N θ

(N x ) 0

Analytical 

Multiple-Domain(FD/FE)

Multiple-Domain(FE/FE)

Edge of

Cutout

Interface

Figure 5.4. Longitudinal Stress Distribution along Midwidth and Midlength for Infinite

Plate with Central Circular Cutout.
180

(a) Single-Domain Model (b) Multiple-Domain Model

Figure 5.5. Displacement Magnitude Distribution for Infinite Plate with Central Cutout.

(a) Single-Domain Model (b) Multiple-Domain Model

Figure 5.6. Longitudinal Stress Resultant Distribution for Infinite Plate with Central

Cutout.

While the infinite plate analyzed, herein, is an excellent test of the multifunctional

approach, gradients in the deformation and the stress resultants, as indicated in Figure 5.5

and Figure 5.6, are well away from the subdomain interface boundary. Thus, to assess

the accuracy of the approach when the subdomain interface is within a high gradient

region, a second configuration is analyzed.

In the finite-width plate configuration, the length to radius ratio, 2a R , and the

width to radius ratio, 2b R, are 4 and 2, respectively, and the plate is considered to be

Interface

Interface
181

finite. The aluminum material system and the thickness that was used for the infinite

plate is used here for the finite-width plate. The finite-width effects on the stress

concentration factor for isotropic plates with cutouts have been reported by Peterson50.

By including finite-width effects, the stress concentration factor is reduced from the value

of three for an infinite plate. The stress concentration factor should be applied to the

nominal stresses, which are based on the net cross-sectional area associated with the load

application. For the case of a finite-width plate with a cutout, the net cross-sectional area

corresponds to

( ) 



 



 = − = − b

R Anet 2b 2R0 h 2bh 1

where h is the plate thickness, and the nominal longitudinal stress for an uniaxial load, P,

can be expressed as

( )

net

x nom A

P

σ = and () () N h x nom x nom = σ .

The geometry definition for the finite-width plate, herein, gives a stress concentration

factor of 2.16 reported by Peterson.

Multiple-domain homogeneous and heterogeneous modeling approaches are used

for the finite-width plate. A refined model is used in the near-field domain, and a lessrefined model is used in the far-field domain. The single-domain model and the multipledomain model are shown in Figure 5.7. In the multiple-domain homogeneous modeling

approach, finite element (FE) discretization is used in each domain. In the multipledomain heterogeneous modeling approach, finite difference (FD) discretization is used in

the far-field domain, and finite element (FE) discretization is used in the near-field

domain around the cutout.
182

 (a) Single-Domain Model (b) Multiple-Domain Model

Figure 5.7. Finite Element Models for One Quadrant of Finite-Width Plate with Central

Cutout.

Stress concentration factors obtained using the multifunctional approach with

homogeneous (multiple-domain FE/FE) and heterogeneous (multiple-domain FD/FE)

modeling are 2.19 and 2.73, respectively. These factors are higher by 1.4% and 26.4%,

respectively, than the values given in Peterson46. Note that the solution obtained using

the heterogeneous modeling approach with finite difference and finite element

discretizations is nearly 30% in error. This error is likely due to the inaccuracy of the

finite difference method in the high gradient region and to the constraint conditions along

the interface.

To delineate this error, additional heterogeneous analyses are performed using

finite difference domains with grid spacing in the transverse direction of one half (i.e.,

9×9 mesh of grid points) and one fourth (i.e., 17×17 mesh of grid points) the grid

spacing in the initial finite difference domain (i.e., 5×5 mesh of grid points) (see Figure

5.7(b)). The stress concentration factors obtained with these more refined finite

difference discretizations are 2.42 and 2.31, which are within 12% and 7% of the
183

Peterson’s solution46. The stress distributions of the hoop stress resultant (Nx)

PLGOHQJWK GHQRWHGDVOLQH&'LQFigure 5.2) normalized by the nominal stress

resultant (Nx)nom, are shown in Figure 5.8 as a function of the distance from the plate

center normalized by cutout radius, R. The analytical solution reported by Howland49 is

denoted by the thick solid line. The stress distribution obtained using 5×5, 9×9 and

17×17 mesh of grid points are denoted by the short dashed line, the thin solid line, and

the dashed and dotted line, respectively. The results shown in Figure 5.8 indicate that the

error decreases as the finite difference grid density increases, and the error decreases

away from the edge of the cutout.

The stress distributions of the hoop stress resultant (Nx) DORQJWKHPLGZLGWK 

(denoted as line AB in Figure 5.2DQGPLGOHQJWK GHQRWHGDVOLQH&'LQFigure

5.2) normalized by the nominal stress resultant (Nx)nom, are shown in Figure 5.8 as a

function of the distance from the plate center normalized by cutout radius, R. The

analytical solution reported by Howland49 is denoted by the solid lines. This analytical

solution is valid for distances, r, away from the cutout of less than the plate half-width, b.

Thus, for this configuration the solution along the midwidth is valid only for r ≤ 2R. The

stress distribution for the multifunctional analysis using homogeneous modeling with

finite element discretization in each of the domains is denoted by the open circles in the

figure. The stress distribution for the multifunctional analysis using heterogeneous

modeling with combined finite difference and finite element discretizations is denoted by

the open squares in the figure. For the heterogeneous modeling approach, the

distribution is given for the most refined finite difference discretization (i.e., 17×17 mesh

of grid points). The stress distributions obtained with the multifunctional approach using
184

homogeneous and heterogeneous discretization are in excellent agreement with the

reported solution.

Contour plots of the magnitude of the displacement vector superimposed on the

deformed shape and the longitudinal stress resultant, Nx, are shown in Figure 5.10 and

Figure 5.11, respectively. Results for the multiple-domain homogeneous modeling

approach using finite element discretization in each of the subdomains are shown in the

figures. While not shown, the results for the multiple-domain heterogeneous modeling

approach are nearly identical to those shown in Figure 5.10 and Figure 5.11, and thus

have not been included. Note that the deformation has been magnified by 10% of the

maximum domain dimension. The displacement contour plots reveal a deviation from

the nearly linear variation observed in the far-field region of the infinite plate, and the

deformation at the cutout is more pronounced. The contour plots illustrate further the

excellent correlation of the deformation (primary variable) patterns predicted using the

multifunctional approach with the single-domain solution even with the interface

boundary domain in a high-gradient region. The stress resultant (secondary variable)

patterns predicted using the multifunctional approach are also in excellent agreement.

The slight discontinuity in the stress resultant at the subdomain boundary (i.e., interface)

is due to the derivation of the nodal stress resultant values from the element quantities.

The stress resultants are recovered at the finite element nodes by extrapolating the

stresses at the integration points to the nodes. A single nodal value of the stress resultant

is obtained by averaging the stress resultants of the adjacent elements. In the multipledomain analyses, the stress is not averaged across the subdomain boundary; thus, any

gradient across the interface is not considered.
185

Figure 5.8. Convergence of Longitudinal Stress Distribution along Midlength for FiniteWidth Plate with Central Circular Cutout.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

r/R

N θ

(N x ) nom

Analytical Solution

Multiple-Domain(FD/FE)

Multiple-Domain(FE/FE)

Edge of

Cutout

Interface

Figure 5.9. Longitudinal Stress Distribution along Midwidth and Midlength for

Finite- Width Plate with Central Circular Cutout.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1.0 1.5 2.0

r/R

N θ

(N x ) nom

Analytical Solution

FD/FE-5x5

FD/FE-9x9

FD/FE-17x17

Edge of

Cutout

Interface
186

(a) Single-Domain Model (b) Multiple-Domain Model

Figure 5.10. Displacement Magnitude Distribution for Finite-Width Plate with Central

Circular Cutout.

(a) Single-Domain Model (b) Multiple-Domain Model

Figure 5.11. Longitudinal Stress Resultant Distribution for Finite-Width Plate with

Central Circular Cutout.

5.3. PLANE FLOW PROBLEM

The flow of a viscous incompressible material squeezed between two long

parallel plates41 is considered to illustrate the applicability and performance of the

multifunctional approach to a representative vector-field problem in fluid mechanics.

Interface

Interface
187

The geometric configuration and the associated boundary conditions of the problem are

indicated in Figure 5.12.

A state of plane flow exists when the length of the bounding plates is very large

compared to the width of and distance between the plates. Assuming the conditions of

plane flow, the velocity and pressure fields are determined for a fixed distance between

the plates. The plates are moving toward each other with a velocity, v0, and the width of

and distance between the two plates is given by 2a and 2b, respectively. For this

configuration, the ratio of the plate width and the distance between the plates, 2a/2b, is 3.

Due to the double symmetry present in the problem, one quadrant of the domain was

analyzed. The viscosity, μ, of the fluid is 1 lb-sec/in2

. The penalty finite element

model32 is used in the analysis. The penalty function formulation (see Eq. (3.61))

involves treating the continuity equation as a constraint among velocity components. A

10 × 6 nonuniform mesh (10 elements in the x-direction and 6 elements in the ydirection) of four-node bilinear elements is used for the single-domain analysis (i.e.,

reference model in Figure 5.13(a)). The nonuniform mesh, with smaller elements near

the free surface at x=a, is used to delineate the singularity in the shear stress at the point,

x=a, y=b. This singularity and the associated necessity for nonuniform mesh refinement

make this problem ideal for demonstrating the multifunctional approach with detailed

local modeling. The finite element models for the single- and multiple-domain analyses

are shown in Figure 5.13(a) and Figure 5.13(b), respectively. In the multiple-domain

analysis, homogeneous spatial modeling with finite element discretization is used. In this

analysis, more elements are used in the region near x=a, y=b than in the single-domain

analysis (see Figure 5.13). This local modeling yields a more complex configuration of
188

the subdomain common boundary. That is, the interface between the subdomains

consists of two non-collateral segments.

y

x

2a

2b

u=0, v=-v0=-1

u=0

Tt=0

v=Tt=0

Tn=0

Tt=0

bounding

 plate

bounding

plate

Figure 5.12. Geometric Configuration for Fluid Squeezed Between Parallel Plates.

 

 (a) Single-Domain Model (b) Multiple-Domain Model

Figure 5.13. Finite Element Models for Fluid Squeezed Between Two Parallel Plates.

An approximate analytical solution to this two-dimensional problem is provided

by Nadai51 and is given by

x=2a/3 x=5a/6 x=a y

x
189















 = − − 













 = − 2

2

0

2

2

0 3

2

1 2

3

b

y

b

v y

v

b

y

b

v x

u ;

( ) 3

2

0

0

2 2 2

2

0

2

3 

b

v a

a y x P

a

P

P μ = − − ; = −

( ) 2 2 2 2

2

0 2 x 3y a 2b

a

P

P

x

u

x − = − − +

∂

∂

σ = μ

( ) 2 2 2 2

2

0 2 x y a 2b

a

P

P

y

v

y − = + − − ∂

∂

σ = μ

and

2

2 0

a

P xy

x

v

y

u

xy

− =















∂

∂

+

∂

∂

τ = μ .

Note that this approximate solution does not satisfy the traction-free conditions (Tn x=0

and Tt  xy=0) on the free edge (i.e., x=a). Likewise, these traction-free conditions are

not imposed in the finite element analysis; thus, the conditions are not identically

satisfied. The horizontal velocity, u, as a function of y, at three representative locations,

x=2a/3, x=5a/6 (along the vertical interface), and x=a, is shown in Figure 5.14(a), Figure

5.14(b), and Figure 5.14(c), respectively. The analytical solution of Nadai51 is

represented by the solid line in the figure. Finite element solutions obtained using a

single-domain spatial discretization are represented by the dashed lines in the figure. The

multiple-domain results for the homogeneous spatial modeling approach using finite

element discretization in each of the subdomains are also shown in the figures, and these

results are represented by the open circles. The results for the horizontal velocity

component obtained from the single- and multiple-domain analyses are in excellent
190

agreement with each other, and the results are in overall good agreement with the

analytical solution.

The pressure, P, as a function of x, near the centerline for the flow (i.e., y = b/16 -

the centroids of the first row of finite elements in Figure 5.13), is shown in Figure 5.15.

The analytical solution is denoted by the solid line. The solutions obtained from the

single- and multiple-domain analyses are denoted by the dashed line and open circles,

respectively. The results obtained from the multiple-domain analysis are in excellent

agreement with those from the single-domain analysis. These finite element results are

also consistent with the results published in the literature32. However, the finite element

models predict a higher pressure in the center of the flow field (i.e., x=0) than predicted

by the analytical solution.

While the velocity components and pressure field characterize the flow through

the plates, the shear stress distribution illustrates the significance of using a graded

single-domain mesh and a locally-refined multiple-domain mesh. The shear stress, τxy, as

a function of x, near the upper bounding plate (i.e., y = 15b/16 - the centroids of the last

row of finite elements in Figure 5.13), is computed at the center of the finite elements and

is shown in Figure 5.16. Again, the single-domain (dashed line in the figure) and

multiple-domain (open circles in the figure) results are in excellent agreement with the

approximate solution of Nadai51 (solid line in the figure) away from the free-edge. In

addition, because of the local refinement at the free edge, the multiple-domain results for

x≥5a/6 correspond to the shear stress located at y = 31b/32 (the centroids of the last row

of elements in the refined region). These results illustrate the better representation of the

gradient in the shear stress at the free edge than either the single-domain analysis or the
191

analytical solution. The approximate nature of the analytical solution is highlighted by

these results since the solution given does not delineate the gradient on the boundary.

)RUFRPSOHWHQHVVWKHORQJLWXGLQDO xDQGWUDQVYHUVHVWUHVV y, distributions are

shown at y = 15b/16 and y = 31b/32, respectively, in Figure 5.17 and Figure 5.18. In

general, the stresses predicted by the single- and multiple-domain finite element analyses

have a larger value than those obtained by the analytical solution. However, the

analytical solution is an approximate solution, and the finite element solutions predict the

VDPHRYHUDOOWUHQGVLQWKHVWUHVVGLVWULEXWLRQV7KHORQJLWXGLQDOVWUHVVGLVWULEXWLRQ x,

reveals the oscillatory nature of the finite element solution at the free-edge. The

wavelength of the oscillations decreases as the mesh is increased in the local region at the

free edge as indicated by the results from the multiple-domain analysis. In addition, the

value of the peak stress at the free edge increases as the finite element mesh is refined.

Overall, the results obtained with the multifunctional discretization approach are

in excellent agreement with the single-domain analysis results and with the analytical

solution given in the literature. These successful comparisons indicate the effectiveness

of the method and its applicability to the vector-field problem, specifically that of the

fluid flow problem.
192

 (a) Velocity, u, at x=2a/3

(b) Velocity, u, at x=5a/6

(c) Velocity, u, at x=a

Figure 5.14. Horizontal Velocity for the Flow Between Two Parallel Plates.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0

u(a,y/b)

y

b

Analytical Solution

Single-Domain Solution

Multiple-Domain Solution

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0

u(5a/6,y)

y

b

Analytical Solution

Single-Domain Solution

Multiple-Domain Solution

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0

u(2a/3,y/b)

y

b

Analytical Solution

Single-Domain Solution

Multiple Domain Solution
193

Figure 5.15. Pressure Distribution Near Centerline for the Flow Between Two Parallel

Plates.

Figure 5.16. Shear Stress Distribution Near Plate Boundary for the Flow Between Two

Parallel Plates.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 0.2 0.4 0.6 0.8 1.0

x/a

 -P(x/a,31b/32)

Analytical Solution

Single-Domain Solution

Multiple-Domain Solution

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 0.2 0.4 0.6 0.8 1.0

x/a

τxy (x/a,15b/16)

Analytical Solution

Single-Domain Solution

Multiple-Domain Solution
194

Figure 5.17. Longitudinal Stress Distribution Near Plate Boundary for the Flow Between

Two Parallel Plates.

Figure 5.18. Transverse Stress Distribution Near Centerline for the Flow Between Two

Parallel Plates.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0.0 0.2 0.4 0.6 0.8 1.0 x/a

σy (x/a,31b/32)

Analytical Solution

Single-Domain Solution

Multiple-Domain Solution

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0.0 0.2 0.4 0.6 0.8 1.0

x/a

−σ x (x/a,15b/16)

Analytical Solution

Single-Domain Solution

Multiple-Domain Solution
195

5.4. EXTENSIONS TO MULTIPLE DISCIPLINES

In the present work, the multifunctional capability has been demonstrated on

scalar- and vector-field problems applicable to the general field of engineering science

and mechanics. While the demonstrations have illustrated the capability within different

disciplines (i.e., solid mechanics, fluid mechanics, and heat transfer), the method’s use

has not been demonstrated for multidisciplinary analysis. Extensions to simultaneous

multiple disciplines are discussed here.

The term multidisciplinary or coupled systems refers to two or more systems that

interact with each other, with the independent solution of any one system being

impossible without simultaneous solution of the others52. In general, coupled systems

and formulations, such as the multifunctional methodology presented in this work, are

those applicable to multiple domains and dependent variables which usually describe

different physical phenomena, and in which (1) neither domain can be analyzed

independently; and (2) neither set of dependent variables can be explicitly eliminated at

the differential equation level. The class of coupling problems that are the focus of this

work can be categorized by coupling that occurs on domain interfaces via the boundary

conditions imposed on that interface. Generally, the domains describe different physical

situations, but it is possible to consider coupling between domains that are physically

similar in which different discretization strategies have been used. Fluid-structure and

thermal-structure interaction problems are typical examples that involve different

disciplines in different but adjacent domains. Structure-structure or fluid-fluid interaction

problems are examples where the interface divides arbitrarily chosen regions in which

different mathematical approximations and/or spatial discretization procedures are used.
196

Single discipline interaction problems have been demonstrated extensively in this work.

The extension of the multifunctional approach to multiple disciplines is illustrated using

the fluid-structure interaction problem.

Different methodologies have been developed for the computational analysis of

the fluid-structure interaction problem, and different terminology has been used to

describe the extent to which the disciplines are coupled. In this work, two classes of

coupling are outlined; namely, fully coupled and loosely coupled methods. Fully coupled

methods reformulate the governing equations so both the fluid and structural equations

are combined into one set of equations, coupling the solutions only at the boundary

interfaces between the fluid and the structure36. These new governing equations are

solved and integrated in time simultaneously. Loosely coupled methods make use of

independent computational fluid dynamic (CFD) and computational structural mechanics

(CSM) software modules. The coupling is accounted for by the exchange of data at the

interface between the fluid and the structure. This coupling approach takes full

advantage of the numerical procedures of individual disciplines such as finite difference

approximations for fluids and finite element approximations for structures. In addition,

software development efforts are simplified and software modularity is preserved. An

alternate to the coupling approaches is to solve both the structures and fluids problems in

a single computational domain. The major disadvantage of this methodology is the illconditioned matrices associated with the two physical domains. A secondary

disadvantage is the inability to use existing CFD codes because they do not account for

the interaction with the structure. In addition, the codes can not be readily extended to

include this interaction. Thus, the method does not take full advantage of these
197

specialized and well-trusted programs. The extensions of the multifunctional capability

will focus on the loosely coupled method.

The procedure for a loosely coupled method is given by (1) advance the structural

system under the fluid-induced load, (2) transfer the motion on the wet boundary (e.g.,

the fluid-structure interface) of the structure to the fluid system, (3) update the fluid

dynamic mesh accordingly, (4) advance the fluid system and compute new pressure and

fluid stress fields, and (5) convert the pressure and stresses into structural loads. The

multifunctional approach is applicable to steps two and five in the procedure outlined.

These steps are concerned with the transfer of data from a CFD grid to a CSM grid. Data

transfer is complicated by the fact that there are basic differences between the nature of

the solution methods. CFD analyses are concerned with the flow field surrounding the

surface exposed to the flow. Thus, a CFD grid is very fine around the exterior of an

airfoil, wherever changes in the flow field characteristics (i.e., boundary layer effects) are

expected to be maximum. Conversely, CSM methods examine airloads on the surface

and how these loads affect the internal structure. CSM grids lie on the surface within the

airfoil and are oriented to the structural components. Thus, CFD and CSM grids differ in

grid density and data transfer requires extrapolation and interpolation of disciplinespecific field variables.

Smith et al.53 evaluated computational algorithms to interface CFD and CSM

grids. In this reference, several candidate algorithms for passing information from the

fluid regime to the structural regime were evaluated and the disadvantages of each were

discussed. In addition, a load and motion transfer method based on the conservation of

momentum and energy has been developed by Farhat54. In this reference, a conservative
198

algorithm for computing the loads induced by a fluid on a structure is discussed. This

algorithm was shown to be accurate, robust and reliable for transferring data from a CFD

grid to a CSM grid not only when the discretization differed, but also when the grids did

not share the same geometry as in beam or wing-box geometric models (see Figure 5.19).

In the figure, the structural surface is denoted by ΓS and the fluid surface is denoted by

ΓF. The beam model is representative of the use of a beam finite element model to

idealize the structural component within the airflow. The wing-box model is

representative of a plate and shell finite element model to idealize the component in the

flow. The multifunctional methodology developed herein provides an alternate

conservative algorithm for transferring data from the CFD grid to a CSM grid. In

general, the methodology can be used to transfer data among many different disciplines.

Further development of the methodology to a two-dimensional (surface) interface is

required. This development follows the approach presented by Aminpour et al.55 for

coupling three-dimensional finite element meshes.

ΓS ΓS

ΓF ΓF

Figure 5.19. Beam and Wing-box Structural Models.

The governing equations for multifunctional analysis of vector-field problems

have been developed in Chapter III and are given in Eqs. (3.32) and (3.34). Discretized
199

equations are given for solid mechanics in Eq. (3.57) and for fluid mechanics in Eqs.

(3.60) and (3.61). In these systems of equations, the third equation represents the

subdomain discretization mapping from one subdomain to another subdomain. This

equation is given by

 or [ ]

2

1

I I I 1 2 K 0 K K = 0











 = (5.1)

where the variables with a subscript 1 represent a solid subdomain and the variables with

a subscript 2 represent a fluid subdomain. At this point, consider that the loads, 2 , on

the CFD grid are known. Eq. (5.1) can be used to solve for the unknown structural loads,

1, provided that matrix KI

1

 is square and invertible (i.e., the number of pseudo nodes

used to describe the generalized displacement along the interface is equal to the number

of Lagrange multipliers). Therefore,

I 2

-1

I 1 2 1

= −K K (5.2)

and

2 2

I -1

I

2

1

I

2 1 A K K = 













− =











 = .

Moreover, it can be shown that KIA = 0 56. That is, the matrix A spans the null space of

matrix KI.

 The fourth and fifth partitioned equations of the system of equations, given in

Eqs. (3.57), (3.60) and (3.61), may be used to interpolate the structural deformations to

the fluid grid. Recall that these equations are associated with the generalized

displacements on the interface and thus, the generalized displacement vector may be

partitioned as
200

















=

















= i

o

i

o

2

2

2

1

1

1 and similarly 

u

u

u

u

u

u

where the subscripts, i and o, represent generalized deformations on the interface and

within subdomain 1 or 2 (e.g., not on the interface). As such, the fourth and fifth

equations are given as

K u + K u = 0 K u + K uI = 0 T

2 I I p

T

1 I p 2 2 1 1 and i i

or

I

T Kpu + KI u = 0 i .

Premultiplying this equation by AT yields

A K u + A K uI = 0 T

I

T

p

T i

Since A K = 0 T

I

T ,

A K u = 0 A K u + A K u = 0 i i i

p 2

T

p 1

T

p

T

1 2 and 

or

K u + K u = 0 i i

1 1 2 2 (5.3)

The variables, i

1u , are associated with the known structural deformations from the

structures grid, and the variables, i

2 u , are associated with the unknown deformations to

be imposed on the fluid grid. Given that the matrix K2 is square and invertible, Eq.

(5.3) can be solved to obtain the unknown deformations. Therefore,

1 1

-1

2 2

i i u = −K K u (5.4)
201

The values, i

2 u , can now be used in the CFD code to update the surface deformation and

to calculate a new set of surface loads. With Eqs. (5.2) and (5.4), the multifunctional

methodology described herein may be extended to the multiple-domain analyses of

different disciplines.

5.5. SUMMARY

In this chapter, the multifunctional methodology has been described and

demonstrated for vector-field problems in engineering science. The selected problems

included problems of solid mechanics and fluid mechanics. The governing equation in

each case is the equation of linear momentum. In addition, for fluid mechanics,

continuity conditions are required. The analyses performed have demonstrated the

effectiveness and accuracy of the solutions obtained for the respective problems. In all

cases, the results obtained using the multifunctional methodology were in overall good

agreement with the reported analytical or reference solution.

Based on the findings for the vector-field problems, extensions of the

multifunctional collaborative methodology to multiple-domain analyses of different

disciplines have been briefly investigated. An exploratory examination of the extensions

illustrates the applicability of the methodology to loosely coupled multiple-discipline

applications.
202

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

6.1. GENERAL

Multifunctional methodologies and analysis procedures have been formulated for

interfacing diverse domain idealizations including multi-fidelity modeling methods and

multiple-discipline analysis methods. The methods, based on the method of weighted

residuals, ensure accurate compatibility of primary and secondary variables across the

domain interfaces. Methods have been developed for scalar-field and vector-field

problems. The methods have been rigorously developed for multiple-domain

applications, and the robustness and accuracy has been illustrated. Multi-fidelity

modeling approaches have been developed that include both homogeneous (i.e., the same

discretization method in each domain) and heterogeneous (i.e., different discretization

methods in each domain) discretization approaches. Results have been presented for the

scalar- and vector-field multifunctional formulation using representative test problems.

Associated computational issues are also discussed. In addition, the extension to

multiple-domain analysis with different disciplines has been discussed.

6.2. CONCLUSIONS

The multi-fidelity modeling of domains has been developed for homogeneous and

heterogeneous discretization approaches for both scalar- and vector-field problems. The

finite element and finite difference methods and combinations thereof have been used in

each of the discretization approaches.
203

Multi-fidelity modeling

Several general conclusions regarding the multi-fidelity modeling approaches can

be made. First, each of the multiple-domain approaches leads to a non-positive definite

system of equations, which impacts the solution strategy. Second, modeling flexibility in

the multiple-domain method is increased at the expense of additional degrees of freedom

introduced to the system of equations. However, the modeling advantage gained

outweighs the computational expense due to the additional degrees of freedom, and the

impact of the increased number of degrees of freedom due to the interface constraints is

reduced as the overall problem size is increased. Third, while the multifunctional method

encompasses heterogeneous discretization approaches using the finite difference method,

the limitations regarding its use in the presence of complex boundary conditions and

configurations restrict the method’s general-purpose use. Fourth, in general, the

homogeneous and heterogeneous multiple-domain approaches using the finite difference

discretization in one or both domains yield systems of equations that are not symmetric.

This lack of symmetry is due to the use of the Dirac delta function as the weight function

in the formulation. This function is introduced in the constraint integral used to form the

coupling matrix in the upper triangular part of the system matrix. The finite difference

“shape function” is used in the corresponding constraint integral used to form the

coupling matrix in the lower triangular part of the system matrix. In fact, in the finite

difference method, there may be a lack of symmetry in each of the independent

subdomain “stiffness” matrices due to the imposition of the boundary conditions.
204

Scalar-field problems

Conclusions regarding the multiple-domain modeling approach for the scalar-field

problem include the following statements. First, scalar-field problems introduce many of

the computational issues associated with the multifunctional approach. Second,

satisfaction of the boundary conditions for the scalar-field problem using finite difference

discretization is more straightforward than for the vector-field problem. The five-point

template used to approximate the derivatives does not introduce difficulties at the corners

of the domain, as is the case with the nine-point template used in the vector-field

problem. Third, fictitious nodes are avoided by evaluating the governing equations only

at the interior grid points of the domain. The essential and natural boundary conditions

are applied at the boundary nodes with higher-order forward and backward difference

approximations used for the first derivatives present in the natural boundary condition

equations. Fourth, the governing equation is evaluated at the nodes along the subdomain

common boundary. Straightforward central difference approximations are used at the

interface to represent the interface tractions, which in turn are used to eliminate the

fictitious nodes at the common boundary.

Vector-field problems

Based on the studies of the multiple-domain modeling approach for the vectorfield problem, the following conclusions are drawn. First, the use of the finite difference

method for the vector-field problem (e.g., plane stress problem) was far more

complicated than for the scalar-field problem. The traction and displacement boundary

conditions and the necessity to introduce and eliminate fictitious nodes outside the

domain boundary greatly complicate the development. Second, the nine-point template
205

required in the finite difference approximation of the governing equations of the

continuum introduces the need for alternative higher-order forward and backward

difference approximations of the cross-derivatives present in the equations. Third,

because of the difficulties associated with the first and second conclusions, the

homogeneous and heterogeneous modeling approach using the finite difference method

in one or both subdomains is not as attractive for vector-field problems as for scalar-field

problems. Fourth, the governing equation is evaluated at the nodes along the subdomain

common boundary. Complex manipulation of the nine-point template is required using

forward and backward difference approximations of the cross-derivatives in order to limit

the introduction of the fictitious nodes to the node along the common boundary at which

the governing equation is being evaluated. This requirement is automatically satisfied in

the scalar-field problem by the five-point template. The interface tractions are used to

eliminate the fictitious nodes at the common boundary.

Limitations

While a rigorous multifunctional formulations has been presented, there are

limitations in the implementation. Note that the purpose of the implementation described

herein was to demonstrate the capabilities of the multifunctional approach on a set of

representative benchmark problems. With this in mind, the limitations of the current

implementation are as follows:

• The nodes or grid points at the ends of the common subdomain boundary of each of

the subdomains must coincide.

• In the finite difference method used, at least three nodes are required in each of the

coordinate directions where traction boundary conditions are imposed.
206

• Extreme care must be taken to perform accurate input and output using data-exchange

files (in this work, double-precision floating-point accuracy).

• The development of the interface routines in MATLAB limits the size of problem

that may be analyzed.

• Cubic splines are used on the subdomain common boundary, which requires at least

four unique nodes along this boundary.

• The implementation is limited to one-dimensional straight or curved common

subdomain boundaries.

• The geometry is assumed to be conforming. That is, each of the subdomains describe

the same geometry along the common boundary.

In this work, the benchmark vector-field problems illustrated require only C0

continuity (continuity of the primary variable). Thus, continuity of the primary variable

is maintained along the subdomain common boundary through the interface constraint.

For plate bending problems using classical plate theory, C1 continuity is required. In this

case, continuity of the primary variable and its derivative is maintained along the

common subdomain boundary. Here, the derivatives are approximated in the same

manner as the primary variable. That is, cubic spline functions are used to approximate

the generalized variables along the common subdomain boundary. Results for a wider

range of problems including a plate bending problem have been given in reference 25.

Summary of Results

Results were presented for the scalar- and vector-field developments using

example patch test problems. In addition, results for torsion, heat conduction and

potential flow problems have been presented to demonstrate further the effectiveness of
207

the scalar-field development. Results for plane stress and plane flow problems have been

presented for the vector-field development. Results for all problems presented are in

overall good agreement with the exact or reference configuration by which they were

evaluated.

The multifunctional methodology presented provides an effective mechanism by

which domains with diverse idealizations can be interfaced. This capability promises to

provide rapidly the high-fidelity data needed in the early design phase. Moreover, the

capability is applicable to the problems in the general field of engineering science and

mechanics. Hence, the methodology provides a collaborative capability that accounts for

discipline interactions among many disciplines.

6.3. RECOMMENDATIONS FOR FUTURE WORK

Future studies related to the present work are recommended. The present work

provides a starting point for the following additional studies:

1. Explore the use of a finite difference energy method, which alleviates many

of the issues associated with the proper identification of boundary conditions

and the use of irregular grids.

2. Evaluate the performance of the methodology for the analysis of more

complex structures and fluid flow problems.

3. Extend and implement the multiple-discipline capability.

4. Develop other analysis capabilities including thermal analysis, modal and

buckling analysis, dynamic analysis, and nonlinear analysis.

5. Develop other heterogeneous multiple-domain discretization approaches such

as the use of the finite element and boundary element methods.
208

6. Develop strategy to exploit massively parallel processing (MPP) computer

systems.

7. Incorporate computationally intelligent strategies to identify where and when

homogeneous or heterogeneous approaches should be used.
209

APPENDIX A

OVERVIEW OF STEPS IN ANALYSIS AND SIMULATION

Multifunctional collaborative methods should address four typical steps of

analysis and design, namely, (1) representation or modeling of the geometry, (2)

knowledge-based selection and development of appropriate mathematical models (i.e.,

idealization/discretization), (3) solution of the mathematical model (continuous and/or

discrete), and (4) interrogation/assessment of the results. These steps provide the

foundation for enhanced integrated design and analysis tools, and the steps are briefly

outlined in this appendix.

Geometry Modeling

To represent the structural geometry (geometry modeling) a geometric model is

created to represent the size and shape of a system component. In aerodynamic and

structural analyses, a common three-dimensional parameterized description of the

airframe is shared. Geometry modeling is the starting point of the product design and

manufacture process and is the first step in using a computer-aided design/computeraided manufacturing (CAD/CAM) system57. The accuracy of the geometric model and

the way in which it is structured has far-reaching effects on other CAD functions such as

finite element analysis, drafting, and numerical control (NC) part programming.

CAD/CAM systems can be utilized to develop a design and monitor and control the

manufacturing process from start to finish. Numerous CAD software packages58 for

defining the geometry of structural systems are commercially available.

Computer-aided engineering (CAE) has facilitated the assimilation of the

engineer/analyst earlier in the design stage as an engineer in-the-loop. Typically, this
210

cycle leads from the design engineer to the analyst and back to the designer. A critical

aspect of this cycle is the time required to generate analysis models, perform the analysis

and decide if changes are needed. However, new trends in modeling and simulation are

redefining the roles of the designer and the analyst. Many companies are now turning

designers into analysts. The underlying philosophy guiding this paradigm shift is the

desire to give designers the tools needed to predict a design’s performance early in the

process, rather than just to define its geometry. These tools also embody a knowledge

base to guide the designer through various analysis steps. Moreover, this new paradigm

allows the highly specialized analysts to impact the design by performing more complex

analyses to determine the structural integrity, the potential failure mechanisms and the

complex response characteristics (i.e., material or geometric nonlinearity), and

multidisciplinary characteristics of the design.

This role redefinition can succeed only if enough analyses are performed early in

the design process to identify critical design parameters, evaluate their interactions, and

determine the best overall design. To expedite this process developers of computer-aided

design (CAD) and analysis software have integrated the CAD and analysis functions.

Such software integration and database coupling frequently enables designers to perform

analyses directly on geometry, thus reducing the time required to prepare analysis

models.

Idealization/Discretization

To develop discretized mathematical models of aerospace systems, several

approximate numerical analysis methods have evolved over the years. The most

commonly used discretization methods are the finite difference method and the finite
211

element method. The finite difference method of a configuration gives a pointwise

approximation to the governing equations. While finite difference techniques are widely

used in fluid dynamics and can treat fairly complex problems, they become hard to use

when irregular geometrical shapes or unusual boundary conditions are encountered. This

adverse attribute is particularly significant in structural analysis. In contrast, the finite

element method is widely used for the analysis of many engineering problems involving

static, dynamic and thermal stresses of structures. Typical input for a finite element

analysis program consists of the geometric idealization, the material properties, the

loading, and boundary conditions. The area of greater difficulty in the finite element

technique lies in the geometric idealization, that is, representing the geometry of the

structure by a suitable finite element mesh. Element aspect ratio, taper, and skew are

characteristics that adversely affect the performance of many finite elements in use today

and thus are factors in determining the suitability of a mesh. As the complexity of

structural configurations and material systems being modeled with the finite element

method has increased, manual mesh generation has become extremely tedious, timeconsuming, expensive and consequently, intractable. This limitation is alleviated through

the development of automatic mesh generators, which are typically integrated within the

finite element modeling software and often integrated within the CAD system. These

mesh generators are powerful tools for discretizing complex structural configurations.

Issues associated with idealization still arise such as whether to use solid finite elements

or shell finite elements. However, if the CAD and analysis engines are not driven from

the same geometry, the translation of geometry may introduce errors in analytical models.

In addition, due to the geometric complexity of such configurations, even the most robust
212

automatic mesh generator can often require analyst interaction to establish a suitable

mesh and to provide engineering insight into the proper finite element to be used in the

analysis. For example, some automatic mesh generators place three-dimensional models

where two-dimensional shells should be used, which may distort the results.

Response Prediction

To solve the discrete system of simultaneous equations resulting from the

discretization process and subsequent finite element assembly operations, myriad solution

strategies have been developed for obtaining efficiently the unknown nodal values of the

field variable or the primary unknowns. Two families of methods for solving linear

systems of algebraic equations can be distinguished: direct and iterative equation solvers.

The former can be defined as leading to the solution of a linear system in one step, while

the latter will require many iterative steps. If the equations are linear, a number of

standard solution techniques may be used which generally include either an iterative or

direct solver. If the equations are nonlinear, their solution is more difficult to obtain. All

approaches will necessarily be repeated solution of linearized equations. A common

solution method used to solve nonlinear systems of equations is the Newton-Raphson

incremental-iterative solution procedure, which is accurate and converges for highly

nonlinear behavior. High-performance equation solvers are a key component of solution

strategies for linear and nonlinear structural response calculations for static, dynamic and

eigenvalue problems in finite element analysis. There has been a plethora of research in

the area of equation solvers for large-scale aerospace structures with only representative

works referenced herein. Matrices resulting from discretization of structural systems are

generally real, symmetric, positive definite, banded, and sparse. The performance of
213

iterative and direct equation solvers has been compared to identify the most appropriate

tool for the solution of equations arising from structures systems59. This work identified

advantages and disadvantages of both types of solvers. The study concluded that the

relative performance of solvers depends on the amount of computations as well as the

rate at which operations can be carried out on a given computer.

Direct sparse solvers were found to be most attractive for models composed of

higher-order finite elements, where they benefit most from a greatly reduced operation

count. Sparse direct techniques are efficient improvements over first-generation direct

methods that require more operation counts and larger memory capacity60. The number

of operations in a sparse method are significantly reduced through reordering and storage

strategies that effectively compress the global stiffness matrix into a format that exhibits

a greater degree of nonsparsity prior to factorization and thus substantially reduces the

associated computational costs. Iterative methods require much less memory than direct

solvers, but their effective use depends on a fast convergence rate, which has been found

to be best for finite elements with low aspect ratios. Skyline and variable band linear

equation solvers have been developed to exploit the matrix characteristics of structural

systems and to exploit the full capabilities of parallel and vector supercomputers61. More

recently, general-purpose equation solvers have been developed for complex,

nonsymmetric, indefinite, and dense matrix characteristics, which are prevalent in

disciplines such as electromagnetic and acoustic analysis62. Over the years, equation

solvers have been developed to take advantage of the rapidly increasing computational

power afforded by vector and parallel high-performance computers. These ultra-rapid

equation solvers coupled with the major advances in computational power now available
214

in desktop personal computers and workstations have made it feasible to perform highfidelity analyses in the preliminary design stage. However, additional developments are

required to perform real-time large-scale analyses within an interactive virtual reality

analysis and design environment. More intensive reviews of equation solvers may be

found in the open literature (e.g., references 63, 64, and 65).

Assessment of Results

The fourth and final step in the analysis and simulation process is the

interrogation of the results. In years past, the engineer would spend an enormous amount

of time plowing through pages of computer output while waiting for results from

additional analyses. With the increased speed and efficiency of today’s equation solvers,

the rapid interrogation of results becomes decidedly more significant. It is at this step of

interpretation of results that the engineer must be integrally involved. Powerful pre- and

post-processing tools coupled with state-of-the-art computational technology provide the

engineer with a comprehensive tool set for creating and discretizing complex geometries,

performing analyses and visualizing results. Some software provide novel capability to

enhance the designer-computer interaction while interrogating results. Engineers can

view the results of parametric studies in a series of windows to identify or compare

important design parameters. In addition, analysis results from different design

approaches may be viewed in different windows and assessed to determine the most

feasible design. This and other such visualization capabilities facilitate the rapid

interpretation of analysis results, thus improving productivity of higher-order analyses

and providing an opportunity for the engineer/analyst to be an integral part of the design

process from concept to manufacture. Recently, immersive virtual reality environments
215

for visualization and interpretation of geographically dispersed results have been

proposed as part of the NASA Intelligent Synthesis Environment (ISE) Initiative that

promises to revolutionize the design process66, 67. Immersive environments are humanscale computer-generated projection systems that allow users to interact directly with

their data in three spatial dimensions. Emerging advanced engineering environments68

will provide visual, auditory, and haptic feedback to further aid the engineer in detailed

assessment of results.
216

APPENDIX B

CUBIC SPLINE INTERPOLATION MATRICES

The interpolation matrices used in the deformation and geometry assumptions of

the multifunctional approach are outlined in this appendix. Given a series of points xi

( ) i = 0,1,...,n which are generally not evenly spaced, and the corresponding function

values f(xi), the cubic spline function denoted g(x) may be written as

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )



 





∆

−  +



 





∆

− +















 − ∆ − ∆

− +















 − ∆ − ∆

− =

+

+

+

+

+

i

i

i

i

i

i

i i

i

i xx i

i i

i

i xx i

x

x x f x

x

x x f x

x x x

x

g x x x

x x x

x

g x x x

g x

1

1

3

1

1

3

1

6 6

, ,

 (B.1)

where ∆x=xi+1 – xi and g,xx denotes differentiation twice with respect to x. This equation

provides the interpolating cubics over each interval for i = 0,1,...,n −1 and may be given

in matrix form as

 T f ˆ T g, ˆ g = 1 xx + 2 (B.2)

For each of the k values of x at which the spline function is to be evaluated, xixk xi+1,

k =1,2 ... p, and p is the number of evaluation points. The T1

ˆ and T2

ˆ matrices may be

written in the form

() () ()

() () ()

() () ()

() () ()

() () ()

() () () 



























=





























=

+

+

+

+

+

+

t t t

t t t

t t t

t t t

t t t

t t t

p p p n

n

n

p p p n

n

n

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ

2 2 2

2 2 2

2 2 2

 and 

1 1 1

1 1 1

1 1 1

,1 ,2 , 1

2,1 2,2 2, 1

1,1 1,2 1, 1

2

,1 ,2 , 1

2,1 2,2 2, 1

1,1 1,2 1, 1

1

L

M M O M

L

L

L

M M O M

L

L

T T

where
217

( ) ( ) ( )

( ) ( ) 

























≤ ≤ = +















 − ∆ − ∆

−

≤ ≤ = +















 − ∆ − ∆

−

< >

=

+

+ +

+

+

 for and 2 

 for and 1

0 for or 

1

3

1 1

3

1

1

1

x x x x x x j i

x

x x

x x x x x x j i

x

x x

x x x x

t

i k i i k i

i

k i

i i k i k i

i

i k

k i k i

k j ˆ , ,

( ) ( )

( )























≤ ≤ = +

∆

−

≤ ≤ = +

∆

−

< >

=

+

+

+

+

 for and 2 

 for and 1

0 for or 

1

1

1

1

2

x x x j i

x

x x

x x x j i

x

x x

x x x x

t

i k i

i

k i

i k i

i

i k

k i k i

k j ˆ , ,

and

( )

( )

( )



























=

xx n

xx

xx

xx

g x

g x

g x

,

,

,

g, M

1

0

, and 

( )

( )

( ) 



























=





























=

n nf

f

f

f x

f x

f x

M M

1

0

1

0

f .

Note that there are, at most, two nonzero coefficients in each row of the ˆ T 1 and ˆ T 2

matrices given above.

Applying additional smoothness conditions (i.e., equating the first and second

derivatives of adjacent interpolating cubics at xi) yields a set of simultaneous equations of

the form

( ) ( ) ( ) [ ] ( )

( ) ()

( )

() ( )

( )( ) 















∆ ∆

− −

∆

− =

 +



 





∆

−  +



 





∆

∆

−

+ −

+

+ − −

1

1

2

1

1

1

1

1

6

1 2

i i

i i

i

i i

xx i xx i

i

i i

xx i

i

i

x x

f x f x

x

f x f x

g x g x

x

x x

g x

x

x , , ,

 i =1,2,...,n −1. (B.3)

If the xi are evenly separated with spacing ∆x, then the Eq. (B.3) becomes
218

[ ] ( ) [ ] ( ) [ ] ( )

( ) () ( )

( ) 















∆

− + =

+ +

+ −

− +

2

1 1

1 1

2

6

1 4 1

x

f x f x f x

g x g x g x

i i i

xx i xx i xx i , , ,

. (B.4)

Eqs. (B.3) and (B.4) may be written as

Ag, = Pf xx (B.5)

The coefficients of matrices A and P are dependent upon the end conditions, which are

discussed in the following section.

End Conditions

Whether the equations are of the form of Eq. (B.3) or Eq. (B.4), there are n-1

equations in the n+1 unknowns ( ) () ( ) xx xx xx n g, x , g, x , ..., g, x 0 1 . The two necessary

additional equations are obtained by specifying conditions on g,xx (x0) and g,xx (xn). For a

natural spline, g,xx (x0)= g,xx (xn) =0. However, in this work, these second derivatives are

calculated by differentiating (twice) a cubic function which passes through the first four

pseudo-nodes along the interface path and another cubic function that passes through the

last four pseudo-nodes along the interface path. Evaluating this cubic function,

g x( ) = a0 + a1x + a2x

2 + a3x3 , and at the first four points gives

( )

( )

( )

( ) 























































=





























3

2

1

0

3

3

2

3 3

3

2

2

2 2

3

1

2

1 1

3

0

2

0 0

3

2

1

0

1

1

1

1

a

a

a

a

x x x

x x x

x x x

x x x

g x

g x

g x

g x

 or Na = g (B.6)

Solving for the coefficients yields a N g -1 = or
219

( )

( )

( )

( )



















































=





























3

2

1

0

41 42 43 44

31 32 33 34

21 22 23 24

11 12 13 14

3

2

1

0

g x

g x

g x

g x

n n n n

n n n n

n n n n

n n n n

a

a

a

a

 (B.7)

From the cubic function, g ( ) x a a x xx = 2 2 + 6 3 , where a2 and a3 are determined

from Eq. (B.7). Equation (B.7) is valid for evenly spaced as well as arbitrarily spaced

points. Similar expressions are obtained for the cubic function passing through the last

four points where coefficients of the inverted matrix similar to those in Eq. (B.7) are

denoted n kl for k,l =1,...,4. With these end conditions, the matrices of Eq. (B.5) are

given for equally-spaced points as

( )

A

+1 x +1









































=

n n 0 0 1

1 4 1

1 4 1

1 4 1

1 0 0

L

L

O

L

L

and 

( )

P

+1 x +1 















































∆ ∆

−

∆

∆ ∆

−

∆

=

n n p p p p

x x x

x x x

p p p p

1 2 3 4

1 2 3 4

6 12 6

6 12 6

L

L O

O

L

L

L

where pk = 2n3k + 6n4k and pk = 2n 3k + 6n 4k for k,l =1,...,4. For unevenly spaced

points, the tridiagonal A and P matrices may readily be obtained from Eq. (B.3).
220

Expressing g(x) in Terms of Functional Values f(xi)

In Eq. (B.2), the spline function g(x) is expressed in terms of the functional values

f(xi) as well as second derivatives of the spline function, g,xx (xi). However, it is desirable

to express g(x) in terms of the function values f(xi) only. This manipulation is done by

solving for g,xx (xi) in Eq. (B.5) yielding

 g, A Pf −1

xx = (B.8)

Substituting in Eq. (B.2) yields

 ( ) ( T )f Tf Tˆ A P ˆ T f Tˆ A Pf ˆ g = + = + = − − 2

1

2 1

1

1 x . (B.9)

Derivatives of the spline function are obtained by differentiating Eq. (B.9) yielding

( ) () ( ) () ( ) [ T , ]f T, f A P ˆ T , ˆ T , f A Pf ˆ T , ˆ , x x x x x x g x = + = + = − − 2

1

2 1

1

1 (B.10)

where

( )

() () ()

() () ()

() () ()

( )

() () ()

() () ()

() () () 



























=





























=

′ ′ ′

′ ′ ′

′ ′ ′

′ ′ ′

′ ′ ′

′ ′ ′

+

+

+

+

+

+

2 2 2

2 2 2

2 2 2

, ˆ and

1 1 1

1 1 1

1 1 1

, ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

,1 ,2 , 1

2,1 2,2 2, 1

1,1 1,2 1, 1

2

,1 ,2 , 1

2,1 2,2 2, 1

1,1 1,2 1, 1

1

t t t

t t t

t t t

t t t

t t t

t t t

p p p n

n

n

x

p p p n

n

n

x

L

M M O M

L

L

L

M M O M

L

L

T T

( ) ( )

( )



























≤ ≤ = +















 − ∆

∆

−

≤ ≤ = +

















+ ∆

∆

− −

< >

=

+

+

+

+

′

 for and 2 3

 for and 1 3

0 for or 

1

1

2

1

2

1

1

x x x x j i

x

x x

x x x x j i

x

x x

x x x x

i i k i

i

k i

i i k i

i

i k

k i k i

k j t

ˆ , ,

and
221

( )





















≤ ≤ = +

∆

≤ ≤ = +

∆

−

< >

=

+

+

+

′

 for and 2 1

 for and 1 1

0 for or 

2

1

1

1

x x x j i

x

x x x j i

x

x x x x

i k i

i

i k i

i

k i k i

k j t

ˆ , ,

Again, note that there are, at most, two nonzero coefficients in the ( ) x T , ˆ

1 and ( ) x T , ˆ2

matrices. In this derivation, x has been used as the independent variable. However, in

the context of the interface definition herein, s is the independent variable and is

substituted for x in the derivation in Appendix C. For the displacement assumption, the

matrices developed for equally spaced points were used. For the geometry assumption,

matrices for unequally spaced points were used.
222

APPENDIX C

 DERIVATION OF INTERFACE GEOMETRY

C.1. GENERAL

In the initial development outlined in reference 25WKHLQWHUIDFHSDWK I, was

defined by piecewise linear segments. For curved interfaces, this definition only

approximates the true curved geometry. The error in this approximation is a function of

the interface path curvature and the number and location of the subdomain nodes along

the interface. In addition, the interface path was computed along each subdomain

independently, thus producing two different interface geometry definitions. For a

structure with mild curvature, the error in the interface path definition did not influence

the accuracy of the solution obtained in the analysis25. However, for problems with

moderate to large curvature, this error may be large and adversely influence the accuracy

of the interface element analysis.

In the present work, the element interface geometry is determined in one of two

ways: (1) by specifying the function that represents the exact geometry of the interface

(i.e., the linear interface is the trivial case) or (2) by passing a spline of the desired order

(typically a cubic spline) through the specified coordinate data points to determine the

function representing the geometry. In either case, the specified or computed function is

parameterized and its first derivative is used to determine the arc length along the

interface geometry of the subdomains as well as the interface boundary. Thus, in contrast

to the earlier work, the interface geometry definition is a more accurate representation of
223

an arbitrarily curved geometry. In addition, only one interface path geometry is defined,

and all the finite element nodes along that interface lie on that geometry.

For a curved geometry, the most general way of determining the interface path of

the two approaches mentioned previously is by using the latter approach (i.e., passing a

cubic spline through the specified coordinates). In this case, a smooth curve is fit to the

set of spatial coordinates by computing three cubic spline functions (one for each

coordinate direction) expressing the coordinates as functions of a chordal distance

parameter. The derivatives of these functions are obtained by differentiating the

interpolating function. These derivatives are used in the parametric definition for the

length of the arc between two points to compute the arc length between each of the

specified coordinates. The spatial coordinates of the finite element nodes along each

subdomain boundary provide the input for the interface geometry definition. These nodal

coordinates are used to construct the function representing the curved geometry and to

determine the arc length of the path. The associated variable, s, is computed along the

subdomain boundaries. The number of evenly-spaced pseudo-nodes is determined

internally or from the used-specified value after which the path variable, s, is computed

along the interface path. See Appendix B for a brief discussion of the cubic spline used

as the basis for the geometry representation.

C.2. GEOMETRY REPRESENTATION

The arc length or interface path is derived in this appendix. The spatial

coordinates of finite element node i are given by xi, yi, and zi. The curve may be

represented parametrically by
224

x = x( )r

y = y r( )

z = z r( )

where ri = xi+1 − xi ( )2+ yi+1 − yi ( )2+ zi+1 − zi ( )2 . Smooth cubic splines are fit

through each of these coordinate functions. These coordinate functions are then

expressed as

x( )r = Txs

y r( ) = Tys

z r( ) = Tzs

where T is a matrix of interpolation functions (see Eq. B.9 in Appendix B) and is

evaluated at the points ri. The vectors xs, ys, and zs contain the sorted nodal coordinates,

xi, yi, and zi, along the interface (i.e., the concatenation of the nodes from each of the

subdomains to which the interface element is attached).

The length of the arc between each of the points along the interface may be

calculated immediately as

s ri ( ) = dx

dr

 

   

2

+ dy

dr

 

   

2

+

dz

dr

 

   

2

ri−1

ri

∫

and

( )

( )

( ) r r s

r r s

r r s

z r

dr

dz

y r

dr

dy

x r

dr

dx

, T, z

, T, y

, T, x

= =

= =

= =
225

where r T, is obtained by differentiation of the interpolation matrix T with respect to the

independent variable, r, (see Eq. B.10 in Appendix B) and is evaluated at points, ri. The

variable, s, is called the parameter of the arc length or the path variable herein. This

variable measures the distance along the curve given by the parametric equations above.

Thus, the arc length, s(ri) is obtained by numerical integration using Gaussian quadrature

with four quadrature points. The path variable, as previously defined, is associated with

the coordinates of the finite element nodes along the interface. The path variable, s, for

the pseudo-nodes is computed by dividing the total arc length into equal segments. This

total arc length is determined by summing the arc length between each set of two points,

ri-1 and ri, over the total interface path to obtain the total arc length. In addition to the

path variable, s, at the j pseudo-nodes, the coordinate location of these pseudo-nodes is

also desired.

Moreover, in general, a computational coordinate frame is established along the

interface; thus, the tangent to the interface path is desired. These calculations are

addressed in the following discussion.

Upon obtaining the path variable at the finite element nodes along the interface,

the coordinate functions may now be expressed as

x = x( )s = Txs

y = y s( ) = Tys

z = z s( ) = Tzs

Here, the interpolation matrix T is evaluated at the path coordinates, s, of the pseudonodes yielding desired x, y, and z coordinates. The unit tangent vector to the interface
226

path is obtained by differentiating the coordinate functions with respect to the path

variable, s, and is given by

( )

( )

( ) s s s

s s s

s s s

z s

y s

x s

, T, z

, T, y

, T, x

=

=

=

where s T, is evaluated at the path coordinate, s, of the pseudo-nodes and the finite

element nodes. The tangent vector is then given by

( ) ( ) ( ) k

r

z s j

r

y s i

r

x s s s s ˆ , ˆ , ˆ , tv = + +

where ( ) () ( ) ( ) () ( ) 2 2 2

r x s y s z s s s s = , + , + , .
227

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4. TITLE AND SUBTITLE

On Multifunctional Collaborative Methods in Engineering Science

5. FUNDING NUMBERS

WU 706-85-10-01

6. AUTHOR(S)

Jonathan B. Ransom 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

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 L-18104

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Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in

Engineering Mechanics, Old Dominion University, Norfolk, Virginia, May 2001.

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13. ABSTRACT (Maximum 200 words)

Multifunctional methodologies and analysis procedures are formulated for interfacing diverse subdomain

idealizations including multi-fidelity modeling methods and multi-discipline analysis methods. These methods,

based on the method of weighted residuals, ensure accurate compatibility of primary and secondary variables

across the subdomain interfaces. Methods are developed using diverse mathematical modeling (i.e., finite

difference and finite element methods) and multi-fidelity modeling among the subdomains. Several benchmark

scalar-field and vector-field problems in engineering science are presented with extensions to multidisciplinary

problems. Results for all problems presented are in overall good agreement with the exact analytical solution or

the reference numerical solution. Based on the results, the integrated modeling approach using the finite element

method for multi-fidelity discretization among the subdomains is identified as most robust. The multiplemethod approach is advantageous when interfacing diverse disciplines in which each of the method’s strengths

are utilized.

14. SUBJECT TERMS

Global/Local Analysis; Multiple Domain Analysis; Engineering Science;

15. NUMBER OF PAGES

245

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