bluuebunny/crossref_metadata_2025 huggingface dataset:
Instantaneous response of elastic thin‐walled structures to rapid heating
Authors: Chen Xiaoqin , Mohan Ram V. , Tamma Kumar K.
Published: August 2005

A generalized modelling and analysis approach of thermally induced coupled vibrations of elastic thin‐walled configurations with arbitrary open cross‐sections are presented in conjunction with a unified implicit transient methodology. Limited research which takes into account the influence of rapid thermal heating effects on structures involving various forms of coupling appears in the literature. As a consequence, the dynamic response of such thin‐walled structures of arbitrary open cross‐section to rapid heating are described here. Effects involving triple, double, and no coupling between bending and torsional vibrations caused by sudden heating on these structures are examined. Numerical test cases are presented which describe the influence of sudden heating on elastic thin‐walled structures of arbitrary open cross‐sections.
Combined micro‐ and macromechanical considerations of layered composite shells
Authors: Rammerstorfer Franz G. , Starlinger Alois , Dorninger Konrad
Published: August 2005

This contribution deals with a computational treatment of the non‐linear behaviour of composite sandwich shells with extremely thin face layers and a relatively thick orthotropic core, and multi‐layered fiber reinforced polymer or metal matrix composite shells. Special emphasis is laid on algorithms which allow the consideration of local effects, like face layer wrinkling in sandwich shells and progressive damage or microplasticity in fibre composites, by semi‐analytical approaches implemented into a finite element formulation. This leads to the advantage that the shell structure must be discretized only up to an element mesh density which is sufficient for describing the global deformation and stability behaviour.
Wavelet–Galerkin solutions for one‐dimensional partial differential equations
Authors: Amaratunga Kevin , Williams John R. , Qian Sam , Weiss John
Published: August 2005

In this paper we describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multi‐grid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies1) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem.

In order to demonstrate the wavelet technique, we consider the one‐dimensional counterpart of Helmholtz's equation. By comparison with a simple finite difference solution to this problem with periodic boundary conditions, we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund2 and others. The convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method, at least for problems with simple geometries.
A least‐squares front‐tracking finite element method analysis of phase change with natural convection
Authors: McDaniel David J. , Zabaras Nicholas
Published: August 2005

This paper focuses on the numerical modelling of phase‐change processes with natural convection. In particular, two‐dimensional solidification and melting problems are studied for pure metals using an energy preserving deforming finite element model. The transient Navier–Stokes equations for incompressible fluid flow are solved simultaneously with the transient heat flow equations and the Stefan condition. A least‐squares variational finite element method formulation is implemented for both the heat flow and fluid flow equations. The Boussinesq approximation is used to generate the bulk fluid motion in the melt. The mesh motion and mesh generation schemes are performed dynamically using a transfinite mapping. The consistent penalty method is used for modelling incompressibility. The effect of natural convection on the solid/liquid interface motion, the solidification rate and the temperature gradients is found to be important. The proposed method does not possess some of the false diffusion problems associated with the standard Galerkin formulations and it is shown to produce accurate numerical solutions for convection dominated phase‐change problems.
Finite element formulation for a baffled, fluid‐loaded, finite cylindrical shell
Authors: Grosh Karl , Pinsky Peter M. , Malhotra Manish , Rao Vinay S.
Published: August 2005

A new method for obtaining the response of a baffled, fluid‐loaded, finite cylindrical shell using the finite element method is presented. The Galerkin finite element formulation for this problem utilizes a Neumann Green's function representation of the pressure loading on the shell. By representing the pressure loading in this manner, only the structure need be discretized. Both 𝒞0 and 𝒞1 finite element discretizations of the shell are considered. The finite element results are compared with the analytic solutions of the fluid‐loaded cylinder which are obtained using an expansion of the displacement in terms of the in vacuo modes of the Naghdi–Cooper and Flügge shell theories.
Hybrid finite element formulation for periodic piezoelectric arrays subjected to fluid loading
Authors: Chin Liang‐Chi , Varadan Vasundara V. , Varadan Vijay K.
Published: August 2005

The effect of fluid loading on a structure embedded with a periodic distribution of piezoelectric transducers is considered in a hybrid formulation that involves a finite element formulation for the structure and transducers that is coupled to an analytical series representation of the acoustic field in the fluid. The passive fluid loading redefines the stiffness matrix via a radiation impedance term. Modal analysis of the array can be effected for elastic vibrations, open and close circuited electrical boundary conditions. General damping in the stiffness as well as dielectric constants is considered. Sensor and actuator equations are then formulated. The actuator equations permit computation of the displacement and stress distribution in the transducer and structure when a known voltage is applied to the transducer electrodes. The sensor equation is illustrated for the case of a known incident plane harmonic wave obliquely incident on the array. The voltage induced in the transducer elements can then be computed.
The domain composition method applied to Poisson's equation in two dimensions
Authors: Charlesworth W. W. , Cox J. J. , Anderson D. C.
Published: August 2005

Domain composition, a recently described method for formulating continuum field problems, removes certain restrictions on the construction of finite element models such that it is possible to solve a finite element problem without using a global compatible mesh. The domain composition method couples or otherwise constrains meshes in local regions to obtain a solution equivalent to that produced by conventional finite element methods. In particular, the domain composition method enables finite element models to be formulated with overlapping elements. Several advantages come from this, including an ability to automatically generate a finite element model from a solid geometric model, an ability to use a variety of element types in a single finite element model and an ability to exactly match element boundaries to the local geometry. This paper shows in detail a finite element formulation of Poisson's equation using domain composition and presents certain key algorithms that incorporate the domain composition method into well‐established finite element procedures.
On the application of multi‐step integration methods to infinitesimal elastoplasticity
Authors: Papadopoulos Panayiotis , Taylor Robert L.
Published: August 2005

This article proposes an implicit algorithm for time integration of the differential/algebraic equations of small‐deformation elastoplasticity in strain space. The algorithm employs a two‐step backwards differentiation formula and a standard operator‐split procedure. Second‐order accuracy and non‐linear stability are established under appropriate constitutive restrictions. The algorithm is implemented for the model of J2‐plasticity and numerical simulations are conducted that illustrate its applicability and accuracy.
Enhanced superconvergent patch recovery incorporating equilibrium and boundary conditions
Authors: Wiberg Nils‐Erik , Abdulwahab Fethi , Ziukas Saulius
Published: August 2005

Patch recovery based on superconvergent derivatives and equilibrium (SPRE), an enhancement of the Superconvergent Patch Recovery (SPR), is studied for linear elasticity problems. The paper also presents a further improvement for recovery of derivatives near boundaries, SPREB, where either tractions or displacements are prescribed. This is made by inclusion of weighted residual errors at boundary points in the patch recovery. A pronounced improvement in the post processed gradients of the finite element solution is observed by this method.
A postprocessed error estimate and an adaptive procedure for the semidiscrete finite element method in dynamic analysis
Authors: Wiberg N.‐E. , Li X. D.
Published: August 2005

In the paper we present a postprocessed type of a posteriori error estimate and a h‐version adaptive procedure for the semidiscrete finite element method in dynamic analysis. In space the super‐convergent patch recovery technique is used for determining higher‐order accurate stresses and, thus, a spatial error estimate. In time a postprocessing technique is developed for obtaining a local error estimate for one step time integration schemes (the HHT‐α method). Coupling the error estimate with a mesh generator, a h‐version adaptive finite element procedure is presented for two‐dimensional dynamic analysis. It updates the spatial mesh and time step automatically so that the discretization errors are controlled within specified tolerances. Numerical studies on different problems are presented for demonstrating the performances of the proposed adaptive procedure.
Inelastic post‐buckling analysis of truss structures by dynamic relaxation method
Authors: Ramesh G. , Krishnamoorthy C. S.
Published: August 2005

The paper presents the inelastic post‐buckling analysis of truss structures by the Dynamic Relaxation (DR) method. A simplified inelastic finite element formulation for truss element and new algorithms are proposed for Elastic Post‐Buckling (EPB) analysis and Inelastic Post‐Buckling (IEPB) analysis using the DR method. The post‐buckling paths for elastic, EPB and IEPB analyses are completely traced using the variable‐arc‐length method. Four numerical examples are presented to illustrate the application of the proposed algorithms.
Optimal shape design for thin‐walled beam cross‐sections
Authors: Schramm Uwe , Pilkey Walter D.
Published: August 2005

Problems of shape optimization for thin‐walled beam cross‐sections are investigated. In particular, shape optimization under torsional loading is treated. The shape of the beam cross‐section is described by non‐uniform rational B‐splines (NURBs). Biquadratic and p‐version finite elements are used to calculate the warping function. The definition of the element geometry is carried out by isoparametric and B‐spline mappings. Both methods are discussed and compared. For structural shape optimization the B‐spline mapping in connection with the p‐version finite elements is very efficient. The optimization is implemented using convex linearisation and a dual method of mathematical programming. Sensitivity analysis is performed analytically. Sample calculations are included.
Spatial stability and free vibration of shear flexible thin‐walled elastic beams. II: Numerical approach
Authors: Kim Moon‐Young , Chang Sung‐Pil , Kim Sung‐Bo
Published: August 2005

In a companion paper,1 equations of motion and closed‐form solutions for spatial stability and free vibration analysis of shear flexible thin‐walled elastic beams were analytically derived from the linearized Hellinger–Reissner principle. In this paper, elastic and geometric stiffness matrices and consistent mass matrix for finite element analysis are evaluated by using isoparametric and Hermitian interpolation polynomials. Isoparametric interpolation functions with 2, 3 and 4 nodes per element are utilized in isoparametric beam elements, and in Hermitian beam elements, the third‐ and fifth‐order Hermitian polynomials including shear deformation effects are newly derived and applied for the calculation of element matrices. In order to verify the validity of the finite element formulation, both analytic and numerical solutions for spatial buckling and free vibration problems including shear effects are presented and compared.
Dynamic stability of stiffened laminated composite plates and shells subjected to in‐plane pulsating forces
Authors: Liao Chung‐Li , Cheng Chu‐Ren
Published: August 2005

The dynamic stability of laminated composite stiffened or non‐stiffened plates and shells due to periodic in‐plane forces at boundaries is investigated in this paper. A three‐dimensional (3‐D) degenerated shell element and a 3‐D degenerated curved beam element are used to model plates/shells and stiffeners, respectively. The characteristic equations to find the natural frequencies, buckling loads and their corresponding mode shapes are obtained from the finite element equation of motion. Then, the method of Hill's infinite determinants or the method of multiple scales is applied to analyse the dynamic instability regions. Numerical results are presented to demonstrate the effects of various parameters, such as skew angle, lamination scheme, stiffened scheme, in‐plane force type and curvature of cylindrical shell, on the dynamic stability of stiffened and non‐stiffened plates and shells subjected to in‐plane pulsating forces at boundaries.
A finite element analysis of the singular stress fields in anisotropic materials loaded in antiplane shear
Authors: Pageau Stephane S. , Joseph Paul F. , Biggers Sherrill B.
Published: August 2005

A finite element formulation is developed to determine the order and angular variation of singular stress states at material and geometric discontinuities in anisotropic materials subject to antiplane shear loading. The displacement field of the sectorial element is quadratic in the angular co‐ordinate direction and asymptotic in the radial direction measured from the singular point. The formulation of Yamada and Okumura14 for in‐plane problems is adapted for this purpose. The simplicity and accuracy of the formulation are demonstrated by comparison to several analytical antiplane shear solutions for both isotropic and anisotropic multi‐material wedges and junctions with and without disbonds. The nature and speed of convergence of the eigensolution suggests that the solution presented here could be used in developing enriched elements for accurate and computationally efficient evaluation of stress intensity factors in problems having complex global geometries.
A modification of the Petrov–Galerkin method for the transient convection–diffusion equation
Authors: Cardle James A.
Published: August 2005

A variation of the Petrov–Galerkin method of solution of a partial differential equation is presented in which the weight function applied to the time derivative term of the transient convection–diffusion equation is different from the weight function applied to the special derivatives. This allows for the formulation of fourth‐order explicit and centred difference schemes. Comparison with analytic solutions show that these methods are able to capture steep wave fronts. The ability of the explicit method to capture wave fronts increases as the amount of convective transport increases.
Hierarchal triangular elements using orthogonal polynomials
Authors: Webb J. P. , Abouchacra R.
Published: August 2005

Hierarchal elements are finite elements which have the useful property that elements with different polynomial orders can be used together in the same mesh without causing discontinuities. This paper introduces a new hierarchal triangular element in which the basis functions are constructed from orthogonal polynomials—Jacobi polynomials. The resulting element is shown to be better conditioned than the earlier hierarchal element of Rossow and Katz.1 Recursive formulas allow the complete set of basis functions for an element to be efficiently evaluated at a given point. In addition, the formulas can be used to generate pre‐computed (universal) matrices. Examples are given of universal matrices, up to order 4, for the generalized Helmholtz equation. An electromagnetic problem involving a length of transmission line is used to show the usefulness of the new elements.
An algorithm for optimization of non‐linear shell structures
Authors: Ringertz Ulf Torbjörn
Published: August 2005

An algorithm for optimal design of non‐linear shell structures is presented. The algorithm uses numerical optimization techniques and nonlinear finite element analysis to find a minimum weight structure subject to equilibrium conditions, stability constraints and displacement constraints. A barrier transformation is used to treat an apparent non‐smoothness arising from posing the stability constraints in terms of the eigenvalues of the Hessian of the potential energy of the structure. A sequential quadratic programming strategy is used to solve the resulting non‐linear optimization problem. Matrix sparsity in the constraint Jacobian is exploited because of the large number of variables. The usefulness of the proposed algorithm is demonstrated by minimizing the weight of a number of stiffened thin shell structures.
Flaw identification using the boundary element method
Authors: Mellings S. C. , Aliabadi M. H.
Published: August 2005

In this paper a new boundary element formulation is presented for the identification of the location and size of internal flaws in two‐dimensional structures. An introduction to inverse analysis is given, with special reference to methods of flaw identification, along with a brief review of the optimization methods employed. Both the standard boundary element and the dual boundary element method are presented, with the dual boundary element method proposed as the basis for the new formulation. The flaw identification method is presented, along with the computation of the boundary displacement and traction derivatives and the specialized analytical integration used for cracked boundaries. Examples are given to demonstrate the accuracy of the sensitivity values and the performance of flaw location.
A locally analytic technique applied to grid generation by elliptic equations
Authors: Montgomery Matthew , Fleeter Sanford
Published: August 2005

One technique for obtaining grids for irregular geometries is to solve sets of elliptic partial differential equations. The solution of the partial differential equations yields a grid which discretizes the physical solution domain and also a transformation for the irregular physical domain to a regular computational domain. Expressing the governing equation of interest in the computational domain requires the derivatives of the physical to computational domain transformation, i.e., the metrics. These metrics are typically determined by numerical differentiation, which is a potential source of error. The locally analytic method uses the analytic solution of the locally linearized equation to develop numerical stencils. Thus, the locally analytic method allows numerical differentiation with no loss of accuracy. In this paper, the locally analytic method is applied to the solution of the Poisson and Brackbill–Saltzman equations. Comparison with an exact solution shows the locally analytic method to be more accurate than the finite difference method, both in solving the partial differential equation and evaluating the metrics. However, it is more computationally expensive.
Optimized partitioning of unstructured finite element meshes
Authors: Vanderstraeten Denis , Keunings Roland
Published: August 2005

We address the problem of automatic partitioning of unstructured finite element meshes in the context of parallel numerical algorithms based on domain decomposition. A two‐step approach is proposed, which combines a direct partitioning scheme with a non‐deterministic procedure of combinatorial optimization. In contrast with previously published experiments with non‐deterministic heuristics, the optimization step is shown to produce high‐quality decompositions at a reasonable compute cost. We also show that the optimization approach can accommodate complex topological constraints and minimization objectives. This is illustrated by considering the particular case of topologically one‐dimensional partitions, as well as load balancing of frontal subdomain solvers. Finally, the optimization procedure produces, in most cases, decompositions endowed with geometrically smooth interfaces. This contrasts with available partitioning schemes, and is crucial to some modern numerical techniques based on domain decomposition and a Lagrange multiplier treatment of the interface conditions.
An inverse problem of simultaneously estimating contact conductance and heat transfer coefficient of exhaust gases between engine's exhaust valve and seat
Authors: Huang Cheng‐Hung , Ju Tzann‐Ming
Published: August 2005

The conjugate gradient method using two search step sizes is used to solve the inverse problem of simultaneously estimating the periodic thermal contact conductance, hc(t), and the heat transfer coefficient of the exhaust gases, hg(t), between the exhaust valve and seat in an internal combustion engine. The importance of the determination of hc(t) and hg(t) lie in that they are the critical factors for designing the cooling system and the insulation of the exhaust valve.

The inverse analysis is based on the temperature measurements taken from the sensors placed in both the valve and seat regions during the transient process of operation. In this study two unknown timewise‐varying functions hc(t) and hg(t) are to be estimated at the same time, thus two search step sizes with each one corresponding to each unknown function are derived. The results show that the CPU time for the inverse solutions using two search step sizes are greatly reduced than using just one search step size1 for the determination of two unknowns, besides, it also shows that the inverse solutions are reliable even when the measurement errors are considered.

The advantage of the conjugate gradient method is that no a priori information is needed on the variation of the unknown quantities, since the solution automatically determines the functional form over the domain specified. The successful development of the present technique can be applied to any kind of two‐dimensional periodic contact problems, such as the determination of a two‐dimensional contact conductance problem2 and the temperature or heat flux behaviour on the inside wall of internal combustion engines3.
Mesh generation and adaptivity for the solution of compressible viscous high speed flows
Authors: Hassan O. , Probert E. J. , Morgan K. , Peraire J.
Published: August 2005

An implicit–explicit procedure for the solution of the compressible Navier–Stokes equations on unstructured triangular and tetrahedral meshes is outlined. A procedure for constructing continuous lines, made up of edges in the mesh, is employed and the implicit equation system is solved via line relaxation. The problem of generating, and adapting, unstructured meshes for viscous flow simulations is addressed. A number of examples are included which demonstrate the numerical performance of the proposed procedures.
A hybrid micro–macro BEM formulation for micro‐crack clusters in elastic components
Authors: Chandra A. , Huang Y. , Wei X. , Hu K. X.
Published: August 2005

Local analysis schemes capable of detailed representations of the micro‐features of a problem are integrated with a macro‐scale BEM technique capable of handling complex finite geometries and realistic boundary conditions. The micro‐scale effects are introduced into the macro‐scale BEM analysis through an augmented fundamental solution obtained from an integral equation representation of the micro‐scale features. The proposed hybrid micro‐macro BEM formulation allows decomposition of the complete problem into two sub‐problems, one residing entirely at the micro‐level and the other at the macro‐level. This allows for investigations of the effects of the micro‐structural attributes while retaining the macro‐scale geometric features and actual boundary conditions for the component or structure under consideration. As a first attempt, elastic fracture mechanics problems with interacting cracks at close spacings are considered. The numerical results obtained from the hybrid BEM analysis establish the accuracy and effectiveness of the proposed micro–macro computational scheme for this class of problems. The proposed micro–macro BEM formulation can easily be extended to investigate the effects of other micro‐features (e.g. interfaces, short or continuous fibre reinforcements, voids and inclusions, in the context of linear elasticity) on macroscopic failure modes observed in structural components.
A self‐adaptive fe system based on recursive spatial decomposition and multigrid analysis
Authors: Pressburger Yoram , Perucchio Renato
Published: August 2005

This paper introduces an automatic self‐adaptive finite element system for 3‐D stress analysis of homogeneous elastic solids described in a Solid Modelling System (SMS). This system reduces the role of the analyst into two basic tasks: defining the problem (i.e. geometry, material properties and boundary conditions) and specifying the desired accuracy in the analysis. The approach followed in this work is based on Recursive Spatial Decomposition (RSD) of solids defined in a SMS. The system includes procedures for meshing, analysis, error estimation, and adaptive remeshing and reanalysis. The various procedures share the same hierarchical data structure built on the RSD.
Non‐linear dynamics of three‐dimensional rods: Exact energy and momentum conserving algorithms
Authors: Simo J. C. , Tarnow N. , Doblare M.
Published: August 2005

The long‐term dynamic response of non‐linear geometrically exact rods under‐going finite extension, shear and bending, accompanied by large overall motions, is addressed in detail. The central objective is the design of unconditionally stable time‐stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum, the total angular momentum and, for the Hamiltonian case, the total energy. This objective is accomplished in two steps. First, a class of algorithms is introduced which conserves linear and angular momentum. This result holds independently of the definition of the algorithmic stress resultants. Second, an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved. Conventional schemes exhibiting no numerical dissipation, symplectic algorithms in particular, are shown to lead to unstable solutions when the high frequencies are not resolved. Compared to conventional schemes there is little, if any, additional computational cost involved in the proposed class of energy–momentum methods. The excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations.
A material based model for topology optimization of thermoelastic structures
Authors: Rodrigues H. , Fernandes P.
Published: August 2005

This paper presents the development of a computational model for the topology optimization problem, using a material distribution approach, of a 2‐D linear‐elastic solid subjected to thermal loads, with a compliance objective function and an isoperimetric constraint on volume. Defining formally the augmented Lagrangian associated with the optimization problem, the optimality conditions are derived analytically. The results of analysis are implemented in a computer code to produce numerical solutions for the optimal topology, considering the temperature distribution independent of design. The design optimization problem is solved via a sequence of linearized subproblems. The computational model developed is tested in example problems. The influence of both the temperature and the finite element model on the optimal solution obtained is analysed.
A general formulation in rigid multibody dynamics
Authors: Franchi Claudio G.
Published: August 2005

The dynamics of rigid multibodies is traditionally formulated by means of either minimal or redundant co‐ordinates methods. An alternative approach is here proposed whereby a highly redundant set of coordinates is adopted. As a result, the equations of motion of the constrained bodies are decoupled. Several meaningful parameters are directly available and the constraint conditions are enforced in a very natural way. The first part of the paper presents the basic meanings and the theoretical developments of the formulation. The second develops a numerical approximation for the methodology proposed in the first part. The non‐linear system of differential‐algebraic equations governing the motion of the multibody is reduced to its weak form. It is linearized by applying a Newton–Raphson procedure and approximated through the method of finite elements in time. The details of the numerical application of this method are discussed and a solution procedure is presented. Finally, some numerical examples involving tree and closed loop topologies prove the capability of the present formulation in handling multibody dynamics.
Explicit determination of element stiffness matrices in the hybrid stress method
Authors: Saether Erik
Published: August 2005

The hybrid stress method has demonstrated many improvements over conventional displacement‐based formulations. A main detraction from the method, however, has been the higher computatational cost in forming element stiffness coefficients due to matrix inversions and manipulations as required by the technique. By utilizing permissible field transformations of initially assumed stresses, a spanning set of orthonormalized stress modes can be generated which simplify the matrix equations and allow explicit expressions for element stiffness coefficients to be derived. The developed methodology is demonstrated using several selected 2‐D quadrilateral and 3‐D hexahedral elements.
A formulation for the 4‐node quadrilateral element
Authors: Hueck Ulrich , Wriggers Peter
Published: August 2005

A formulation for the plane 4‐node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second‐order Taylor series in the physical co‐ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices.

For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix.

The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non‐linear analysis. There, an application is given for the calculation of inelastic problems in physically non‐linear elasticity.

The element is efficient to implement and it is frame invariant. Locking effects and zero‐energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.
Formulation and solution of the non‐linear, damped eigenvalue problem for skeletal systems
Authors: Smith H. A. , Singh R. K. , Sorensen D. C.
Published: August 2005

This paper presents and discusses an Arnoldi‐based eigensolution technique for evaluating the complex natural frequencies and mode shapes from frequency dependent quadratic eigenproblems associated with vibration analysis of damped structures. The new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices. This modelling provides the ability to represent a frequency dependent damping matrix in vibration analysis of skeletal systems.

The eigensolution methodology presented here is based upon the ability to evaluate a specific set of parametrized curves for the non‐linear eigenvalue problem at given values of the parameter. Numerical examples illustrate that this method, used in conjunction with a secant interpolation, accurately evaluates the complex natural frequencies and modes of the quadratic non‐linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models.
Dynamic response of finite sized elastic runways subjected to moving loads: A coupled BEM/FEM approach
Authors: Pan G. , Atluri S. N.
Published: August 2005

The transient response of a finite elastic plate, resting on an elastic half‐space, and subjected to moving loads is considered here. Both the cases of an elastic foundation alone, as well as a finite sized elastic plate resting on an elastic foundation are considered. The numerical methods employed are: (1) the time‐domain boundary element method for the elastic foundation and (2) a combination of the time‐domain boundary element method for the soil and the semi‐discrete finite element method for the finite sized elastic plate. Both constant as well as linear‐time‐interpolation schemes are included in the BEM. The integration is carried out analytically in time. The analytical solution for a moving point load on an infinite elastic plate resting on an elastic half‐space is derived here. This is used as a benchmark against which the present numerical solution is compared with. The accuracy of the numerical method is also verified by comparing the solutions with some existing numerical results; the comparison with the solutions based on a Winkler foundation model reveals the limitations of the applicability of such a model, especially in the cases of high velocities of the moving load. This is because neither the inertia of the foundation, nor the behaviour of the foundation as a continuum, can be properly accounted for in Winkler's model. A parametric study is conducted, and the influences of velocity of the moving load, load distribution, etc. on the dynamic response of the soil/runway system are investigated. Furthermore, the present computational method is applied to the problem of a transport airplane taxiing on a concrete pavement resting on a typical soil. The responses of pavements are presented for different taxiing velocities.
A generalized fictitious crack model based on plastic localization and discontinuous approximation
Authors: Larsson Ragnar
Published: August 2005

Modelling of crack behaviour is an important topic when establishing realistic macroscopic constitutive relations for brittle materials, such as ceramics, concrete, rock, etc. Upon introducing discontinuous approximation of the continuum, a crack model is derived on the basis of the Rankine plasticity model and the fictitious crack concept of Hillerborg et al.1 The derivation results in a crack band relation, whose constitutive properties are based on failure kinematics that resemble the classical shear band concept. It appears that it is, essentially, the acoustic tensor that governs the constitutive relation for the crack band. This tensor has previously been analysed extensively to determine conditions for loss of ellipticity and possibility for discontinuous bifurcation in the rate of deformation field. In the present approach a similar analysis is carried out to determine conditions that characterize fracture initiation and crack band orientation. The model is implemented together with a mesh realignment strategy, and a solution advancement formulation, that ensures the crack propagates along critical interfacial directions. These directions depend on the actual stress state and are kept fixed after plastic localization (or cracking) has occurred. Hence, a ‘fixed crack’ model is obtained. Numerical results involving successive crack development as well as failure of concrete beams are presented.
Axisymmetric elastodynamic infinite elements for multi‐layered half‐space
Authors: Yun Chung‐Bang , Kim Jae‐Min , Hyun Chang‐Hun
Published: August 2005

This paper presents three axisymmetric infinite elements for the elastodynamic problems in a multi‐layered half‐space. They are the horizontal, the vertical and the corner infinite elements that are developed by using the wave functions in the function spaces derived from approximate expressions of the analytical solutions. An efficient integration procedure is proposed for calculating the element matrices involving multiple wave components. Numerical example analyses are presented for rigid disks on homogeneous and layered half‐spaces, and for an embedded caisson. The numerical results obtained show the effectiveness of the proposed infinite elements.
On the computation of mode I and II thermal shock stress intensity factors using a boundary‐only element method
Authors: Katsareas D. E. , Anifantis N. K.
Published: August 2005

A time domain boundary‐only element method is used for the analysis of fractured planar bodies, subjected to thermal shock type loads. The uncoupled quasistatic thermoelasticity equations are solved without the need for domain discretization. The singular behaviour of the temperature and displacement fields, in the vicinity of the crack tip, is modelled by quarter‐point elements. Transient thermal stress and heat flux intensity factors are evaluated from computed nodal values, using the well‐known displacement and traction formulas. The accuracy of the method, the stability of the solution and the dependence on space‐time discretization is investigated through comparison of present results with analytical and computational ones, taken from the literature. For mode I and mixed mode two‐dimensional problems examined, the method proved to be a potent tool for fracture analysis in presence of severe thermal shock loads, even in the microcrack size range. Potent in the sense that, it is very stable and cost effective from the space‐time discretization point of view.
The natural occurrence of flux splitting in a hyperbolic setting
Authors: Cooke C. H. , Smith Philip W.
Published: August 2005

Classical derivations of the so‐called Riemann invariants for hyperbolic partial differential equations have depended upon the strong‐solution concept. Thus, invariance may rigorously be guaranteed only in regions of smooth flow. In general, this is as much as can be said. However, by restricting attention to linear hyperbolic systems, it further emerges that the Riemann invariant fully justifies its title. By using distribution‐theoretical arguments based on the weak‐solution concept. Riemann invariants of a more generalized nature are studied. For a particular weak solution u there exists, among the equivalence class [u] of weak solutions that differ from u at most on a set of measure zero, a weak solution u whose Riemann invariant corresponding to characteristic direction λ is constant on lines C: dx/dt = λ. Moreover, every piecewise‐smooth weak solution has Riemann invariants that are continuous across a finite jump discontinuity. This result is used to establish for a certain Riemann problem that the one‐sided time derivative at a point of discontinuity exhibits a character usually regarded in the literature as flux splitting. This result sheds light upon the validity of some upstream‐biased approximation techniques for the numerical solution of hyperbolic systems.
The global element method and its extension to parabolic problems
Authors: Botha J. F. , Bakkes G. N.
Published: August 2005

This article describes the derivation of the global element method, originally proposed by Delves and Hall, using the classical Euler–Lagrange theory. The method is then extended so as to be applicable to parabolic partial differential equations. The main advantages of the method, the ease of adjusting its accuracy and applying it to problems with discontinuous coefficients and point singularities, are also discussed.
High‐order difference schemes for two‐dimensional elliptic equations
Authors: Gupta Murli M. , Manohar Ram P. , Stephenson John W.
Published: August 2005

A high‐order finite‐difference approximation is proposed for numerical solution of linear or quasilinear elliptic differential equation. The approximation is defined on a square mesh stencil using nine node points and has a truncation error of order h4. Several test problems, including one modeling convection‐dominated flows, are solved using this and existing methods. The results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.
Unified approach to numerical methods. III. Finite differences and ordinary differential equations
Authors: Herrera Ismael , Chargoy Lucia , Alduncin Gonzalo
Published: August 2005

This is the third of a sequence of articles devoted to the presentation of a direct method of analysis recently developed by one of the authors. The approach is quite general, since it is applicable to any linear operator, symmetric or nonsymmetric, regardless of its type. In particular, the theory includes steady‐state and time‐dependent problems. In this article the method is applied to ordinary differential equations, which constitute a very convenient illustration of its application because the analysis can be carried out in an exhaustive manner. General algorithms which yield the exact values of the solution or its derivatives are obtained in this manner.
Variable trial functions and the CVBEM
Authors: Hromadka T. V.
Published: August 2005

The Complex Variable Boundary Element Method; or CVBEM will be developed with respect to a variable trial function definition over each boundary element. The benefits in using this technique are that the modeling error in matching the prescribed boundary conditions (there is no error in satisfying the Laplace equation) is reduced without the addition of nodal points to the problem discretization. Consequently, the n × n matrix requirements are not increased when using this new approach.
Two‐phase laminar axisymmetric jet flow: Explicit, implicit, and split‐operator approximations
Authors: Aggarwal S. K. , Fix G. J. , Sirignano W. A.
Published: August 2005

An hybrid Eulerian‐Lagrangian numerical scheme is developed for a two‐phase problem and four finite‐difference schemes are compared. For this purpose, the problem of hydrodynamic and thermal interactions between a fuel spray and a mixing region of two laminar, unconfined axisymmetric jets is formulated in terms of a set of parabolic differential equations for the gas phase and a set of Lagrangian ordinary differential equations for the condensed phase. Consistent, second‐order accurate hybrid numerical schemes, with the exception of the explicit scheme with an accuracy between linear and quadratic, are used to solve these equations. The subset of gas‐phase equations has been solved by four different numerical methods: a predictor‐corrector explicit method, a sequential implicit method, a block implicit method, and a symmetric operator‐splitting method. The subsystem of liquid‐phase equations is solved along the droplet trajectories by a second‐order Runge‐Kutta scheme. The computations have been made to predict the hydro‐dynamic and thermal mixing regions of the gas phase as well as the trajectories of each individual group of droplets. In addition, the size, velocity and temperature associated with each group are predicted along these trajectories. The relative merits of the above four difference‐schemes are discussed by constructing effectiveness curves. At low error tolerances, the sequential implicit method gives the best results, where for large error tolerances, the explicit and operator splitting give better results. The block implicit scheme is the least effective at all accuracy requirements.
On operator splitting of the Euler equations consistent with Harten's second‐order accurate TVD scheme
Authors: Cooke C. H.
Published: August 2005

Operator splitting is necessary in order to apply Harten's second‐order accurate, total‐variation‐diminishing (TVD) method for shock capturing to higher‐dimensional problems. Here two such splittings of the Euler equations are considered. By analytical means, one splitting is shown to be inconsistent; the other, scheme satisfies a fundamental necessary condition for consistency. Numerical results from a typical blast wave calculation employing this scheme are exhibited.
Nonreflecting upwind boundaries for hyperbolic equations
Authors: Vichnevetsky R. , Pariser E. C.
Published: August 2005

It is commonly, but erroneously, assumed that the best way to treat upwind (closed) boundaries in numerical approximations of hyperbolic equations consists in a literal transcription, letting the numerical value be equal to the prescribed value. This results in a total reflection of spurious solutions that may arrive at the boundary from the computing domain. Those reflected solutions cannot be distinguished from consistent solutions, and they may seriously degrade the overall accuracy. We show that modifications of this treatment of the boundary may result in the absorption of spurious solutions. The effect of the absorbing properties of these boundary schemes is analyzed in Fourier space. We also analyze their numerical stability properties, and their effect on the accuracy of solutions generated in response to a time dependent boundary condition.
Numerical methods for the linear Boltzmann transport equation in slab geometry
Authors: Pao C. V.
Published: August 2005

An iterative method for computing numerical solutions of a finite‐difference system corresponding to the linear Boltzmann equation in slab geometry is presented. This iterative scheme gives a straightforward marching process starting from the given boundary and initial conditions. It is shown that with a suitable initial iteration the sequence of iterations converges monotonically to a unique solution of the finite‐difference system. This monotone convergence leads to improved upper and lower bounds of the solution in each iteration, and to the well‐posedness of the discrete system in the sense of Hadamard. It also leads to the convergence of the discrete system to the continuous system as the mesh size of the space–velocity–time variables approaches to zero. Under a mild restriction on the time‐increment the discrete system is numerically stable, independent of the mesh‐size of the space and velocity. An error estimate for the computed solution due to simultaneous initial and iteration error is obtained. Also given are some numerical results for the time‐dependent and the steady‐state solutions.