bluuebunny/crossref_metadata_2025 huggingface dataset:
Stochastic elastic–viscoplastic analysis for discontinuous rock masses
Authors: Chen S. H. , Shen B. K. , Huang M. H.
Published: August 2005

Both the first‐order second moment method and the Monte–Carlo method are introduced to elastic–viscoplastic block theory in this paper. The safety index of blocks, block combinations and the whole rock–block system can be obtained by establishing adequate limit state equation. Some slope safety problems have been studied with the proposed methods.
A three‐dimensional layer‐wise constant shear element for general anisotropic shell‐type structures
Authors: Zinno Raffaele , Barbero Ever J.
Published: August 2005

This paper deals with the development of a new three‐dimensional element with two‐dimensional kinematic constraints capable of analysing the mechanical behaviour of the laminated anisotropic shell‐type structures. This element, originally developed for the linear analysis of plates, is extended for the linear analysis of laminated composite shells. The element can represent arbitrarily curved shells with variable number of layers and thicknesses, including ply drop‐off problems. The element was validated in a previous work by the patch test. All the analytical details necessary to make possible the shell analysis are presented here. Examples are reported to show the capability of the element to predict the behaviour of complex structures and a refined computation of the stresses is carried out by integrating the equilibrium equations.
Mathematical boundary integral equation analysis of an embedded shell under dynamic excitations
Authors: Pak Ronald Y. S. , Ji Feng
Published: August 2005

A boundary integral equation method is presented for the analysis of a thin cylindrical shell embedded in an elastic half‐space under axisymmetric excitations. By virtue of a set of ring‐load Green's functions for the shell and a group of dynamic fundamental solutions for the semi‐infinite medium, the structure–medium interaction problem of wave propagation is shown to be reducible to a set of coupled boundary integral equations. Through the analysis of an auxiliary pair of Cauchy integral equations, the singularities of the contact stress distributions arc rendered explicit. With a direct incorporation of such analytical features into the formulation, an effective computational procedure is developed which involves an interpolation of regular functions only. Typical results for the dynamic contact load distributions, displacements, and complex compliance functions are included as illustrations. In addition to furnishing quantities of direct engineering interest, this treatment is apt to be useful as a foundation for further rigorous as well as approximate developments for various related physical problems and boundary integral methods.
Finite element analysis of contact–impact between a shell of revolution and a rigid wall
Authors: Gautham B. P. , Ganesan N.
Published: August 2005

In this paper an attempt is made to use a shell theory that includes transverse normal stresses to study the contact–impact behaviour involving thick shells. The problem is solved using the finite element method. The contact criterion is imposed by means of displacement and force constraints. The Lagrangian multipliers technique is used to impose these constraints. The case of a spherical shell impacting a rigid wall with a constant velocity is considered for the analysis. It is shown that the method produces accurate results with less amount of computation than the usual three‐dimensional analyses. The effects of the thickness of shell, velocity of impact and modulus of the material are studied.
Subdomain methods in structural mechanics
Authors: Radi B. , Gelin J. C. , Perriot A.
Published: August 2005

The principle of subdomain methods has been introduced many years ago1 and such methods have been used with success in recent years. In this paper, we discuss some methods used to solve the condensed system resulting from the Schur complement method. We will present the hybrid method and a mechanical application: the contact problem between two deformable solids. In the case of such a problem, we propose two methods: one for the contact with friction and one for the contact without friction. Finally, the results of validation tests for both proposed approaches are given.
A new formulation for the direct dynamic simulation of flexible mechanisms based on the Newton–Euler inverse method
Authors: Verlinden Olivier , Dehombreux Pierre , Conti Calogero , Boucher Serge
Published: August 2005

This paper is concerned with the direct dynamic analysis of mechanisms comprising flexible members.

It is first recalled how the time integration of the dynamic equations governing the motion of a flexible mechanism is conditioned by the availability of the so‐called iteration matrix. The latter is obtained not only from the mass matrix, which is sufficient in the rigid case, but also from the stiffness and damping tangent matrices.

Hence, in a context based upon relative co‐ordinates where the deformation of the flexible bodies is described by component modes, two formulations, a classical one and a new one, which permit the determination of this iteration matrix are presented. The classical formulation is based upon the Lagrange's theorem. The new one consists in a direct calculation of the residues of the dynamic equations by using the inverse method of Newton–Euler. In this case, the iteration matrix is obtained by a numerical derivation technique which proves well‐conditioned.

A first simple example is treated in order to compare the two formulations. A more complex one illustrates the real capabilities of the new formulation.
Ray tracing with a space‐filling finite element mesh
Authors: Favre Jean , Löhner Rainald
Published: August 2005

A new ray‐tracing technique is presented which does not work with ray–object intersections per se, but is based on the traversal of an unstructured tetrahedral mesh providing a convex enclosure of a scene of polyhedral objects. The tetrahedral mesh provides tight bounding and an adaptive subdivision of space. This non‐hierarchical data structure is traversed adaptively until one is led directly and unconditionally to the first object intersected. Rendering times are directly related to the average thickness of the enclosing mesh since all tetrahedra are traversed in constant time.

Since the proposed algorithm operates directly with volume elements, it allows for volumetric rendering effects. Volume rendering or anisotropic media can be implemented without any further effort. This is an important advantage as compared to usual techniques, which only operate on surface data.

Timings for several examples show that the use of this type of ray‐tracing technique, which is more suitable for general purpose visualization codes than traditional techniques, results in CPU times that are comparable with the best ray‐tracing techniques presently used. This is an unexpected and important result, as the vectorization and parallclization of the proposed technique are straightforward, in contrast with traditional ray‐tracing techniques.
Inflation of hyperelastic cylindrical membranes as applied to blow moulding. Part II. Non‐axisymmetric case
Authors: Khayat R. E. , Derdouri A.
Published: August 2005

The non‐axisymmetric confined inflation of an isotropic homogeneous cylindrical membrane is considered in this study. The material is assumed to obey the Mooney–Rivlin constitutive model. The continuum‐based formulation is adopted, with the resulting equilibrium partial differential equations, governing the deformation field, solved using a Galerkin based finite element procedure. Upon contact, the inflating material is assumed to stick to the solid boundary. The dependent variables corresponding to nodes which have contacted are then fixed for subsequent deformation. In an attempt to simulate the inflation stage of the blow‐moulding process, the method is illustrated through some examples of moulded tapered rectangular bottles with centered and off‐centred neck. Examples of initial membranes of uniform and non‐uniform thickness distribution are examined. Experiments are carried out in an attempt to validate the theory, in particular, to compare the predicted and measured final thickness distributions of blow‐moulded containers.
Implicit objective integration for sensitivity analysis in non‐linear solid mechanics
Authors: Leu Liang‐Jenq , Mukherjee Subrata
Published: August 2005

Incrementally objective integration schemes are proposed for the accurate and efficient determination of design sensitivity coefficients (DSCs) for solid mechanics problems with both material and geometrical non‐linearities. The derivation of these schemes are based on the direct differentiation of objective schemes that are used in stress analysis for problems of this class. Two widely used objective stress rates, the Jaumann rate and the Green–Naghdi rate, are considered here within the same setting with only minor changes of the integration scheme. Numerical results are presented for a simple shear problem with different material constitutive laws, including a hypoelastic model and a unified isotropic viscoplastic model, for these two objective rates. The numerical results are compared with analytical solutions or direct integration solutions. The close agreement among these solutions demonstrates the accuracy and efficiency of the proposed scheme.
Solution of the slump test using a finite deformation elasto‐plastic Drucker‐Prager model
Authors: Famiglietti C. M. , Prevost J. H.
Published: August 2005

Development of a finite deformation elasto‐plasticity model based on the multiplicative decomposition of the deformation gradient is presented and discussed in detail. The formulation presented in this paper includes the derivation of the full set of equations for the Drucker‐Prager yield criterion. The equations, which are not available elsewhere, are developed within a framework using a spectral decomposition approach. Further, expressions for the consistent (algorithmic) tangent moduli in the finite strain regime are developed. Since the finite deformation framework employed to obtain the expressions presented here collapses to the classical infinitesimal plasticity framework when the finite strain assumption is no longer necessary, the finite deformation consistent tangent moduli are compared to the consistent tangent moduli valid for use with infinitesimal plasticity. Validation of the implemented finite deformation elasto‐plastic Drucker‐Prager model is performed through the solution of the concrete slump test. Comparisons between an existing approximate analytical solution and experimental data are presented, and results are discussed in detail.
A qualitative analysis of the behaviour of the Galerkin equations relevant to non‐linear eddy current problems
Authors: Miano Giovanni
Published: August 2005

The Galerkin equations relevant to the eddy currents induced in an iron body are considered. These equations are obtained by formulating the field problem in terms of the magnetic vector potential and by applying the Galerkin method. They are shown to have a unique steady‐state solution if a certain condition on the magnetic constituitive relationship is satisfied. In particular a T‐periodic source gives rise to a unique T‐periodic solution to which all other solutions converge asymptotically independently from the initial conditions. Under the same condition the exponential decay of the ‘transients’ is shown, and an explicit lower and upper bound for its rate is given. These structural properties allow us to exclude a priori that qualitatively different asymptotic behaviours, including even chaotic solutions, may occur. Numerical simulation, when based on qualitative information of this type, enables us to obtain the quantitative properties in an efficient manner. In order to demonstrate the practical use of these results some numerical experiments are presented.
A robust refined quadrilateral plane element
Authors: Wanji Chen , Cheung Y. K.
Published: August 2005

Based on a rational choice of the internal incompatible displacement function and a special formulation of the a priori elimination of the internal non‐conforming displacement parameters, a new refined quadrilateral plane element RQ4 has been developed. The present element can be shown to be computationally efficient, accurate and free from locking, and is better than other elements such as the Plan's element HS, the generalized hybrid element QCS6, and the refined hybrid element RGH4, etc. Several numerical examples are given to show the superior performances of the present element RQ4.
Multi‐grid domain decomposition approach for solution of Navier–Stokes equations in primitive variable form
Authors: Ku Hwar‐Ching , Ramaswamy Bala
Published: August 2005

A new multi‐grid (two‐grid) pseudospectral element method has been carried out for solution of incompressible flow in terms of primitive variable formulation. The main objective of the proposed method is to apply the multi‐grid technique solving the incompressible flow problems associated with three commonly encountered multi‐grid environments. In domain decomposition terminology, it includes (I) partially overlapped subdomains, each of which has same types of grids; (II) partially overlapped subdomains, each of which has different types of grids; (III) local adaptive subdomains fully overlapped with the original computational domain (composite grids).

The approach for flow problems, complex geometry or not, is to first divide the computational domain into a number of subdomains with the inter‐overlapping area (partially or fully overlapped). In categories I and II, the fine‐grid or coarse‐grid subdomains can be defined by their representation, while in category III the fine‐grid or coarse‐grid subdomains are defined as usual. Next, implement the Schwarz Alternating Procedure (SAP) to exchange the data among subdomains, where the coarse‐grid correction is used to remove the high frequency error that occurs when the data interpolation from the fine‐grid subdomain to the coarse‐grid subdomain is conducted. The strategy behind the coarse‐grid correction is to adopt the operator of the divergence of velocity field, which intrinsically links the pressure equation, into this process. The solution of each subdomain can be efficiently solved by the direct (or iterative) eigenfunction expansion technique or preconditioned method with the least storage requirement, i.e. O(N2) in 2‐D.

Numerical results of (i) driven cavity flow (Re = 100,400) with Cartesian grids (category I) in each subdomain, (ii) driven cavity flow (Re = 3200) with local adaptive grids (category III) in each subdomain, and (iii) flow over a cylinder (Re = 250) with ‘O’ grids in one subdomain and Cartesian grids in another (category II) will be presented in the paper to account for the versatility of the proposed multi‐grid method.
A posteriori error estimates and adaptivity for finite element solutions in finite elasticity
Authors: Mücke Roland , Whiteman J. R.
Published: August 2005

Methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems. In contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context. In the paper a brief outline of continuum theory of finite elasticity is first given. Using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally. The proposed error estimator is physically interpreted in the energy sense. We then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences.
A geometrically non‐linear quasi‐conforming nine‐node quadrilateral degenerated solid shell element
Authors: Guan Yupu , Tang Limin
Published: August 2005

In this paper, a geometrically non‐linear quasi‐conforming nine‐node quadrilateral degenerated solid shell element has been constructed from generalization of a linear quasi‐conforming nine‐node quadrilateral degenerated solid shell element,10 based upon the non‐linear quasi‐conforming finite element method. First, the geometrical description and displacement model of the nine‐node quadrilateral degenerated solid shell element is given. Then, the non‐linear quasi‐conforming finite element method is discussed and assumed strains and weighed functions are given. Last, analysis of numerical examples for a clamped square plate, a circular arch, a spherical shell, two cylindrical shells, the patch test and a simply supported square plate is carried out. The results are compared with exact and other finite element solutions. The results indicate that the shell element in the paper is accurate and efficient.
Automatic mesh generation over intersecting surfaces
Authors: Lo S. H.
Published: August 2005

A fully automatic general scheme is proposed to determine and analyse the intersections between two groups of surfaces composed of triangular facets. By carefully maintaining the spatial position of the lines of intersection in regenerating mesh locally around surface intersections, pieces of surfaces can be arbitrarily combined and merged. A great variety of objects can be easily created by selectively putting together different surface parts derived from surface intersections. The new algorithm is best applied in conjunction with an existing surface mesh generator to enhance its general capability in dealing with objects built from intersecting surfaces. In fact, it is a powerful surface mesh manipulator, and through the repeated use of the process, complex structures can be rapidly and accurately constructed.
On the solution of non‐linear drying problems in capillary porous media through integral transformation of Luikov equations
Authors: Ribeiro J. W. , Cotta R. M.
Published: August 2005

The generalized integral transform technique is employed to provide hybrid numerical–analytical solutions to the non‐linear Luikov equations, that govern drying within capillary porous media. The coupled equations for moisture and temperature distributions are integral transformed to eliminate the spatial co‐ordinates, yielding an ordinary differential system for the time variation of the transformed potentials, which is readily solved through scientific subroutine libraries with automatic error control schemes. The analytical inversion formula is then recalled to provide explicit expressions for the original potentials at any desired position. The convergence behaviour of the proposed eigenfunction expansions is illustrated, and a parametric study is performed, for the drying of a slab subjected to non‐linear radiative boundary conditions.
The complementary functions method for the element stiffness matrix of arbitrary spatial bars of helicoidal axes
Authors: Haktanir Vebi̇l
Published: August 2005

The statical behaviour of a spatial bar of an elastic and isotropic material under arbitrary distributed loads having a non‐circular helicoidal axis and cross‐section supported elastically by single and/or continuous supports is studied by the stiffness matrix method based on the complementary functions approach. By considering the geometrical compatibility conditions together with the constitutive equations and equations of equilibrium, a set of 12 first‐order differential equations having variable coefficients is obtained for spatial elements of helicoidal axes. The stiffness matrix and the element load vector of a helicoidal bar with a non‐circular axis and arbitrary cross‐section are obtained taking into consideration both the presence of an elastic support and the effects of the axial and shear deformations. For helicoidal staircases, the significance of both axial and shear deformations and eccentricities existing in wide and shallow sections are also investigated. The developed model has been coded in Fortran‐77, which has been applied to various example problems available in the relevant literature, and the results have been compared.
Analysis and optimization of weakly coupled thermoelastoplastic systems with applications to weldment design
Authors: Michaleris Panagiotis , Tortorelli Daniel A. , Vidal Creto A.
Published: August 2005

A systematic approach for the design of weakly coupled thermoelastoplastic systems is presented. The Newton–Raphson iteration method is used in the solution process so that analytic design sensitivity formulations may be efficiently derived via the direct differentiation technique. The derived formulations are suitable for finite element implementations. Analysis and sensitivity analysis capabilities are combined with numerical optimization to form an optimum design algorithm. To demonstrate the algorithm, we optimally design a weldment with respect to manufacturing and service life aspects.
Plate spectral elements based upon Reissner–Mindlin theory
Authors: Zrahia Uzi , Bar‐Yoseph Pinhas
Published: August 2005

Plate bending spectral elements based upon the Reissner–Mindlin theory is developed for a rectangular plate. Numerical comparison studies for static and eigenvalue problems, demonstrate the excellent performance and robustness of the present method. The improvement in accuracy and computational efficiency in comparison with low order plate bending elements is remarkable and suggests that the proposed approach is an alternative to existing schemes.
Control of the freezing interface morphology in solidification processes in the presence of natural convection
Authors: Zabaras Nicholas , Nguyen T. Hung
Published: August 2005

This paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q0 in the fixed mold wall, δΩ0, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history qms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous.

The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, ΩL(t), before solving an ill‐posed inverse design conduction problem in the solid phase in an a priori known growing region, ΩS(t). The direct convection problem is used to evaluate the flux qml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux qml can be used together with the Stefan condition to provide the freezing interface flux qms in the solid side of the front. As such, two boundary conditions (flux qms and freezing temperature θm) are especified along the (known) freezing interface δΩI(t). The developed design technique uses the adjoint method to calculate in L2 the derivative of the cost functional, ∥θm – θ(x, t; q0)∥, that expresses the square error between the calculated temperature θ(x, t; q0) in the solid phase along δΩI(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.
A novel finite element for treating inhomogeneous solids
Authors: Meguid S. A. , Zhu Z. H.
Published: August 2005

This study is concerned with the development and implementation of a novel finite element which is capable of treating the problem of interacting circular inhomogeneities in heterogeneous solids. The general form of the element, which is constructed from a cell containing a single circular inhomogeneity in a surrounding matrix, is derived explicitly using the complex potentials of Muskhelishvili and the Laurent series expansion method. The strength of the proposed eight‐noded plane element is demonstrated by its ability to treat arbitrarily and periodically located multiple inhomogeneities under general loading conditions using a limited number of elements. Assessment of the accuracy and efficiency of the devised element is obtained by comparing its performance against existing analytical and traditional finite element attempts. The current element enables the determination of the local and effective elastic properties of composite materials with relative ease.
A hybrid finite element method for heterogeneous materials with randomly dispersed rigid inclusions
Authors: Zhang J. , Katsube N.
Published: August 2005

A hybrid finite element approach is proposed for the mechanical response of two‐dimensional heterogeneous materials with linearly elastic matrix and randomly dispersed rigid circular inclusions of arbitrary sizes. In conventional finite element methods, many elements must be used to represent one inclusion. In this work, each inclusion is embedded inside a polygonal element and only one element is required to represent one inclusion.

In numerically approximating stress and displacement distributions around the inclusion, classical elasticity solutions for a multiply‐connected region are employed. A modified hybrid functional is used as the basis of the element formulation where the displacement boundary conditions of the element are automatically considered in a variational sense.

The accuracy and efficiency of the proposed method are demonstrated by two boundary value problems. In one example, the results based on the proposed method with only 64 hybrid elements (450 degrees of freedom) are shown to be almost identical to those based on the traditional method with 2928 conventional elements (5526 degrees of freedom).
Non‐conforming mesh gluing in the finite elements method
Authors: Quiroz Luis , Beckers Pierre
Published: August 2005

A methodology to connect two or more non‐conforming meshes generated from a geometric model is presented. The problem leads to a variational problem subject to constraints. The emphasis of the work is on constraints construction. Two methods, namely, discrete connection and continuous connection, are developed. The relation with hybrid formulations is considered and two application examples are presented.
An application of Trefftz method to the sensitivity analysis of two‐dimensional potential problem
Authors: Kita Eisuke , Kamiya Norio , Ikeda Youichi
Published: August 2005

This paper presents a new formulation for the sensitivity analysis with the help of the Trefftz boundary‐type solution procedure. Direct differentiation of the expressions of the field variables leads to the expressions of the sensitivities. Since the field variables are approximated by linear combination of regular T‐complete functions, the analytical expressions of the sensitivities are also regular. Sensitivity analysis schemes with respect to variations of dimensions of the object and boundary conditions are explained in detail. Two‐dimensional potential problems in the bounded region are considered as a numerical example in order to confirm its validity.
Finite element evaluation of free‐edge singular stress fields in anisotropic materials
Authors: Pageau Stephane S. , Biggers Sherrill B.
Published: August 2005

A finite element formulation based on the work of Yamada and Okumura14 is presented to determine the order of singularity and angular variation of the stress and displacement fields surrounding a singular point on a free edge of anisotropic materials. Emphasis is placed on the computational aspects of this method when applied to configurations including fully bonded multi‐material junctions intersecting a free edge as well as materials containing cracks intersecting a free edge. The study shows that the singularity of the three‐dimensional stress field may be accurately determined with a relatively small number of elements only when a proper level of numerical integration is used. The method is applied to isotropic and orthotropic materials with a crack intersecting a free edge and an anisotropic three‐material junction intersecting a free edge. The efficiency and accuracy of the method indicates it could be used to develop a numerical solution for the singular field that could in turn be used to create free‐edge enriched finite elements.
The green element method
Authors: Taigbenu Akpofure E.
Published: August 2005

This paper discusses an element‐by‐element approach of implementing the Boundary Element Method (BEM) which offers substantial savings in computing resource, enables handling of a wider range of problems including non‐linear ones, and at the same time preserves the second‐order accuracy associated with the method. Essentially, by this approach, herein called the Green Element Method (GEM), the singular integral theory of BEM is retained except that its implementation is carried out in a fashion similar to that of the Finite Element Method (FEM). Whereas the solution procedure of BEM couples the information of all nodes in the computational domain so that the global coefficient matrix is dense and full and as such difficult to invert, that of GEM, on the other hand, involves only nodes that share common elements so that the global coefficient matrix is sparse and banded and as such easy to invert. Thus, GEM has the advantage of being more computationally efficient than BEM. In addition, GEM makes the singular integral theory more flexible and versatile in the sense that GEM readily accommodates spatial variability of medium and flow parameters (e.g., flow in heterogeneous media), while other known numerical features of BEM—its second‐order accuracy and ability to readily handle problems with singularities are retained by GEM.

A number of schemes is incorporated into the basic Green element formulation and these schemes are examined with the goal of identifying optimum schemes of the formulation. These schemes include the use of linear and quadratic interpolation functions on triangular and rectangular elements. We found that linear elements offer acceptable accuracy and computational effort. Comparison of the modified fully implicit scheme against the generalized two‐level scheme shows that the modified fully implicit scheme with weight of about 1·25 offers a marginally better approximation of the temporal derivative. The Newton–Raphson scheme is easily incoporated into GEM and provides excellent results for the time‐dependent non‐linear Boussinesq problem. Comparison of GEM with conventional BEM is done on various numerical examples, and it is observed that, for comparable accuracy, GEM uses less computing time. In fact, from the numerical simulations carried out, GEM uses between 15 and 45 per cent of the simulation time of BEM.
Axisymmetric three‐ and four‐node finite elements for large strain elastoplasticity
Authors: Heiduschke K.
Published: August 2005

Within the framework of the finite element method an application of the logarithmic strain space formulation of large strain elastoplasticity is illustrated for the examples of axisymmetric three‐node triangular and four‐node quadrilateral finite elements. The formulation of the large strain elastoplasticity is based on a strain space formulation in conjunction with logarithmic (or Hencky) strain tensors with respect to the reference configuration. It is therefore—from a material point of view—a full Lagrangian formulation. The use of logarithmic strains enables an additive split of finite dilatation and distortion, which are given by the logarithmic strain trace and deviator. As a consequence of the strain space formulation no stress tensors are involved in order to describe the plasticity. The stress which is work‐conjugate to the logarithmic strain follows from the stress‐strain relations and may be transformed to Cauchy stress. The desired finite element matrices are derived via the principle of virtual work applied to the Cauchy stress distribution of the current configuration. It should be noted that our considerations are not restricted to axisymmetry and that they remain valid for isoparametric, position‐ (displacement‐) based finite elements in general.
Numerical resolution of the boundary integral equations for elastic scattering by a plane crack
Authors: Alves Carlos , Duong Tuong Ha
Published: August 2005

The problem of wave scattering by a plane crack is solved, either in the case of acoustic waves or in the case of elastic waves incidence using the boundary integral equation method. A collocation method is often used to solve that equation, but here we will use a variational method, first writing the problem of Fourier variables, and then writing the associated integrals in the sesquilinear form with weak singularity kernels. This representation is used in the numerical approach, made with a finite element method in the surface of the crack. Numerical tests were made with circular and elliptical cracks, but this method can be extended to other shapes, with the same convergence profiles. Extensive results are given concerning the crack opening displacement, the scattering cross‐section, the back‐scattered amplitude and far‐field patterns.
Remarks on the Gauss quadrature rule for a particular class of finite‐part integrals
Authors: Ioakimidis Nikolaos I.
Published: August 2005

The method of finite‐part (Hadamard‐type or hypersingular) integrals is already a standard approach in applied numerical methods in engineering and, mainly, in acoustics, fluid dynamics, elasticity and fracture mechanics. Here we study the convergence of the non‐classical (possessing a negative node, outside the integration interval, as well as a negative weight corresponding to this node) Gauss‐Legendre quadrature rule for finite‐part integrals on the interval [0,1] with the weight function 1/y and we prove its validity for integrands possessing a continuous first derivative. The case of integrands possessing a higher‐order derivative is also considered and the rate of convergence is established. These results are based on the derivation of a bound for the negative node in the above quadrature rule and, further, on the theory of convergence of the classical Gauss quadrature rule. Numerical results, verifying the results of this paper, are also displayed. The present results can be generalized to the Gauss‐Laguerre similarly non‐classical quadrature rule for finite‐part integrals (on the interval [0], ∞)) as well as to the two special, but important in engineering applications, non‐classical Gaussian quadrature rules for Cauchy‐type principal‐value integrals and Mangler‐type finite‐part integrals have been proposed by Tsamasphyros and Dimou.
A new 12 DOF quadrilateral element for analysis of thick and thin plates
Authors: Jirousek J. , Wróblewski A. , Szybiński B.
Published: August 2005

This paper presents a simple quadrilateral 12 DOF plate bending element based on a modified version of the hybrid‐Trefftz approach. This element makes use of two independent fields of generalized displacements:

a non‐conforming field (11 Trefftz terms for transverse displacement w and the corresponding rotations Θx, Θy) satisfying the governing differential equations of Reissner‐Mindlin theory;

an auxiliary conforming field with displacements w̃ linked to rotations \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \Theta _x,\tilde \Theta _y$\end{document}, by the requirement of constant boundary distribution of the corresponding tangential component S̃t, of the transverse shear. This allows quadratic w̃ and linear \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \Theta _x,\tilde \Theta _y $\end{document}, at the element boundary to be obtained with only 3 DOF at the corner nodes.

The resulting element, denoted by Q̃21–11, is robust and free of shear locking in the thin limit. The numerical assessment involves comparison with several recently presented 12 DOF thick plate quadrilaterals as well as with the standard 16 DOF hybrid‐Trefftz quadrilateral, Q21‐15S, with 15 Trefftz terms and independently interpolated w̃ and \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \Theta _x,\tilde \Theta _y $\end{document}.
An algorithm to generate quadrilateral or triangular element surface meshes in arbitrary domains with applications to crack propagation
Authors: Potyondy David O. , Wawrzynek Paul A. , Ingraffea Anthony R.
Published: August 2005

A new hybrid algorithm for automatically generating either an all‐quadrilateral or an all‐triangular element mesh within an arbitrarily shaped domain is described. The input consists of one or more closed loops of straight‐line segments that bound the domain. Internal mesh density is inferred from the boundary density using a recursive spatial decomposition (quadtree) procedure. All‐triangular element meshes are generated using a boundary contraction procedure. All‐quadrilateral element meshes are generated by modifying the boundary contraction procedure to produce a mixed element mesh at half the density of the final mesh and then applying a polygon‐splitting procedure. The final meshes exhibit good transitioning properties and are compatible with the given boundary segments which are not altered. The algorithm can support discrete crack growth simulation wherein each step of crack growth results in an arbitrarily shaped region of elements deleted about each crack tip. The algorithm is described and examples of the generated meshes are provided for a representative selection of cracked and uncracked structures.
A uniform deformation gradient hexahedron element with artificial hourglass control
Authors: Bonet J. , Bhargava P.
Published: August 2005

An 8‐noded linear hexahedron element for large strain hyperelastic analysis is presented in this paper. The element is based on a constant deformation gradient interpolation and is formulated using a mixed variational principle of the Hu–Washizu type. Volumetric and isochoric components of the deformation are treated independently to ensure the correct evaluation of the element volume. A simple procedure to control the propagation of spurious hourglass deformation modes is also discussed. This is based on the addition of artificial hourglass forces which vanish under constant deformation gradient conditions, thereby ensuring that the element passes a non‐linear version of the patch test. Applications in the field of neo‐Hookean materials and superplastic forming processes are also considered.
On augmented Lagrangian algorithms for thermomechanical contact problems with friction
Authors: Zavarise G. , Wriggers P. , Schrefler B. A.
Published: August 2005

The detailed discretization of contact zones with contact stiffness based on real physical characteristics of contact surfaces can produce stiffness terms which induce ill‐conditioning of the global stiffness matrix. Moreover the consistent treatment of frictional behaviour generates non‐symmetric tangent stiffness matrices due to the non‐associativity of the slip phase. Other non‐symmetries are due to the coupling terms and to the dependencies on various parameters that can be involved. To overcome these difficulties almost consistent techniques based on two‐step algorithms have been proposed in the past. Here an augmentation technique is proposed which takes into account micro‐mechanical effects, and permits the symmetrization of the tangent stiffness during frictional slip phase.
Shape optimal design of elastic planar frames with non‐linear response
Authors: Kegl M. , Butinar B. J. , Oblak M. M.
Published: August 2005

This paper describes an approach to shape optimal design of elastic planar frames with non‐linear response. The foundation of the proposed approach forms an appropriate strategy of shape representation of the structure, based on the design element technique. A frame structure is treated as to be assembled from several frame design elements, which in turn may consist of several appropriately joined beam finite elements. The shape of each frame design element is defined by convenient functions involving Bezier blending polynomials. The original formulation of the beam finite element, proposed by Saje, is modified in order to fit nicely into the context of the frame design element technique. The formulation of the shape optimal design problem in a form of a problem of non‐linear mathematical programming and its solution by employing gradient‐based methods of mathematical programming are discussed briefly. The theory is illustrated with two numerical examples.
The effect of grid irregularity on truncation error for discretizations of Laplaces's equation
Authors: Montgomery Matthew , Fleeter Sanford
Published: August 2005

The discretization of differential equations to obtain numerical solutions introduces truncation error (TE). The form of the TE depends on the differential equation, the discretization scheme, and the discretization grid in a fairly complicated fashion. The character and magnitude of the TE determines in part the discrepancy between the approximate and exact solutions to the PDE. In this study, a symbolic manipulator was used to investigate the geometrical dependence of the TE in discrete approximations to Laplace's equation. The effect of various grid irregularities on the leading TE terms was investigated for two discretization schemes. A nine point centered stencil was varied in aspect ratio, skewness, and non‐uniformity (irregular spacing). The first scheme was the common centered finite difference (FD) method, applied by transforming to a uniform, orthogonal computational space. The second scheme was the implicit interpolation method (II), developed for discretization on irregular grids. Both discretization methods had leading TE that was second order (inversely proportional to the number of grid points squared) and had a sharp rise in TE for skewness angles beyond sixty degrees. The FD method skewness errors were greater than those for the II method, but high aspect ratio errors were less. For non‐linear transformations the FD method TE contained second derivatives, the same order as the governing differential equation, while the II method's lowest order TE derivatives were third order. Thus while both methods have the same order of accuracy, the nature of the numerical error in the discrete solution would be different.
A numerical green's function approach for boundary elements applied to fracture mechanics
Authors: Telles J. C. F. , Castor G. S. , Guimarães S.
Published: August 2005

The most accurate boundary element formulation to deal with fracture mechanics problems is obtained with the implementation of the associated Green's function acting as the fundamental solution. Consequently, the range of applications of this formulation is dependent on the availability of the appropriate Green's function for actual crack geometry. Analytical Green's functions have been presented for a few single crack configurations in 2‐D applications and require complex variable theory. This work extends the applicability of the formulation through the introduction of efficient numerical means of computing the Green's function components for single or multiple crack problems, of general geometry, including the implementation to 3‐D problems as a future development. Also, the approach uses real variables only and well‐established boundary integral equations.
Coarse/fine mesh preconditioners for the iterative solution of finite element problems
Authors: Dracopoulos M. C. , Crisfield M. A.
Published: August 2005

A class of preconditioners built around a coarse/fine mesh framework is presented. The proposed method involves the reconstruction of the stiffness equations using a coarse/fine mesh idealization with relative degrees‐of‐freedom derived from the element shape functions. This approach leads naturally to effective preconditioners for iterative solvers which only require a factorization involving coarse mesh variables. A further extension is the application of the proposed method to super‐elements in conjunction with substructuring (domain decomposition) techniques. The derivation of the coarse/fine mesh discretization via the use of transformation matrices, allows a straightforward implementation of the proposed techniques (as well as multigrid type procedures) within an existing finite element system.
An incremental formulation for the linear analysis of thin viscoelastic structures using generalized variables
Authors: Ghazlan G. , Caperaa S. , Petit C.
Published: August 2005

This paper is concerned with the development of an incremental formulation in the time domain for the displacement and stress analysis of quasistatic, linear, thin viscoelastic structures undergoing mechanical deformation. By representing the viscoelastic behaviour of a material by a discrete creep spectrum and by an incremental constitutive equations, the difficulty of retaining the stress history in computer solutions is avoided. A complete general formulation of linear viscoelastic stress analysis is developed in terms of increments of midsurface strains and curvatures and corresponding stress resultants.
Hexahedral mesh generation by medial surface subdivision: Part I. Solids with convex edges
Authors: Price M. A. , Armstrong C. G. , Sabin M. A.
Published: August 2005

A method is presented for subdividing a large class of solid objects into topologically simple subregions suitable for automatic finite element meshing with hexahedral elements. The technique uses a geometric property of a solid, its medial surface, to define the necessary subregions. The subregions are defined explicitly to be one of only 13 possible types. The subdividing cuts are between parts of the object in geometric proximity and produce good quality meshes of hexahedral elements. The method as introduced here is applicable to solids with convex edges and vertices, but the extension to complete generality is feasible.
An augmented Lagrangian quasi‐Newton solver for constrained nonlinear finite element applications
Authors: Laursen T. A. , Maker B. N.
Published: August 2005

A solution scheme is presented for constrained non‐linear equations of evolution that result, for example, from the finite element discretization of mechanical contact problems. The algorithm discussed utilizes a quasi‐Newton non‐linear equation solving strategy, with constraints enforced by an augmented Lagrangian iteration procedure. Through presentation of a simple model problem and its generalization, it is shown that the iterations associated with both the quasi‐Newton algorithm and the augmentation procedure can be interwoven to produce a highly efficient and robust solution strategy.
Modelling and simulation of high‐speed machining
Authors: Marusich T. D. , Ortiz M.
Published: August 2005

A Lagrangian finite element model of orthogonal high‐speed machining is developed. Continuous remeshing and adaptive meshing are the principal tools which we employ for sidestepping the difficulties associated with deformation‐induced element distortion, and for resolving fine‐scale features in the solution. The model accounts for dynamic effects, heat conduction, mesh‐on‐mesh contact with friction, and full thermo‐mechanical coupling. In addition, a fracture model has been implemented which allows for arbitrary crack initiation and propagation in the regime of shear localized chips. The model correctly exhibits the observed transition from continuous to segmented chips with increasing tool speed.