3D fracture analysis by the symmetric Galerkin BEM

 The subject of this paper is the formulation and the implementation of the symmetric Galerkin BEM for three-dimensional linear elastic fracture mechanics problems. A regularized version of the displacement and traction equations in weak form is adopted and the integration techniques utilized for the evaluation of the double surface integrals appearing in the discretized equations are detailed. By using quadratic isoparametric quadrilateral and triangular elements, some example crack problems are solved to assess the efficiency and robustness of the method. Received 6 November 2000     

Integration‐free Coons macroelements for the solution of 2D Poisson problems

Large isoparametric macroelements with closed‐form cardinal global shape functions under the label ‘Coons‐patch macroelements’ (CPM) have been previously proposed and used in conjunction with the finite element method and the boundary element method. This paper continues the research on the performance of CPM in conjunction with the collocation method. In contrast to the previous CPM that was based on a Galerkin/Ritz formulation, no domain integration is now required, a fact that justifies the name ‘integration‐free Coons macroelements’. Therefore, in addition to avoiding mesh generation, and saving human effort, the proposed technique has the additional advantage of further reducing the computer effort. The theory is supported by five test cases concerning Poisson and Laplace problems within 2D smooth quadrilateral domains. Copyright © 2008 John Wiley & Sons, Ltd.     

Non-linear analysis for anisotropic materials

An elasto-plastic analysis of anisotropic work-hardening materials, using the finite element method is presented. The analysis is based on the generalized Huber-Mises yield criterion extended by Hill for anisotropic materials. General expressions for the anisotropic parameters in the yield criterion have been derived both for initial yielding as well as subsequent yielding in the case of work-hardening materials. The isoparametric ‘quadratic’ quadrilateral elements have been used for the analysis and the ‘initial stress technique’ has been adopted for the iterative solution of the non-linear problems. The results of the various numerical examples have been compared with the available solutions.     

Error estimates for mixed finite element approximations of nonlinear problems

The finite element methods have proved a very effective tool for the numerical solutions of nonlinear problems arising in elasticity and other related engineering sciences. Relative to linear elliptic theory, little is known about the accuracy and convergence properties of mixed finite element approximation of nonlinear elliptic boundary value problems. The nonlinear problems are much more complicated, since each problem has to be treated individually. This is one of the reasons that there is no unified and general theory for the nonlinear problems. In this paper, the application of the mixed finite element method to a highly nonlinear Dirichlet problem, which arises in the field of oceanography and elasticity is studied and new results involving the error estimates are derived. In fact, some of the results and methods to be described in this paper may be extended to more complicated problems or problems with other boundary conditions. As a special case, we obtain the well known error estimates for the corresponding linear and mildly nonlinear elliptic boundary value problems.     

A new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method

The purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto-plastic solids, respectively, by the direct boundary element method (BEM). Triangle polar co-ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares. The method was subsequently extended to the cubature of singular integrals over three-dimensional internal cells as occur in applications of the BEM to three-dimensional elasto-plasticity. For this purpose so-called tetrahedron polar co-ordinates were introduced. Singular boundary integrals stretching over either linear, triangular, or quadratic quadilateral, isoparametric boundry elements and singular volume integrals extending over either linear, tetrahedral, or quadratic, hexahedral, isoparametric internal cells are treated. In case of higher order isoparametric boundary elements and internal cells, division into a number of subelements and subcells, respectively, is necessary. The analytical investigation is followed by a numerical study restricted to the use of quadratic, quadrilateral, isoparametric boundary elements. This is justified by the fact that such elements, as opposed to linear elements, yield singular boundary integrals which cannot be integrated analytically. The results of the numerical investigation demonstrate the potential of the developed concept.     

Oscillation limits for weighted residual methods applied to convective diffusion equations

Convection–diffusion equations are difficult to solve when the convection term dominates because most solution methods give solutions which oscillate in space. Previous criteria based on the one-dimensional convection–diffusion equation have shown that finite difference and Galerkin (linear or quadratic basis functions) will not give oscillatory solutions provided the Peclet number times the mesh size (Pe Δx) is below a critical value. These criteria are based on the solution at the nodes, and ensure that the nodal values are monotone. Similar criteria are developed here for other methods: quadratic Galerkin with upwind weighting, cubic Galerkin, orthogonal collocation on finite elements with quadratic, cubic or quartic polynomials using Lagrangian interpolation, cubic or quartic polynominals using Hermite interpolation, and the method of moments. The nodal values do not oscillate for collocation or moments methods with Hermite cubic polynomials regardless of the value of Pe Δx. A new criterion is developed for all methods based on the monotonicity of the solutions throughout the domain. This criterion is more restrictive than one based only on the nodal values. All methods that are second order (Δx²) or better in truncation error give oscillatory solutions (based on the entire domain) unless Pe Δx is below a critical value. This value ranges from 2 for finite difference methods to 4·6 for Hermite, quartic, collocation methods.     

A time-domain collocation-Galerkin BEM for transient dynamic crack analysis in anisotropic solids

This paper presents a time-domain boundary element method (BEM) for transient elastodynamic crack analysis in homogeneous and linear elastic solids of general anisotropy. A finite crack subjected to a transient loading is investigated. Two-dimensional (2D) generalized plane-strain or plane-stress condition is considered. The initial-boundary value problem is described by a set of hypersingular time-dependent traction boundary integral equations (BIEs), in which the crack-opening displacements (CODs) are unknown quantities. The hypersingular time-domain BIEs are first regularized to weakly singular ones by using spatial Galerkin method, which transfers the derivatives of the fundamental solutions to the unknown CODs and the weight functions. To solve the time-domain BIEs numerically, a time-stepping scheme is developed. The scheme applies the collocation method for temporal discretization of the time-domain BIEs. As spatial shape-functions, two different functions are implemented. For elements away from crack-tips, linear spatial shape-function is used, while for elements near the crack-tips a special ‘crack-tip shape-function’ is applied to describe the local ‘square-root’ behavior of the CODs at the crack-tips properly. Special attention of the analysis is devoted to the numerical computation of the transient elastodynamic stress intensity factors for cracks in general anisotropic and linear elastic solids. Numerical examples are presented to verify the accuracy of the present time-domain BEM.     

The collapsed cubic isoparametric element as a ingular element for crack probblems

For the 12?node quadrilateral isoparametric elements, it is shown that the inverse square root singularity of the strain field at the crack tip can be obtained by the simple technique of collapsing the quadrilateral elements into triangular elements around the crack tip and placing the two side nodes of each side of the triangles at 1/9 and 4/9 of the length of the side from the tip. This is analogous to placing the mid?side nodes at quarter points in the vicinity of the crack tip for the quadratic isoparametric elements. The advantages of this method are that the displacement compatibility is satisfied throughout the region and that there is no need of special crack tip elements. The stress intensity factors can be accurately obtained by using general purpose programs having isoparametric elements such as NASTRAN.     

A boundary element model for acoustic-elastic interaction with applications in ultrasonic NDE

The boundary integral equation method is applied to a class of time-harmonic acoustic scattering problems where the bounded elastic scatterer is submerged in a fluid. An exact mathematical model is presented for a finite scatterer with a closed surface, where the surface integral equations are exclusively used to represent the fluid-solid or acoustic-elastic interaction of the scattering process. The numerical procedure involves application of point collocation with quadratic isoparametric approximations that reduce the integral equations to a discrete set of complex linear algebraic equations. Examples emphasize the potential of the method to solve three-dimensional problems of practical interest. Limitations of the formulations and the extension to the case of a semi-infinite plane and curved fluid-solid interface are discussed in the latter part of the paper.     

Discontinuous Galerkin methods for non‐linear elasticity

This paper presents the formulation and a partial analysis of a class of discontinuous Galerkin methods for quasistatic non‐linear elasticity problems. These methods are endowed with several salient features. The equations that define the numerical scheme are the Euler–Lagrange equations of a one‐field variational principle, a trait that provides an elegant and simple derivation of the method. In consonance with general discontinuous Galerkin formulations, it is possible within this framework to choose different numerical fluxes. Numerical evidence suggests the absence of locking at near‐incompressible conditions in the finite deformations regime when piecewise linear elements are adopted. Finally, a conceivable surprising characteristic is that, as demonstrated with numerical examples, these methods provide a given accuracy level for a comparable, and often lower, computational cost than conforming formulations. Stabilization is occasionally needed for discontinuous Galerkin methods in linear elliptic problems. In this paper we propose a sufficient condition for the stability of each linearized non‐linear elastic problem that naturally includes material and geometric parameters; the latter needed to account for buckling. We then prove that when a similar condition is satisfied by the discrete problem, the method provides stable linearized deformed configurations upon the addition of a standard stabilization term. We conclude by discussing the complexity of the implementation, and propose a computationally efficient approach that avoids looping over both elements and element faces. Several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous Galerkin method within the class. Copyright © 2006 John Wiley & Sons, Ltd.     

Explicit expressions for 3D boundary integrals in potential theory

On employing isoparametric, piecewise linear shape functions over a flat triangular domain, exact expressions are derived for all surface potentials involved in the numerical solution of three‐dimensional singular and hyper‐singular boundary integral equations of potential theory. These formulae, which are valid for an arbitrary source point in space, are represented as analytic expressions over the edges of the integration triangle. They can be used to solve integral equations defined on polygonal boundaries via the collocation method or may be utilized as analytic expressions for the inner integrals in the Galerkin technique. In addition, the constant element approximation can be directly obtained with no extra effort. Sample problems solved by the collocation boundary element method for the Laplace equation are included to validate the proposed formulae. Published in 2008 by John Wiley & Sons, Ltd.     

The dual boundary element formulation for elastoplastic fracture mechanics

In this paper the extension of the dual boundary element method (DBEM) to the analysis of elastoplastic fracture mechanics (EPFM) problems is presented. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. In order to avoid collocation at crack tips, crack kinks and crack-edge corners, both crack surfaces are discretized with discontinuous quadratic boundary elements. The elastoplastic behaviour is modelled through the use of an approximation for the plastic component of the strain tensor on the region expected to yield. This region is discretized with internal quadratic, quadrilateral and/or triangular cells. This formulation was implemented for two-dimensional domains only, although there is no theoretical or numerical limitation to its application to three-dimensional ones. A centre-cracked plate and a slant edge-cracked plate subjected to tensile load are analysed and the results are compared with others available in the literature. J-type integrals are calculated.     

Galerkin lumped parameter methods for transient problems

In this paper we propose a general methodology to obtain lumped parameter models for systems governed by parabolic partial differential equations which we call Galerkin lumped parameter methods. The idea consists of decomposing the computational domain into several subdomains connected through so‐called ports. Then a time‐independent adapted reduced basis is introduced by numerically solving several elliptic problems in each subdomain. The proposed lumped parameter model is the Galerkin approximation of the original problem in the space spanned by this basis. The relationship of these methods with classical lumped parameter models is analyzed. Numerical results are shown as well as a comparison of the solution obtained with the lumped model and the ‘exact’ one computed by standard finite element procedures. Copyright © 2011 John Wiley & Sons, Ltd.     

Evaluation of ‘specia’ finite elements with non-conventional Gaussian quadratures

Finite elements with ‘Special’ basis functions have been proposed to develop better approximations for problems where the behaviour is known to be non-polynomial. This paper discusses extension of Gaussian quadrature integration procedures of non-conventional form to the evaluation of the finite element matrices for ‘Special’ elements. Algorithms for general ‘Special’ elements are discussed. The techniques are applied to ‘Special’ one- and two-dimensional elements for spherically symmetric potential flow. The accuracy of the new ‘Special’ element is shown to be superior to linear and quadratic elements for spherically symmetric potential flow problems.     

A class of data structures for 2-D and 3-D adaptive mesh refinement

A ‘family’ of tree data structures for adaptive mesh refinement is described and details concerning the associated logic are provided. The data structures encompass triangular elements and quadrilateral elements in two dimensions and quadrilateral bricks in three dimensions. Furthermore, both linear (bilinear) and quadratic (biquadratic) element types, respectively, are developed. Representative refinement results are given for the bilinear, trilinear and biquadratic types and associated performance studies made for the refinement procedure.     

On Using Collocation in Three Dimensions and Solving a Model Semiconductor Problem

A research code has been written to solve an elliptic system of coupled nonlinear partial differential equations of conservation form on a rectangularly shaped three-dimensional domain. The code uses the method of collocation of Gauss points with tricubic Hermite piecewise continuous polynomial basis functions. The system of equations is solved by iteration. The system of nonlinear equations is linearized, and the system of linear equations is solved by iterative methods. When the matrix of the collocation equations is duly modified by using a scaled block-limited partial pivoting procedure of Gauss elimination, it is found that the rate of convergence of the iterative method is significantly improved and that a solution becomes possible. The code is used to solve Poisson’s equation for a model semiconductor problem. The electric potential distribution is calculated in a metal-oxide-semiconductor structure that is important to the fabrication of electron devices.     

Two refined non‐conforming quadrilateral flat shell elements

Two refined quadrilateral flat shell elements named RSQ20 and RSQ24 are constructed in this paper based on the refined non‐conforming element method, and the elements can satisfy the displacement compatibility requirement at the interelement of the non‐planar elements by introducing the common displacements suggested by Chen and Cheung. A refined quadrilateral plate element RPQ4 and a plane quadrilateral isoparametric element are combined to obtain the refined quadrilateral flat shell element RSQ20, and a refined quadrilateral flat shell element RSQ24 is constructed on the basis of a RPQ4 element and a quadrilateral isoparametric element with drilling degrees of freedom. The numerical examples show that the present method can improve the accuracy of shell analysis and that the two new refined quadrilateral flat shell elements are efficient and accurate in the linear analysis of some shell structures. Copyright © 2000 John Wiley & Sons, Ltd.     

Modified crack closure integral using six-noded isoparametric quadrilateral singular elements

Six-noded, isoparametric serendipity type quadrilateral regular/singular elements are used for the estimation of stress intensity factors (SIF) in linear elastic fracture mechanics (LEFM) problems involving cracks in two-dimensional structural components. The square root singularity is achieved in the six-noded elements by moving the in-side nodes to the quarter point position. The modified crack closure integral (MCCI) method is adopted which could generate accurate estimates of SIF for a relatively coarse mesh. The equations for strain energy release rate and SIF are derived for mixed mode situations using six-noded quadrilateral elements at the crack tip. The model is validated by numerical studies for a centre crack in a finite plate under uniaxial tension, a single edge notched specimen under uniaxial tension, an inclined crack in a finite rectangular plate and cracks emanating from a pin-loaded lug (or lug attachment). The results compare very well with reference solutions available in the literature.     

四边形单元面积坐标理论

本文建立了四边形单元面积坐标的系统理论，包括：（１）给出四边形单元两个特征参数ｇ１，ｇ２的定义以及四边形退化为平行四边形（含矩形），梯形，三角形时相应的特征条件；（２）给出四边形单元面积坐标的定义及其与直角坐标和四边形等参坐标之间的变换关系；（３）给出四边形单元四个面积坐标分量之间应满足的两个恒等式并予以证明；（４）给出相关的一些重要公式。可以看出，四边形面积坐标是构造四边形单元的有效工具。它既是自然坐标，具有不变性；同时它与直角坐标之间为线性关系，易于得出单元刚度矩阵的积分显式，无需依赖于数值积分。     

Symmetric‐Galerkin BEM simulation of fracture with frictional contact

A symmetric‐Galerkin boundary element framework for fracture analysis with frictional contact (crack friction) on the crack surfaces is presented. The algorithm employs a continuous interpolation on the crack surface (utilizing quadratic boundary elements) and enables the determination of two important quantities for the problem, namely the local normal tractions and sliding displacements on the crack surfaces. An effective iterative scheme for solving this non‐linear boundary value problem is proposed. The results of test examples are compared with available analytical solutions or with those obtained from the displacement discontinuity method (DDM) using linear elements and internal collocation. The results demonstrate that the method works well for difficult kinked/junction crack problems. Copyright © 2003 John Wiley & Sons, Ltd.