Symmetric Galerkin boundary formulations employing curved elements

Accounts of the symmetric Galerkin approach to boundary element analysis (BEA) have recently been published. This paper attempts to add to the understanding of this method by addressing a series of fundamental issues associated with its potential computational efficiency. A new symmetric Galerkin theoretical formulation for both the (harmonic) heat conduction and the (biharmonic) elasticity problem that employs regularized singular and hypersingular boundary integral equations (BIEs) is presented. The novel use of regularized BIEs in the Galerkin context is shown to allow straightforward incorporation of curved, isoparametric elements. A symmetric reusable intrinsic sample point (RISP) numerical integration algorithm is shown to produce a Galerkin (i.e. double) integration strategy that is competitive with its counterpart (i.e. singular) integration procedure in the collocation BEA approach when the time saved in the symmetric equation solution phase is also taken into account. This new formulation is shown to be capable of employing hypersingular BIEs while obviating the requirement of C¹ continuity, a fact that allows the employment of the popular continuous element technology. The behaviour of the symmetric Galerkin BEA method with regard to both direct and iterative equation solution operations is also addressed. A series of example problems are presented to quantify the performance of this symmetric approach, relative to the more conventional unsymmetric BEA, in terms of both accuracy and efficiency. It is concluded that appropriate implementations of the symmetric Galerkin approach to BEA indeed have the potential to be competitive with, if not superior to, collocation-based BEA, for large-scale problems.     

Preconditioned Krylov solvers for BEA

The performance of a number of preconditioned Krylov methods is analysed for a large variety of boundary element formulations. Low- and high-order element, two-dimensional (2-D) and three-dimensional 3-D, regular, singular and hypersingular, collocation and symmetric Galerkin, single- and multi-zone, thermal and elastic, continuous and discontinuous boundary formulations with and without condensation are considered. Preconditioned Conjugate Gradient (CG) solvers in standard form and a form effectively operating on the normal equations (CGN), Generalized Minimal Residual (GMRES), Conjugate Gradient Squared (CGS) and Stabilized Bi-conjugate Gradient (Bi-CGSTAB) Krylov solvers are employed in this study. Both the primitive and preconditioned matrix operators are depicted graphically to illustrate the relative amenability of the alternative formulations to solution via Kryiov methods, and to contrast and explain their computational performances. A notable difference between 2-D and 3-D BEA operators is readily visualized in this manner. Numerical examples are presented and the relative conditioning of the various discrete BEA operators is reflected in the performance of the Krylov equation solvers. A preconditioning scheme which was found to be uncompetitive in the collocation BEA context is shown to make iterative solution of symmetric Galerkin BEA problems more economical than employing direct solution techniques. We conclude that the preconditioned Krylov techniques are competitive with or superior to direct methods in a wide range of boundary formulated problems, and that their performance can be partially correlated with certain problem characteristics.     

Iterative solution techniques in boundary element analysis

Iterative techniques for the solution of the algebraic equations associated with the direct boundary element analysis (BEA) method are discussed. Continuum structural response analysis problems are considered, employing single- and multi-zone boundary element models with and without zone condensation. The impact on convergence rate and computer resource requirements associated with the sparse and blocked matrices, resulting in multi-zone BEA, is studied. Both conjugate gradient and generalized minimum residual preconditioned iterative solvers are applied for these problems and the performance of these algorithms is reported. Included is a quantification of the impact of the preconditioning utilized to render the boundary element matrices solvable by the respective iterative methods in a time competitive with direct methods. To characterize the potential of these iterative techniques, we discuss accuracy, storage and timing statistics in comparison with analogous information from direct, sparse blocked matrix factorization procedures. Matrix populations that experience block fill-in during the direct decomposition process are included. With different degrees of preconditioning, iterative equation solving is shown to be competitive with direct methods for the problems considered.     

Nonlinear thermal analysis with a boundary element zone condensation technique

This paper presents a substantially more economical technique for the boundary element analysis (BEA) of a large class of nonlinear heat transfer problems including those with temperature dependent conductivity, temperature dependent convection coefficients, and radiation boundary conditions. The technique involves an exact static condensation of boundary element zones in a multi-zone boundary element model. The condensed boundary element zone contributions to be overall sparse blocked boundary element system matrices are formed once in the first step of the iterative nonlinear solution process and subsequently reused as the nonlinear parts of the overall problem are evolved to a convergent solution. Through a series of example problems it is demonstrated that the zone condensation technique facilitates the use of highly convergent iterative strategies for the solution of the nonlinear heat transfer problem involving modification and subsequent factorization of the overall boundary element system left had side matrix. For heat transfer problems with localized nonlinear effects, the condensation technique is shown to allow for the solution of nonlinear problems in less than half the CPU time required by methods that do not employ condensation.     

Symmetric coupling of multi‐zone curved Galerkin boundary elements with finite elements in elasticity

The coupling of Finite Element Method (FEM) with a Boundary Element Method (BEM) is a desirable result that exploits the advantages of each. This paper examines the efficient symmetric coupling of a Symmetric Galerkin Multi‐zone Curved Boundary Element Analysis method with a Finite Element Method for 2‐D elastic problems. Existing collocation based multi‐zone boundary element methods are not symmetric. Thus, when they are coupled with FEM, it is very difficult to achieve symmetry, increasing the computational work to solve the problem. This paper uses a fully Symmetric curved Multi‐zone Galerkin Boundary Element Approach that is coupled to an FEM in a completely symmetric fashion. The symmetry is achieved by symmetrically converting the boundary zones into equivalent ‘macro finite elements’, that are symmetric, so that symmetry in the coupling is retained. This computationally efficient and fast approach can be used to solve a wide range of problems, although only 2‐D elastic problems are shown. Three elasticity problems, including one from the FEM‐BEM literature that explore the efficacy of the approach are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

A fully symmetric multi‐zone Galerkin boundary element method

This paper examines the efficient integration of a Symmetric Galerkin Boundary Element Analysis (SGBEA) method with multi‐zone resulting in a fully symmetric Galerkin multi‐zone formulation. In a previous approach, a Galerkin multi‐zone method was developed where the interfacial nodes are assigned degrees of freedom globally so that the displacement and traction continuity across the zonal interfaces are addressed directly. However, the method was only block symmetric. In the present paper, two new approaches are derived. In the first approach, the degrees of freedom for a particular zone are assigned locally, independent of the other zones. The usual linear set of equations, from the symmetric Galerkin approach, are augmented with an additional set of equations generated by the Galerkin form of hypersingular boundary integrals along the interfaces. Zonal continuity is imposed externally through Lagrange's constraints. This approach is also only block symmetric. The second approach derived from the first, uses the continuity constraints at the zonal assembly level to achieve full symmetry. These methods are compared to collocation multi‐zone and an earlier formulation, on two elasticity problems from the literature. It was found that the second method is much faster than the collocation method for medium to large scale problems, primarily due to its complete symmetry. It is also observed that these methods spend marginally more time on integration than the previous Galerkin multi‐zone method but are better suited to parallel processing. Copyright © 1999 John Wiley & Sons, Ltd.     

Symmetric collocation BEM/FEM coupling procedure for 2-D dynamic structural–acoustic interaction problems

 This paper presents a symmetric collocation BEM (SCBEM)/FEM coupling procedure applicable to 2-D time domain structural–acoustic interaction problems. The use of symmetry for BEM not only saves memory storage but also enables the employment of efficient symmetric equation solvers, especially for BEM/FEM coupling procedure. Compared with symmetric Galerkin BEM (SGBEM) where double boundary integration should be carried out, SCBEM can reduce significantly the computing cost. Two numerical examples are included to illustrate the effectiveness and accuracy of the proposed method. Received: 2 November 2001 / Accepted: 27 May 2002     

Boundary element based formulations for crack shape sensitivity analysis

The present paper addresses several BIE-based or BIE-oriented formulations for sensitivity analysis of integral functionals with respect to the geometrical shape of a crack. Functionals defined in terms of integrals over the external boundary of a cracked body and involving the solution of a frequency-domain boundary-value elastodynamic problem are considered, but the ideas presented in this paper are applicable, with the appropriate modifications, to other kinds of linear field equations as well. Both direct differentiation and adjoint problem techniques are addressed, with recourse to either collocation or symmetric Galerkin BIE formulations. After a review of some basic concepts about shape sensitivity and material differentiation, the derivative integral equations for the elastodynamic crack problem are discussed in connection with both collocation and symmetric Galerkin BIE formulations. Building upon these results, the direct differentiation and the adjoint solution approaches are then developed. In particular, the adjoint solution approach is presented in three different forms compatible with boundary element method (BEM) analysis of crack problems, based on the discretized collocation BEM equations, the symmetric Galerkin BEM equations and the direct and adjoint stress intensity factors, respectively. The paper closes with a few comments.     

A boundary integral Trefftz formulation with symmetric collocation

The Galerkin and collocation methods are combined in the implementation of a boundary integral formulation based on the Trefftz method for linear elastostatics. A finite element approach is used in the derivation of the formulation. The domain is subdivided in regions or elements, which need not be bounded, simply connected or convex. The stress field is directly approximated in each element using a complete solution set of the governing Beltrami condition. This stress basis is used to enforce on average, in the Galerkin sense, the compatibility and elasticity conditions. The boundary of each element is, in turn, subdivided into boundary elements whereon the displacements are independently approximated using Dirac functions. This basis is used to enforce by collocation the static admissibility conditions, which reduce to the Neumann conditions as the stress approximation satisfies locally the domain equilibrium condition. The resulting solving system is symmetric and sparse. The coefficients of the structural matrices and vectors are defined either by regular boundary integral expressions or determined by direct collocation of the trial functions.     

Analytical integrations in 2D BEM elasticity

In the context of two‐dimensional linear elasticity, this paper presents the closed form of the integrals that arise from both the standard (collocation) boundary element method and the symmetric Galerkin boundary element method. Adopting polynomial shape functions of arbitrary degree on straight elements, finite part of Hadamard, Cauchy principal values and Lebesgue integrals are computed analytically, working in a local coordinate system. For the symmetric Galerkin boundary element method, a study on the singularity of the external integral is conducted and the outer weakly singular integral is analytically performed. Numerical tests are presented as a validation of the obtained results. Copyright © 2001 John Wiley & Sons, Ltd.     

Essential boundary condition enforcement in meshless methods: boundary flux collocation method

Element‐free Galerkin (EFG) methods are based on a moving least‐squares (MLS) approximation, which has the property that shape functions do not satisfy the Kronecker delta function at nodal locations, and for this reason imposition of essential boundary conditions is difficult. In this paper, the relationship between corrected collocation and Lagrange multiplier method is revealed, and a new strategy that is accurate and very simple for enforcement of essential boundary conditions is presented. The accuracy and implementation of this new technique is illustrated for one‐dimensional elasticity and two‐dimensional potential field problems. Copyright © 2001 John Wiley & Sons, Ltd.     

An alternative collocation boundary element method for static and dynamic problems

A collocation boundary element formulation is presented which is based on a mixed approximation formulation similar to the Galerkin boundary element method presented by Steinbach (SIAM J Numer Anal 38:401–413, 2000) for the solution of Laplace’s equation. The method is also applicable to vector problems such as elasticity. Moreover, dynamic problems of acoustics and elastodynamics are included. The resulting system matrices have an ordered structure and small condition numbers in comparison to the standard collocation approach. Moreover, the employment of Robin boundary conditions is easily included in this formulation. Details on the numerical integration of the occurring regular and singular integrals and on the solution of the arising systems of equations are given. Numerical experiments have been carried out for different reference problems. In these experiments, the presented approach is compared to the common nodal collocation method with respect to accuracy, condition numbers, and stability in the dynamic case.     

Trefftz boundary elements—multi‐region formulations

This paper is concerned with an effective numerical implementation of the Trefftz boundary element method, for the analysis of two‐dimensional potential problems, defined in arbitrarily shaped domains. The domain is first discretized into multiple subdomains or regions. Each region is treated as a single domain, either finite or infinite, for which a complete set of solutions of the problem is known in the form of an expansion with unknown coefficients. Through the use of weighted residuals, this solution expansion is then forced to satisfy the boundary conditions of the actual domain of the problem, leading thus to a system of equations, from which the unknowns can be readily determined. When this basic procedure is adopted, in the analysis of multiple‐region problems, proper boundary integral equations must be used, along common region interfaces, in order to couple to each other the unknowns of the solution expansions relative to the neighbouring regions. These boundary integrals are obtained from weighted residuals of the coupling conditions which allow the implementation of any order of continuity of the potential field, across the interface boundary, between neighbouring regions. The technique used in the formulation of the region‐coupling conditions drives the performance of the Trefftz boundary element method. While both of the collocation and Galerkin techniques do not generate new unknowns in the problem, the technique of Galerkin presents an additional and unique feature: the size of the matrix of the final algebraic system of equations which is always square and symmetric, does not depend on the number of boundary elements used in the discretization of both the actual and region‐interface boundaries. This feature which is not shared by other numerical methods, allows the Galerkin technique of the Trefftz boundary element method to be effectively applied to problems with multiple regions, as a simple, economic and accurate solution technique. A very difficult example is analysed with this procedure. The accuracy and efficiency of the implementations described herein make the Trefftz boundary element method ideal for the study of potential problems in general arbitrarily‐shaped two‐dimensional domains. Copyright © 1999 John Wiley & Sons, Ltd.     

Symmetric Galerkin boundary element method for quasi-brittle-fracture and frictional contact problems

The analysis of elastic quasi-brittle structures containing cohesive cracks and contacts with friction is given a unitary formulation in the framework of incremental plasticity. Integral equations for displacements and tractions are enforced by a weighted-residual Galerkin approach so that symmetry is preserved in the key operators (in contrast to collocation BE approaches) and cracks (either internal or edge cracks) can be dealt with by a single-domain BE formulation. The space-discrete problem in rates is expressed as a linear complementarity problem centered on a symmetric matrix or, equivalently, as a quadratic programming problem in variables pertaining to the displacement discontinuity locus only. Criteria for overall instabilities and bifurcations are derived from this formulation. The BE approach proposed and implemented by a suitable time-stepping technique, is comparatively tested by numerical solutions of cohesive-crack propagation problems.     

A solution to linear elasticity using locally supported RBF collocation in a generalised finite-difference mode

This work presents a novel meshless numerical approach for the solution of linear elasticity problems, using locally supported RBF collocation. The Kansa (unsymmetric) RBF collocation method is used to form local collocation systems, which enforce the PDE governing and boundary operators. With the displacement values acting as the unknowns in the system, a sparse global system is formed. This global matrix is formed in a manner analogous to a finite difference method, with the displacement values at each internal node defined in terms of the displacements at other nodes within the local stencil.In contrast to traditional finite difference methods, here the RBF collocation assumes the role traditionally played by polynomial interpolants. The RBF collocation does itself satisfy the governing PDE operator at some collocation points, and therefore allows for a significantly more accurate reconstruction than is found from simple polynomial interpolants. In addition, the boundary operators (for applied displacement and applied surface traction) are enforced directly within the local RBF collocation systems, rather than being enforced at the global matrix. In contrast to traditional finite difference methods based on polynomial interpolation, the RBF collocation does not require a regular arrangement of nodes. Therefore, the proposed numerical method is directly applicable to unstructured datasets.     

Performance evaluation of algorithms for mildly nonlinear elliptic problems

A number of numerical methods for mildly nonlinear elliptic boundary value problems on general domains is presented. The discretization procedures considered are: a fourth-order FFT-type method, collocation using Hermite bicubic splines and Galerkin with linear triangular as well as quadratic quadrilateral isoparametric elements. The linearized collocation and Galerkin equations are solved by various direct methods available in the ELLPACK system. A comparative study of the above equation solvers is presented for different domain geometries and compilers. The evaluation of software for the general mildly nonlinear elliptic equations is performed over 36 instances from a population of 16 parametrized problems with ‘real world’ and ‘mathematical’ behaviour. The performance data suggests that collocation is an effective method for such general problems, while Galerkin with quadratic quadrilateral isoparametric elements is uniformly superior to the one with linear elements.     

A galerkin symmetric boundary-element method in elasticity: Formulation and implementation

Static discontinuities (i.e. distributions of forces along a line or a surface, implying a jump of tractions across it) and kinematic (displacement) discontinuities are considered simultaneously as sources acting on the unbounded elastic space Ω∞ along the boundary Γ of a homogeneous elastic body Ω embedded in Ω∞. The auxiliary elastic state thus generated in the body is associated with the actual elastic state by a Betti reciprocity equation. Using suitable discretizations of actual and fictitious boundary variables, a symmetric Galerkin formulation of the direct boundary element method is generated. The following topics are addressed: reciprocity relations among kernels with particular attention to the role of singularities; conditions to be satisfied by the boundary field modelling in order to achieve the symmetry of the coefficient matrix; variational properties of the solution. With reference to two-dimensional problems, a technique based on a complex-variable formalism is proposed to perform the double integrations involved in this approach. An implementation of this technique for elastic analysis is described assuming straight elements, with continuous linear displacements and piecewise-constant tractions; all the double integrations are carried out analytically. Comparisons, from the computational standpoint, with the traditional non-symmetric method based on collocation and single integration, demonstrate the effectiveness of the present approach.     

A symmetric Galerkin boundary element method for 3d linear poroelasticity

In this paper, a symmetric Galerkin boundary element formulation for 3D linear poroelasticity is presented. By means of the convolution quadrature method, the time domain problem is decoupled into a set of Laplace domain problems. Regularizing their kernel functions via integration by parts, it is possible to compute all operators for rather general discretizations, only requiring the evaluation of weakly singular integrals. At the end, some numerical results are presented and compared with a collocation BEM. Throughout these studies, the symmetric Galerkin BEM performs better than the collocation method, especially for not optimal discretizations parameters, i.e. a bad relation of mesh to time-step size. The most obvious advantages can be observed in the fluid flux results. However, these advantages are obtained at a higher numerical cost.     

A collocation finite element method for potential problems in irregular domains

A potentially powerful numerical method for solving certain boundary value problems is developed. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees-of-freedom per node are used. A subparametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representation of irregular boundaries. It is shown that by taking advantage of the boundary conditions, a minimum number of collocation points can be used. The method is particularly suitable for potential and mass transport problems where a C¹ continuous solution is required. In contrast to the Galerkin approach, it does not require the evaluation of basis function products and numerical integration, also the coefficient matrix contains only about half as many non-zero terms as the corresponding Galerkin coefficient matrix. This results in approximately a 90 per cent reduction in formulation and a 50 per cent reduction in solution operation, as compared with the Galerkin finite element method, for this type of problem. Examples show that the accuracy of the collocation solution is as good as or better than that of the Galerkin solution.     

Integrationsverfahren zur Galerkin-Randeiementmethode für dreidimensionale Elastizitätsprobleme

The mixed boundary value problem in three-dimensional linear elasticity is solved via a system of singular boundary integral equations. This procedure is an alternative to the finite element method and has the main advantage that expensive volume mesh generation is omitted and only a surface mesh is sufficient. The integral equations are discretized by the Galerkin-type boundary element method, which has essential advantages compared to the widely used collocation method. At present the Galerkin method is almost never used in engineering, because this method leads to an unacceptably high effort for the computation of singular double integrals if traditional integration methods are used. The main result of this paper is a new method for the computation of such singular double integrals. The integration procedure leads to simple regular integrand functions also in the case of curved boundary elements. This result simplifies the implementation of the Galerkin-type boundary element method and makes this method applicable in mechanical engineering. Furthermore, the integration of regular double integrals is explained. Numerical tests for model problems in linear elasticity are discussed. Quadrature and discretization errors are analyzed.