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.     

A symmetric Galerkin multi-zone boundary element formulation

The recent development of the symmetric Galerkin approach to boundary element analysis (BEA) has been demonstrated to be superior to the collocation method for medium to large problems. This fact has been shown in both heat conduction and elasticity. Accounts of collocation multi-zone analysis techniques have also been prevalent in the literature, dealing with multiple boundary integral relations associated with portions of overall objects. This technique results in overall system matrices with a blocked, sparse, but unsymmetric character. It has been shown that multi-zone techniques can produce smaller solution times than a single zone analysis for large problems. These techniques are useful for multi-material problems as well. This paper presents an approach for combining the benefits of both techniques resulting in a Galerkin multi-zone method, that is overall unsymmetric but contains a significant amount of block symmetry. A condensation technique in the multi-zone solver is shown to exploit the symmetry of the Galerkin formulation as well as the blocked sparsity of the multi-zone technique. This method is compared to collocation multi-zone on two elasticity problems from the literature. It is concluded that an appropriate implementation of the symmetric Galerkin multi-zone BEA indeed has the potential to be superior to the collocation based multi-zone BEA, for medium to large-scale elasticity problems. © 1997 John Wiley & Sons, Ltd.     

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.     

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.     

Generic domain decomposition and iterative solvers for 3D BEM problems

In the past two decades, considerable improvements concerning integration algorithms and solvers involved in boundary‐element formulations have been obtained. First, a great deal of efficient techniques for evaluating singular and quasi‐singular boundary‐element integrals have been, definitely, established, and second, iterative Krylov solvers have proven to be advantageous when compared to direct ones also including non‐Hermitian matrices. The former fact has implied in CPU‐time reduction during the assembling of the system of equations and the latter fact in its faster solution. In this paper, a triangle‐polar‐co‐ordinate transformation and the Telles co‐ordinate transformation, applied in previous works independently for evaluating singular and quasi‐singular integrals, are combined to increase the efficiency of the integration algorithms, and so, to improve the performance of the matrix‐assembly routines. In addition, the Jacobi‐preconditioned biconjugate gradient (J‐BiCG) solver is used to develop a generic substructuring boundary‐element algorithm. In this way, it is not only the system solution accelerated but also the computer memory optimized. Discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions. Several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving. In the discussion of results, expressed in terms of accuracy and CPU time, advantages and potential applications of the BE code developed are highlighted. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical experiments of preconditioned Krylov subspace methods solving the dense non-symmetric systems arising from BEM

Discretization of boundary integral equations leads, in general, to fully populated non-symmetric linear systems of equations. An inherent drawback of boundary element method (BEM) is that, the non-symmetric dense linear systems must be solved. For large-scale problems, the direct methods require expensive computational cost and therefore the iterative methods are perhaps more preferable. This paper studies the comparative performances of preconditioned Krylov subspace solvers as bi-conjugate gradient (Bi-CG), generalized minimal residual (GMRES), conjugate gradient squared (CGS), quasi-minimal residual (QMR) and bi-conjugate gradient stabilized (Bi-CGStab) for the solution of dense non-symmetric systems. Several general preconditioners are also considered and assessed. The results of numerical experiments suggest that the preconditioned Krylov subspace methods are effective approaches solving the large-scale dense non-symmetric linear systems arising from BEM.     

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     

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.     

Preconditioned Krylov subspace methods for boundary element solution of the Helmholtz equation

Discretization of boundary integral equations leads, in general, to fully populated complex valued non-Hermitian systems of equations. In this paper we consider the efficient solution of these boundary element systems by preconditioned iterative methods of Krylov subspace type. We devise preconditioners based on the splitting of the boundary integral operators into smooth and non-smooth parts and show these to be extremely efficient. The methods are applied to the boundary element solution of the Burton and Miller formulation of the exterior Helmholtz problem which includes the derivative of the double layer Helmholtz potential—a hypersingular operator. © 1998 John Wiley & Sons, Ltd.     

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.     

Performance of a Petrov–Galerkin algebraic multilevel preconditioner for finite element modeling of the semiconductor device drift‐diffusion equations

This study compares the performance of a relatively new Petrov–Galerkin smoothed aggregation (PGSA) multilevel preconditioner with a nonsmoothed aggregation (NSA) multilevel preconditioner to accelerate the convergence of Krylov solvers on systems arising from a drift‐diffusion model for semiconductor devices. PGSA is designed for nonsymmetric linear systems, Ax=b, and has two main differences with smoothed aggregation. Damping parameters for smoothing interpolation basis functions are now calculated locally and restriction is no longer the transpose of interpolation but instead corresponds to applying the interpolation algorithm to A^T and then transposing the result. The drift‐diffusion system consists of a Poisson equation for the electrostatic potential and two convection–diffusion‐reaction‐type equations for the electron and hole concentration. This system is discretized in space with a stabilized finite element method and the discrete solution is obtained by using a fully coupled preconditioned Newton–Krylov solver. The results demonstrate that the PGSA preconditioner scales significantly better than the NSA preconditioner, and can reduce the solution time by more than a factor of two for a problem with 110 million unknowns on 4000 processors. The solution of a 1B unknown problem on 24 000 processor cores of a Cray XT3/4 machine was obtained using the PGSA preconditioner. Copyright © 2010 John Wiley & Sons, Ltd.     

Generalized boundary element method for galerkin boundary integrals

A meshless approach to the Boundary Element Method in which only a scattered set of points is used to approximate the solution is presented. Moving Least Square approximations are used to build a Partition of Unity on the boundary and then used to construct, at low cost, trial and test functions for Galerkin approximations. A particular case in which the Partition of Unity is described by linear boundary element meshes, as in the Generalized Finite Element Method, is then presented. This approximation technique is then applied to Galerkin boundary element formulations. Finally, some numerical accuracy and convergence solutions for potential problems are presented for the singular, hypersingular and symmetric approaches.     

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 comparison between a symmetric and a non-symmetric Galerkin finite element—boundary integral equation coupling for the two-dimensional exterior Stokes problem

Symmetric and non-symmetric Galerkin formulations are presented for the coupling of a finite element modelled interior region to a boundary integral supported exterior region for the two-dimensional steady state exterior Stokes problem. Both single and double-layer hydrodynamic potentials are used allowing a well conditioned symmetric matrix structure for the entire interior–exterior, velocity–pressure system when the exterior velocity boundary integral equation (VBIE) is augmented by a traction boundary integral equation (TBIE) with the pressure determined everywhere purely through the imposition of the divergence-free velocity condition. Corresponding non-symmetric formulations are obtained by additionally discretizing an associated pressure boundary integral equation (PBIE), where the associated kernel functions have singularities an order higher than in the VBIE, with a simple regularization of the new hyper-singular pressure kernel. Comparable solution convergence with mesh refinement for the symmetric and non-symmetric schemes is shown for stabilized and mixed velocity–pressure conforming finite element pairs using Lagrangian shape functions.     

The Steklov‐Poincaré technique for data completion: Preconditioning and filtering

This paper presents a study of primal and dual Steklov‐Poincaré approaches for the identification of unknown boundary conditions of elliptic problems. After giving elementary properties of the discretized operators, we investigate the numerical solution with Krylov solvers. Different preconditioning and acceleration strategies are evaluated. We show that costless filtering of the solution is possible by postprocessing Ritz elements. Assessments are provided on a 3D mechanical problem.     

On the removal of the non‐uniqueness in the solution of elastostatic problems by symmetric Galerkin BEM

A study of the removal of the non‐uniqueness in the solution of elastostatic problems by means of the symmetric Galerkin boundary element method is presented. The paper focuses on elastic problems defined on domains with cavities, where cavity boundaries are subjected to traction boundary conditions. A simple method consisting in a direct application of support conditions and several methods based on the Fredholm theory of linear operators are introduced, implemented and analysed. Numerical examples demonstrate the performance of the proposed methods and accuracy of their results, a comparative evaluation of the methods developed being finally presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations

We present new iterative solvers for large‐scale linear algebraic systems arising from the finite element discretization of the elasticity equations. We focus on the numerical solution of 3D elasticity problems discretized by quadratic tetrahedral finite elements and we show that second‐order accuracy can be obtained at very small overcost with respect to first‐order (linear) elements. Different Krylov subspace methods are tested on various meshes including elements with small aspect ratio. We first construct a hierarchical preconditioner for the displacement formulation specifically designed for quadratic discretizations. We then develop efficient tools for preconditioning the 2 × 2 block symmetric indefinite linear system arising from mixed (displacement‐pressure) formulations. Finally, we present some numerical results to illustrate the potential of the proposed methods. Copyright © 2010 John Wiley & Sons, Ltd.     

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.     

A monolithic geometric multigrid solver for fluid‐structure interactions in ALE formulation

We present a monolithic geometric multigrid solver for fluid‐structure interaction problems in Arbitrary Lagrangian Eulerian coordinates. The coupled dynamics of an incompressible fluid with nonlinear hyperelastic solids gives rise to very large and ill‐conditioned systems of algebraic equations. Direct solvers usually are out of question because of memory limitations, and standard coupled iterative solvers are seriously affected by the bad condition number of the system matrices. The use of partitioned preconditioners in Krylov subspace iterations is an option, but the convergence will be limited by the outer partitioning. Our proposed solver is based on a Newton linearization of the fully monolithic system of equations, discretized by a Galerkin finite element method. Approximation of the linearized systems is based on a monolithic generalized minimal residual method iteration, preconditioned by a geometric multigrid solver. The special character of fluid‐structure interactions is accounted for by a partitioned scheme within the multigrid smoother only. Here, fluid and solid field are segregated as Dirichlet–Neumann coupling. We demonstrate the efficiency of the multigrid iteration by analyzing 2d and 3d benchmark problems. While 2d problems are well manageable with available direct solvers, challenging 3d problems highly benefit from the resulting multigrid solver. Copyright © 2015 John Wiley & Sons, Ltd.