Symmetric Boundary Element Method Research Articles

Fast multipole acceleration of the MEG/EEG boundary element method

The accurate solution of the forward electrostatic problem is an essential first step before solving the inverse problem of magneto- and electroencephalography (MEG/EEG). The symmetric Galerkin boundary element method is accurate but cannot be used for very large problems because of its computational complexity and memory requirements. We describe a fast multipole-based acceleration for the symmetric boundary element method (BEM). It creates a hierarchical structure of the elements and approximates far interactions using spherical harmonics expansions. The accelerated method is shown to be as accurate as the direct method, yet for large problems it is both faster and more economical in terms of memory consumption.

Loading-carrying welds joints analysis using boundary element method

The behavior of loading-carrying welds joints has been studied extensively using boundary element method. The symmetric boundary element method for multiple cracks problem is derived using Betti’s reciprocal theorem. The analysis can be performed effectively in single domain. High order element is proposed to solve the double integrals. The stress intensity factors are calculated for weld root and weld toe and the critical value for class F2 and W are obtained. Formulation for the critical value is obtained for the engineering design. The results are compared with those obtained by dual boundary element method and derived from the code of practice. Finally, the theoretical fatigue life of such joints is estimated.

Analysis of non-loading carrying fillet welds joint using SGBEM

Fatigue failure is probably the most common type of failure in welded construction. It is usually initiates at a stress concentration area within the structure. The fatigue behavior of non-load carrying cruciform fillet welded joint has been studied extensively using boundary element method. The symmetric boundary element method for multiple cracks problem is derived using Betti's reciprocal theorem in auxiliary fictitious state. High order element is proposed to solve the double integrals. The analysis demonstrates that symmetrical Galerkin boundary element method (SGBEM) can be used effectively for analyzing non-load carrying fillet welded cruciform joints containing any number of surface and embedded cracks. The stress intensity factor and the magnification factors Mk are analyzed. General formulation for this kind of fatigue life estimation is derived for engineering design purpose. The fatigue life estimate results are consistent with the code of practice and those of other researchers. This can provide a good method for engineering design under fatigue loading conditions.

Crack propagation analysis with Galerkin boundary element method

AbstractThis paper describes the application of symmetric Galerkin boundary element methods (SGBEM) for the analysis of a 2D crack propagation problem. The sub‐domain SGBEM for crack problem is derived. The coefficient matrix is completely symmetric. Cohesive crack model is used to simulate crack propagation. The increment control method for crack propagation and the method for unknown crack propagation path have been derived for high‐order element. Two‐stage interpolation method called the ‘quasi‐higher order element method’ (QHOEM) is then proposed to solve the double integrals. In the initial stage, it uses higher order elements to interpolate the field variables, and for the numerical integration involved, it further uses interpolation functions to decompose the higher order elements into lower order elements so that the existing analytical integration can be applied. A finite rectangular plate containing a centre crack growth and four‐points bending beam problem have been analysed to check the accuracy of the proposed method. For actual application, a dam buttress with an edge crack has been analysed and the results are found to be in agreement with the other numerical and experimental results. Copyright © 2004 John Wiley & Sons, Ltd.

Efficient implementation of a generalized Cholesky factorization for Symmetric Galerkin Boundary Element Methods

In the present paper a natural extension of the Cholesky factorization, adapted to the peculiar matrix-structure of the solving system of Symmetric Galerkin Boundary Element Methods is presented. The implementation is described in some detail and preliminary timing figures are given, showing that the proposed approach is considerably faster than the Bunch-Kaufman factorization for non-definite matrices. Thread-based parallelism is exploited to show the scalability of the algorithm on moderately parallel shared memory multiprocessors.

A Symmetric Boundary Element Method/Finite Element Method Coupling Procedure for Two-Dimensional Elastodynamic Problems

In this paper, a symmetric collocation boundary element method (SCBEM)/finite element method (FEM) coupling procedure is given and applied to a two-dimensional elastodynamic problem. The use of symmetry for the boundary element method not only saves memory storage but also enables the employment of efficient symmetric equation solvers. This is especially important for BEM/FEM coupling procedure. Compared with the symmetric Galerkin boundary element method (SGBEM) where double-space integration should be carried out, SCBEM is easier and faster.

Symmetric BEM formulations for 2-D steady-state and transit potential problems

This paper presents a new concept for symmetric boundary element method (SBEM) applicable to 2-D steady-state and transit potential problems. Two kinds of SBEM formulations are derived. Symmetry is obtained simply through matrix manipulation, and no hypersingularity appears. Therefore, SBEM is much easier than the traditional symmetric Galerkin BEM. Compared with the traditional asymmetric BEM, the present SBEM can reduce the computational cost for time domain problems only. However, when applied to BEM/FEM coupling procedure, SBEM can reduce the computational cost for both steady-state and time domain problems. Three numerical examples are included to illustrate the effectiveness and accuracy of the present formulations.

A symmetric boundary element method for the Stokes problem in multiple connected domains

AbstractWe consider a symmetric Galerkin boundary element method for the Stokes problem with general boundary conditions including slip conditions. The boundary value problem is reformulated as Steklov–Poincaré boundary integral equation which is then solved by a standard approximation scheme. An essential tool in our approach is the invertibility of the single layer potential which requires the definition of appropriate factor spaces due to the topology of the domain. Here we describe a modified boundary element approach to solve Dirichlet boundary value problems in multiple connected domains. A suitable extension of the standard single layer potential leads to an operator which is elliptic on the original function space. Copyright © 2003 John Wiley & Sons, Ltd.

Domain decomposition in the symmetric boundary element analysis

Recent developments in the symmetric boundary element method (SBEM) have shown a clear superiority of this formulation over the collocation method. Its competitiveness has been tested in comparison to the finite element method (FEM) and is manifested in several engineering problems in which internal boundaries are present, i.e. those in which the body shows a jump in the physical characteristics of the material and in which an appropriate study of the response must be used. When we work in the ambit of the SBE formulation, the body is subdivided into macroelements characterized by some relations which link the interface boundary unknowns to the external actions. These relations, valid for each macroelement and characterized by symmetric matricial operators, are similar in type to those obtainable for the FEM. The assembly of the macroelements based on the equilibrium conditions, or on the compatibility conditions, or on both of these conditions leads to three analysis methods: displacement, force, and mixed-value methods, respectively. The use of the fundamental solutions involves the punctual satisfaction of the compatibility and of the equilibrium inside each macroelement and it causes a stringent elastic response close to the actual solution. Some examples make it possible to perform numerical checks in comparison with solutions obtainable in closed form. These checks show that the numerical solutions are floating ones when the macroelement geometry obtained by subdividing the body changes.

Symmetric boundary element method versus finite element method

The paper examines the effectiveness of the symmetric boundary element formulation when the continuum body is subdivided into large elements called macro-elements. The approach proposed combines a strong reduction of variables with an elastic solution close to the real response. Indeed, if the displacement method is used, this approach permits one to determine for every macro-element a relationship connecting the weighted traction vector defined on the sides of the interface boundary with the node displacement vector of the same boundary and with the external action vector. Such a strategy is very similar to that followed through the finite element method, but with the advantages of having the punctual satisfaction of the equilibrium and compatibility by using the fundamental solution, and of employing large elements whose discretization is performed on the boundary only. Some examples are shown using a general computing programme developed by the present authors.

Numerical treatment of acoustic problems with the hybrid boundary element method

The symmetric hybrid boundary element method in the frequency and time domain is introduced for the computation of acoustic radiation and scattering in closed and infinite domains. The hybrid stress boundary element method in a frequency domain formulation is based on the dynamical Hellinger–Reissner potential and leads to a Hermitian, frequency-dependent stiffness equation. As compared to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix are communicated. On the other hand, the hybrid displacement boundary element method for time domain starts out from Hamilton's principle formulated with the velocity potential. The field variables in both formulations are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions, generated by Dirac distributions, and generalized loads, that are time dependent in the transient case. The domain is modified such that small spheres centered at the nodes are subtracted. Then the property of the Dirac distribution, now acting outside the domain, cancels the remaining domain integral in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. In the time domain formulation, an analytical transformation is employed to transform the remaining domain integral into a boundary one. This approach results in a linear system of equations with a symmetric stiffness and mass matrix. Earlier 2D results are generalized in the present paper by a 3D implementation. Numerical results of transient pressure wave propagation in a closed domain are presented.

Additive Schwarz preconditioners for a fully-discrete and symmetric boundary element method

We discuss various additive Schwarz preconditioners for a fully-discrete and symmetric boundary element method when used to solve a Dirichlet problem in the plane. These preconditioners work in the same way as when they are used for the Galerkin boundary element method: the condition numbers of the preconditioned stiffness matrices grow at most logarithmically with the degree of freedom. Several numerical results are presented to support the theory.

Spectral properties of Galerkin boundary element matrices

The motivation for the present article is to review some key features of the Symmetric Boundary Element Method from the point of view of the algebraic properties of the matrices arising from the Galerkin discretization of the displacement- and traction-Somigliana identities. The focus is on showing which features of these linear pseudo-differential operators are preserved and which are lost due to discretization.

On a hybrid boundary element method

In this paper we study a symmetric boundary element method based on a hybrid discretization of the Steklov–Poincare operator well suited for a symmetric coupling of finite and boundary elements. The representation used involves only single and double layer potentials and does not require the discretization of the hypersingular integral operator as in the symmetric formulation. The stability of the hybrid Galerkin discretization is based on a BBL–like stability condition for the trial spaces. Numerical examples confirm the theoretical results.

Implementation of a symmetric boundary element method in transient heat conduction with semi-analytical integrations

A time-convolutive variational hypersingular integral formulation of transient heat conduction over a 2-D homogeneous domain is considered. The adopted discretization leads to a linear equation system, whose coefficient matrix is symmetric, and is generated by double integrations in space and time. Assuming polynomial shape functions for the boundary unknowns, a set of compact formulae for the analytical time integrations is established. The spatial integrations are performed numerically using very efficient formulae just recently proposed. The competitiveness from the computational point of view of the symmetric boundary integral equation approach proposed herein is investigated on the basis of an original computer implementation. Copyright © 1999 John Wiley & Sons, Ltd.

A symmetric boundary element method for contact problems with friction

A symmetric boundary element method for the static contact problem with Coulomb friction is presented. The method is based on the penalty approximation, a smooth approximation of the Coulomb law of friction and a symmetric discretization of the Steklov–Poincaré operator. The convergence of the proposed algorithm is proved and a numerical example is given.

A symmetric Galerkin multi-zone boundary element method for cohesive crack growth

In the present paper, a fully symmetric Galerkin multi-zone boundary element method (GMZBEM) is proposed for the cohesive crack growth. For interface problems, the system matrix of each subdomain formulated by GMZBEM is not completely symmetric. Based on regarding the coefficient matrix as the sum of a symmetric part and a non-symmetric one, this problem is solved by adopting a proper assembly. According to this approach, the cohesive crack growth is simulated by marching one of craze-tips step-by-step and by using a special iterative algorithm. During the fracture process, the crack is simulated by means of the cohesive crack model, and the materials in this paper follow linear-elastic and isotropic models. The iterative enforcement of the cohesive crack involves the solution from boundary element equations at each iteration, but these equations in each subdomain need be factored only once for the entire problem unless the interfaces are modified. Numerical results obtained by using the present method are compared with the results from previous methods, which shows that GMZBEM is accurate and effective.

Teaching boundary element methods in acoustics

The boundary element method (BEM) provides a powerful tool for the numerical solution of sound fields, because (1) it maps the domain equation on the boundary and thus reduces the problem dimension by one, (2) it provides a nonreflecting boundary on an arbitrary radiating surface, and (3) pressure, velocity, and intensity can be calculated at every field point from the boundary data. The author’s concept of teaching BEM in a graduate engineering course is to provide a link with subjects taught in undergraduate engineering mechanics such as rod vibrations, reciprocity principle, influence functions, and finite elements (FE). These subjects are generalized to the field equations of acoustics and elastodynamics. The associated integral equations are derived from the dynamic reciprocity principle. Their kernels are fundamental solutions which turn out to be influence functions of a full space. Boundary formulations of rods and beams recall well-known FE matrices. The fluid–structure interaction of interior and exterior acoustics is formulated by adopting the unsymmetric allocation approach of BEM and a symmetric hybrid BEM which is based on a multifield varational principle. Waves and vibrations in a flexible pipe and the sound radiation from a vibrating tire are treated as examples.

Fast solution techniques for the symmetric boundary element method in linear elasticity

We use the symmetric formulation of boundary integral equations to solve mixed boundary value problems based on a Galerkin discretization. This leads to a skew-symmetric and positive definite system of linear equations, which may be solved via iterative schemes. Toeplitz type preconditioners are used to bound the spectral condition number of the discrete system. Numerical examples confirm the theoretical results.

A NEW SYMMETRIC AND POSITIVE DEFINITE BOUNDARY ELEMENT FORMULATION FOR LATERAL VIBRATIONS OF PLATES

A new symmetric and positive definite boundary element method in the time domain is presented for the dynamic analysis of thin elastic plates. The governing equations of the problem are obtained from a variational principle in which a hybrid modified functional is employed. The functional is expressed in terms of the domain and boundary basic variables in plate bending, assumed to be independent of each other. In the discretized model the boundary variables are expressed by nodal values, whereas the internal displacement field is modelled by a superposition of static fundamental solutions. The equations of motion are deduced from the functional stationarity conditions and they constitute a linear system of ordinary differential equations expressed in terms of nodal displacements. The structural operators are frequency independent and preserve the symmetry and definiteness properties of the continuum. Moreover these operators are calculated by performing boundary integrations of regular kernels only. Some applications to free vibrations and self-excited vibrations under aerodynamic loads have been worked out to test the accuracy of the proposed method. The numerical results have been found in very good agreement with those obtained by using other solution techniques by which the accuracy of the present boundary element model is demonstrated.