Galerkin Boundary Element Method Research Articles

A general regularization of the hypersingular integrals in the symmetric Galerkin boundary element method

AbstractThe symmetric Galerkin boundary element method is used to solve boundary value problems by keeping the symmetric nature of the matrix obtained after discretization. The matrix elements are obtained from a double integral involving the double derivative of Green's operator, which is highly singular. The paper presents a regularization of the hypersingular integrals which depend only on the properties of Green's tensor. The method is presented in the case of Laplace's operator, with an example of application. The case of elasticity is finally addressed theoretically, showing an easy extension to any case of anisotropy. Copyright © 2009 John Wiley & Sons, Ltd.

Combined equivalent charge formulations and fast wavelet Galerkin BEM for 3‐D electrostatic analysis

AbstractWe describe a new equivalent charge formulation (ECF), i.e. the combined ECF (CECF), for electrostatic analysis of structures consisting of conductors and dielectrics. The CECF uses a weighted combination of the single‐ and the adjoint double‐layer operators to account for the potential on the conductor–dielectric surface and is found to have better conditioning than the standard ECF. A perturbation approach is presented to insure that the capacitances are computed accurately even when the permittivity ratios of the dielectrics are vary large. Unlike the original perturbation approach, this new approach uses only one system matrix with different right‐hand sides.A wavelet Galerkin boundary element method (WGBEM) for solving the ECF and CECF is developed based on the new variable‐order WGBEM introduced in (J. Numer. Math. 2004; 12(3):233–254). The wavelets are directly constructed on the usual boundary element triangulation. This enables the proposed WGBEM to solve electrostatic problems in complicated geometries, unstructured meshes and comparatively coarse discretizations. The quasi‐vanishing moment wavelets introduced in (Comput. Methods Appl. Mech. Engng 2008; 197:4000–4006) are used to further reduce the memory and CPU time requirements.Several numerical examples show that the proposed CECF converges faster than the ECF, and that the WGBEM, using both the ECF and CECF, has almost linear complexity in solving large‐scale 3‐D electrostatic problems. Moreover, since the truncated non‐standard form is computed once and then stored, the WGBEM is very suitable for solving problems with multiple right‐hand sides, like the perturbation approach in this paper. Copyright © 2009 John Wiley & Sons, Ltd.

Fast multipole boundary element method for electrostatic field computations

PurposeA wide range of micro‐electro‐mechanical‐systems are based on the electrostatic principle, and for their design the computation of the electric capacities is of great importance. The purpose of this paper is to efficiently compute the capacities as a function of all possible positions of the two electrode structures within the transducer by an enhanced boundary element method (BEM).Design/methodology/approachA Galerkin BEM is developed and the arising algebraic system of equations is efficiently solved by a CG‐method with a multilevel preconditioner and an appropriate fast multipole algorithm for the matrix‐vector operations within the CG‐iterations.FindingsIt can be demonstrated that the piecewise linear and discontinuous trial functions give an approximation, which is almost as good as the one of the piecewise constant trial functions on the refined mesh, at lower computational costs and at about the same memory requirements.Originality/valueThe paper can proof mathematically and demonstrate in practice, that a higher order of convergence is achieved by using piecewise linear, globally discontinuous basis functions instead of piecewise constant basis functions. In addition, an appropriate preconditioner (artificial multilevel boundary element preconditioner, which is based on the Bramble Pasciak Xu like preconditioner) has been developed for the fast iterative solution of the algebraic system of equations.

Coupling BEM with FEM by a direct computation of the boundary stiffness matrix

The “boundary stiffness matrix” characterizing in a discretized form the response of the boundary to given Dirichlet boundary conditions allows the coupling between BEM and FEM. The paper presents an approach based on the direct computation of the stiffness matrix from the potential function related to a given Dirichlet boundary condition. The method produces a symmetric stiffness matrix as for the Singular Galerkin boundary element method, but does not need to compute hypersingular integrals. In addition, the method uses only the nodal values of the boundary potential, but does not need discretized values of the normal gradient at the boundary, as for usual boundary element methods, which reduces the number of matrices to be constructed. The regularization of the integrals is achieved by means used for (simply) singular integrals introduced for the collocation boundary element method. An example of application is produced in the case of the solution of Laplace equation within a plane domain.

Efficient coupling of finite elements and boundary elements---adaptive procedures and preconditioners

Efficient coupling of finite elements and boundary elements---adaptive procedures and preconditioners

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.

유한요소 교호법을 이용한 삼차원 균열의 탄소성 J 적분 해석

SGBEM(Symmetric Galerkin Boundary Element Method)-FEM alternating method has been proposed by Nikishkov, Park and Atluri. In the proposed method, arbitrarily shaped three-dimensional crack problems can be solved by alternating between the crack solution in an infinite body and the finite element solution without a crack. In the previous study, the SGBEM-FEM alternating method was extended further in order to solve elastic-plastic crack problems and to obtain elastic-plastic stress fields. For the elastic-plastic analysis the algorithm developed by Nikishkov et al. is used after modification. In the algorithm, the initial stress method is used to obtain elasticplastic stress and strain fields. In this paper, elastic-plastic J integrals for three-dimensional cracks are obtained using the method. For that purpose, accurate values of displacement gradients and stresses are necessary on an integration path. In order to improve the accuracy of stress near crack surfaces, coordinate transformation and partitioning of integration domain are used. The coordinate transformation produces a transformation Jacobian, which cancels the singularity of the integrand. Using the developed program, simple three-dimensional crack problems are solved and elastic and elastic-plastic J integrals are obtained. The obtained J integrals are compared with the values obtained using a handbook solution. It is noted that J integrals obtained from the alternating method are close to the values from the handbook.

A convolution quadrature Galerkin boundary element method for the exterior Neumann problem of the wave equation

AbstractThe numerical solution of the Neumann problem of the wave equation on unbounded three‐dimensional domains is calculated using the convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The mathematical analysis that has been built up for the Dirichlet problem is extended and developed for the Neumann problem, which is important for many modelling applications. Numerical examples are then presented for one of these applications, modelling transient acoustic radiation. Copyright © 2009 John Wiley & Sons, Ltd.

Rapid Solution of the Wave Equation in Unbounded Domains

In this paper we propose and analyze a new, fast method for the numerical solution of time domain boundary integral formulations of the wave equation. We employ Lubich's convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The coefficient matrix of the arising system of linear equations is a triangular block Toeplitz matrix. Possible choices for solving the linear system arising from the above discretization include the use of fast Fourier transform (FFT) techniques and the use of data-sparse approximations. By using FFT techniques, the computational complexity can be reduced substantially while the storage cost remains unchanged and is, typically, high. Using data-sparse approximations, the gain is reversed; i.e., the computational cost is (approximately) unchanged while the storage cost is substantially reduced. The method proposed in this paper combines the advantages of these two approaches. First, the discrete convolution (related to the block Toeplitz system) is transformed into the (discrete) Fourier image, thereby arriving at a decoupled system of discretized Helmholtz equations with complex wave numbers. A fast data-sparse (e.g., fast multipole or panel-clustering) method can then be applied to the transformed system. Additionally, significant savings can be achieved if the boundary data are smooth and time-limited. In this case the right-hand sides of many of the Helmholtz problems are almost zero, and hence can be disregarded. Finally, the proposed method is inherently parallel. We analyze the stability and convergence of these methods, thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature. We also present numerical results which illustrate the predicted convergence behavior.

The boundary element method with Lagrangian multipliers

AbstractOn open surfaces, the energy space of hypersingular operators is a fractional order Sobolev space of order 1/2 with homogeneous Dirichlet boundary condition (along the boundary curve of the surface) in a weak sense. We introduce a boundary element Galerkin method where this boundary condition is incorporated via the use of a Lagrangian multiplier. We prove the quasi‐optimal convergence of this method (it is slightly inferior to the standard conforming method) and underline the theory by a numerical experiment.The approach presented in this article is not meant to be a competitive alternative to the conforming method but rather the basis for nonconforming techniques like the mortar method, to be developed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Boundary Element Computations in the Forward and Inverse Problems of Electrocardiography: Comparison of Collocation and Galerkin Weightings

In electrocardiographic imaging, epicardial potentials are reconstructed computationally from electrocardiographic measurements. The reconstruction is typically done with the help of the boundary element method (BEM), using the point collocation weighting and constant or linear basis functions. In this paper, we evaluated the performance of constant and linear point collocation and Galerkin BEMs in the epicardial potential problem. The integral equations and discretizations were formulated in terms of the single- and double-layer operators. All inner element integrals were calculated analytically. The computational methods were validated against analytical solutions in a simplified geometry. On the basis of the validation, no method was optimal in all testing scenarios. In the forward computation of the epicardial potential, the linear Galerkin (LG) method produced the smallest errors. The LG method also produced the smallest discretization error on the epicardial surface. In the inverse computation of epicardial potential, the electrode-specific transfer matrix performed better than the full transfer matrix. The Tikhonov 2 regularization outperformed the Tikhonov 0. In the optimal modeling conditions, the best BEM technique depended on electrode positions and chosen error measure. When large modeling errors such as omission of the lungs were present, the choice of the basis and weighting functions was not significant.

A comparison of FE–BE coupling schemes for large‐scale problems with fluid–structure interaction

AbstractTo predict the sound radiation of structures, both a structural problem and an acoustic problem have to be solved. In case of thin structures and dense fluids, a strong coupling scheme between the two problems is essential, since the feedback of the acoustic pressure onto the structure is not negligible. In this paper, the structural part is modeled with the finite element (FE) method. An interface to a commercial FE package is set up to import the structural matrices. The exterior acoustic problem is efficiently modeled with the Galerkin boundary element (BE) method. To overcome the well‐known drawback of fully populated system matrices, the fast multipole method is applied. Different coupling formulations are investigated. They are either based on the Burton–Miller approach or use a mortar coupling scheme. For all cases, iterative solvers with different preconditioners are used. The efficiency with respect to their memory consumption and computation time is compared for a simple model problem. At the end of the paper, a more complex structure is simulated. Copyright © 2008 John Wiley & Sons, Ltd.

On fast matrix–vector multiplication in wavelet Galerkin BEM

So far, the wavelet boundary element method (BEM) has been recognized as a different kind of fast BEM from the methods based on low-rank approximation, such as fast multipole method, panel clustering method and H 2 -matrices. We consider the matrix–vector multiplication in the wavelet Galerkin BEM [Tausch J. A variable order wavelet method for the sparse representation of layer potentials in the non-standard form. J Numer Math 2004;12(3):233–54]. We show that the system matrix of the wavelet Galerkin BEM can be transformed into a matrix with hierarchical structure by combining the forward and inverse wavelet transform in matrix–vector multiplications phase. The new system matrix only involves the data about the scaling functions. Any operations concerning the wavelet functions are thus avoided. A new version of matrix–vector multiplication scheme is proposed. We prove that the complexity of the new scheme never exceeds that of the old scheme. This work: (1) simplifies the implementation wavelet Galerkin BEM; (2) bridges a link between the wavelet BEMs and the methods of low-rank approximation.

Convolution quadrature method‐based symmetric Galerkin boundary element method for 3‐d elastodynamics

AbstractThe boundary integral equations in 3‐d elastodynamics contain convolution integrals with respect to the time. They can be performed analytically or with the convolution quadrature method. The latter time‐stepping procedure's benefit is the usage of the Laplace‐transformed fundamental solution. Therefore, it is possible to apply this method also to problems where analytical time‐dependent fundamental solutions might not be known.To obtain a symmetric formulation, the second boundary integral equation has to be used which, unfortunately, requires special care in the numerical implementation since it involves hypersingular kernel functions. Therefore, a regularization for closed surfaces of the Laplace‐transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one. Supplementarily, a weakly singular formulation of the Laplace‐transformed elastodynamic double layer potential is presented. This results in a time domain boundary element formulation involving at least only weakly singular integral kernels.Finally, numerical studies validate this approach with respect to different spatial and time discretizations. Further, a comparison with the wider used collocation method is presented. It is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods. Copyright © 2008 John Wiley & Sons, Ltd.

Singularity-Reduced Integral Representations for Discontinuity in Smart Materials

This paper presents a complete set of singularity-reduced integral relations for isolated discontinuity embedded in a three-dimensional infinite medium. The development is carried out in a broad context such that the constitutive law governing the material behavior assumes a general form and the discontinuity surface possesses general configuration and jump distribution. The former feature allows the treatment of a well-known class of smart materials (e.g. piezoelectric and piezomagnetic materials) as a special case while the latter renders the treatment of particular types of discontinuity such as cracks and dislocations possible. The key elements utilized in the regularization procedure are special decompositions of two involved kernels in a form well-suited for integration by parts to be performed via Stokes’ theorem. The weakly singular kernels appearing in these representations are obtained in a concise form appropriate for numerical evaluation. A set of integral relations is subsequently specialized to cracks and dislocations. For dislocations, the field quantities such as state variables, the body flux, and the generalized interaction energy are given in terms of line integral representations. The obtained expressions are fundamental and useful in the context of dislocation mechanics and modeling. For cracks, a weakly singular, weak-form integral equation for the surface flux is established. Such integral equation constitutes a basis for a wellknown numerical procedure, a symmetric Galerkin boundary element method (SGBEM). The crucial feature of using the derived integral equation as the key governing equation is its weakly singular nature that allows low order interpolations to be used in the numerical approximation.

A weakly-singular SGBEM for analysis of cracks in 3D anisotropic media

A weakly-singular, symmetric Galerkin boundary element method (SGBEM) is developed for analysis of fractures in three-dimensional, anisotropic linearly elastic media. The method constitutes a generalization of that by Li et al. [S. Li, M.E. Mear, L. Xiao, Symmetric weak-form integral equation method for three-dimensional fracture analysis, Comput. Methods Appl. Mech. Engrg. 151 (1998) 435–459] for isotropic media, and is based upon a pair of weak-form displacement and traction integral equations recently established by Rungamornrat and Mear [J. Rungamornrat, M.E. Mear, Weakly-singular, weak-form integral equations for cracks in three-dimensional anisotropic media, Int. J. Solids Struct. 45 (2008) 1283–1301]. The formulation involves only weakly-singular kernels and, as a consequence, standard C 0 elements can be employed in the numerical discretization. The kernels possess a relatively simple form (for general anisotropy) which involves an equatorial line integral like that associated with the displacement fundamental solution. Despite their simple form, it is still necessary to evaluate these kernels efficiently in the numerical implementation; to do so, certain symmetry properties associated with the specific material type under consideration are exploited, and a simple yet accurate interpolation strategy is introduced. Another important aspect of the numerical treatment is the use of a special crack-tip element which allows highly accurate mixed-mode stress intensity factor data to be extracted as a function of position along the crack front. This crack-tip element was originally introduced by Li et al. [S. Li, M.E. Mear, L. Xiao, Symmetric weak-form integral equation method for three-dimensional fracture analysis, Comput. Methods Appl. Mech. Engrg. 151 (1998) 435–459] in their treatment of isotropic media, and here it is extended to allow evaluation of the stress intensity factors for generally anisotropic media. The element involves “extra” degrees of freedom associated with the nodes along the crack front (with these quantities being directly related to the stress intensity factors), and it is capable of representing the asymptotic behavior in the vicinity of the crack front to a sufficiently high order that relatively large crack-tip elements can be utilized. Various examples are treated for cracks in an unbounded domain and for both embedded and surface-breaking cracks in a finite domain, and it is demonstrated that very accurate results can be obtained even with relatively coarse meshes.

Two variational approaches for domain decomposition problems solved by SGBEM with nonconforming discretizations

Two original approaches for the solution of elastic boundary value problems with domain decomposition (DDBVP) using the symmetric Galerkin boundary element method (SGBEM) are presented. Each approach is based on a variational principle, a difference between them consisting in the treatment of the coupling conditions which connect the solutions through an interface. The computer codes developed are able to deal with curved interfaces in a domain decomposition problem discretized by nonmatching meshes of linear elements along the interfaces. The effectiveness of the methods is documented by a numerically solved example.

Analysis of fractures in 3D piezoelectric media by a weakly singular integral equation method

A weakly singular, symmetric Galerkin boundary element method (SGBEM) is established to compute stress and electric intensity factors for isolated cracks in three-dimensional, generally anisotropic, piezoelectric media. The method is based upon a weak-form integral equation, for the surface traction and the surface electric charge, which is established by means of a systematic regularization procedure; the integral equation is in a symmetric form and is completely regularized in the sense that its integrand contains only weakly singular kernels of \({\mathcal{O}(1/r)}\) (hence allowing continuous interpolations to be employed in the numerical approximation). The weakly singular kernels which appear in the weak-form integral equation are expressed explicitly, for general anisotropy, in terms of a line integral over a unit circle. In the numerical implementation, a special crack-tip element is adopted to discretize the region near the crack front while the remainder of the crack surface is discretized by standard continuous elements. The special crack-tip element allows the relative crack-face displacement and electric potential in the vicinity of the crack front to be captured to high accuracy (even with relatively large elements), and it has the important feature that the mixed-mode intensity factors can be directly and independently extracted from the crack front nodal data. To enhance the computational efficiency of the method, special integration quadratures are adopted to treat both singular and nearly singular integrals, and an interpolation strategy is developed to approximate the weakly singular kernels. As demonstrated by various numerical examples for both planar and non-planar fractures, the method gives rise to highly accurate intensity factors with only a weak dependence on mesh refinement.

A symmetric Galerkin BEM for plate bending analysis

Abstract The Symmetric Galerkin Boundary Element Method is employed in thin plate bending analysis in accordance with the Love–Kirchhoff kinematical assumption. The equations are obtained through the stationary conditions of the total potential energy, written for a plate whose boundary is discretized in boundary elements. Since the matrix coefficients are made up as double integrals with high order singularities, a strategy is shown to compute these coefficients in closed form. Furthermore, in order to model the kinematical discontinuities and to weight the mechanical quantities along the boundary elements, the Lagrangian quadratic shape functions, rather than C 1 type (spline, Hermitian), are employed. The effectiveness of the matrix coefficients is shown through the rigid body movement technique.

Sparse convolution quadrature for time domain boundary integral formulations of the wave equation

Many important physical applications are governed by the wave equation. The formulation as time domain boundary integral equations involves retarded potentials. For the numerical solution of this problem, we employ the convolution quadrature method for the discretization in time and the Galerkin boundary element method for the space discretization. We introduce a simple a priori cut-off strategy where small entries of the system matrices are replaced by zero. The threshold for the cut-off is determined by an a priori analysis which will be developed in this paper. This analysis will also allow to estimate the effect of additional perturbations such as panel clustering and numerical integration on the overall discretization error. This method reduces the storage complexity for time domain integral equations from O(M 2 N) to O(M 2 N 1/2 log M), where N denotes the number of time steps and M is the dimension of the boundary element space.