Volume integration in the hypersingular boundary integral equation

The evaluation of volume integrals that arise in conjunction with a hypersingular boundary integral formulation is considered. In a recent work for the standard (singular) boundary integral equation, the volume term was decomposed into an easily computed boundary integral, plus a remainder volume integral with a modified source function. The key feature of this modified function is that it is everywhere zero on the boundary. In this work it is shown that the same basic approach is successful for the hypersingular equation, despite the stronger singularity in the domain integral. Specifically, the volume term can be directly evaluated without a body-fitted volume mesh, by means of a regular grid of cells that cover the domain. Cells that intersect the boundary are treated by continuously extending the integrand to be zero outside the domain. The method and error results for test problems are presented in terms of the three-dimensional Poisson problem, but the techniques are expected to be generally applicable.     

Galerkin boundary integral analysis for the axisymmetric Laplace equation

The boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green's function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric‐Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non‐singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post‐processing of the surface gradient. Published in 2005 by John Wiley & Sons, Ltd.     

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.     

An accelerated surface discretization‐based BEM approach for non‐homogeneous linear problems in 3‐D complex domains

For non‐homogeneous or non‐linear problems, a major difficulty in applying the boundary element method (BEM) is the treatment of the volume integrals that arise. An accurate scheme that requires no volume discretization is highly desirable. In this paper, we describe an efficient approach, based on the precorrected‐FFT technique, for the evaluation of volume integrals resulting from non‐homogeneous linear problems. In this approach, the 3‐D uniform grid constructed initially to accelerate surface integration is used as the baseline mesh for the evaluation of volume integrals. As such, no volume discretization of the interior problem domain is necessary. Moreover, with the uniform 3‐D grid, the matrix sparsification techniques (such as the precorrected‐FFT technique used in this work) can be extended to accelerate volume integration in addition to surface integration, thus greatly reducing the computational time. The accuracy and efficiency of our approach are demonstrated through several examples. A 3‐D accelerated BEM solver for Poisson equations has been developed and has been applied to a 3‐D multiply‐connected problem with complex geometries. Good agreement between simulation results and analytical solutions has been obtained. Copyright © 2005 John Wiley & Sons, Ltd.     

A Galerkin boundary element formulation with dual reciprocity for elastodynamics

In this paper, a Galerkin boundary integral equation method for two‐dimensional elastodynamic problems is presented. The formulation makes use of a static fundamental solution to weight the dynamic equilibrium equations. Following the Galerkin approach, the equations are weighted again with the interpolation functions used in the discretization of the unknowns. For the numerical integration, a regularization process is followed to deal with the integrands containing strong singularities. The implementation of the dual reciprocity method to transfer the domain integrals to the boundary is also presented in the context of the Galerkin formulation. Finally, the Hubolt integration scheme was used for the time‐marching process. Several numerical examples are presented to demonstrate the accuracy of the method. Copyright © 2000 John Wiley & Sons, Ltd.     

Flux and traction boundary elements without hypersingular or strongly singular integrals

The present paper deals with a boundary element formulation based on the traction elasticity boundary integral equation (potential derivative for Laplace's problem). The hypersingular and strongly singular integrals appearing in the formulation are analytically transformed to yield line and surface integrals which are at most weakly singular. Regularization and analytical transformation of the boundary integrals is done prior to any boundary discretization. The integration process does not require any change of co‐ordinates and the resulting integrals can be numerically evaluated in a simple and efficient way. The formulation presented is completely general and valid for arbitrary shaped open or closed boundaries. Analytical expressions for all the required hypersingular or strongly singular integrals are given in the paper. To fulfil the continuity requirement over the primary density a simple BE discretization strategy is adopted. Continuous elements are used whereas the collocation points are shifted towards the interior of the elements. This paper pretends to contribute to the transformation of hypersingular boundary element formulations as something clear, general and easy to handle similar to in the classical formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

Evaluation of supersingular integrals: second‐order boundary derivatives

The boundary integral representation of second‐order derivatives of the primary function involves second‐order (hypersingular) and third‐order (supersingular) derivatives of the Green's function. By defining these highly singular integrals as a difference of boundary limits, interior minus exterior, the limiting values are shown to exist. With a Galerkin formulation, coincident and edge‐adjacent supersingular integrals are separately divergent, but the sum is finite, while the individual hypersingular integrals are finite. Moreover, the cancellation of the supersingular divergent terms only requires a continuous interpolation of the surface potential, and there is no continuity requirement on the surface flux. The algorithm is efficient, the non‐singular integrals vanish and the singular integrals are computed entirely analytically, and accurate values are obtained for smooth surfaces. However, it is shown that a (continuous) linear interpolation is not appropriate for evaluation at boundary corners. Published in 2006 by John Wiley & Sons, Ltd.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

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 Grid-based integral approach for quasilinear problems

For non-homogeneous and nonlinear problems, a major difficulty in applying the Boundary Element Method is the treatment of the volume integrals that arise. A recent proposed method, the grid-based integration method (GIM), uses a 3-D uniform grid to reduce the complexity of volume discretization, i.e., the discretization of the whole domain is avoided. The same grid is also used to accelerate both surface and volume integration. The efficiency of the GIM has been demonstrated on 3-D Poisson problems. In this paper, we report our work on the extension of this technique to quasilinear problems. Numerical results of a 3-D Helmholtz problem and a quasilinear Laplace problem on a multiply-connected domain with Dirichlet boundary conditions are presented. These results are compared with analytic solutions. The performance of the GIM is measured by plotting the L₂-norm error as a function of the overall CPU time and is compared with the auxiliary domain method in the Helmholtz problem.     

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials,with an application to acoustic scattering problems

This paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.     

Continuity requirements for density functions in the boundary integral equation method

Two methods of forming regular or hypersingular boundary integral equations starting from an interior integral representations are discussed. One method involves direct treatment of the singularities such as Cauchy principal value and/or finite-part interpretation of the integrals and the other does not. By either approach, theory places the same restrictions on the smoothness of the density function for the integrals to exist, assuming sufficient smoothness of the geometrical boundary itself. Specifically, necessary conditions on the smoothness of the density function for meaningful boundary integral formulas to exist as required for the collocation procedure are established here. Cases for which such conditions may not be sufficient are also mentioned and it is understood that with Galerkin techniques, weaker smoothness requirements may pertain. Finally, the bearing of these issues on the choice of boundary elements, to numerically solve a hypersingular boundary integral equation, is explored and numerical examples in 2D are presented.     

Buckling analysis of shear deformable plates by boundary element method

In this paper, the derivation and numerical implementation of boundary integral equations for the buckling analysis of shear deformable plates are presented. Plate buckling equations are derived as a standard eigenvalue problem. The formulation is formed by coupling boundary element formulations of shear deformable plate and two dimensional plane stress elasticity. The eigenvalue problem of plate buckling yields the critical load factor and buckling modes. The domain integrals which appear in this formulation are treated in two different ways: initially the integrals are evaluated using constant cells, and next, they are transformed into equivalent boundary integrals using the dual reciprocity method (DRM). Several examples with different geometry, loading and boundary conditions are presented to demonstrate the accuracy of the formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Accurate integration of singular kernels in boundary integral formulations for Helmholtz equation

Boundary integral formulations for the 2D Helmholtz equation involve kernels in the form of modified Bessel functions. Accurate schemes for evaluating integrals of the kernels and their derivatives are presented. Special attention is paid to integrals involving singular and near singular kernels. Both boundary and domain integrals are considered. It is shown that, with the use of series expansion functions for the modified Bessel functions, the boundary integrals can be evaluated analytically in the neighbourhood of the singularity. For domain integrals, the behaviour of the kernels in the vicinity of the singularity is used to construct accurate numerical quadrature schemes. A transient heat conduction problem is formulated as a Helmholtz equation, solved, and compared against analytic solution to demonstrate the effectiveness of these schemes in relation to traditional methods. References are made to previous work to advocate the utility of the boundary integral method for non-linear and time-transient problems.     

A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity

In this paper, a new and simple boundary‐domain integral equation is presented for heat conduction problems with heat generation and non‐homogeneous thermal conductivity. Since a normalized temperature is introduced to formulate the integral equation, temperature gradients are not involved in the domain integrals. The Green's function for the Laplace equation is used and, therefore, the derived integral equation has a unified form for different heat generations and thermal conductivities. The arising domain integrals are converted into equivalent boundary integrals using the radial integration method (RIM) by expressing the normalized temperature using a series of basis functions and polynomials in global co‐ordinates. Numerical examples are given to demonstrate the robustness of the presented method. Copyright © 2005 John Wiley & Sons, Ltd.     

A symmetric Galerkin formulation and dual reciprocity for 2D elastostatics

In this paper, a symmetric Galerkin boundary integral equation including body force terms is presented. The implementation of the dual reciprocity method to transfer the domain integrals to the boundary is presented in the context of the Galerkin formulation. Several numerical examples involving self-weight and centrifugal body forces are studied to demonstrate the efficiency of the method.     

A comparison of transformation methods for evaluating two‐dimensional weakly singular integrals

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of Laplace's equation in three dimensions and Poisson's equation in two dimensions. Both are two‐dimensional weakly singular integrals and are evaluated using (in a product fashion) methods which have recently been used for evaluating one‐dimensional weakly singular integrals arising in the boundary element method. The methods used are based on various polynomial transformations of conventional Gaussian quadrature points where the transformation polynomial has zero Jacobian at the singular point. Methods which split the region of integration into sub‐regions are considered as well as non‐splitting methods. In particular, the newly introduced and highly accurate generalized composite subtraction of singularity and non‐linear transformation approach (GSSNT) is applied to various two‐dimensional weakly singular integrals. A study of the different methods reveals complex relationships between transformation orders, position of the singular point, integration kernel and basis function. It is concluded that the GSSNT method gives the best overall results for the two‐dimensional weakly singular integrals studied. Copyright © 2002 John Wiley & Sons, Ltd.     

Galerkin boundary integral method for evaluating surface derivatives

A Galerkin boundary integral procedure for evaluating the complete derivative, e.g., potential gradient or stress tensor, is presented. The expressions for these boundary derivatives involve hypersingular kernels, and the advantage of the Galerkin approach is that the integrals exist when a continuous surface interpolation is employed. As a consequence, nodal derivative values, at smooth surface points or at corners, can be obtained directly. This method is applied to the problem of electromigration-driven void dynamics in thin film aluminum interconnects. In this application, the tangential component of the electric field on the boundary is required to compute the flux of atoms at the void surface.     

An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis

An efficient algorithm is employed to evaluated hyper and super singular integral equations encountered in boundary integral equations analysis of engineering problems. The algorithm is based on multiple subtractions and additions to separate singular and regular integral terms in the polar transformation domain, primarily established in Refs. (Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the numerical solution of hypersingular boundary integral equations. Trans ASME 1992;59:604–614; Guiggiani M, Casalini P. Direct computation of Cauchy principal value integral in advanced boundary element. Int J Numer Meth Engng 1987;24:1711–1720. Guiggiani M, Gigante A. A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. J Appl Mech Trans ASME 1990;57:906–915). It can be proved that the regular terms have finite analytical solutions in the range of integration, and the singular terms will be replaced by special periodic kernels in the integral equations. The subtractions involve to multiple derivatives of analytical kernels and the additions require some manipulation to separate the remaining regular terms from singular ones. The regular terms are computed numerically. Three examples on numerical evaluation of singular boundary integrals are presented to show the efficiency and accuracy of the algorithm. In this respect, strongly singular and hypersingular integrals of potential flow problems are considered, followed by a supersingular integral which is extracted from the partial differentiation of a hypersingular integral with respect to the source point.