A pure boundary element method approach for solving hypersingular boundary integral equations

This paper is concerned with discretization and numerical solution of a regularized version of the hypersingular boundary integral equation (HBIE) for the two-dimensional Laplace equation. This HBIE contains the primary unknown, as well as its gradient, on the boundary of a body. Traditionally, this equation has been solved by combining the boundary element method (BEM) together with tangential differentiation of the interpolated primary variable on the boundary. The present paper avoids this tangential differentiation. Instead, a “pure” BEM method is proposed for solving this class of problems. Dirichlet, Neumann and mixed problems are addressed in this paper, and some numerical examples are included in it.     

Thermoelastic fracture mechanics with regularized hypersingular boundary integral equations

This paper is concerned with thermoelastic fracture mechanics, in three-dimensional linear elasticity, using hypersingular boundary integral equations (HBIEs). The HBIEs are regularized by employing modes of deformation (or “simple solutions”). In addition to rigid-body and linear displacement modes, which have been used before for isothermal linear elastic fracture mechanics (LEFM), a new mode is employed here. This new mode is a thermal one in which the body is elevated to a uniform temperature but fully constrained (i.e., zero prescribed displacements) on its bounding surface. An existing isothermal LEFM computer code called BES is extended in this work to include thermoelastic terms. Some numerical results are presented from this new code.     

Evaluation of gradients on the boundary using fully regularized hypersingular boundary integral equations

Summary The evaluation of the gradient of the primary variable on the boundary for the Laplace problem, and the stress for the elasticity problem, involves hypersingular boundary integrals (HBIEs). To obtain any meaningful results from these integrals, an appropriate regularization scheme needs to be developed. We present an elegant way of calculation of gradients on the boundary of a body, starting from HBIEs regularized by using simple solutions or modes. Our method is currently limited to the calculation of gradients at regular points on the boundary at which the gradients of the primary variables are continous. Theiterative scheme developed in this paper is shown to be extremely robust for the calculations of the gradients. The method is tested on two Laplace problems and two problems in linear elasticity. This method does not involve any limiting process and can be easily extended to 3-dimensions. The approach developed in this paper can also be extended to other problems like acoustics and elastodynamics.     

A fast spectral Galerkin method for hypersingular boundary integral equations in potential theory

This research is focused on the development of a fast spectral method to accelerate the solution of three-dimensional hypersingular boundary integral equations of potential theory. Based on a Galerkin approximation, the fast Fourier transform and local interpolation operators, the proposed method is a generalization of the precorrected-FFT technique to deal with double-layer potential kernels, hypersingular kernels and higher-order basis functions. Numerical examples utilizing piecewise linear shape functions are included to illustrate the performance of the method. The US Government retains a nonexclusive royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for US Government purposes.     

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.     

Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method

This paper presents a new set of boundary integral equations for three dimensional acoustic shape sensitivity analysis based on the direct differentiation method. A linear combination of the derived equations is used to avoid the fictitious eigenfrequency problem associated with the conventional boundary integral equation method when solving exterior acoustic problems. The strongly singular and hypersingular boundary integrals contained in the equations are evaluated as the Cauchy principal values and Hadamard finite parts for constant element discretization without using any regularization technique in this study. The present boundary integral equations are more efficient to use than the usual ones based on any other singularity subtraction technique and can be applied to the fast multipole boundary element method more readily and efficiently. The effectiveness and accuracy of the present equations are demonstrated through some numerical examples.     

On the use of conjugate gradient-type methods for boundary integral equations

This paper deals with a number of iterative methods for solving matrix equations that result from boundary integral equations. The matrices are non-sparse, and in general neither positive definite nor symmetric. Traditional methods like Gauss-Seidel do not give satisfactory results, therefore the use of conjugate gradient- and Krylov-type methods is investigated based on work of Kleinman and Van den Berg, who presented a general framework for these methods. Eleven of these algorithms are given and their performance (without preconditioning) is compared in a test case involving four different integral operators arising in potential theory. For all four matrix equations the Generalized Minimal Residual method (GMRES) outperforms all other iterative methods in both computation time per iteration and total computation time. For the Fredholm equations of the first kind this method also is the fastest with respect to the number of iterations. The Bi-conjugate gradient method (Bi-CG) and the Quasi-minimal residual method (QMR) are the best alternatives. For the Fredholm equations of the second kind more methods can be used efficiently besides GMRES. The Conjugate Gradient Squared method (CGS), the Bi-conjugate gradient method (Bi-CG), its stabilized version (Bi-CGSTAB) and the Quasi Minimal Residual method (QMR) are efficient alternatives.     

On direct numerical treatment of hypersingular integral equations arising in mechanics and acoustics

Summary.  In this paper, we present a treatment of hypersingular integral equations, which have relevant applications in many problems of wave dynamics, elasticity and fluid mechanics with mixed boundary conditions. The main goal of the present work is the development of an efficient direct numerical collocation method. The paper also includes two examples taken from fracture mechanics and acoustics: a single crack in a linear isotropic elastic medium, and diffraction of a plane acoustic wave by a thin rigid screen. Received July 15, 2002; revised November 13, 2002 Published online: May 8, 2003 The paper has been supported by Italian Ministry for Instructions, University and Research (M.I.U.R.) through its national and local projects.     

An improved form of the hypersingular boundary integral equation for exterior acoustic problems

An improved form of the hypersingular boundary integral equation (BIE) for acoustic problems is developed in this paper. One popular method for overcoming non-unique problems that occur at characteristic frequencies is the well-known Burton and Miller (1971) method [7], which consists of a linear combination of the Helmholtz equation and its normal derivative equation. The crucial part in implementing this formulation is dealing with the hypersingular integrals. This paper proposes an improved reformulation of the Burton–Miller method and is used to regularize the hypersingular integrals using a new singularity subtraction technique and properties from the associated Laplace equations. It contains only weakly singular integrals and is directly valid for acoustic problems with arbitrary boundary conditions. This work is expected to lead to considerable progress in subsequent developments of the fast multipole boundary element method (FMBEM) for acoustic problems. Numerical examples of both radiation and scattering problems clearly demonstrate that the improved BIE can provide efficient, accurate, and reliable results for 3-D acoustics.     

Complex hypersingular integrals and integral equations in plane elasticity

Summary Complex hypersingular (finite-part) integrals and integral equations are considered in the functional class of N. Muskhelishvili. The appropriate definition is given. Three regularization (equivalence) formulae follow from this definition. They reduce hypersingular integrals to singular ones and allow to derive hypersingular analogues for Sokhotsky-Plemelj's formulae and for conditions that are necessary and sufficient for the function to be piecewise holomorphic. Two approaches to get and investigate complex hypersingular equations follow from these results: one of them is based on the equivalence formulae; as to the other, it is based on above-mentioned conditions. As an example, authors' equation for plane elasticity is studied. The existence of a unique solution is stated and some advantages over singular equations are outlined. To solve hypersingular equations the quadrature rules are presented. The accuracy of different quadrature formulae is compared, the examples being used. They confirm the need to take into account asymptotics and to carry out a thorough analytical investigation to get safe numerical results.     

Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method

This paper presents a gradient field representation using an analytical regularization of a hypersingular boundary integral equation for a two-dimensional time harmonic wave equation called the Helmholtz equation. The regularization is based on cancelation of the hypersingularity by considering properties of hypersingular elements that are adjacent to a singular node. Advantages to this regularization include applicability to evaluate corner nodes, no limitation for element size, and reduced computational cost compared to other methods. To demonstrate capability and accuracy, regularization is estimated for a problem about plane wave propagation. As a result, it is found that even at a corner node the most significant error in the proposed method is due to truncation error of non-singular elements in discretization, and error from hypersingular elements is negligibly small.     

On the implementation of 3D Galerkin boundary integral equations

In this article, a reverse contribution technique is proposed to accelerate the construction of the dense influence matrices associated with a Galerkin approximation of hypersingular boundary integral equations of mixed-type in potential theory. In addition, a general-purpose sparse preconditioner for boundary element methods has also been developed to successfully deal with ill-conditioned linear systems arising from the discretization of mixed boundary-value problems on non-smooth surfaces. The proposed preconditioner, which originates from the precorrected-FFT method, is sparse, easy to generate and apply in a Krylov subspace iterative solution of discretized boundary integral equations. Moreover, an approximate inverse of the preconditioner is implicitly built by employing an incomplete LU factorization. Numerical experiments involving mixed boundary-value problems for the Laplace equation are included to illustrate the performance and validity of the proposed techniques.     

Differentiability of strongly singular and hypersingular boundary integral formulations with respect to boundary perturbations

In this paper, we establish that the Lagrangian-type material differentiation formulas, that allow to express the first-order derivative of a (regular) surface integral with respect to a geometrical domain perturbation, still hold true for the strongly singular and hypersingular surface integrals usually encountered in boundary integral formulations. As a consequence, this work supports previous investigations where shape sensitivities are computed using the so-called direct differentiation approach in connection with singular boundary integral equation formulations. Communicated by T. Cruse, 6 September 1996     

Dual boundary integral equations for exterior problems

A dual integral formulation for the interior problem of the Laplace equation with a smooth boundary is extended to the exterior problem. Two regularized versions are proposed and compared with the interior problem. It is found that an additional free term is present in the second regularized version of the exterior problem. An analytical solution for a benchmark example in ISBE is derived by two methods, conformal mapping and the Poisson integral formula using symbolic software. The potential gradient on the boundary is calculated by using the hypersingular integral equation except on the two singular points where the potential is discontinuous instead of failure in ISBE benchmarks. Based on the matrix relations between the interior and exterior problems, the BEPO2D program for the interior problem can be easily reintegrated. This benchmark example was used to check the validity of the dual integral formulation, and the numerical results match the exact solution well.     

On the boundary integral equations approach to a semi-infinite strip investigation

Summary An orthotropic semi-infinite strip under arbitrary boundary conditions is considered. By means of Fourier transforms, boundary integral relations of special type with moving and motionless singularities of the Cauchy type are obtained. These relations lead to a system of singular integral equations corresponding to the various mixed boundary value problem. The power of singularities at the corner points, stresses and stress intensity factors are calculated for different loads and various material properties.     

Exact boundary integral transformation of the thermoelastic domain integral in BEM for general 2D anisotropic elasticity

In the direct formulation of the boundary element method, body-force and thermal loads manifest themselves as additional volume integral terms in the boundary integral equation. The exact transformation of the volume integral associated with body-force loading into surface ones for two-dimensional elastostatics in general anisotropy, has only very recently been achieved. This paper extends the work to treat two-dimensional thermoelastic problems which, unlike in isotropic elasticity, pose additional complications in the formulation. The success of the exact volume-to-surface integral transformation and its implementation is illustrated with three examples. The present study restores the application of BEM to two-dimensional anisotropic elastostatics as a truly boundary solution technique even when thermal effects are involved.     

Notes on boundary integral equations for three-dimensional magnetostatics

Several methods of formulating two-region magnetostatics problems with boundary integral equations and scalar potentials are discussed. These integral equations contain single- (i.e., monopole) and/or double- (i.e., dipole) layer source distributions. If only one type of source is used for both regions, certain numerical difficulties occur, which are discussed. For numerical accuracy a combined approach is adopted: it is sufficient to choose single layers in the exterior or less permeable region and to use double layers in the interior high-permeability region and at the interface. As an example, the combined method is applied to a recording head energized with a current loop.     

Reproducing polynomial particle methods for boundary integral equations

Since meshless methods have been introduced to alleviate the difficulties arising in conventional finite element method, many papers on applications of meshless methods to boundary element method have been published. However, most of these papers use moving least squares approximation functions that have difficulties in prescribing essential boundary conditions. Recently, in order to strengthen the effectiveness of meshless methods, Oh et al. developed meshfree reproducing polynomial particle (RPP) shape functions, patchwise RPP and reproducing singularity particle (RSP) shape functions with use of flat-top partition of unity. All of these approximation functions satisfy the Kronecker delta property. In this paper, we report that meshfree RPP shape functions, patchwise RPP shape functions, and patchwise RSP shape functions effectively handle boundary integral equations with (or without) domain singularities.