An improved boundary integral equation method for Helmholtz problems

The eigenvalue problem for the Laplace operator is numerical investigated using the boundary integral equation (BIE) formulation. Three methods of discretization are given and illustrated with numerical examples.     

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.     

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.     

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.     

A low-frequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems

A low-frequency fast multipole boundary element method (FMBEM) for 3D acoustic problems is proposed in this paper. The FMBEM adopts the explicit integration of the hypersingular integral in the dual boundary integral equation (BIE) formulation which was developed recently by Matsumoto, Zheng et al. for boundary discretization with constant element. This explicit integration formulation is analytical in nature and cancels out the divergent terms in the limit process. But two types of regular line integrals remain which are usually evaluated numerically using Gaussian quadrature. For these two types of regular line integrals, an accurate and efficient analytical method to evaluate them is developed in the present paper that does not use the Gaussian quadrature. In addition, the numerical instability of the low-frequency FMBEM using the rotation, coaxial translation and rotation back (RCR) decomposing algorithm for higher frequency acoustic problems is reported in this paper. Numerical examples are presented to validate the FMBEM based on the analytical integration of the hypersingular integral. The diagonal form moment which has analytical expression is applied in the upward pass. The improved low-frequency FMBEM delivers an algorithm with efficiency between the low-frequency FMBEM based on the RCR and the diagonal form FMBEM, and can be used for acoustic problems analysis of higher frequency.     

A local boundary integral equation method for potential problems

Boundary integral equation (boundary element) methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed. Further benefits would be obtained if attention could be confined to information at one small part of the boundary, the particular region of interest in a given problem. A local boundary integral equation method based on a Taylor series expansion of the unknown function is developed to do this for two-dimensional potential problems governed by Laplace's equation. Very accurate local values of the function and its derivatives can be obtained. The method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques.     

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.     

An ancillary boundary integral equation for magnetostatic analysis

The boundary element method (BEM) has been established as an effective means for magnetostatic analysis. Direct BEM formulations for the magnetic vector potential have been developed over the past 20 years. There is a less well-known direct boundary integral equation (BIE) for the magnetic flux density which can be derived by taking the curl of the BIE for the magnetic vector potential and applying properties of the scalar triple product. On first inspection, the ancillary boundary integral equation for the magnetic flux density appears to be homogeneous, but it can be shown that the equation is well-posed and non-homogeneous using appropriate boundary conditions. In the current research, the use of the ancillary boundary integral equation for the magnetic flux density is investigated as a stand-alone equation and in tandem with the direct formulation for the magnetic vector potential.     

Perfectly matched layers for frequency-domain integral equation acoustic scattering problems

Simulations of acoustic wavefields in inhomogeneous media are always performed on finite numerical domains. If contrasts actually extend over the domain boundaries of the numerical volume, unwanted, non-physical reflections from the boundaries will occur. One technique to suppress these reflections is to attenuate them in a locally reflectionless absorbing boundary layer enclosing the spatial computational domain, a perfectly matched layer (PML). This technique is commonly applied in time-domain simulation methods like finite element methods or finite-difference time-domain, but has not been applied to the integral equation method. In this paper, a PML formulation for the three-dimensional frequency-domain integral-equation-based acoustic scattering problem is derived. Three-dimensional acoustic scattering configurations are used to test the PML formulation. The results demonstrate that strong attenuation (a factor of 200 in amplitude) of the scattered pressure field is achieved for thin layers with a thickness of less than a wavelength, and that the PMLs themselves are virtually reflectionless. In addition, it is shown that the integral equation method, both with and without PMLs, accurately reproduces pressure fields by comparing the obtained results with analytical solutions.     

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.     

A regularized boundary integral equation method for elastodynamic crack problems

This paper presents a double layer potential approach of elastodynamic BIE crack analysis. Our method regularizes the conventional strongly singular expressions for the traction of double layer potential into forms including integrable kernels and 0th, 1st and 2nd order derivatives of the double layer density. The manipulation is systematized by the use of the stress function representation of the differentiated double layer kernel functions. This regularization, together with the use of B-spline functions, is shown to provide accurate numerical methods of crack analysis in 3D time harmonic elastodynamics.     

Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks

New complex hypersingular integral equation (CHSIE) is derived for the half-plane containing the inclusions (which can have the different elastic properties), holes, notches and cracks of the arbitrary shape. This equation is obtained by superposition of the equations for each homogeneous region in a half-plane. The last equations follow from the use of complex analogs of Somigliana's displacement and stress identities (SDI and SSI) and Melan's fundamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homogeneous plane is extended on case of a half plane. The unknown functions are approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips are taken into account. Only two types of the boundary elements (straight segments and circular arcs) are used to approximate the boundaries. All the integrals involved in CHSIE are evaluated in a closed form. A wide range of elasticity problems for a half-plane with cracks, openings and inclusions are solved numerically.     

Numerical solutions of a hypersingular integral equation for antiplane elastic curved crack problems of circular regions

Summary. In this paper, a hypersingular integral equation for the antiplane elasticity curved crack problems of circular regions is suggested. The original complex potential is formulated on a distribution of the density function along a curve, where the density function is the COD (crack opening displacement). The modified complex potential can also be established, provided the circular boundary is traction free or fixed. Using the proposed modified complex potential and the boundary condition, the hypersingular integral equation is obtained. The curve length method is suggested to solve the integral equation numerically. By using this method, the usual integration rule on the real axis can be used to the curved crack problems. In order to prove that the suggested method can be used to solve more complicated cases of the curved cracks, several numerical examples are given.     

An advanced boundary integral equation method for three-dimensional thermoelasticity

The features of an advanced numerical solution capability for boundary value problems of linear, homogeneous, isotropic, steady-state thermoelasticity theory are outlined. The influence on the stress field of thermal gradient, or comparable mechanical body force, is shown to depend on surface integrals only. Hence discretization for numerical purposes is confined to body surfaces. Several problems are solved, and verification of numerical procedures is obtained by comparison with accepted results from the literature.     

The universal algorithm based on complex hypersingular integral equation to solve plane elasticity problems

The effective numerical algorithm to solve a wide range of plane elasticity problems is presented. The method is based on the use of the complex hypersingular boundary integral equation (CHBIE) for blocky systems and bodies with cracks and holes. The BEM technique is employed to solve this equation. The unknown functions (displacement discontinuities (DD) or tractions) are approximated by Lagrange polynomials of the arbitrary degree. For the tip elements the asymptotics for the DD are taken into account. The boundaries of the blocks, cracks and holes are approximated by the arcs of the circles and the straight elements. In this case all the integrals (hypersingular, singular and regular) involved in this equation are evaluated in a closed form. Numerical results are given and compared either to the ones obtained by the other authors or to analytical solutions.This work was supported by the International Science Foundation (Grants R3X000, R3X300) funded by George Soros. The part of this work was performed when the author was a visiting research fellow at the University of Minnesota. I am especially grateful to Prof. A. Linkov of the Institute of Problems of Mechanical Engineering (Russian Academy of Science) for the fruitful discussion and Prof. T. Cruse of the University of Vanderbilt for much communication on the contents of this paper.

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On the validity of conforming BEM algorithms for hypersingular boundary integral equations

The widely held notion that the use of standard conforming isoparametric boundary elements may not be used in the solution of hypersingular integral equations is investigated. It is demonstrated that for points on the boundary where the underlying field is C ^1,α continuous, a class of rigorous nonsingular conforming BEM algorithms may be applied. The justification for this class of algorithms is interpreted in terms of some recent criticism. It is shown that the numerical integration in these conforming BEM algorithms using relaxed regularization represents a finite approximation to the standard two-sided Hadamard finite part interpretation of hypersingular integrals. It is also shown that the integration schemes in this class of algorithms are not based upon one-sided finite part interpretations. As a result, the attendant ambiguities associated with the incorrect use of the one-sided interpretations in boundary integral equations pose no problem for this class of algorithms. Additionally, the distinction is made between the analytic discontinuities in the field which place limitations on the applicability of the conforming BEM and the discontinuities resulting from the use of piece-wise C ^1,α interpolations.     

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.     

Axisymmetric formulation for boundary integral equation methods in scalar potential problems

Axisymmetric geometries often appear in electromagnetic device studies. The authors present an original formulation for Boundary Integral Equation methods in scalar potential problems. This technique requires only 2D boundary in the r-z plane and evaluation of the equations only on those boundaries.     

The boundary integral equation method for the solution of problems involving elastic slabs

Three boundary integral equations for the solution of an important class of elastic slab problems are considered. Some numerical examples are examined in order to illustrate the application of the integral equations to particular boundary-value problems.