Standard Boundary Integral Equation Research Articles

Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem

We consider the Helmholtz transmission problem with piecewise-constant material coefficients, and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur, and propose modified boundary integral equations that are not affected by them.

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm <1/2 as an operator on the natural trace space H^{1/2}(Gamma ) whenever Gamma is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in H^{1/2}(Gamma ) for any sequence of finite-dimensional subspaces ({{mathcal {H}}}_N)_{N=1}^infty that is asymptotically dense in H^{1/2}(Gamma ). Long-standing open questions are whether the essential norm is also <1/2 for D as an operator on L^2(Gamma ) for all Lipschitz Gamma in 2-d; or whether, for all Lipschitz Gamma in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators pm frac{1}{2}I+D are compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces ({{mathcal {H}}}_N)_{N=1}^infty that is asymptotically dense in L^2(Gamma ). We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is ge 1/2, and examples with Lipschitz constant two for which the operators pm frac{1}{2}I +D are not coercive plus compact. We also give, for every C>0, examples of Lipschitz polyhedra for which the essential norm is ge C and for which lambda I+D is not a compact perturbation of a coercive operator for any real or complex lambda with |lambda |le C. We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the L^2(Gamma ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on C(Gamma ), equivalent to the standard supremum norm, for which the essential norm of D on C(Gamma ) is <1/2.

Perfectly Matched Layer Boundary Integral Equation Method for Wave Scattering in a Layered Medium

For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive, since they are formulated on lower-dimension boundaries or interfaces, and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on the Green's function of the background medium must evaluate the expensive Sommefeld integrals. Alternative BIE methods based on the free-space Green's function give rise to integral equations on unbounded interfaces which are not easy to truncate, since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. The PMLs are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space. Our PML-based BIE method uses the Green's function of the PML-transformed free space to define the boundary integral operators. The method is efficient, since the Green's function of the PML-transformed free space is easy to evaluate and the PMLs are very effective in truncating the unbounded interfaces. Numerical examples are presented to validate our method and demonstrate its accuracy.

A Hybrid Method for Systems of Closely Spaced Dielectric Spheres and Ions

We develop an efficient, semianalytical, spectrally accurate, and well-conditioned hybrid method for the study of electrostatic fields in composites consisting of an arbitrary distribution of dielectric spheres and ions that are loosely or densely (close to touching) packed. We first derive a closed-form formula for the image potential of a general multipole source of arbitrary order outside a dielectric sphere. Based on this formula, a hybrid method is then constructed to solve the boundary value problem by combining these analytical methods of image charges and image multipoles with the spectrally accurate mesh-free method of moments. The resulting linear system is well conditioned and requires many fewer unknowns on material interfaces as compared with standard boundary integral equation methods, in which the formulation becomes increasingly ill-conditioned and the number of unknowns also increases sharply as the spheres approach each other or ions approach the spheres due to the geometric and physical...

A semi-analytical approach to Green׳s functions for heat equation in regions of irregular shape

Initial-boundary-value problems are considered for the classical two-dimensional heat equation in regions of irregular configuration. A semi-analytical algorithm is proposed to accurately compute profiles of Green׳s function for such problems. The algorithm is based on a modification of the standard boundary integral equation method. To make the modification efficient, analytical representations of Green׳s functions are required for relevant regularly shaped regions. These are obtained in a closed form and employed then as kernels of the corresponding heat potentials, reducing the problem to a regular integral equation on a part of a boundary of the considered region.

Seismic oscillations of an elastic rectangular structure protected by a viscoelastic stratum: in-plane problem

We study the problem of seismic protection of a rectangular elastic construction by mean of a viscoelastic stratum made of material of Kelvin–Voigt type. In particular we consider the in-plane problem. By using the standard boundary integral equations method (BIEs), the dynamic problem is reduced to a system of BIEs over the set of boundary lines. In order to reduce the dimension of the problem in its discrete form, for wave processes in the stratum and in the half-space foundation we use a special Green’s function satisfying the stress-free boundary conditions over the boundary of the half-space. A numerical algorithm is used to solve the problem after discretization and finally we discuss the physical meaning of the results and the efficiency of the seismic protection by the viscoelastic stratum.

Modeling of sound propagation in the vicinity of rigid porous ground by boundary element method

A standard boundary integral equation (BIE) technique takes advantage of the well-known Green’s function for the sound fields above an impedance ground. It leads to a simplified solution that enables the development of efficient two-dimensional boundary element numerical codes for computing sound fields above a locally reacting ground. The method is particularly useful if there are multiple reflecting surfaces and the presence of a mixed impedance ground in the proximity of sources and receivers. However, the BIE formulation cannot be applied readily for calculating the sound fields above a non-locally reacting ground because the corresponding Green’s function is generally not available. The so-called two-domain approach was frequently used to model the sound propagation inside the porous material by assuming it as a dissipative fluid medium. By matching the particle velocities and pressures at the interface, a two-domain BIE formulation can be developed for modeling the sound propagation above a rigid porous ground. Use of the two-domain approach has a significant impact on the required computational resources. The current paper presents an alternative method aiming to reduce the computational time by exploring the use of an accurate Green’s function for the development of a BIE formulation above a non-locally reacting ground.

Improvement of Unified Boundary Integral Equation Method in Magnetostatic Shielding Analysis

When using the indirect method in magnetostatic analysis, the magnetic material is represented by the double-layer charge on the material surface. The double-layer charge offers an integral form of scalar potential to give a boundary integral equation (BIE). A unified integral form of excitation potential has been derived, resulting in a BIE capable of solving generic problems. When we apply the standard BIE to analyze magnetic shields by thin shells with high permeability, we usually cannot get accurate results in the shielded space. To accurately solve shielding problems using rough meshes, in this paper we propose a new modeling approach by using a concept of the boundary element method.

Volume integration in the hypersingular boundary integral equation

Abstract 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.

Dynamic element discretization method for solving 2D traction boundary integral equations

A sufficient condition for the existence of element singular integral of the traction boundary integral equation for elastic problems requires that the tangential derivatives of the boundary displacements are Hölder continuous at collocation points. This condition is violated if a collocation point is at the junction between two standard conforming boundary elements even if the field variables themselves are Hölder continuous there. Various methods are proposed to overcome this difficulty. Some of them are rather complicated and others are too different from the conventional boundary element method. A dynamic element discretization method to overcome this difficulty is proposed in this work. This method is novel and very simple: the form of the standard traction boundary integral equation remains the same; the standard conforming isoparametric elements are still used and all collocation points are located in the interior of elements where the continuity requirements are satisfied. For boundary elements with boundary points where the field variables themselves are singular, such as crack tips, corners and other boundary points where the stress tensors are not unique, a general procedure to construct special elements has been developed in this paper. Highly accurate numerical results for various typical examples have been obtained.

Hyper singular boundary element formulation for the Grad-Shafranov equation as an axisymmetric problem

The Grad-Shafranov equation describes the magnetic flux distribution of plasma in an axisymmetric system such as a tokamak-type nuclear fusion device. This paper presents a scheme to solve the hyper singular boundary integral equation (HBIE) corresponding to this Grad-Shafranov equation. All hyper and strong singularities caused by differentials of the complete elliptic integrals have been regularized up to the level of the Cauchy principal value integral. Test calculations commonly using discontinuous boundary elements have been made to compare the HBIE solutions with the solutions of the standard boundary integral equation (SBIE).

Regularized p-version collocation BEM algorithms for two-dimensional heat conduction

Based on conditioning, expectations should be that the boundary element method (BEM) is at least as sensitive to local modeling error as the finite element method (FEM), thus requiring comparable surface mesh densities when using the same order approximations. While the FEM appears most efficient as a sparse, low order, large scale approximation, the dense BEM formulation is more attractive as a small scale, high order approximation. Results show that p-version refinement in the BEM using interpolations as high as sixteenth order can reduce errors by five orders of magnitude for a given system size and can reduce system sizes by an order of magnitude for a given level of accuracy, thus reducing computational expense by three orders of magnitude. The algorithm presented includes a regularized form for the gradient boundary integral equation (BIE) and a super-regularized form for the standard BIE along with a comparison of the convergence behavior for the two forms. Examples are taken from two-dimensional heat conduction.

The extended boundary node method for three-dimensional potential theory

The boundary node method (BNM) [Mukherjee YX, Mukherjee S. The boundary node method for potential problems. Int J Numer Methods Eng 1997;40:797–815] is a boundary-only mesh-free method that combines the moving least-squares (MLS) interpolation scheme with the standard boundary integral equations (BIEs). Curvilinear boundary co-ordinates were originally proposed and used in this method—for both two [Mukherjee YX, Mukherjee S. The boundary node method for potential problems. Int J Numer Methods Eng 1997;40:797–815] and three-dimensional [Mukherjee S, Mukherjee YX. Boundary methods—elements, contours and nodes. Boca Raton, FL: CRC Press, in press] problems in potential theory and in linear elasticity. Li and Aluru [Li G, Aluru NR. Boundary cloud method: a combined scattered point/boundary integral approach for boundary-only analysis. Comput Methods Appl Mech Eng 2002;191:2337–70; Li G. Aluru NR. A boundary cloud method with a cloud-by-cloud polynomial basis. Eng Anal Boundary Elem 2003;27:57–71] have recently proposed an elegant improvement to the BNM (called the boundary cloud method (BCM)) that allows the use of Cartesian co-ordinates. Their novel variable basis BCM [Li G. Aluru NR. A boundary cloud method with a cloud-by-cloud polynomial basis. Eng Anal Boundary Elem 2003;27:57–71] has several advantages relative to the original BCM. It does, however, have a drawback in that continuous approximants are used for all boundary variables, even across corners. It is well known, for example, that the normal derivative of the potential function in potential theory, or the traction in linear elasticity, often suffers jump discontinuities across corners in two-dimensional (2-D) and across edges and corners in three-dimensional (3-D) problems. The present authors [Telukunta S, Mukherjee S. An extended boundary node method for modeling normal derivative discontinuities in potential theory across edges and corners. Eng Anal Boundary Elem 2004;28:1099–110] have recently proposed a further improvement to the BNM and the variable basis BCM. This new approach is called the extended BNM (EBNM). This method employs Cartesian co-ordinates with variable bases, together with appropriate approximants for the normal derivative across edges and corners that can model discontinuities in this variable. Two-dimensional problems in potential theory are presented in [Telukunta S, Mukherjee S. An extended boundary node method for modeling normal derivative discontinuities in potential theory across edges and corners. Eng Anal Boundary Elem 2004;28:1099–110]. The present paper is concerned with far more challenging problems—3-D problems in potential theory.

Electrostatic BEM for MEMS with thin beams

AbstractMicro‐electro‐mechanical (MEM) and nano‐electro‐mechanical (NEM) systems sometimes use beam‐ or plate‐shaped conductors that can be very thin—with h/L≈𝒪(10−2–10−3) (in terms of the thickness h and length L of a beam or the side of a square pate). Conventional boundary element method (BEM) analysis of the electric field in a region exterior to such thin conductors can become difficult to carry out accurately and efficiently—especially since MEMS analysis requires computation of charge densities (and then surface tractions) separately on the top and bottom surfaces of such objects. A new boundary integral equation (BIE) is derived in this work that, when used together with the standard BIE with logarithmically singular kernels, results in a powerful technique for the BEM analysis of such problems with thin beams. This new approach, in fact, works best for very thin beams. This thin beam BEM is derived and discussed in this work. Copyright © 2005 John Wiley &amp; Sons, Ltd.

Accuracy of desingularized boundary integral equations for plane exterior potential problems

In this article, computational results from boundary integral equations and their normal derivatives for the same test cases are compared. Both kinds of formulations are desingularized on their real boundary. The test cases are chosen as a uniform flow past a circular cylinder for both the Dirichlet and Neumann problems. The results indicate that the desingularized method for the standard boundary integral equation has a much larger convergence speed than the desingularized method for the hypersingular boundary integral equation. When uniform nodes are distributed on a circle, for the standard boundary integral formulation the accuracies in the test cases reach the computer limit of 10 −15 in the Neumann problems; and O(N −3) in the Dirichlet problems. However, for the desingularized hypersingular boundary integral formulation, the convergence speeds drop to only O(N −1) in both the Neumann and Dirichlet problems.

Tangential residual as error estimator in the boundary element method

In this paper a new error estimator based on tangential derivative Boundary Integral Equation residuals for 2D Laplace and Helmholtz equations is shown. The direct problem for general mixed boundary conditions is solved using standard and hypersingular boundary integral equations. The exact solution is broken down into two parts: the approximated solution and the error function. Based on theoretical considerations, it is shown that tangential derivative Boundary Integral Equation residuals closely correlate to the errors in the tangential derivative of the solution. A similar relationship is shown for nodal sensitivities and tangential derivative errors. Numerical examples show that the tangential Boundary Integral Equation residual is a better error estimator than nodal sensitivity, because of the accuracy of the predictions and the lesser computational effort.

Electrostatic BEM for MEMS with thin conducting plates and shells

Micro-electro-mechanical systems (MEMS) sometimes use beam or plate shaped conductors that can be very thin-with h / L ≈ O ( 10 − 2 − 10 − 3 ) (in terms of the thickness h and length L of the side of a square pate). Conventional Boundary Element Method (BEM) analysis of the electric field in a region exterior to such thin conductors can become difficult to carry out accurately and efficiently—especially since MEMS analysis requires computation of charge densities (and then surface tractions) separately on the top and bottom surfaces of such plates. A new boundary integral equation (BIE) is derived in this work that, when used together with the standard BIE with weakly singular kernels, results in a powerful technique for the BEM analysis of such problems. This new approach, in fact, works best for very thin plates. This thin plate BEM is derived and discussed in this work. Numerical results, from several BEM based methods, are presented and compared for the model problem of a parallel plate capacitor.

An extended boundary node method for modeling normal derivative discontinuities in potential theory across edges and corners

The Boundary Node Method (BNM) [Int. J. Numer. Meth. Engng 40 (1997) 797] is a boundary-only meshfree method that combines the Moving Least squares (MLS) interpolation scheme with the standard Boundary Integral Equations (BIEs) for a boundary value problem. Curvilinear boundary co-ordinates were originally proposed and used in this method—for both two- [Int. J. Numer. Meth. Engng 40 (1997) 797] and three-dimensional [Boundary methods—elements contours and nodes, 2004] (2D and 3D) problems. Li and Aluru [Comput. Meth. Appl. Mech. Engng 191 (2002) 2337; Engng Anal. Bound. Elem. 27 (2003) 57] have recently proposed an elegant improvement to the BNM (called the Boundary Cloud Method, BCM) that allows the use of Cartesian co-ordinates. Their novel variable basis BCM [Engng Anal. Bound. Elem. 27 (2003) 57] has several advantages relative to the original BCM. It does, however, have a drawback in that continuous approximants are used for all boundary variables, even across corners. It is well known, for example, that the normal derivative of the potential function in potential theory, or the traction in linear elasticity, often suffers jump discontinuities across corners in 2D and across edges and corners in 3D problems. The present paper describes a further improvement to the BNM and the variable basis BCM. This new approach is called the Extended Boundary Node Method (EBNM). This method employs Cartesian co-ordinates with variable bases, together with appropriate approximants for the normal derivative across edges and corners that can model discontinuities in this variable. This new formulation is presented below in detail for 2D problems in potential theory. Numerical results from the EBNM, for 2D potential theory, are compared with those from the variable basis BCM for selected examples.

Green’s functions for the bending of thin plates under various boundary conditions and applications: a review

Green’s functions of a point dislocation for the bending problem of an infinite thin plate containing an arbitrarily shaped hole under displacement, stress, and mixed boundary conditions and for a bimaterial problem are stated and the review of the related works is given. The point dislocation is defined by the discontinuity of the plate deflection angle. The original formulation for these problems consists of two parts: a principal part containing singular and multi-valued terms, and a complementary part containing only holomorphic terms. The Green’s functions were obtained in closed forms by using the complex stress function method in junction with the rational mapping function. The applications of the Green’s functions are demonstrated in studying the interaction of a hole or inclusion with a line crack in an infinite plate subjected to bending. The Green’s functions can also be used as the kernels for boundary integral representations in boundary element method analysis, where they can notably simplify the procedure of the standard boundary integral equations.

Error estimation using hypersingular integrals in boundary element methods for linear elasticity

A natural measure of the error in the boundary element method rests on the use of both the standard boundary integral equation (BIE) and the hypersingular BIE (HBIE). An approximate (numerical) solution can be obtained using either one of the BIEs. One expects that the residual, obtained when such an approximate solution is substituted to the other BIE is related to the error in the solution. The present work is developed for vector field problems of linear elasticity. In this context, suitable ‘hypersingular residuals’ are shown, under certain special circumstances, to be globally related to the error. Further, heuristic arguments are given for general mixed boundary value problems. The calculated residuals are used to compute element error indicators, and these error indicators are shown to compare well with actual errors in several numerical examples, for which exact errors are known. Conclusions are drawn and potential extensions of the present error estimation method are discussed.