Hypersingular Boundary Integral Equations Research Articles

Numerical treatment of finite part integrals in 2-D boundary element analysis with application infracture mechanics

For the solution of problems in fracture mechanics by the boundary element method usually the subregion technique is employed to decouple the crack surfaces. In this paper a different procedure is presented. By using the displacement boundary integral equation on one side of the crack surface and the hypersingular traction boundary integral equation on the opposite side, one can renounce the subregion technique.

Hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions

A weakly singular form of the hypersingular boundary integral equation (BIE) (traction equation) for 3-D elastic wave problems is developed in this paper. All integrals involved are at most weakly singular and except for a stronger smoothness requirement on boundary elements, regular quadrature and collocation procedures used for conventional BIEs are sufficient for the discretization of the original hypersingular BIE. This weakly singular form of the hypersingular BIE is applied to the composite BIE formulation which uses a linear combination of the conventional BIE and the hypersingular BIE to remove the fictitious eigenfrequencies existing in the conventional BIE formulation for elastic wave problems. Numerical examples employing different types of boundary elements clearly demonstrate the effectiveness and efficiency of the developed formulation.

Symmetric Galerkin boundary formulations employing curved elements

AbstractAccounts of the symmetric Galerkin approach to boundary element analysis (BEA) have recently been published. This paper attempts to add to the understanding of this method by addressing a series of fundamental issues associated with its potential computational efficiency. A new symmetric Galerkin theoretical formulation for both the (harmonic) heat conduction and the (biharmonic) elasticity problem that employs regularized singular and hypersingular boundary integral equations (BIEs) is presented. The novel use of regularized BIEs in the Galerkin context is shown to allow straightforward incorporation of curved, isoparametric elements. A symmetric reusable intrinsic sample point (RISP) numerical integration algorithm is shown to produce a Galerkin (i.e. double) integration strategy that is competitive with its counterpart (i.e. singular) integration procedure in the collocation BEA approach when the time saved in the symmetric equation solution phase is also taken into account. This new formulation is shown to be capable of employing hypersingular BIEs while obviating the requirement of C1 continuity, a fact that allows the employment of the popular continuous element technology. The behaviour of the symmetric Galerkin BEA method with regard to both direct and iterative equation solution operations is also addressed. A series of example problems are presented to quantify the performance of this symmetric approach, relative to the more conventional unsymmetric BEA, in terms of both accuracy and efficiency. It is concluded that appropriate implementations of the symmetric Galerkin approach to BEA indeed have the potential to be competitive with, if not superior to, collocation‐based BEA, for large‐scale problems.

Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity

AbstractBoth the hypersingular and nearly singular integrals, which appear in the hypersingular boundary integralequations and integral representations of the secondary fields, respectively, are regularized by the application of the superposition principle. Two kinds of the non‐singular formulations, namely, those with the strongly singular and weakly singular kernels, are presented in this paper. The formulations are given in terms of the relevant boundary quantities and the collocation at element junctions is possible. Two‐ and three‐dimensional problems are analysed simultaneously in a unique way for either internal or external problems of the potential theory and elasticity.

A hypersingular-boundary integral equation method for the solution of an elastic multiple interacting crack problem

The authors formulate the antiplane problem concerning a finite elastic body with an arbitrary number of coplanar cracks in its interior in terms of a system of boundary integral equations coupled with Hadamard finite-part singular integral equations. These integral equations are readily amenable to numerical treatment and, for the purpose of illustrations, they are employed to obtain numerical values of the stress intensity factors for certain specific problems.

Three‐dimensional fracture simulation with a single‐domain, direct boundary element formulation

AbstractAn efficient hypersingular boundary integral equation method for three‐dimensional fracture mechanics was presented in a previous paper. The details of the numerical implementation of this method are further discussed herein. In particular, an algorithm for achieving the required differentiability of the crack surface displacement function is discussed. To illustrate the utility of the method, computational results for several strongly interacting multiple‐crack geometries are presented. The calculated stress intensity factors are in excellent agreement with those obtained by an approximate analytical method due to Kachanov and Laures.

On a Full Discretization Scheme for a Hypersingular Boundary Integral Equation over Smooth Curves

In this paper we consider a quadrature method for a hypersingular integral equation arising from a boundary integral formulation for the Neumann problem in a two-dimensional domain with smooth boundary. We prove that this method is equivalent to the trigonometric.Galerkin method. Consequently, we obtain stability and asymptotic error estimates in the scale of Sobolev spaces. Finally, we present some numerical tests.

A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations

The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.

A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems

The composite boundary integral equation (BIE) formulation, using a linear combination of the conventional BIE and the hypersingular BIE, emerges as the most effective and efficient formula for acoustic wave problems in an exterior medium which is free of the well-known fictitious eigen-frequency difficulty. The crucial part in implementing this formulation is dealing with the hypersingular integrals. Various regularization procedures in the literature give rise, in general, to integrals which are still difficult and/or extremely time consuming to evaluate or are limited to the use of special, usually flat, boundary elements. In this paper, a general form of the hypersingular BIE is developed for 3-D acoustic wave problems, which contains at most weakly singular integrals. This weakly singular form can be derived by employing certain integral identities involving the static Green's function. It is shown that the discretization of this weakly singular form of the hypersingular BIE is straightforward and the same collocation procedures and regular quadrature as that used for conventional BIEs are sufficient to compute all the integrals involved. Computing times are only slightly longer than with conventional BIEs. The C 1 smoothness requirement imposed on the density function for existence of the hypersingular BIEs and the possibility of relaxing this requirement are discussed. Three kinds of boundary elements, having different smoothness features, are employed. Numerical results are given for scattering from a rigid sphere at the fictitious frequencies, for values of wavenumber from π to 5π. In essence, with the methodology in this paper the fictitious eigenfrequency difficulty, long associated with the BIE for exterior problems, should no longer be a troublesome issue.

Numerical computation of hypersingular integrals and application to the boundary integral equation for the stress tensor

This paper describes a method for the numerical computation of hypersingular integrals as they appear in the evaluation of the stress tensor by hypersingular boundary integral equations in two-dimensional linear elastostatics. The method is not restricted to this type of problem however and may be easily applied to other hypersingular boundary integral equations. It allows the use of linear and circular representation of the boundary geometry and arbitrary approximation of the boundary value functions. The main advantage of the proposed method is the numerical computation of the hypersingular integrals by Kutt's quadrature formulas, thus requiring little analytic pre-work. The calculated examples reveal a high level of accuracy provided the necessary continuity requirements are fulfilled, which may be achieved by Overhauser spline elements.

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.

The use of simple solutions in the regularization of hypersingular boundary integral equations

The normal derivative of the boundary integral equation pertaining to a harmonic function, or a hypersingular boundary integral equation, is regularized by the use of two simple solutions on an arbitrary domain. This regularization produces a form convenient for computation and requires no explicit evaluation for finite part integrals. The equation is specialized for the case of linear elements, and some sample problems are given to illustrate the accuracy of the technique compared to the regular boundary element method. The procedure is easily extended to other, more complex problems.

On the convergence of the multigrid method for a hypersingular integral equation of the first kind

We present a multigrid method to solve linear systems arising from Galerkin schemes for a hypersingular boundary integral equation governing three dimensional Neumann problems for the Laplacian. Our algorithm uses damped Jacobi iteration, Gauss-Seidel iteration or SOR as preand post-smoothers. If the integral equation holds on a closed, Lipschitz surface we prove convergence ofV- andW-cycles with any number of smoothing steps. If the integral equation holds on an open, Lipschitz surface (covering crack problems) we show convergence of theW-cycle. Numerical experiments are given which underline the theoretical results.

A boundary element Galerkin method for a hypersingular integral equation on open surfaces

AbstractA hypersingular boundary integral equation of the first kind on an open surface piece Γ is solved approximately using the Galerkin method. As boundary elements on rectangles we use continuous, piecewise bilinear functions which vanish on the boundary of Γ. We show how to compensate for the effect of the edge and corner singularities of the true solution of the integral equation by using an appropriately graded mesh and obtain the same convergence rate as for the case of a smooth solution. We also derive asymptotic error estimates in lower‐order Sobolev norms via the Aubin–Nitsche trick. Numerical experiments for the Galerkin method with piecewise linear functions on triangles demonstrate the effect of graded meshes and show experimental rates of convergence which underline the theoretical results.

Hypersingular Boundary Integral Equations: Some Applications in Acoustic and Elastic Wave Scattering

The properties of hypersingular integrals, which arise when the gradient of conventional boundary integrals is taken, are discussed. Interpretation in terms of Hadamard finite-part integrals, even for integrals in three dimensions, is given, and this concept is compared with the Cauchy Principal Value, which, by itself, is insufficient to render meaning to the hypersingular integrals. It is shown that the finite-part integrals may be avoided, if desired, by conversion to regular line and surface integrals through a novel use of Stokes’ theorem. Motivation for this work is given in the context of scattering of time-harmonic waves by cracks. Static crack analysis of linear elastic fracture mechanics is included as an important special case in the zero-frequency limit. A numerical example is given for the problem of acoustic scattering by a rigid screen in three spatial dimensions.

On hypersingular boundary integral equations for certain problems in mechanics

Improper or weakly singular integrals, once thought to be a handicap in computations, are now accepted as a source of effectiveness and stability in the numerical solution of many problems in mechanics. Such integrals naturally arise, for example, in the Boundary Integral Equation (BIE) method, and the literature contains many examples of successful computations based on Boundary Element-type (EEM) solutions of the BIEs (e.g. [1,2]). For some vector problems, the BIE/BEM process gives rise to a stronger type of singular integral which exists in the sense of the Cauchy Principal Value (CPV) (cf. [3]). This integral also has been treated numerically with success. Less commonly, but with growing frequency it seems, the gradient or normal derivative of such boundary integrals is taken, especially in the formulation of mechanics problems involving cracks. Then integrals more (hyper) singular than the CPV can explicitly arise. Usually, however, rather than confront such hypersingular integrals directly, a process of regularization (e.g. [4,5,6,7]) is employed to lower the singularity of the integrands. Such regularizatlon usually carries a formulational complexity and computational cost if it is even possible. However, the alternatives to regularization seem to be divergent integrals or numerical computation with integrals more singular than the CPV with, perhaps, even questionable definition.

On boundary integral equations for crack problems

A ubiquitous linear boundary-value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle. One typical example is the scattering of scalar waves by a curved crack or rigid strip (Neumann boundary condition) in two dimensions. This problem can be reduced to an integrodifferential equation, which is often regularized. We adopt a more direct approach, and prove that the problem can be reduced to a hypersingular boundary integral equation. (Similar reductions will obtain in more complicated situations.) Com­putational schemes for solving this equation are described, with special emphasis on smoothness requirements. Extensions to three-dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed.