Review of Dual Boundary Element Methods With Emphasis on Hypersingular Integrals and Divergent Series

Abstract

This article provides a perspective on the current status of the formulations of dual boundary element methods with emphasis on the regularizations of hypersingular integrals and divergent series. A simple example is given to show the dual integral representation and the dual series representation for a discontinuous function and its derivative and thereby to illustrate the regularization problems encountered in dual boundary element methods. Hypersingularity and the theory of divergent series are put under the framework of the dual representations, their relation and regularization techniques being examined. Applications of the dual boundary element methods using hypersingularity and divergent series are explored. This review article contains 249 references.

Degenerate scale problem when solving Laplace's equation by BEM and its treatment

“…*n x *n s (13) in which n x is the normal vector for the field point x. Equations (8) and (11) are coined the dual boundary regular integral equations for the domain point x. The explicit forms of the kernel functions can be found in Reference [37].…”

Stress computations on perforated polygonal domains

“…If one extends one's search to Fredholm equations of the first kind it is easier to find candidates. In fact, many popular integral equation used for fracture mechanics problems [8,37,44] (hypersingular extensions of the Somigliana identity [34], also known as standard BEM) are equations of this type. Schemes based on first kind equations retain some of the advantages of second kind integral equation schemes, such as ease of meshgeneration.…”

“…Others are lost. Hypersingularity, which appears when double layer potentials are differentiated in the dual boundary boundary element method [8,44], is in some sense similar to differentiation-compare the Cauchy integral formula for an analytic function f ðzÞ and z inside G 1 2pi…”

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

“…Previous methods to obtain regularization of hypersingular integrals for second order partial differential equation are reviewed by Tanaka et al [7] and Chen et al [8], and for the Helmholtz equation are summarized in literatures elsewhere; e.g., by Hwang [9], Yang [10], and Yan et al [11]. The regularizations applied in past studies are classified into three approaches.…”