The evaluation of Cauchy principal value integrals in the boundary element method - a review
type: Item
instance of: scholarly article
title: The evaluation of Cauchy principal value integrals in the boundary element method - a review
zbMATH Open document ID: 0724.65019
DOI: 10.1016/0895-7177(91)90063-D
author: M. Guiggiani
published in: Mathematical and Computer Modelling
publication date: 1991-01-01T00:00:00Z
review text: The concept of the Cauchy principle value (CPV) integral \(\oint^{b}_{a}g(x)dx:=\lim_{\epsilon \to 0^+}[\int^{s- \epsilon}_{a}g(x)dx+\int^{b}_{s+\epsilon}g(x)dx]\) and its relationship to the boundary integral equations from boundary element methods are surveyed. The evaluation of CPV integrals can be classified as indirect or direct approaches. An indirect approach is problem dependent since each different kind of problem requires a different elementary solution (if any). Rigid body translations, axisymmetric problems, regularized boundary integral equations, and regularization using two different sets of integral equations are examples of indirect approaches. Direct approaches convert CPV integrals into regular integrals plus simple additional terms. This can be done for 2-D, axisymmetric, and 3-D boundary element formulations for any specific class of problem and for any order of the boundary elements used. The results for plane and axisymmetric problems are summarized. The resulting integrals can be approximated successfully using ordinary Gauss-Legendre quadrature formulae. Theoretical and numerical evidence shows that CPV integrals can be accurately and efficiently computed in a direct numerical way. This well-written survey includes 28 references, mostly recent.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 65N38
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 4195024
zbMATH Keywords: Cauchy principal value integrals
zbMATH Keywords: research survey
zbMATH Keywords: boundary integral equations
zbMATH Keywords: boundary element methods
zbMATH Keywords: regularization
zbMATH Keywords: Gauss-Legendre quadrature formulae
MaRDI profile type: Publication
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full work available at URL: 0895-7177(91)90063-d
OpenAlex ID: W2064543506
description: scientific article; zbMATH DE number 4195024