The evaluation of Cauchy principal value integrals in the boundary element method - a review
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.
scientific article; zbMATH DE number 4195024
scientific article