Supplementing the numerical solution of singular/hypersingular integral equations/inequalities with parametric inequality constraints with applications to crack problems

Nikolaos I. Ioakimidis

doi:10.24423/cames.202

Nikolaos I. Ioakimidis University of Patras

DOI: http://dx.doi.org/10.24423/cames.202

Abstract

Singular and hypersingular integral equations appear frequently in engineering problems. The approximate solution of these equations by using various numerical methods is well known. Here we consider the case where these equations are supplemented by inequality constraints-mainly parametric inequality constraints, but also the case of singular/hypersingular integral inequalities. The approach used here is simply to employ the computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica and derive the related set of necessary and sufficient conditions for the validity of the singular/hypersingular integral equation/inequality together with the related inequality constraints. The present approach is applied to singular integral equations/inequalities in the problem of periodic arrays of straight cracks under loading- and fracture-related inequality constraints by using the Lobatto-Chebyshev method. It is also applied to the hypersingular integral equation/inequality of the problem of a single straight crack under a parametric loading by using the collocation and Galerkin methods and parametric inequality constraints.

Keywords

singular integral equations/inequalities, hypersingular integral equations/inequalities, boundary integral equations/inequalities, parametric inequality constraints, numerical methods, crack problems,

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