A matrix function solution for the scaled boundary finite-element equation in statics

Abstract

The scaled boundary finite-element method is a fundamental-solution-less boundary element method based on finite elements. It leads to semi-analytical solutions for displacement and stress fields, which permits problems with singularities or in infinite domains to be handled conveniently and accurately. However, the present eigenvalue method for solving the scaled boundary finite-element equation requires additional treatments for multiple eigenvalues with parallel eigenvectors, which results from logarithmic terms in the solutions. A matrix function solution for the scaled boundary finite-element equation in statics is presented in this paper. It is numerically stable for multiple or near-multiple eigenvalues as it is based on real Schur decomposition. Power functions, logarithmic functions and their transitions as occurring in fracture mechanics, composites and two-dimensional unbounded domains, are represented semi-analytically. No a priori knowledge on the types and orders of singularity are required when simulating stress singularities.