An iterative radial simplex method for elastostatic and elastodynamic boundary elements

The boundary element method is a powerful numerical technique that can be employed for the accurate analysis of physical systems. A difficulty encountered with the method arises from the need to evaluate to high accuracy the many forms of singular and non-singular integral that are allied with the method. The singular integrals involved can be categorised into weakly singular, Cauchy principal value singular, hypersingular along with higher-order singular integrals introduced in the paper, which are denoted as supersingular. The principal distinction between the types of singular integral is that with the first two, with or without an appropriate limiting process, are finite but the latter two are unbounded and require the removal of the sources of unboundeness to obtain a finite part. In this paper the iterated radial simplex method is introduced for the evaluation of these singular integrals. The method involves the careful employment of multiple integration where the inner integral is performed along a radial direction. Evaluation of the radial-inner integral on a simplex of dimension n provides n +1 integrals on simplex domains of dimension n -1. On each simplex in the lower dimension the procedure is repeated and so on until the final dimension is 0 which yields an analytical solution or the singularity is sufficiently remote to facilitate numerical integration. Attention in the paper is restricted to dimensions 3, 2 and 1 with integration performed on tetrahedrons, triangles and closed intervals. In addition the super-singular integral equation is introduced obtained from the differentiation of the hyper-singular integral equation. The super-singular integral equation has been developed for the purpose of analysing surface waves, which requires continuity of inter-element stresses, which is not present with the standard forms of boundary elements. Illustrated in the paper through detailed examples is the method's ability to evaluate to high accuracy the severe singular integrals involved in elastostatics and elastodynamics.