Assessment of the spectral properties of the double-layer potential matrix H

Abstract

The double-layer potential matrix H of the conventional, collocation boundary element method (CBEM) is singular, as referred to a static problem in a bounded continuum. This means that no rigid body displacements can be transformed between two different reference systems -and that unbalanced forces are to be excluded from a consistent linear algebra contragradient transformation. The properties of H T may become quite informative, as they reflect the topology (concavities, notches, cracks, holes) of the discretized domain as well as material non-homogeneities, which comes from the fact that local stress gradients can be represented by fundamental solutions only in a global sense. Symmetries and antisymmetries are also evidenced in N (H T ) as well as the spectral properties related to simple polynomial solutions that one may propose as patch tests. In the usual implementations of the CBEM with real fundamental solutions, all eigenvalues λ of H are real, λ ∈ R, 0 ≥ λ < 1 for a bounded domain. This means that H is a contraction -a paramount mechanical feature that comes up naturally in the frame of a virtual work investigation of Kelvin's (singular) fundamental solution and is resorted to in a simplified variational implementation of the boundary element method.

The hypersingular boundary element method revisited

“…It can be now seen, according to the previous Section, that, for a 2D problem with an isotropic material, eqn (13) holds, as stated, exactly for a general linear displacement field, which is not the case of the hitherto proposed boundary element implementations [9]. For a 3D problem, eqn (6) provides only conditionally an exact statement for a linear displacement field, but is nevertheless a simpler and more accurate formulation. The single-layer potential matrix G is in general rectangular, as stated in eqn (13), since the traction forces depend on the outward normal to the boundary.…”

“…Since then, several numerical implementations reported in the literature apparently attest the appropriateness of the proposed developments. However, there still are some conceptual issues to be addressed as a result of the theoretical inquiries carried out starting from variational principles and consistency checks of the classical, collocation boundary element method [4][5][6]. Motivation of the present developments is a variationally-based application of the boundary element method in the strain gradient elasticity [7,8], from which it turns out that the interpolation function proposed in Reference [5] for the evaluation of the single-layer potential matrix is not just a subtle improvement of the boundary element method, but a decisive requirement when dealing with hypersingularity and curved boundaries.…”