Extracting edge flux intensity functions for the Laplacian

Abstract

SUMMARYThe solution to the Laplace operator in three-dimensional domains in the vicinity of straight edges is presented as an asymptotic expansion involving eigenpairs with their coe cients called edge ux intensity functions (EFIFs). The eigenpairs are identical to their two-dimensional counterparts over a plane perpendicular to the edge. Extraction of EFIFs however, cannot be obtained in a straightforward manner over this two-dimensional plane. A method based on L 2 projection and Richardson extrapolation is presented for point-wise extraction of EFIFs from p-ÿnite element solutions and illustrated by examples. A similar but more e cient method, based on 'energy projection', for extracting EFIFs is proposed.

A Quasi-Dual Function Method for Extracting Edge Stress Intensity Functions

“…It has the following advantages over (6.1): the continuity of the coefficients is no longer necessary; the basic function K m [α, p; b] is easier to determine (one-dimensional problems on (0, ω)) and less singular than Ψ Formula (6.5) is closer to the idea of the quasi-dual formulas, since it is no longer necessary to solve three-dimensional problems for the determination of the dual functionals, but it does require the knowledge of the solution u. Still The works [34] in two dimensions and [36] in three dimensions also introduce an extraction method based on integration over a circular arc of radius R, followed by Richardson extrapolation in R. They are successfully implemented in an engineering stress analysis code. In a certain sense, they are precursory to our present method, with the following important distinction: In these two references the antisymmetric duality pairing J[R] is replaced by a simple scalar product involving only the angular part of the singular functions.…”

“…In a certain sense, they are precursory to our present method, with the following important distinction: In these two references the antisymmetric duality pairing J[R] is replaced by a simple scalar product involving only the angular part of the singular functions. This possibility exists only for the Laplace operator due to its natural separation of variables (see [36]) and for the Lamé equations in two dimensions (see [34]). In order to reach a wide generality, we are led to deal with the universal duality pairing J [R].…”

“…In order to reach a wide generality, we are led to deal with the universal duality pairing J [R]. On the other hand, the extraction done in [36] yields pointwise values of the coefficients. Extracting moments is more suitable to the regularity properties of the edge coefficients near corners and to the approximation by finite elements.…”

“…A first step towards an algorithm suitable for implementation in an engineering stress analysis code is described in [36], where point values of edge coefficients are computed in the case of the Laplace equation near a straight edge. Very special orthogonality conditions of the Laplace edge singular functions are used to construct extraction formulas that are essentially two-dimensional.…”

“…Whereas this idea cannot be extended directly to more general geometrical and physical situations like Lamé equations in a polyhedral domain, our paper is an extension of [36] to such situations in the practical sense of suitability for implementation in engineering codes.…”