Flux intensity functions for the Laplacian at axisymmetric edges

Abstract

We derive explicit representation formulas for the computation of flux intensity functions for mixed boundary value problems for the Poisson equation in axisymmetric domains trueω̂MathClass-rel⊂double-struckR3 with edges. We rely on the decomposition of the boundary value problems in three dimensions by means of partial Fourier analysis with respect to the rotational angle into boundary value problems in the two‐dimensional meridian domain of trueω̂. Utilizing smooth cutoff functions, the solutions of the reduced problems are analyzed semi‐analytically near corners of the plane meridian domain, and the edge flux intensity functions are constructed via Fourier synthesis and convergence analysis. The formulas are also applicable in the case of crack fronts. The constructive nature of the formulas provides in a straightforward way an efficient strategy for the accurate computation of edge flux intensity functions in axisymmetric domains. A demonstration example that illustrates the application of the formulas is presented. Copyright © 2012 John Wiley & Sons, Ltd.

Singular solutions of the Poisson equation at edges of three‐dimensional domains and their treatment with a predictor–corrector finite element method

“…It has been shown both theoretically and computationally to be an efficient procedure for reconstructing the coefficients of the edge singularities. (4) In [23, 25], the case of the Poisson equation in domains with rotationally symmetric edges is considered and the edge coefficients are represented explicitly by converging Fourier series. Thus the coefficients can be approximated very efficiently by combining Fourier approximation with a suitable quadrature formula.…”