Flux intensity functions for the Laplacian at polyhedral edges
Source abstract
Abstract
We present explicit representation formulas for the coefficients of the singularities associated with mixed boundary value problems for the Poisson equation in two-dimensional domains with corners and three-dimensional domains with straight edges including cracks. We rely on partial Fourier analysis of the boundary value problem in the vicinity of the singularities to derive an asymptotic expansion of the solution. Hence, the edge flux intensity functions are expressed in terms of Fourier series and we give integral expressions from which the associated Fourier coefficients can be computed independently and in parallel. Our method does not require the knowledge of the dual singular solutions of the adjoint problems. Moreover, the constructive nature of the formulas provides in a straight forward way a strategy for the construction of efficient numerical methods for the accurate computation of the stress intensity factors and the edge flux intensity functions. Also, the formulas can be used to construct postprocessing algorithms for the standard finite element approximation of the solution of the boundary value problem to improve the accuracy. Examples that illustrate the efficiency and robustness of the formulas are also given.
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Research context
OpenAlex topics: Numerical methods in engineering · Advanced Numerical Methods in Computational Mathematics · Electromagnetic Simulation and Numerical Methods
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References (33)
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