Flux intensity functions for the Laplacian at polyhedral edges

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.

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

“…Here we will describe the edge behavior of the weak solution u ∈ V 0 (Ω) of the boundary value problem (2.1) as in [22]. However, the main results here are the representation formulas for the coefficients of the singularities, see Theorem 2.2.Theorem For each f ∈ L 2 (Ω), let u ∈ V 0 (Ω) be the unique weak solution of the boundary value problem (2.1).…”

“…The predictor–corrector finite element method presented in the next section relies on explicit computable formulas for the coefficients T j ( r j , z j ) * Ψ j , k ( z j ) of the singularities in (2.6). In [22, pp. 179–182], these coefficients are characterized as follows:Theorem The coefficients Φ j , k ( x j , y j , z j ) ≔ T j ( r j , z j ) * Ψ j , k ( z j ) of the singularities in (2.6) can be represented by Fourier series in the variable z j and with respect to the orthogonal system { Z j , n ( z j ) : n ∈ ℕ 0 } ( see (2.5)) namely , …”

The Laplace equation in 3D domains with cracks: dual singularities with log terms and extraction of corresponding edge flux intensity functions

“…The QDFM has been successfully applied for the extraction of EFIFs associated with the non‐integer eigenvalues from finite element (FE) solutions, but it does not apply to EFIFs associated with the integer eigenvalues . Other methods for extracting EFIFs are available (e.g., ); however, these methods do not apply to integer eigenvalues either.…”

Flux intensity functions for the Laplacian at axisymmetric edges

“…This paper is the second in a series that is dedicated to the analysis and computation of coefficients of singularities of solutions of boundary value problems for the Poisson equation in three‐dimensional domains with edges. In the first paper entitled ‘Flux intensity functions for the Laplacian at polyhedral edges’ , we derived explicit extraction formulas for the coefficients of singularities for boundary value problems for the Poisson equation in three‐dimensional domains with straight edges and gave an illustrative example for effective implementation.…”