Singularities and treatments of elliptic boundary value problems

Li, Z. C.; Lu, Tzon‐Tzer

doi:10.1016/s0895-7177(00)00062-5

Cited by 70 publications

(50 citation statements)

References 210 publications

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“…Since the singular functions W k (j) satisfy exactly the boundary conditions on S A and S B and considering the boundary conditions (8), the boundary integral in (30) is identically zero along S A and S B . The discretized equations (30) …”

Section: The Sfbim In the 3-d Casementioning

confidence: 99%

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The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity

Elliotis¹

2016

PAMJ

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Abstract:In this study the Singular Function Boundary Integral Method (SFBIM) is implemented in the case of a planar elliptic boundary value problem in Mechanics, with a point boundary singularity. The method is also extended in the case of a typical problem of Solid Mechanics, concerning the Laplace equation problem in three dimensions, defined in a domain with a straight edge singularity on the surface boundary. In both the 2-D and 3-D cases, the general solution of the Laplace equation is approximated by the leading terms (which contain the singular functions) of the local asymptotic solution expansion. The singular functions are used to weight the governing equation in the Galerkin sense. For the 2-D Laplacian model problem of this study, which is defined over a domain with a re-entrant corner, the resulting discretized equations are reduced to boundary integrals by means of Green's second identity. For the 3-D model problem of this work, the volume integrals of the discretized equations are reduced to surface integrals by implementing Gauss' divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers. The values of the latter are calculated together with the singular coefficients, in the 2-D case or the Edge Flux Intensity Functions (EFIFs), in the 3-D model problem, which appear in the local solution expansion. For the planar problem, the numerical results are favorably compared with the analytic solution. Especially for the extension of the method in three dimensions, the preliminary numerical results compare favorably with available postprocessed finite element results.

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Section: The Sfbim In the 3-d Casementioning

confidence: 99%

“…[1-3, 5, 24, 25]). An extensive survey of the treatment of singularities in elliptic 2-D boundary value problems is provided in [8]. Of course, in the absence of a boundary singularity, there are many other efficient numerical techniques available in the literature (e.g.…”

Section: Introductionmentioning

confidence: 99%

The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity

Elliotis¹

2016

PAMJ

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“…As described in Section 2.5, the total laser field U laser is divided into incoming and scattered parts, U inc laser and U sc laser , respectively, with U inc laser being the radiation in laser traveling from minus infinity to l along e x . We now require a description for the trace of U sc laser (x, y) on l , u sc l (y) as defined in Equation (29). Coefficients {d q } q∈N 0 can be obtained by taking the inner product between U sc laser with a mode h p , p ∈ N 0 along any line with constant x.…”

Section: Eigenfunction Expansion At the Laser Output Lmentioning

confidence: 99%

Hybrid FEM/BEM modeling of finite‐sized photonic crystals for semiconductor laser beams

Jerez-Hanckes

Durán

Guarini

2010

Numerical Meth Engineering

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SUMMARYWe propose a 2-D finite element/boundary element hybrid method for calculating the spatial distribution and frequency response of electromagnetic waves coming from a semiconductor laser when interacting with a finite-sized photonic crystal. We thus provide a flexible tool for the design of novel optical and microwave devices, among other applications. In opposition to current methodologies, we simultaneously take into account the laser modes, the finiteness of the crystal, and the unboundedness of the isotropic medium in which the crystal is embedded. At the laser output, instead of approximating reflected and transmitted beams by plane waves, we use the more realistic Hermite-Gauss functions. In the isotropic medium, we set an artificial boundary encircling the crystal and define exterior and interior domains. Radiating solutions for the scattered far field over the exterior are derived analytically through a series of Hankel polynomials. The interior domain is described by a finite element formulation coupled with Dirichlet-to-Neumann maps enforcing laser and far-field behaviors. Results and error analyses are provided in view of future improvements.

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“…An exhaustive survey of treatment of singularities is provided in the recent papers by Li and Lu [4], Dosiyev [5], Xenophontos et al [6] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On solving the cracked‐beam problem by block method

Dosiyev¹,

Buranay²

2007

Commun. Numer. Meth. Engng.

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SUMMARYAn extremely accurate solution is obtained for the cracked-beam problem by one-block version of the block method (BM). The obtained numerical results demonstrate the exponential convergence of the BM with respect to the number of quadrature nodes. A simple and high accurate formula to compute the stress intensity factor is given. The comparisons with other high accurate results in the literature have been carried out.

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Singularities and treatments of elliptic boundary value problems

Cited by 70 publications

References 210 publications

The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity

The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity

Hybrid FEM/BEM modeling of finite‐sized photonic crystals for semiconductor laser beams

On solving the cracked‐beam problem by block method

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