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

Miltiades C. Elliotis

doi:doi:10.11648/j.pamj.20160506.14

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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

Miltiades C. Elliotis

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 6)

Received: 3 October 2016 Accepted: 15 October 2016 Published: 10 November 2016

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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 post-processed finite element results.

Published in	Pure and Applied Mathematics Journal (Volume 5, Issue 6)
DOI	10.11648/j.pamj.20160506.14
Page(s)	192-204
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Laplace Equation, Boundary Singularity, Straight Edge Singularity, Singular Coefficients, Edge Flux Intensity Functions, Singular Function Boundary Integral Method

References

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APA Style

Miltiades C. Elliotis. (2016). The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity. Pure and Applied Mathematics Journal, 5(6), 192-204. https://doi.org/10.11648/j.pamj.20160506.14

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ACS Style

Miltiades C. Elliotis. The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity. Pure Appl. Math. J. 2016, 5(6), 192-204. doi: 10.11648/j.pamj.20160506.14

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AMA Style

Miltiades C. Elliotis. The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity. Pure Appl Math J. 2016;5(6):192-204. doi: 10.11648/j.pamj.20160506.14

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@article{10.11648/j.pamj.20160506.14,
  author = {Miltiades C. Elliotis},
  title = {The Singular Function Boundary Integral Method for the  2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {6},
  pages = {192-204},
  doi = {10.11648/j.pamj.20160506.14},
  url = {https://doi.org/10.11648/j.pamj.20160506.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160506.14},
  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 post-processed finite element results.},
 year = {2016}
}

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TY  - JOUR
T1  - The Singular Function Boundary Integral Method for the  2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity
AU  - Miltiades C. Elliotis
Y1  - 2016/11/10
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160506.14
DO  - 10.11648/j.pamj.20160506.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 192
EP  - 204
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160506.14
AB  - 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 post-processed finite element results.
VL  - 5
IS  - 6
ER  -

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Author Information

Miltiades C. Elliotis

Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus

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APA Style

Miltiades C. Elliotis. (2016). The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity. Pure and Applied Mathematics Journal, 5(6), 192-204. https://doi.org/10.11648/j.pamj.20160506.14

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ACS Style

Miltiades C. Elliotis. The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity. Pure Appl. Math. J. 2016, 5(6), 192-204. doi: 10.11648/j.pamj.20160506.14

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AMA Style

Miltiades C. Elliotis. The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity. Pure Appl Math J. 2016;5(6):192-204. doi: 10.11648/j.pamj.20160506.14

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@article{10.11648/j.pamj.20160506.14,
  author = {Miltiades C. Elliotis},
  title = {The Singular Function Boundary Integral Method for the  2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {6},
  pages = {192-204},
  doi = {10.11648/j.pamj.20160506.14},
  url = {https://doi.org/10.11648/j.pamj.20160506.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160506.14},
  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 post-processed finite element results.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The Singular Function Boundary Integral Method for the  2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity
AU  - Miltiades C. Elliotis
Y1  - 2016/11/10
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160506.14
DO  - 10.11648/j.pamj.20160506.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 192
EP  - 204
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160506.14
AB  - 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 post-processed finite element results.
VL  - 5
IS  - 6
ER  -

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