A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods
Most implementations of meshless BEMs use a circular integration contours (spherical in 3D) embedded into a local interpolation stencil with the so-called Companion Solution (CS) as a kernel, in order to eliminate the contribution of the single layer potential. However, the Dirichlet Green's Fu...
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Elsevier
2017
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| Online Access: | https://eprints.nottingham.ac.uk/41736/ |
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| author | Power, H. Caruso, N. Portapila, M. |
| author_facet | Power, H. Caruso, N. Portapila, M. |
| author_sort | Power, H. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Most implementations of meshless BEMs use a circular integration contours (spherical in 3D) embedded into a local interpolation stencil with the so-called Companion Solution (CS) as a kernel, in order to eliminate the contribution of the single layer potential. However, the Dirichlet Green's Function (DGF) is the unique Fundamental Solution that is identically zero at any given close surface and therefore eliminates the single layer potential. One of the main objectives of this work is to show that the CS is nothing else than the DGF for a circle collocated at its origin. The use of the DGF allows the collocation at more than one point, permitting the implementation of a P-adaptive scheme in order to improve the accuracy of the solution without increasing the number of subregions. In our numerical simulations, the boundary conditions are imposed at the interpolation stencils in contact with the problem boundary instead of at the corresponding integration surfaces, permitting always the use of circular integration contours, even in regions near or in contact with the problem domain where the densities of the integrals are reconstructed from the interpolation formulae that already included the problem boundary conditions. |
| first_indexed | 2025-11-14T19:46:27Z |
| format | Article |
| id | nottingham-41736 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:46:27Z |
| publishDate | 2017 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-417362020-05-04T18:28:21Z https://eprints.nottingham.ac.uk/41736/ A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods Power, H. Caruso, N. Portapila, M. Most implementations of meshless BEMs use a circular integration contours (spherical in 3D) embedded into a local interpolation stencil with the so-called Companion Solution (CS) as a kernel, in order to eliminate the contribution of the single layer potential. However, the Dirichlet Green's Function (DGF) is the unique Fundamental Solution that is identically zero at any given close surface and therefore eliminates the single layer potential. One of the main objectives of this work is to show that the CS is nothing else than the DGF for a circle collocated at its origin. The use of the DGF allows the collocation at more than one point, permitting the implementation of a P-adaptive scheme in order to improve the accuracy of the solution without increasing the number of subregions. In our numerical simulations, the boundary conditions are imposed at the interpolation stencils in contact with the problem boundary instead of at the corresponding integration surfaces, permitting always the use of circular integration contours, even in regions near or in contact with the problem domain where the densities of the integrals are reconstructed from the interpolation formulae that already included the problem boundary conditions. Elsevier 2017-02-01 Article PeerReviewed Power, H., Caruso, N. and Portapila, M. (2017) A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods. Engineering Analysis with Boundary Elements, 75 . pp. 57-64. ISSN 0955-7997 DRM; Companion Solution; Green's function http://www.sciencedirect.com/science/article/pii/S0955799716304970 doi:10.1016/j.enganabound.2016.12.002 doi:10.1016/j.enganabound.2016.12.002 |
| spellingShingle | DRM; Companion Solution; Green's function Power, H. Caruso, N. Portapila, M. A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods |
| title | A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods |
| title_full | A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods |
| title_fullStr | A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods |
| title_full_unstemmed | A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods |
| title_short | A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods |
| title_sort | note on the use of the companion solution (dirichlet green's function) on meshless boundary element methods |
| topic | DRM; Companion Solution; Green's function |
| url | https://eprints.nottingham.ac.uk/41736/ https://eprints.nottingham.ac.uk/41736/ https://eprints.nottingham.ac.uk/41736/ |