A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods

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• 作者：H. Power^a ; ^{henry.power@nottingham.ac.uk} ; N. Caruso^b ; ^c ; M. Portapila^b ; ^c
• 关键词：DRM ; Companion Solution ; Green's function
• 刊名：Engineering Analysis with Boundary Elements
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：75
• 期：Complete
• 页码：57-64
• 全文大小：571 K
• 卷排序：75

文摘

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.