On the free terms of the dual BEM for the two and three-dimensional Laplace problems

Abstract

A dual integral formulation for the Laplace problem with a smooth boundary is derived by using the contour approach surrounding the singularity. It is found that using the contour approach the jump terms come half and half from the free terms in the L and Mkernel integrations for the two-dimensional case, which is different from the limiting process by approaching an interior point to a boundary point where the jump terms come totally from the L kernel only. The definition of the Hadamard principal value for hypersingular integral at the collocation point of a smooth boundary is extended to a generalized sense for both the tangent and normal derivatives of double-layer potentials in comparison with the conventional definition. For the three dimensional case, the jump terms come one-third and two-thirds from the free terms of L and M kernels, respectively.本文探討Laplace方程式的對偶邊界積分方程之自由項。針對圍繞在平滑邊界上奇異點周圍的半圓(二維)或半球面(三維)積分可導得自由項。在二維問題中，自由項的來源有二，一半來自Ｌ核函數，另一半來自Ｍ核函數。意即自由項分別由Ｌ核函數與Ｍ核函數各貢獻一半。這個過程雖不同于極限方法從域內點逼近到邊界點中，跳躍項完全來自Ｌ核函數所貢獻，但是最終的結果仍然是相同的。在超奇異積分方程中，阿馬主值的觀念在此從雙層勢能的法向微分推廣到切向微分。有趣的是，在三維問題中，我們發現到自由項分別來自Ｌ核函數的三分之一貢獻與Ｍ核函數的三分之二貢獻。