三維位勢問題新的規則化邊界元法

Abstract

本文致力於三維位勢問題的間接變量規則化邊界元法研究，提出了新的規則化邊界元法的理論和方法，構造了與法向量關聯的兩個線性無關的特別切向量，建立與問題基本解有關的量的法向、切向梯度的特性定理，提出轉化域積分方程為邊界積分方程的極限定理，在此基礎上，尋出間接變量規則化邊界積分方程，與廣泛實踐的直接邊界元法比，本文具有優點: (1)降低了密度函數的連續性要求; (2)更適合求解薄體結構問題，因為所給方程中不含超奇異與幾乎超奇異積分，積分的規則化算法更加有效; (3)可計算任何邊界位勢梯度，數值實施時，C0連續單元描述幾何曲面，不連續差值逼近邊界量，針對問題的特殊的邊界曲面，提出一種精確幾何單元。數值算例表明，本文算法穩定、效率高，所得數值結果與精確解相當地吻合。This presentation is mainly devoted to the research on the regularization of indirect boundary integral equations (IBIEs) for three-dimensional problems and establishes the new theory and method of the regularized BEM. The two special tangential vectors, which are linearly independent and associated with the normal vectors, are constructed, and then a characteristics theorem for the contour integrations of the normal and tangential gradients of some quantities, related with the fundamental solutions for 3D potential problems, is presented. A limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) is also proposed. Based on this, together with a novel decomposition technique to the fundamental solution, the regularized BIEs with indirect unknowns, which don't involve the direct calculation of CPV and HFP integrals, are derived for 3D potential problems. Compared with the widely practiced direct regularized BEMs, the presented method has many advantages. First, the continuity requirement for density function in the direct formulation can be reduced here. Second, it is more suitable for solving the structures of thin bodies, considering the solution process for boundary or field quantities doesn't involve the HFP integrals and nearly HFP integrals so the regularization algorithm to the considered singular or nearly singular integrals is more effective. Third, the proposed regularized BIEs can calculate the any potential gradients on the boundary, but not limited to the normal fluxes, and also independent of the potential BIEs. A systematic approach for implementing numerical solutions is proposed by adopting the C0 continuous elements to depict the boundary surface and the discontinuous interpolation to approximate the boundary quantities. Especially, for the boundary value problems with elliptic surfaces or piecewise plane surfaces boundary, the exact elements are developed to model their boundaries with almost no error. The validity of the proposed scheme is demonstrated by several benchmark examples. Excellent agreement between the numerical results and exact solutions is obtained even with using small amounts of element.