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首页> 外文期刊>Applied mathematics and computation >The singular function boundary integral method for 3-D Laplacian problems with a boundary straight edge singularity
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The singular function boundary integral method for 3-D Laplacian problems with a boundary straight edge singularity

机译：具有边界直边奇异性的3-D Laplacian问题的奇异函数边界积分方法

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摘要

Three-dimensional Laplace problems with a boundary straight-edge singularity caused by two intersecting flat planes are considered. The solution in the neighbourhood of the straight edge can be expressed as an asymptotic expansion involving the eigenpairs of the analogous two-dimensional problem in polar coordinates, which have as coefficients the so-called edge flux intensity functions (EFIFs). The EFIFs are functions of the axial coordinate, the higher derivatives of which appear in an inner infinite series in the expansion. The objective of this work is to extend the singular function boundary integral method (SFBIM), developed for two-dimensional elliptic problems with point boundary singularities [G.C. Georgiou, L. Olson, G. Smyrlis, A singular function boundary integral method for the Laplace equation, Commun. Numer. Methods Eng. 12 (1996) 127-134] for solving the above problem and directly extracting the EFIFs. Approximating the latter by either piecewise constant or linear polynomials eliminates the inner infinite series in the local expansion and allows the straightforward extension of the SFBIM. As in the case of two-dimensional problems, the solution is approximated by the leading terms of the local asymptotic solution expansion. These terms are also used to weight the governing harmonic equation in the Galerkin sense. The resulting discretized equations are reduced to boundary integrals by means of the 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 coefficients of the EFIFs. The SFBIM is applied to a test problem exhibiting fast convergence of order k + 1 (k being the order of the approximation of the EFIFs) in the L ~2-norm and leading to accurate estimates for the EFIFs.
机译：考虑了由两个相交的平面引起的边界直边奇异性的三维拉普拉斯问题。直边附近的解可以表示为渐进式展开，包括极坐标中类似二维问题的本征对，其系数为所谓的边通量强度函数（EFIF）。 EFIF是轴坐标的函数，其较高的导数出现在展开的内部无限级数中。这项工作的目的是扩展奇异函数边界积分方法（SFBIM），该方法是针对具有点边界奇点的二维椭圆问题开发的[G.C. Georgiou，L。Olson，G。Smyrlis，拉普拉斯方程式的一个奇异函数边界积分方法，Commun。 Numer。方法工程。为了解决上述问题并直接提取EFIFs，参见J.Am.Chem.Soc.12（1996）127-134]。通过分段常数或线性多项式近似后者，可以消除局部展开中的内部无限级数，并可以直接扩展SFBIM。与二维问题一样，该解可以通过局部渐近解展开的先导项来近似。这些术语还用于加权Galerkin意义上的控制谐波方程。借助于散度定理，将所得离散化方程简化为边界积分。然后通过拉格朗日乘数弱地执行Dirichlet边界条件。后者的值与EFIF的系数一起计算。 SFBIM适用于在L〜2范数中表现出k + 1阶快速收敛（k是EFIF近似的阶数）并导致EFIF准确估计的测试问题。

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