具有边界奇异性的二维和三维力学拉普拉斯方程问题的奇异函数边界积分法

M. Elliotis
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引用次数: 2

摘要

本文采用奇异函数边界积分法(SFBIM)求解具有点边界奇异性的平面椭圆型力学边值问题。将该方法推广到一个典型的固体力学问题,即三维空间的拉普拉斯方程问题,该问题定义在表面边界上有直边奇点的域上。在二维和三维情况下,拉普拉斯方程的通解由局部渐近解展开的前导项(包含奇异函数)逼近。用奇异函数对伽辽金意义上的控制方程进行加权。对于本研究的二维拉普拉斯模型问题,其定义在具有可重入角的区域上,通过格林第二恒等式将得到的离散方程简化为边界积分。对于三维模型问题,利用高斯散度定理将离散方程的体积积分化为曲面积分。然后利用拉格朗日乘子弱地实现狄利克雷边界条件。后者的值与奇异系数一起计算,在二维情况下或在三维模型问题中出现在局部解展开中的边缘通量强度函数(EFIFs)。对于平面问题,数值计算结果与解析解的结果比较良好。特别是将该方法扩展到三维时,初步数值结果与现有的后处理有限元结果比较满意。
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The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity
In this study the Singular Function Boundary Integral Method (SFBIM) is implemented in the case of a planar elliptic boundary value problem in Mechanics, with a point boundary singularity. The method is also extended in the case of a typical problem of Solid Mechanics, concerning the Laplace equation problem in three dimensions, defined in a domain with a straight edge singularity on the surface boundary. In both the 2-D and 3-D cases, the general solution of the Laplace equation is approximated by the leading terms (which contain the singular functions) of the local asymptotic solution expansion. The singular functions are used to weight the governing equation in the Galerkin sense. For the 2-D Laplacian model problem of this study, which is defined over a domain with a re-entrant corner, the resulting discretized equations are reduced to boundary integrals by means of Green’s second identity. For the 3-D model problem of this work, the volume integrals of the discretized equations are reduced to surface integrals by implementing Gauss’ 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 singular coefficients, in the 2-D case or the Edge Flux Intensity Functions (EFIFs), in the 3-D model problem, which appear in the local solution expansion. For the planar problem, the numerical results are favorably compared with the analytic solution. Especially for the extension of the method in three dimensions, the preliminary numerical results compare favorably with available post-processed finite element results.
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期刊介绍: The “Italian Journal of Pure and Applied Mathematics” publishes original research works containing significant results in the field of pure and applied mathematics.
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