三维轴对称曲面上边界积分方程快速求解的直接解法
A Direct Solver for the Rapid Solution of Boundary Integral Equations on Axisymmetric Surfaces in Three Dimensions
AI总结：
提出一种利用傅里叶变换将三维轴对称曲面上的边界积分方程降维为生成曲线上的序列方程，并采用高精度高斯求积和直接求逆的小规模线性系统快速求解的方法。
摘要：
提出了一种快速精确计算三维旋转对称曲面上边界积分方程（BIE）解的方案。该方案利用傅里叶变换将原始定义在曲面上的BIE降维为定义在生成曲线上的一系列BIE。它可以处理不一定旋转对称的载荷。采用Nystrom离散化对生成曲线上的BIE进行离散。所使用的求积法是高阶高斯规则，并在对角线附近进行修正以保持奇异核的高阶精度。维度的降低以及高阶精确求积法的使用，导致线性系统规模较小，可以通过例如高斯消元法直接求逆。这使得该方案在处理多个右端项时特别快速。结果表明，对于与拉普拉斯方程相关的BIE，通过利用勒让德函数的递推关系，可以非常快速地计算简化方程中的核。数值例子说明了该方案的性能；特别地，对于使用320,000个点离散的曲面上的拉普拉斯方程相关BIE，算法的设置阶段在标准台式机上耗时2分钟，然后求解可在0.5秒内完成。
英文摘要：
A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in three dimensions is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nystrom discretization is used to discretize the BIEs on the generating curve. The quadrature used is a high-order Gaussian rule that is modified near the diagonal to retain high-order accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with Laplace's equation, the kernel in the reduced equations can be evaluated very rapidly by exploiting recursion relations for Legendre functions. Numerical examples illustrate the performance of the scheme; in particular, it is demonstrated that for a BIE associated with Laplace's equation on a surface discretized using 320 000 points, the set-up phase of the algorithm takes 2 minutes on a standard desktop, and then solves can be executed in 0.5 seconds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
