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Simulation of two-dimensional steady-state heat conduction problems by a fast singular boundary method

机译：快速奇异边界法模拟二维稳态导热问题

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摘要

The study presents a fast direct algorithm for solutions of systems arising from singular boundary method (SBM) in two-dimensional (2D) steady-state heat conduction problems. As a strong-form boundary discretization collocation technique, the SBM is mathematically simple, easy-to-implement, and free of mesh. Similar to boundary element method (BEM), the SBM generates a dense coefficient matrix, which requires O(N-3) operations and O(N-2) memory to solve the system with direct solvers. In this study, a fast direct solver based on recursive skeletonization factorization (RSF) is applied to solve the linear systems in the SBM. Due to the fact that the coefficient matrix is hierarchically block separable, we can construct a multilevel generalized LU decomposition that allows fast application of the inverse of the coefficient matrix. Combing the RSF and the SBM can dramatically reduce the operations and the memory to O(N). The accuracy and effectiveness of the RSF-SBM are tested through some 2D heat conduction problems including isotropic and anisotropic cases.
机译：该研究提出了一种快速直接算法，用于求解二维（2D）稳态热传导问题中的奇异边界方法（SBM）产生的系统。作为一种强形式的边界离散搭配技术，SBM在数学上简单，易于实现且没有网格。类似于边界元方法（BEM），SBM生成一个密集系数矩阵，该矩阵需要O（N-3）运算和O（N-2）内存才能使用直接求解器求解系统。在这项研究中，基于递归骨架分解（RSF）的快速直接求解器被应用于求解SBM中的线性系统。由于系数矩阵在层次上是块可分离的，因此我们可以构造一个多层次的广义LU分解，从而可以快速应用系数矩阵的逆。将RSF和SBM结合在一起可以极大地减少操作和内存到O（N）。通过一些2D导热问题（包括各向同性和各向异性情况）来测试RSF-SBM的准确性和有效性。

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