Three kinds of fragile points methods based on Petrov-Galerkin weak-forms (PG-FPMs) are proposed for analyzing heat conduction problems in nonhomogeneous anisotropic media. This is a follow-up of the previous study on the original FPM based on a symmetric Galerkin weak-form. The trial function is piecewise-continuous, written as local Taylor expansions at the fragile points. A modified radial basis function-based differential quadrature (RBF-DQ) method is employed for establishing the local approximation. The Dirac delta function, Heaviside step function, and the local fundamental solution of the governing equation are alternatively used as test functions. Vanishing or pure contour integral formulation in subdomains or on local boundaries can be obtained. Extensive numerical examples in 2D and 3D are provided as validations. The collocation method (PG-FPM-1) is superior in transient analysis with arbitrary point distribution and domain partition. The finite volume method (PG-FPM-2) shows the best efficiency, saving 25% to 50% computational time comparing with the Galerkin FPM. The singular solution method (PG-FPM-3) is highly efficient in steady-state analysis. The anisotropy and nonhomogeneity give rise to no difficulties in all the methods. The proposed PG-FPM approaches represent an improvement to the original Galerkin FPM, as well as to other meshless methods in earlier literature.

中文翻译：

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复杂各向异性非均匀介质瞬态热传导问题的基于 Petrov-Galerkin 弱形式的无网格脆弱点方法

提出了三种基于 Petrov-Galerkin 弱形式 (PG-FPM) 的脆弱点方法来分析非均匀各向异性介质中的热传导问题。这是之前基于对称伽辽金弱形式的原始 FPM 研究的后续。试验函数是分段连续的，写成脆弱点的局部泰勒展开式。采用改进的基于径向基函数的微分正交 (RBF-DQ) 方法来建立局部近似。Dirac delta 函数、Heaviside 阶跃函数和控制方程的局部基本解交替用作测试函数。可以获得子域或局部边界上的消失或纯轮廓积分公式。提供了大量 2D 和 3D 数值示例作为验证。搭配方法（PG-FPM-1）在具有任意点分布和域划分的瞬态分析中具有优越性。有限体积法 (PG-FPM-2) 显示出最佳效率，与 Galerkin FPM 相比，可节省 25% 至 50% 的计算时间。奇异解法 (PG-FPM-3) 在稳态分析中非常有效。各向异性和非均匀性在所有方法中都没有引起任何困难。提出的 PG-FPM 方法代表了对原始 Galerkin FPM 以及早期文献中其他无网格方法的改进。奇异解法 (PG-FPM-3) 在稳态分析中非常有效。各向异性和非均匀性在所有方法中都没有引起任何困难。提出的 PG-FPM 方法代表了对原始 Galerkin FPM 以及早期文献中其他无网格方法的改进。奇异解法 (PG-FPM-3) 在稳态分析中非常有效。各向异性和非均匀性在所有方法中都没有引起任何困难。提出的 PG-FPM 方法代表了对原始 Galerkin FPM 以及早期文献中其他无网格方法的改进。