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Three kinds of Fragile Points Methods based on Petrov-Galerkin weak-forms(PG-FPMs) are proposed for analyzing heat conduction problems in nonhomogeneousanisotropic media. This is a follow-up of the previous study on the originalFPM based on a symmetric Galerkin weak-form. The trial function ispiecewise-continuous, written as local Taylor expansions at the Fragile Points.A modified Radial Basis Function-based Differential Quadrature (RBF-DQ) methodis employed for establishing the local approximation. The Dirac delta function,Heaviside step function, and the local fundamental solution of the governingequation are alternatively used as test functions. Vanishing or pure contourintegral formulation in subdomains or on local boundaries can be obtained.Extensive numerical examples in 2D and 3D are provided as validations. Thecollocation method (PG-FPM-1) is superior in transient analysis with arbitrarypoint distribution and domain partition. The finite volume method (PG-FPM-2)shows the best efficiency, saving 25% to 50% computational time comparing withthe Galerkin FPM. The singular solution method (PG-FPM-3) is highly efficientin steady-state analysis. The anisotropy and nonhomogeneity give rise to nodifficulties in all the methods. The proposed PG-FPM approaches represent animprovement to the original Galerkin FPM, as well as to other meshless methodsin earlier literature.

Meshless Fragile Points Methods Based on Petrov-Galerkin Weak-Forms for  Transient Heat Conduction Problems in Complex Anisotropic Nonhomogeneous  Media