  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.
