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A new and effective computational approach is presented for analyzingtransient heat conduction problems. The approach consists of a meshless FragilePoints Method (FPM) being utilized for spatial discretization, and a LocalVariational Iteration (LVI) scheme for time discretization. Anisotropy andnonhomogeneity do not give rise to any difficulties in the presentimplementation. The meshless FPM is based on a Galerkin weak-form formulationand thus leads to symmetric matrices. Local, very simple, polynomial anddiscontinuous trial and test functions are employed. In the meshless FPM,Interior Penalty Numerical Fluxes are introduced to ensure the consistency ofthe method. The LVIM in the time domain is generated as a combination of theVariational Iteration Method (VIM) applied over a large time interval andnumerical algorithms. A set of collocation nodes are employed in each finitelylarge time interval. The FPM + LVIM approach is capable of solving transientheat transfer problems in complex geometries with mixed boundary conditions,including pre-existing cracks. Numerical examples are presented in 2D and 3Ddomains. Both functionally graded materials and composite materials areconsidered. It is shown that, with suitable computational parameters, the FPM +LVIM approach is not only accurate, but also efficient, and has reliablestability under relatively large time intervals. The present methodologyrepresents a considerable improvement to the current state of science incomputational transient heat conduction in anisotropic nonhomogeneous media.

A New Meshless "Fragile Points Method" and A Local Variational Iteration  Method for General Transient Heat Conduction in Anisotropic Nonhomogeneous  Media