  A new and effective computational approach is presented for analyzing
transient heat conduction problems. The approach consists of a meshless Fragile
Points Method (FPM) being utilized for spatial discretization, and a Local
Variational Iteration (LVI) scheme for time discretization. Anisotropy and
nonhomogeneity do not give rise to any difficulties in the present
implementation. The meshless FPM is based on a Galerkin weak-form formulation
and thus leads to symmetric matrices. Local, very simple, polynomial and
discontinuous trial and test functions are employed. In the meshless FPM,
Interior Penalty Numerical Fluxes are introduced to ensure the consistency of
the method. The LVIM in the time domain is generated as a combination of the
Variational Iteration Method (VIM) applied over a large time interval and
numerical algorithms. A set of collocation nodes are employed in each finitely
large time interval. The FPM + LVIM approach is capable of solving transient
heat transfer problems in complex geometries with mixed boundary conditions,
including pre-existing cracks. Numerical examples are presented in 2D and 3D
domains. Both functionally graded materials and composite materials are
considered. It is shown that, with suitable computational parameters, the FPM +
LVIM approach is not only accurate, but also efficient, and has reliable
stability under relatively large time intervals. The present methodology
represents a considerable improvement to the current state of science in
computational transient heat conduction in anisotropic nonhomogeneous media.
