A New and Efficient Meshless Computational Approach for Transient Heat Conduction in Anisotropic Nonhomogeneous Media: Fragile Points Method and Local Variational Iteration Schemeintegration Scheme

In this paper, we present a new and effective computational approach for analyzing 2D and 3D transient heat conduction problems. The proposed approach consists of two parts: a purely meshless Fragile Points Method (FPM) being utilized for spatial discretization, and a Local Variational Iteration (LVI) scheme used to obtain the solution in the time domain. The FPM is based on a Galerkin weak-form formulation. Local, very simple, polynomial and discontinuous trial and test functions are employed. Interior Penalty Numerical Fluxes are introduced to ensure the consistency of the method. The FPM leads to sparse and symmetric matrices, which is good for modeling complex problems with a large number of DoFs. And as discontinuous trial functions are used, it also has advantages in analyzing problems with rupture and fragmentation, e.g., crack development in thermally shocked brittle materials. The LVIM for time discretization is a combination of the Variational Iteration Method (VIM) and a collocation method and numerical discretization in each finitely large time interval. It possesses excellent efficiency in solving nonlinear ODEs and has the potential in being implemented with parallel processing.

The present methodology represents a considerable improvement to the current state of science in computational transient heat conduction in anisotropic nonhomogeneous media. It circumvents the problem of tedious numerical integration caused by the Moving Least Squares (MLS) approximations in the previous Element-Free Galerkin (EFG) Method and the Meshless Local Petrov-Galerkin (MLPG) Method. And the time integration scheme LVIM is considerably superior to the classic finite difference methods. Anisotropy and nonhomogeneity of the media, which have significant influences on the temperature distribution, do not give rise to any difficulties in the present computational approach.

A number of numerical examples in 2D and 3D are provided as validations. The FPM + LVIM approach shows its capability and accuracy in solving transient heat transfer problems in complex geometries with mixed boundary conditions (e.g., essential, natural, convective, and purely symmetric boundary conditions), including pre-existing cracks. The computed solutions are compared with analytical results, equivalent 1D or 2D results, and FEM solutions obtained with ABAQUS. Both functionally graded materials and composite materials are considered. The computing efficiency is extraordinary when the response varies dramatically, or a high accuracy is required. A discussion and recommended range of the penalty parameters and the number of collocation nodes in each time interval in the FPM + LVIM method are also presented. It is shown that, with appropriate computational parameters, the proposed FPM + LVIM approach is not only accurate, but also efficient, and has reliable stability under relatively large time intervals.