Fundamental-Solution-Based Hybrid Element Model for Nonlinear Heat Conduction Problems with Temperature-Dependent Material Properties

Abstract

The boundary-type hybrid finite element formulation coupling the Kirchhoff transformation is proposed for the two-dimensional nonlinear heat conduction problems in solids with or without circular holes, and the thermal conductivity of material is assumed to be in terms of temperature change. The Kirchhoff transformation is firstly used to convert the nonlinear partial differential governing equation into a linear one by introducing the Kirchhoff variable, and then the new linear system is solved by the present hybrid finite element model, in which the proper fundamental solutions associated with some field points are used to approximate the element interior fields and the conventional shape functions are employed to approximate the element frame fields. The weak integral functional is developed to link these two fields and establish the stiffness equation with sparse and symmetric coefficient matrix. Finally, the algorithm is verified on several examples involving various expressions of thermal conductivity and existence of circular hole, and numerical results show good accuracy and stability.

Green's function based finite element formulations for isotropic seepage analysis with free surface

“…Compared to the MFS, which employs the boundary discretization of the entire domain, the present method divides the entire domain into several small elements and in each element, the linear combination of Green's function is employed to represent the interior hydraulic head. Thus, the present method has better interpolation stability than the MFS and the location of the source points can be flexibly arranged outside the element domain (Wang and Qin, 2009;Wang and Qin, 2010a;Wang et al, 2013).…”

“…In this study, another numerical method different to the conventional FEM and BEM (Bathe, 2006;Qin, 1994;1995;2003;Qin and Mai, 2002), called the fundamental solution-based (or Green's function based) hybrid FEM (HFS-FEM), is formulated for solving such problems in two-dimensional isotropic dams and a multi-node element is developed to model the region close to the free surface for simplifying the mesh redefinition. The HFS-FEM was firstly presented by Wang and Qin for heat transfer analysis (Wang and Qin, 2009) and then was extended to analyze elastic stress field (Wang and Qin, 2010b;Wang and Qin, 2011a;Wang and Qin, 2012b), thermal properties of advanced functional/composite materials (Wang and Qin, 2011b;Wang et al, 2012;Wang et al, 2013) and bioheat transfer in biological tissues (Wang and Qin, 2010a;Wang and Qin, 2012a) with general/special elements to achieve the purpose of high accuracy and mesh reduction. It should be mentioned that convergence of the HFS-FEM was fully discussed in these works.…”

The singular boundary method for steady-state nonlinear heat conduction problem with temperature-dependent thermal conductivity

“…One-dimensional nonlinear heat conduction problem has been solved with some semi analytical methods, such as the perturbation method (PM) [11,12], the variational iteration method (VIM) [13], the homotopy analysis method (HAM) [14], the differential transform method (DTM) [3][4][5][6]15,16] and the Adomian decomposition method (ADM) [8,17]. Two-dimensional boundary value problems have been the subject of several studies using the ADM [9], the finite element method (FEM) [18][19][20], the boundary element method (BEM) [21][22][23], the method of fundamental solutions (MFS) [24,25], the fundamental solution-based hybrid finite element method (HFS-FEM) [26], the hybrid Trefftz finite element method (HT-FEM) [27] or the boundary knot method (BKM) [28,29]. In [10], homotopy perturbation sumudu transform method (HPSTM) has been used to evaluate temperature distribution and effectiveness of radial fins with temperature-dependent thermal conductivity and exposed to convection.…”

Orthotropic Seepage Analysis using Hybrid Finite Element Method

“…In this study, another numerical method different to the FEM and BEM mentioned above, called as the fundamental solution-based hybrid finite element method, was extended to two-dimensional seepage problems in orthotropic media. The hybrid finite element formulation with fundamental solutions was firstly proposed by Qin in 2009 (Wang andQin 2009) and then was applied to analyze elastic problems , heat transfer problems Wang et al 2013) and bioheat problems with general or special elements to achieve the purpose of high accuracy and mesh reduction. In the present work, fundamental solutions of the orthotropic seepage governing equation under consideration are employed to construct the element interior hydraulic head field and the conventional shape functions are used for the element frame hydraulic head field.…”