A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials

Abstract

In this article, we propose a novel meshless method for solving two-dimensional stationary heat conduction problems in layered materials. The proposed method is a recently developed boundary-type meshless method which combines the collocation scheme from the method of fundamental solutions (MFS) with the collocation Trefftz method (CTM) to improve the applicability of the method for solving boundary value problems. Particular non-singular basis functions from cylindrical harmonics are adopted in which the numerical approximation is based on the superposition principle using the non-singular basis functions expressed in terms of many source points. For the modeling of multi-layer composite materials, we adopted the domain decomposition method (DDM), which splits the domain into smaller subdomains. The continuity of the flux and the temperature has to be satisfied at the interface of subdomains for the problem. The validity of the proposed method is investigated for several test problems. Numerical applications were also carried out. Comparison of the proposed method with other meshless methods showed that it is highly accurate and computationally efficient for modeling heat conduction problems, especially in heterogeneous multi-layer composite materials.

On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach

“…In the numerical implementation, the terms of the particular solutions, k , was 15. The Dirichlet boundary data were imposed on the domain boundary using three different analytical solutions, as depicted in Equations (27)- (29). The wave number λ was set as π .…”

“…This approach was then used for solving the Laplace-type subsurface flow problem. Xiao et al [29] further applied the method to the problems of heat transfer in heterogeneous multilayer materials in two dimensions. Since the hybrid boundary-type meshless methods are only found to solve Laplace-type problems, it is especially interesting to explore this newly developed method for solving other non-Laplace problems.…”

“…Starting from previous studies [28,29], we conducted pioneering work to solve the modified Helmholtz equation in layered materials. Since it is necessary to derive particular nonsingular basis functions for the two-dimensional modified Helmholtz equation, the specific method in this study is named the multiple source meshfree approach (MSMA).…”

The Method of Fundamental Solutions for Three-Dimensional Nonlinear Free Surface Flows Using the Iterative Scheme

“…Comparing to conventional mesh-based methods, meshless methods are relatively simple because only arbitrary collocation points need to be placed on the physical domain [22]. In particular, the collocation points may be placed only on the boundary for the method of fundamental solutions (MFS) [23][24][25][26][27][28][29] because the basis function is the fundamental solution which satisfies the governing equation.…”

On Solving Two-Dimensional Inverse Heat Conduction Problems Using the Multiple Source Meshless Method

“…The CTM and MFS are therefore referred to boundary-type meshless methods, which usually suffer from the numerical instability due to the ill-posed phenomenon of the meshless method. To improve the applicability of the CTM and MFS, Ku et al [29,30] propose the multiple source meshless method (MSMM). The newly developed MSMM is modified from the CTM and MFS and combines the benefits of both methods.…”