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The particular and homogeneous solutions of two-dimension (2D) thermal-elastic problems are expressed by using the virtual boundary element method (VBEM) and the superposition principle. Accordingly, two matrix equations are needed and formed for particular and homogeneous solutions by the novel nonsingular boundary element method with the radial basis function interpolation (RBFI), namely virtual boundary meshfree Galerkin method (VBMGM). Considering the boundary conditions of heat conduction and combining the Galerkin method, virtual source functions of particular solutions can be solved by first using VBMGM. Applying boundary conditions of transformation, virtual source functions of homogeneous solutions can be obtained by second employing VBMGM. The coefficients of two matrix equations are symmetrical. Formed equations are nonsingular and have merits of virtual boundary element method, meshfree method, and Galerkin method. The detailed expressions, calculation steps, and simplified flow chart are given in detail, that is convenient for other scholars to study other more complex thermo-elastic problems through the method of this paper. Three examples are calculated. Comparing their calculation results with other methods, the stability and the precision of VBMGM for 2D thermal-elastic problems are validated. [ABSTRACT FROM AUTHOR]

Solutions of 3D thermal analysis in inhomogeneous media with variable thermal source are divided into homogeneous and particular solutions by a novel boundary-type element-free method, namely virtual boundary element-free Galerkin method (VBEFGM). The homogeneous solution can be obtained by the virtual boundary element-free method (VBEFM). The virtual source function of the homogeneous solution and the unknown coefficient of the particular solution are constructed by the radial basis function interpolation (RBFI). And VBEFGM for 3D thermal analysis in inhomogeneous media with variable thermal source is real meshfree and does not require the background integral element. Its equation is obtained the Galerkin method of the weighted residual methods. The demonstration of ANSYS Parametric Design Language (APDL) can greatly facilitate the programming for the extraction of the coordinate information and the outer normal cosines on nodes. The calculation scheme and the specific weighted coefficients are deducted in detail for 3D thermal analysis in inhomogeneous media with variable thermal source by VBEFGM. The numerical calculational steps and flow chart are given. Three numerical examples are computed. The results are compared with other numerical methods and the exact solutions. The stability and the accuracy of the proposed method is proved by VBEFGM for 3D thermal analysis in inhomogeneous media with variable thermal source. [ABSTRACT FROM AUTHOR]

The virtual boundary meshfree Galerkin method (VBMGM) is further developed to compute three-dimensional single-domain steady-state heat conduction problem. Using VBMGM, three-dimensional problem is reduced into two-dimensional boundary, whose element information can be obtained by the meshing software, namely ANSYS software. It greatly facilitates the programing implementation of the proposed method that ANSYS Parametric Design Language (APDL) is demonstrated for the element information acquisition of the curved surface. The virtual source function on the virtual boundary element method (VBEM) is interpolated by the radial basis function interpolation (RBFI) of the meshfree method. The calculation equation of three-dimensional single-domain steady-state heat conduction problem by VBMGM is constructed by the Galerkin method of the weighted residual method. Therefore, it has the advantages of the VBEM, the meshfree method and the Galerkin method that the proposed method is employed to computed three-dimensional single-domain steady-state heat conduction problem. It is beneficial to further popularize the proposed method to solve other three-dimensional heat conduction problems that the detailed numerical discrete formula of the proposed method for three-dimensional single-domain steady-state heat conduction problem is obtained. The numerical results of two numerical examples are computed and compared with the other numerical method and the extract solutions. The accuracy and stability of VBMGM for three-dimensional single-domain steady-state heat conduction problem are verified. [ABSTRACT FROM AUTHOR]

The virtual boundary meshfree Galerkin method (VBMGM) is demonstrated to solve the single-domain transient heat conduction with heat source problem. Both the heat source in the computational domain and the virtual source function on the virtual boundary are interpolated by using the radial basis function interpolation. The equation of the proposed method is formed by the Galerkin method. Consequently, VBMGM have the advantages of the Galerkin method, the boundary element method and the meshfree method for the single-domain transient heat conduction with heat source problem. And the numerical definitions of the weighted values in VBMGM are definite, such as the partial derivatives of the temperature, the heat flux and the heat source expression at t + Δ t time. The detailed equation of VBMGM is deducted for the single-domain transient heat conduction with heat source problem. Two numerical examples, the square domain with constant heat source and the square domain with variable boundary condition and variable heat source, are presented. The accuracy and the stability of VBMGM for the single-domain transient heat conduction with heat source problem are proved. [ABSTRACT FROM AUTHOR]

In this article, a new high-precision boundary-type mesh-free method, called the virtual boundary mesh-free Galerkin method (VBMGM), is given for analyzing the heat conduction problem consisting of multiple media. A control equation of the proposed method is established by the the Galerkin method of the weighted residual methods. The virtual source function is approached through the radial basis function interpolation in the mesh-free method. Thus, the main feature of the proposed technique has the advantages of the mesh-free method, the boundary-element method, and the Galerkin method, a symmetrical coefficient matrix. The numerical meanings of the weighted values in the VBMGM equation, such as the partial derivatives the temperature, the heat flux, the temperature connection condition, and the heat flux connection condition, are clear. Numerical examples of a thick-walled cylinder consisting of two media and a two-layered rectangular plate are provided. The feasibility and the accuracy of the proposed technique are proved. [ABSTRACT FROM AUTHOR]