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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]

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]