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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]

A new boundary-type meshfree collection method, namely the virtual boundary element-free Galerkin method (VBEFGM), is demonstrated for three-dimension (3-D) multi-domain steady-state heat conduction problems. The virtual source functions of the virtual nodes are constructed by the radial basis function interpolation. And the virtual and real boundaries are discreted into the virtual and real nodes. 3-D multi-domain steady-state heat conduction problems are solved without the background integral element by VBEFGM. Using the Galerkin method to obtain the formula of VBEFGM, the proposed method has the advantages of the virtual boundary element-free method, the meshfree collection method and the Galerkin method. It is easily programed and extended to solve other more complicated 3-D heat conduction problems, since the detailed calculation scheme is deduced and shown. Two numerical examples of the boundary conditions varying with the space are computed. The computational results are compared with other numerical methods. The accuracy and stability of the proposed method are validated for 3-D multi-domain steady-state heat conduction problems. [ABSTRACT FROM AUTHOR]