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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]

The virtual boundary meshfree Galerkin method (VBMGM) is presented to calculate the single-domain heat conduction with heat source problem. The virtual source function of the virtual boundary and the heat source in the computational domain both are interpolated by the radial basis function interpolation. The equation of the proposed method is obtained by the Galerkin method of the weighted residual methods. Consequently, VBMGM possesses the advantages of Galerkin method, boundary element method and meshfree method. And the numerical meanings of the weighted values in VBMGM, such as the partial derivative the temperature, the heat flux and the heat source expression, are definite. The detailed and specific equation of VBMGM is deducted. The numerical examples, namely the temperatures of the rectangular domain and the square domain, are computed. And the accuracy and the stability of the proposed method for the single-domain heat conduction with heat source problem are validated. [ABSTRACT FROM AUTHOR]