Abstract 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.

中文翻译：

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求解三维单域稳态热传导问题的一种新的边界型无网格方法

摘要 进一步发展了虚拟边界无网格伽辽金方法（VBMGM）来计算三维单域稳态热传导问题。使用VBMGM，将三维问题简化为二维边界，其单元信息可以通过网格划分软件即ANSYS软件获得。ANSYS参数化设计语言(APDL)用于曲面单元信息的获取，极大地方便了所提出方法的编程实现。虚拟边界元法（VBEM）上的虚拟源函数由无网格法的径向基函数插值（RBFI）插值。利用加权残差法的伽辽金法构造VBMGM三维单域稳态热传导问题的计算方程。因此，将所提出的方法用于计算三维单域稳态热传导问题，具有VBEM、无网格方法和Galerkin方法的优点。得到该方法求解三维单域稳态热传导问题的详细数值离散公式，有利于进一步推广该方法求解其他三维热传导问题。计算了两个数值例子的数值结果，并与其他数值方法和提取解进行了比较。