Abstract 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.

中文翻译：

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用虚拟边界无元伽辽金法计算 3-D 多域热传导问题

摘要 针对三维（3-D）多域稳态热传导问题，展示了一种新的边界型无网格收集方法，即虚拟边界无元伽辽金法（VBEFGM）。通过径向基函数插值构造虚拟节点的虚拟源函数。虚实边界离散为虚实节点。VBEFGM 在没有背景积分元素的情况下解决了 3-D 多域稳态热传导问题。利用Galerkin方法得到VBEFGM的公式，该方法具有虚拟边界无元法、无网格集合法和Galerkin法的优点。它易于编程和扩展以解决其他更复杂的 3-D 热传导问题，因为详细的计算方案已经推导并显示出来。计算了随空间变化的边界条件的两个数值示例。计算结果与其他数值方法进行了比较。所提出的方法的准确性和稳定性在 3-D 多域稳态热传导问题上得到了验证。