It is established that the new boundary element-free Galerkin method, namely the virtual boundary element-free Galerkin method (VBEFGM), using dual compactly supported radial basis function interpolation (CSRBFI) for 2D inhomogeneous heat conduction problems with variable heat sources, whose solutions are divided into homogeneous and particular solutions. The virtual loads of the homogeneous solutions and undetermined coefficients of particular solutions are interpolated by CSRBFI. Its formula is established by the Galerkin method. The homogeneous solutions are obtained by the virtual boundary element-free method (VBEFM). The recommended method for 2D inhomogeneous heat conduction problems is real meshfree without background integration elements and exhibits the merits of the meshfree method, Galerkin method, and boundary element method. Its calculation scheme is deduced in detail and easily programmed. The coefficient matrix has symmetrical characteristics. The implementation steps and implementation flow-chart are also shown in detail. Seven numerical examples, such as exponential functionally graded material, are calculated and contrasted with other numerical methods. The precision and stableness of the proposed method for 2D inhomogeneous heat conduction problems are verified.

建立了新的无边界元伽辽金法，即虚拟边界无元伽辽金法（VBEFGM），使用双紧支撑径向基函数插值（CSRBFI）解决具有可变热源的二维非均匀热传导问题，其解分为齐次解和特解。CSRBFI 对齐次解的虚拟载荷和特定解的未定系数进行了插值。其公式由伽辽金法建立。齐次解是通过虚拟边界无元素方法 (VBEFM) 获得的。二维非均匀热传导问题的推荐方法是没有背景积分元素的真正无网格，并展示了无网格方法、伽辽金方法和边界元方法的优点。其计算方案推导出详细，易于编程。系数矩阵具有对称特性。还详细展示了实施步骤和实施流程图。计算了七个数值示例，例如指数函数梯度材料，并与其他数值方法进行了对比。验证了所提出的二维非均匀热传导问题方法的精度和稳定性。