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Exponential, quadratic, and trigonometric 2D thermal analysis in inhomogeneous functionally graded materials (FGMs) are solved through an efficient hybrid boundary-type weak-form meshfree algorithm, which is the virtual boundary meshfree Galerkin method with fundamental solutions (FS-VBMGM). Three types of temperature fundamental solutions for FGMs are introduced. The corresponding thermal flow fundamental solutions are derived in detail. Temperature and thermal flux are obtained by utilizing the virtual boundary element method (VBEM) based on the continuous virtual loads. The continuous virtual loads are constructed through the radial basis function interpolation (RBFI) of meshfree algorithm. Using the Galerkin method, calculating formula of FS-VBMGM for thermal analysis in inhomogeneous FGMs is deducted. The benefits of VBEM, Galerkin method, and meshfree algorithm are thus presented in FS-VBMGM. Exponential, quadratic, and trigonometric inhomogeneous numerical examples of FGMs are calculated and contrasted to the exact solutions as well as other numerical methods. The reliability and exactness are proved. The specific discrete formula of FS-VBMGM for thermal analysis in inhomogeneous FGMs is simple to program and extended to calculate other particularly composite FGMs examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]