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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]

Abstract: This paper is concerned with the virtual boundary meshless least squares collocation method of two dimensional steady-state heat conduction problems in multi-domain. In the proposed method, each region with different heat transfer properties is considered as a piecewise homogeneous in a heterogeneous system. Being different from the conventional virtual boundary element method (VBEM), this method incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Consequently, this method has the advantages of boundary-type meshless methods. In addition, it does not have to deal with singular integral and has the symmetric coefficient matrix, and the pre-processing operation is also very simple. Since the mathematical model for steady-state heat conduction can be represented by a generalized Poisson equation, this problem is solved by finding an approximate particular solution to the Poisson equation and making use of the virtual boundary meshless least squares collocation method to solve the resulting Laplace equation. This method can be used to analyze multi-domain composite structures with each subdomain having different heat transfer properties or materials or geometries. Since the configuration of virtual boundary has a certain preparability, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2D problems) to reduce the complicity and difficulty of calculus without loss of accuracy, while “Vertex Question” existing in BEM can be avoided. In the end, several numerical examples are analyzed using the proposed method and some other commonly used methods for verification and comparison purposes. [Copyright &y& Elsevier]

Abstract: Moving least squares approximation (MLSA) has been widely used in the meshless method. The singularity should appear in some special arrangements of nodes, such as the data nodes lie along straight lines and the distances between several nodes and calculation point are almost equal. The local weighted orthogonal basis functions (LWOBF) obtained by the orthogonalization of Gramm–Schmidt are employed to take the place of the general polynomial basis functions in MLSA. In this paper, MLSA with LWOBF is introduced into the virtual boundary meshless least square integral method to construct the shape function of the virtual source functions. The calculation format of virtual boundary meshless least square integral method with MLSA is deduced. The Gauss integration is adopted both on the virtual and real boundary elements. Some numerical examples are calculated by the proposed method. The non-singularity of MLSA with LWOBF is verified. The number of nodes constructing the shape function can be less than the number of LWOBF and the accuracy of numerical result varies little. [Copyright &y& Elsevier]