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Abstract In this paper, the improved complex variable element-free Galerkin (ICVEFG) method is applied to analyze steady-state heat conduction problems in orthogonal media. The improved complex variable moving least-squares (ICVMLS) method is used to establish the approximation function of two dimensional temperature fields for orthogonal media. By adopting same vector form approximation of field variables as that in the complex variable moving least squares (CVMLS) method, the ICVMLS approximation retains the advantages in improving the computing efficiency of the shape functions for two-dimensional problems. Meanwhile, the precisely defined norm error in the ICVMLS approximation ensures the high accuracy of the approximate solutions. In our numerical implementation, the penalty method is used to apply the essential boundary conditions. The Galerkin integral weak form of steady-state heat conduction in orthotropic medium is derived and the corresponding calculation formula is presented. The compute program is compiled to analyze three example problems of heat conduction in orthogonal media. And the validity of the proposed method is illustrated by comparing with the analytical solutions. The numerical results show that the proposed ICVEFG method can obtain highly accurate temperature fields for orthotropic heat conduction problems. The numerical accuracy and efficiency of the proposed ICVEFG method are also compared with the complex variable element-free Galerkin (CVEFG) method in which the CVMLS approximation is used. For all three examples, it is found that the accuracy of the solutions using the ICVEFG method is much higher than the accuracy of those obtained by the CVEFG method. But the CPU time used for the ICVMLS shape functions is almost the same as that for the CVMLS shape functions. That means, compared with the CVEFG method, the ICVEFG method can greatly improve numerical accuracy and robustness without reducing numerical efficiency. The proposed ICVEFG method for orthotropic steady-state heat conduction problems is proved to be computationally efficient, robust and accurate.
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Received: 17 July 2018
Published: 22 February 2019
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2019, Vol. 40 
