正交各向异性稳态热传导问题的ICVEFG方法
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摘要
论文将改进的复变量无单元Galerkin方法(Improved Complex Variable Element-free Galerkin method,ICVEFG)应用于求解正交各向异性介质中的稳态热传导问题,提出了正交各向异性稳态热传导问题的ICVEFG方法.采用罚函数法引入本质边界条件,推导了正交各向异性介质中的稳态热传导问题的Galerkin积分弱形式.采用改进的复变量移动最小二乘近似(Improved Complex Variable Moving least-squares approximation,ICVMLS)建立二维温度场问题的逼近函数,推导了相应的计算公式.编制了计算程序,对三个正交各向异性介质中的热传导问题进行了分析,说明了论文方法的有效性.
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.
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.
引文
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[2]Li D M,Liew K M,Cheng Y.Analyzing elastoplastic large deformation problems with the complex variable element-free Galerkin method[J].Computational Mechanics,2014,53(6):1149-1162.
[3]Li D M,Liew K M,Cheng Y.An improved complex variable element-free Galerkin method for two-dimensional large deformation elastoplasticity problems[J].Computer Methods in Applied Mechanics and Engineering,2014,269:72-86.
[4]李冬明,彭妙娟,程玉民.弹性大变形问题的复变量无单元Galerkin方法[J].中国科学:物理学力学天文学,2011,41(8):1003-1014.(Li D M,Peng M J,Cheng Y M.The complex variable element-free Galerkin(CVEFG)method for elastic large deformation problems[J].Science China:Physics,Mechanics&Astronomy,2011,41(8):1003-1014.(in Chinese))
[5]Li D M,Bai F,Cheng Y,Liew K M.A novel complex variable element-free Galerkin method for twodimensional large deformation problems[J].Computer Methods in Applied Mechanics and Engineering,2012,233-236:1-10.
[6]Li D M,Zhang Z,Liew K M.A numerical framework for two-dimensional large deformation of inhomogeneous swelling of gels using the improved complex variable element-free Galerkin method[J].Computer Methods in Applied Mechanics and Engineering,2014,274:84-102.
[7]Li D M,Zhang L W,Liew K M.A three-dimensional element-free framework for coupled mechanical-diffusion induced nonlinear deformation of polymeric gels using the IMLS-Ritz method[J].Computer Methods in Applied Mechanics and Engineering,2015,296:232-247.
[8]Li D M,Tian L R.Large deformation analysis of a gel using the complex variable element-free Galerkin method[J].Applied Mathematical Modelling,2018,61:484-497.
[9]孙杰,易红亮,谈和平.求解参与性介质内辐射传递问题的无网格有限差分法[J].工程热物理学报,2016,37(3):653-656.(Sun J,Yi H L,Tan H P.Meshless finite difference method for radiative transfer in participating media[J],Journal of Engineering Thermophysics,2016,37(3):653-656.(in Chinese))
[10] Reutskiy S Y.A meshless radial basis function method for 2Dsteady-state heat conduction problems in anisotropic and inhomogeneous media[J].Engineering Analysis with Boundary Elements,2016,66:1-11.