三维稳态对流扩散问题的无网格求解算法研究

张小华; 邓霁恒

doi:10.3879/j.issn.1000-0887.2014.11.008

三维稳态对流扩散问题的无网格求解算法研究

doi: 10.3879/j.issn.1000-0887.2014.11.008

三峡大学理学院, 湖北宜昌 443002

基金项目: 国家自然科学基金(11102101；11171181)；湖北省教育厅自然科学基金 (Q20111208)

详细信息

作者简介:
张小华(1980—)，男，湖北宜昌人，副教授，博士（通讯作者. E-mail: dengjiheng@foxmail.com）.

中图分类号: TK124;TQ015.9
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出版历程
- 收稿日期: 2014-06-07
- 修回日期: 2014-10-15
- 刊出日期: 2014-11-18

Research on the Meshless Solving Algorithm for 3D Steady ConvectionDiffusion Problems

College of Science, China Three Gorges University, Yichang, Hubei 443002, P.R.China

Funds: The National Natural Science Foundation of China(11102101；11171181)

摘要

摘要: 无网格法是一种不需要生成网格就可模拟复杂形状流场计算的流体力学问题求解算法.为了提高基于Galerkin弱积分形式的无网格方法求解三维稳态对流扩散问题的计算效率，提出了在空间离散上采用基于凸多面体节点影响域的无网格形函数，并通过选取适当节点影响半径因子避免节点搜索问题，同时减少系统刚度矩阵带宽.计算中当节点影响因子为1.01时，无网格方法的形函数近似具有插值特性且本质边界条件的施加与有限元一样简单.三维立方体区域的稳态对流扩散数值算例表明：在保证计算精度的同时，采用凸多面体节点影响域的无网格方法比传统无网格方法最高可节省计算时间42%.因此从计算效率和精度考虑，在运用无网格方法求解三维问题时建议采用凸多面体节点影响域的无网格方法.
- 无网格方法 /
- 三维稳态对流扩散 /
- 计算效率 /
- 凸多面体节点影响域
Abstract: The meshless method is a numerical algorithm for the simulation of flow fields in complicated shapes and solves fluid mechanics problems without grids. In order to improve the computation efficiency of meshless methods based on the Galekrin weak integration form for solving 3D steady convection-diffusion problems, a meshless shape function was proposed based on convex-polyhedral nodal influence domain in the discrete space. Then with a properly selected factor of nodal influence radius, the node-searching process was avoided and the bandwidth of the stiffness matrix for the system was reduced. With a factor of nodal influence radius at 1.01 during the calculation, the shape function of the meshless method almost possesses interpolation properties and the imposition of essential boundary conditions is simplified as that for the FEM. The numerical results of 2 exemplary steady convection-diffusion problems for 3D cubic regions show that: compared with the traditional meshless methods, the present meshless method based on convex-polyhedral nodal influence domain enables the computing time to be reduced by up to 42% without impairing the calculation accuracy. Finally, in the cases that both the computation efficiency and the accuracy are highly demanded, this meshless method based on convex-polyhedral nodal influence domain is suggested for the solution of 3D steady convection-diffusion problems.
- meshless method /
- 3D steady convection-diffusion /
- computation efficiency /
- convex-polyhedral nodal influence domain

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参考文献(11)

[1]	Nguyen V P, Rabczuk T, Bordas S, Duflotd M. Meshless methods: a review and computer implementation aspects[J]. Mathematics and Computers in Simulation,2008,79(3): 763-813.
[2]	张林. 基于多尺度无网格 Galerkin 方法的流动问题模拟[D]. 博士学位论文. 西安: 西北工业大学, 2010.(ZHANG Lin. The numerical simulation of flow problem based on multiscale element free Galerkin method[D]. PhD Thesis. Xi’an: North-Western Polytechnical University, 2010.(in Chinese))
[3]	陶文铨, 吴学红, 戴艳俊. 无网格方法在流动和传热问题中的应用[J]. 中国电机工程学报, 2010,30(8): 1-8.(TAO Wen-quan, WU Xue-hong, DAI Yan-jun. Applications of meshless methods in fluid flow and heat transfer problems[J]. Proceedings of the CSEE,2010,30(8): 1-8.(in Chinese))
[4]	Dolbow J, Belytschko T. Numerical integration of the Galerkin weak form in meshfree method[J]. Computational Mechanics,1999,23(3): 219-230.
[5]	Liu G R, Tu Z H. An adaptive procedure based on background cells for meshless methods[J]. Computer Methods in Applied Mechanics and Engineering,2002,191(17/18): 1923-1943.
[6]	ZHANG Xiang-kun, Kwon K C, Youn S K. Least-squares meshfree method for incompressible Navier-Stokes problems[J]. International Journal for Numerical Methods in Fluids,2004,46(3): 263-288.
[7]	Singh I V. Parallel implementation of the EFG method for heat transfer and fluid flow problems[J]. Computational Mechanic,2004,34(6): 453-463.
[8]	Belytschko T, Krongauz Y, Fleming M, Organ D, Liu W K S. Smoothing and accelerated computations in the element free Galerkin method[J]. Journal of Computational and Applied Mathematics,1996,74(1/2): 111-126.
[9]	HUANG Xiao-ying, Augarde C. Aspects of the use of orthogonal basis functions in the element-free Galerkin method[J]. International Journal for Numerical Methods in Engineering,2010,81(3): 366-380.
[10]	ZHANG Lin, OUYANG Jie, ZHANG Xiao-hua. The variational multisclae element free Galerkin method for MHD flows at high Hartmann numbers[J]. Comput Phys Commun,2013,184(3): 1106-1118.
[11]	Romao E C, Mendes de Moura L F. Galerkin and least squares methods to solve a 3D convection-diffusion-reaction equation with variable coefficients[J]. Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology,2012,61(9): 669-698.