三维薄结构热传导问题分层二次元方程的多水平方法

张申; 肖映雄; 郭瑞奇

doi:10.21656/1000-0887.380035

三维薄结构热传导问题分层二次元方程的多水平方法

doi: 10.21656/1000-0887.380035

湘潭大学土木工程与力学学院, 湖南湘潭 411105

基金项目: 国家自然科学基金(11601462)；湖南省教育厅资助科研项目(15A183)

详细信息

作者简介:
张申（1990—），男，硕士生（E-mail: zhangshenf@126.com）;肖映雄（1970—），男，教授，博士（通讯作者. E-mail: xyx610xyx@xtu.edu.cn）;郭瑞奇（1993—），男，硕士生（E-mail: 191522177@qq.com）.

中图分类号: O343.3;TB115
计量
- 文章访问数: 980
- HTML全文浏览量: 105
- PDF下载量: 868
- 被引次数: 0
出版历程
- 收稿日期: 2017-02-16
- 修回日期: 2017-11-30
- 刊出日期: 2018-06-15

A Multi-Level Method for Hierarchical Quadratic Discretizations of Thin-Walled Structures in 3D Heat Conduction

College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan, Hunan 411105, P.R.China

Funds: The National Natural Science Foundation of China(11601462)

摘要

摘要: 在利用有限元法对三维薄结构进行分析时，为了减少单元数目，常采用六面体薄单元，相应的高阶单元在计算精度、抗畸变程度等方面具有明显优势.但与低阶元相比，高阶单元需要更多的计算机存储空间，离散化线性系统具有更高的计算复杂性，并且系数矩阵是严重病态的，采用通常的求解方法其效率将大大降低.该文针对三维薄结构稳态热传导问题，利用局部块Gauss-Seidel光滑子和基于“距离矩阵”的DAMG法，为其分层二次元离散系统设计了一种具有更好计算效率和鲁棒性（robustness）的多水平方法.由于采用了分层基，程序实现中不再需要建立判定未知数变量指标与所属几何节点类型对应关系的代数判据，网格转换算子的构造也变得非常简单，从而大大提高了运算效率.数值实验结果验证了该方法的有效性和鲁棒性.
- 热传导问题 /
- 薄体结构 /
- 分层二次元 /
- 多水平方法 /
- 代数多重网格法
Abstract: When the finite element method is applied to analyze the 3D thin-walled structures, some thin hexahedral elements are usually used in order to reduce the number of elements, and the corresponding higher-order elements are preferred since they have some obvious advantages in the calculation accuracy, the anti-distortion degree and so on. However, they have much higher computational complexity than the lower-order (e.g., linear) elements and the coefficient matrix of the linear algebraic equation system is severely ill-conditioned. The convergence of the commonly used solvers will deteriorate with the increasing size of the problem. An efficient and robust multi-level method was presented for the hierarchical quadratic discretizations of 3D thin-walled structures through combination of two special local block Gauss-Seidel smoothers and the DAMG algorithm based on the distance matrix. Since a hierarchical basis is used, those algebraic criteria are not needed to judge the relationships between the unknown variables and the geometric node types, and the grid transfer operators are also trivial. This makes it easy to find the coarse level (linear element) matrix derived directly from the fine level matrix, and thus the overall efficiency is greatly improved. The numerical results verify the efficiency and robustness of the proposed method.
- heat conduction problem /
- thin-walled structure /
- hierarchical quadratic element /
- multi-level method /
- algebraic multigrid

HTML全文

参考文献(22)

[1]	BOUZAKIS K D, VIDAKIS N. Prediction of the fatigue behavior of physically vapor deposited coating in the ball-on-rod rolling contact fatigue test, using an elastic-plastic finite elements method simulation[J]. Wear,1997,206(1/2): 197-203.
[2]	DOBRZANSKI L A,LIWA A, KWANY W. Employment of the finite element method for determining stresses in coatings obtained on high-speed steel with the PVD process[J]. Journal of Materials Processing Technology,2005,164/165: 1192-1196.
[3]	BRANDT A, MCCORMICK S, RUGE J. Algebraic multigrid (AMG) for sparse matrix equations[C]// Sparsity and Its Applications.Loughborough, 1983: 257-284.
[4]	BREZINA M, CLEARY A J, FALGOUT R D, et al. Algebraic multigrid based on element interpolation (AMGe)[J]. SIAM Journal on Scientific Computing,2000,22(5): 1570-1592.
[5]	CLEARY A J, FALGOUT R D, HENSON V E, et al. Robustness and scalability of algebraic multigrid[J].SIAM Journal on Scientific Computing,2000,21(5): 1886-1908.
[6]	GRIEBEL M, OELTZ D, SCHWEITZER M A. An algebraic multigrid method for linear elasticity[J]. SIAM Journal on Scientific Computing,2003,25(2): 385-407.
[7]	BAKER A H, KOLEV T V, YANG U M. Improving algebraic multigrid interpolation operators for linear elasticity problems[J]. Numerical Linear Algebra With Applications,2010,17(2/3): 495-517.
[8]	SHU Shi, XU Jinchao, YANG Ying, et al. An algebraic multigrid method for finite element systems on criss-cross grids[J]. Advances in Computational Mathematics,2006,25(1/3): 287-304.
[9]	VANK P, MANDEL J, BREZINA M. Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems[J].Computing,1996,56(3): 179-196.
[10]	NOTAY Y. An aggregation-based algebraic multigrid method[J]. Electronic Transactions on Numerical Analysis,2010,37: 123-146.
[11]	RUGE J, MCCORMICK S, MANTEUFFEL T, et al. AMG for higher-order discretizations of second-order elliptic problems[C]// Abstracts of the Eleventh Copper Mountain Conference on Multigrid Methods.2003.
[12]	HEYS J J, MANTEUFFEL T A, MCCORMICK S F, et al. Algebraic multigrid for higher-order finite elements[J]. Journal of Computational Physics,2005,204: 520-532.
[13]	EL MALIKI A, GUNETTE R, FORTIN M. An efficient hierarchical preconditioner for quadratic discretizations of finite element problems[J]. Numerical Linear Algebra With Applications,2011,18: 789-803.
[14]	SHU S, SUN D, XU J. An algebraic multigrids for higher order finite element discretizations[J]. Computing,2006,77(4): 347-377.
[15]	XIAO Yingxiong, SHU Shi. A robust preconditioner for higher order finite element discretizations in linear elasticity[J]. Numerical Linear Algebra With Applications,2011,18: 827-842.
[16]	谭敏, 肖映雄, 舒适. 一种各向异性四边形网格下的代数多重网格法[J]. 湘潭大学自然科学学报, 2005,27(1): 78-84.(TAN Min, XIAO Yingxiong, SHU Shi. An algebraic multigrid method on anisotropic quadrilateral grid[J]. Natural Science Journal of Xiangtan University,2005,27(1): 78-84.(in Chinese))
[17]	XIAO Yingxiong, SHU Shi, ZHANG Ping, et al. An algebraic multigrid method for isotropic linear elasticity on anisotropic meshes[J]. International Journal for Numerical Methods in Biomedical Engineering,2010,26: 534-553.
[18]	GRAHAM E, FORSYTH P A. Preconditioning methods for very ill-conditioned three-dimensional linear elasticity problems[J]. International Journal for Numerical Methods in Engineering,1999,44(1): 77-99.
[19]	SUMANT P S, CANGELLARIS A C, ALURU N R. A node-based agglomeration AMG solver for linear elasticity in thin bodies[J]. Communications in Numerical Methods in Engineering,2009,25: 219-236.
[20]	YSERENTANT H. On the multi-level spliting of finite element spaces[J]. Numerische Mathematik,1986,49(4): 379-412.
[21]	XIAO Yingxiong, ZHOU Zhiyang, SHU Shi. An efficient AMG method for quadratic discretizations of linear elasticity problems on some typical anisotropic meshes in three dimensions[J]. Numerical Linear Algebra With Applications,2015,22(3): 465-482.
[22]	WU S C, LIU G R, ZHANG H O, et al. A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems[J]. International Journal of Thermal Sciences,2009,48(7): 1367-1376.

施引文献

资源附件(0)

访问统计

点击查看大图

计量

文章访问数: 980
HTML全文浏览量: 105
PDF下载量: 868
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

留言板

三维薄结构热传导问题分层二次元方程的多水平方法

doi: 10.21656/1000-0887.380035

作者简介:
张申（1990—），男，硕士生（E-mail: zhangshenf@126.com）;肖映雄（1970—），男，教授，博士（通讯作者. E-mail: xyx610xyx@xtu.edu.cn）;郭瑞奇（1993—），男，硕士生（E-mail: 191522177@qq.com）.

计量

A Multi-Level Method for Hierarchical Quadratic Discretizations of Thin-Walled Structures in 3D Heat Conduction

计量

目录

留言板

三维薄结构热传导问题分层二次元方程的多水平方法

doi: 10.21656/1000-0887.380035

作者简介: 张申（1990—），男，硕士生（E-mail: zhangshenf@126.com）;肖映雄（1970—），男，教授，博士（通讯作者. E-mail: xyx610xyx@xtu.edu.cn）;郭瑞奇（1993—），男，硕士生（E-mail: 191522177@qq.com）.

计量

出版历程

A Multi-Level Method for Hierarchical Quadratic Discretizations of Thin-Walled Structures in 3D Heat Conduction

计量

出版历程

目录

作者简介:
张申（1990—），男，硕士生（E-mail: zhangshenf@126.com）;肖映雄（1970—），男，教授，博士（通讯作者. E-mail: xyx610xyx@xtu.edu.cn）;郭瑞奇（1993—），男，硕士生（E-mail: 191522177@qq.com）.