一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用

doi:10.12286/jssx.2020.3.298

计算数学 ›› 2020, Vol. 42 ›› Issue (3): 298-309.DOI: 10.12286/jssx.2020.3.298

一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用

洪庆国¹, 刘春梅², 许进超¹

1 宾夕法尼亚州立大学数学系, 宾州 PA 16802, 美国;
2 湖南科技学院理学院, 永州 425199

收稿日期:2020-04-30 出版日期:2020-08-15 发布日期:2020-08-15
基金资助:
作者刘春梅由国家自然科学基金（No.11901189），湖南省教育厅科研项目（No.19A191）支持.

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洪庆国, 刘春梅, 许进超. 一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用[J]. 计算数学, 2020, 42(3): 298-309.

Hong Qingguo, Liu Chunmei, Xu Jinchao. AN ABSTRACT STABILIZATION METHOD WITH APPLICATIONS TO NONLINEAR INCOMPRESSIBLE ELASTICITY[J]. Mathematica Numerica Sinica, 2020, 42(3): 298-309.

AN ABSTRACT STABILIZATION METHOD WITH APPLICATIONS TO NONLINEAR INCOMPRESSIBLE ELASTICITY

Hong Qingguo¹, Liu Chunmei², Xu Jinchao¹

1 Department of Mathematics, Pennsylvania State University, Unverisity Park, PA 16802, USA;
2 College of Science, Hunan University of Science and Engineering, Yongzhou 425199, China

Received:2020-04-30 Online:2020-08-15 Published:2020-08-15
Supported by:
The work of Qingguo Hong was partially supported by Center for Computational Mathematics and Applications, The Pennsylvania State University.

针对非线性不可压缩弹性力学问题，本文提出了一种抽象的稳定化方法并将其应用于非线性不可压缩弹性问题上.在该框架中，我们证明了只要连续的混合问题是稳定的，则可以修正任何满足离散inf-sup条件的混合有限元方法使其是稳定的且最优收敛的.我们将这种抽象的稳定化理论框架应用于非线性不可压缩弹性力学问题，给出了稳定性和收敛性理论结论，并通过数值实验验证了该结论.

关键词： 非线性不可压缩弹性问题, 混合有限元方法, 稳定性

In this paper, we propose and analyze an abstract stabilized mixed finite element framework that can be applied to nonlinear incompressible elasticity problems. In the abstract stabilized framework, we prove that any mixed finite element method that satisfies the discrete inf-sup condition can be modified so that it is stable and optimal convergent as long as the mixed continuous problem is stable. Furthermore, we apply the abstract stabilized framework to nonlinear incompressible elasticity problems and present numerical experiments to verify the theoretical results.

Key words: nonlinear incompressible problem, mixed finite methods, stability

MR(2010)主题分类:

65N30

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[1] Reddy B, Simo J. Stability and convergence of a class of enhanced strain methods[J]. SIAM J. Numer. Anal., 1995, 32(6):1705-1728.

[2] Braess D, Carstensen C, Reddy B. Uniform convergence and a posteriori error estimators for the enhanced strain finite element method[J]. Numer. Math., 2004, 96(3):461-479.

[3] Houston P, Schotzau D, Wihler T. An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity[J]. Comput. Methods Appl. Mech. Engrg., 2006, 195(25-28):3224-3246.

[4] Hansbo P, Larson M. Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method[J]. Comput. Methods Appl. Mech. Engrg., 2002, 191(17-18):1895-1908.

[5] Hong Q, Kraus J, Xu J, Zikatanov L. A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations[J]. Numer. Math., 2016, 132(1):23-49.

[6] Wu S, Gong S, Xu J. Interior penalty mixed finite element methods of any order in any dimension for linear elasticity with strongly symmetric stress tensor[J]. Math. Models Methods in Appl. Sci., 2017, 27(14):2711-2743.

[7] Gong S, Wu S, Xu J. New hybridized mixed methods for linear elasticity and optimal multilevel solvers[J]. Numer. Math., 2019, 141(2):569-604.

[8] Wang F, Wu S, Xu J. A mixed discontinuous Galerkin method for linear elasticity with strongly imposed symmetry[J]. J. Sci. Comput., 2020, 83(1):1-17.

[9] Hong Q, Hu J, Ma L, Xu J. An Extended Galerkin Analysis for Linear Elasticity with Strongly Symmetric Stress Tensor[J]. arXiv preprint arXiv:2002.11664, 2020.

[10] Auricchio F, da Veiga L B, Lovadina C, Reali A. A stability study of some mixed finite elements for large deformation elasticity problems[J]. Comput. Methods Appl. Mech. Engrg., 2005, 194(9-11):1075-1092.

[11] Wriggers P, Reese S. A note on enhanced strain methods for large deformations[J]. Comput. Methods Appl. Mech. Engrg., 1996, 135(3-4):201-209.

[12] Lovadina C, Auricchio F. On the enhanced strain technique for elasticity problems[J]. Comput.& Structures, 2003, 81(8-11):777-787.

[13] Pantuso D, Bathe K. On the stability of mixed finite elements in large strain analysis of incompressible solids[J]. Finite Elem. Anal. Des., 1997, 28(2):83-104.

[14] Eyck A T, Celiker F, Lew A. Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity:Analytical estimates[J]. Comput. Methods Appl. Mech. Engrg., 2008, 197(33-40):2989-3000.

[15] Eyck A T, Celiker F, Lew A. Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity:Motivation, formulation, and numerical examples[J]. Comput. Methods Appl. Mech. Engrg., 2008, 197(45-48):3605-3622.

[16] A. Ten Eyck A, Lew A. An adaptive stabilization strategy for enhanced strain methods in nonlinear elasticity[J]. Internat. J. Numer. Methods Engrg., 2010, 81(11):1387-1416.

[17] Auricchio F, da Veiga L B, Lovadina C, Reali A. The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity:mixed FEMs versus NURBS-based approximations[J]. Comput. Methods Appl. Mech. Engrg., 2010, 199(5-8):314-323.

[18] Hughes T, Cottrell J, Bazilevs Y. Isogeometric analysis:CAD, finite elements, NURBS, exact geometry and mesh refinement[J]. Comput. Methods Appl. Mech. Engrg., 2005, 194(39-41):4135-4195.

[19] Brezzi F, Fortin M, Marini L. Mixed Finite Element Methods with Continuous Stresses[J]. Math. Models Methods Appl. Sci., 1993, 3:275-287.

[20] Xie X, Xu J, Xue G. Uniformly stable finite element methods for Darcy-Stokes-Brinkman models[J]. J. Comput. Math., 2008, 26(3):437-455.

[21] Bonet J, Wood R. Nonlinear continuum mechanics for finite element analysis[M]. Cambridge University Press, New York, 1997.

[22] Brezzi F, Fortin M. Mixed and hybrid finite element methods[M]. Springer-Verlag, New York, 1991.

[23] Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[J]. RAIRO Anal. Numer., 1974, 8(2):129-151.

[24] Arnold D, Brezzi F, Fortin M. A stable finite element for the Stokes equations[J]. Calcolo, 1984, 21(4):337-344.

[25] Girault V, Raviart P. Finite element methods for Navier-Stokes equations:theory and algorithms[M]. Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986.

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摘要

一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用

AN ABSTRACT STABILIZATION METHOD WITH APPLICATIONS TO NONLINEAR INCOMPRESSIBLE ELASTICITY

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