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ISSN 1088-6842(e) ISSN 0025-5718(p)

A finite element method for nearly incompressible elasticity problems

Author(s): Dietrich Braess; Pingbing Ming.
Journal: Math. Comp. 74 (2005), 25-52.
MSC (2000): Primary 65N30, 74S05, 74B20
Posted: April 28, 2004
MathSciNet review: 2085401
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Abstract: A finite element method is considered for dealing with nearly incompressible material. In the case of large deformations the nonlinear character of the volumetric contribution has to be taken into account. The proposed mixed method avoids volumetric locking also in this case and is robust for $\lambda \to \infty$ (with $\lambda$ being the well-known Lamé constant). Error estimates for the $L^{\infty}$ -norm are crucial in the control of the nonlinear terms.

References:

1.: L. Agmon, L. Douglas, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure. Appl. Math. 17 (1964), 35-92. MR 28:5252
2.: I. Babuska and M. Suri, On locking and robustness in the finite element method, SIAM. J. Numer. Anal. 29 (1992), 1261-1293. MR 94c:65128
3.: C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements, SIAM. J. Numer. Anal. 35 (1998), 1893-1916. MR 99g:65107
4.: J. M. Boland and R. A. Nicolaides, Stable and semistable low order finite elements for viscous flows, SIAM. J. Numer. Anal. 22 (1985), 474-492. MR 86m:65139
5.: D. Braess, Stability of saddle point problems with penalty, RAIRO, MAN 30 (1996), 731-742. MR 99k:65099
6.: -, Enhanced assumed strain elements and locking in membrane problems, Comput. Methods Appl. Mech. Eng. 165 (1998), 155-174. MR 2000j:74084
7.: -, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2001. MR 2001k:65002
8.: D. Braess, C. Carstensen, and B. D. Reddy, Uniform convergence and a posteriori error estimators for the enhanced strain finite element method, Numer. Math. 96 (2004) 461-479.
9.: S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994. MR 95f:65001
10.: F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.MR 92d:65187
11.: P. G. Ciarlet, Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978.MR 58:25001
12.: -, Mathematical Elasticity. Volume 1: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988.MR 89e:73001
13.: P. Clément, Approximation by finite element functions using local regularization, RAIRO, Sér. Rouge Anal. Numér. R-2 (1975), 77-84.MR 53:4569
14.: M. C. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340.MR 44:5836
15.: M. Dobrowolski, A mixed finite element method for approximating incompressible materials, SIAM. J. Numer. Anal. 29 (1992), 365-389.MR 93b:65167
16.: R. Durán, R. Nochetto, and J. Wang, Sharp maximum norm error estimates for finite element approximations of the Stokes problem in 2-D, Math. Comp. 51 (1988), 491-506.MR 89b:65261
17.: M. Fortin, An analysis of the convergence of mixed finite element methods, RAIRO, Anal. Numér. 11 (1977), 341-354.MR 57:4473
18.: J. Frehse and R. Rannacher, Eine ${L}^1$ -Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente, Bonner Math. Schriften. 89 (1976), 92-114.MR 57:11104
19.: -, Asymptotic ${L}^{\infty}$ error estimates for linear finite element approximations of quasilinear boundary value problems, SIAM. J. Numer. Anal. 15 (1978), 418-431.MR 58:19224
20.: P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.MR 86m:35044
21.: H. Le Dret, Constitutive laws and existence questions in incompressible nonlinear elasticity, J. Elasticity. 15 (1985), 369-387. MR 87h:73022
22.: -, Incompressible limit behaviour of slightly compressible nonlinear elastic materials, RAIRO, MAN 20 (1986), 315-340. MR 87k:73011
23.: P. Le Tallec, Numerical Methods for Nonlinear Three-dimensional Elasticity, in ``Handbook of Numerical Analysis", VOL. III. (P. G. Ciarlet and J. L. Lions eds.), 465-662. North-Holland, 1994. MR 96b:73093
24.: J. L. Lions and E. Magènes, Problèmes aux limites non homogènes et applications (tome 1), S. A. Dunod, Paris, 1968.MR 40:512
25.: F. Natterer, Über die punktweise Konvergenz finiter Elemente, Numer. Math. 25 (1975), 67-77.MR 57:14514
26.: R. W. Ogden, Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubber-like solids, Proc. Roy. Soc. London, A328 (1972), 567-583.
27.: R. W. Ogden, Volume changes associated with the deformation of rubber-like solids, J. Mech. Phys. Solids 24 (1976), 323-338.
28.: D. Pantuso and K. J. Bathe, On the stability of mixed finite elements in large strain analysis of incompressible solids, Finite Elements in Analysis and Design 28 (1997), 83-104. MR 98i:73060
29.: J. M. Pouyot, Études numérique de quelques problémes raides en mécanique des milieux faiblement compressible, Calcolo 12 (1975), 275-314. MR 54:9252
30.: S. Reese, P. Wriggers, and B.D. Reddy, A new locking-free brick element technique for large deformation problems in elasticity, Computers & Structures 75 (2000), 291-304.MR 2001j:74087
31.: W. C. Rheinboldt, Methods for Solving Nonlinear Equations, vol. 14, CBMS Series, SIAM, Philadelpha, 1974.MR 51:7278
32.: R. Rostamian, Internal constraints in boundary value problems of continuum mechanics, Indiana Math. 27 (1978), 637-656. MR 58:14158
33.: V. Ruas, Existence and stability of asymmetric finite-element approximations in nonlinear elasticity, SIAM J. Numer. Anal. 26 (1989), 1031-1059.MR 90h:65187
34.: A. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959-962. MR 51:9526
35.: J. C. Simo and M. S. Rifai, A class of assumed strain methods and the method of incompatible modes, Int. J. Numer. Meths. Engrg. 29 (1990), 1595-1638.MR 91d:73062
36.: T. Sussman and K. J. Bathe, A finite elment formulation for nonlinear incompressible elastic and noninelastic analysis, Computer & Structures 26 (1987), 357-409.
37.: L. R. Treloar, The Physics of Rubber Elasticity, Oxford University Press, Oxford, 1975.

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