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Applied Mathematics

Gradient Schemes for Linear and Non-linear Elasticity Equations

Profile image of Bishnu LamichhaneBishnu Lamichhane

2014, arXiv (Cornell University)

https://doi.org/10.48550/ARXIV.1402.3866
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20 pages

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Abstract

The Gradient Scheme framework provides a unified analysis setting for many different families of numerical methods for diffusion equations. We show in this paper that the Gradient Scheme framework can be adapted to elasticity equations, and provides error estimates for linear elasticity and convergence results for non-linear elasticity. We also establish that several classical and modern numerical methods for elasticity are embedded in the Gradient Scheme framework, which allows us to obtain convergence results for these methods in cases where the solution does not satisfy the full H 2 -regularity or for non-linear models.

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References (39)

  1. L. Agélas, R. Eymard, and S Lemaire. A locking-free euler-gradient scheme approximation of biots consolidation problem on general meshes. In preparation.
  2. M. A. Barrientos, G. N. Gatica, and E. P. Stephan. A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate. Numer. Math., 91(2):197-222, 2002.
  3. D. Braess. Finite Elements. Theory, fast solver, and applications in solid mechanics. Cam- bridge University Press, Second Edition, 2001.
  4. D. Braess, C. Carstensen, and B.D. Reddy. Uniform convergence and a posteriori error esti- mators for the enhanced strain finite element method. Numerische Mathematik, 96:461-479, 2004.
  5. D. Braess and P.-B. Ming. A finite element method for nearly incompressible elasticity prob- lems. Mathematics of Computation, 74:25-52, 2005.
  6. S.C. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer- Verlag, New York, 1994.
  7. S.C. Brenner and L. Sung. Linear finite element methods for planar linear elasticity. Mathe- matics of Computation, 59:321-338, 1992.
  8. E. Burman and P. Hansbo. A stabilized non-conforming finite element method for incom- pressible flow. Computer Methods in Applied Mechanics and Engineering, 195:2881-2899, 2006.
  9. C. Carstensen and G. Dolzmann. An a priori error estimate for finite element discretizations in nonlinear elasticity for polyconvex materials under small loads. Numerische Mathematik, 97:67-80, 2004.
  10. M. Cervera, M. Chiumenti, and R. Codina. Mixed stabilized finite element methods in non- linear solid mechanics Part II: strain localization. Comput. Methods Appl. Mech. Engrg., 199(37-40):2571-2589, 2010.
  11. P.G Ciarlet. The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978.
  12. P.G. Ciarlet. Mathematical Elasticity Volume I: Three-Dimensional Elasticity. North- Holland, Amsterdam, 1988.
  13. K. Deimling. Nonlinear functional analysis. Springer-Verlag, Berlin, 1985.
  14. J.K. Djoko, B.P. Lamichhane, B.D. Reddy, and B.I. Wohlmuth. Conditions for equivalence between the Hu-Washizu and related formulations, and computational behavior in the incom- pressible limit. Computer Methods in Applied Mechanics and Engineering, 195:4161-4178, 2006.
  15. J. Droniou. Finite volume schemes for fully non-linear elliptic equations in divergence form. M2AN Math. Model. Numer. Anal., 40(6):1069-1100 (2007), 2006.
  16. J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R Herbin. Gradient schemes for elliptic and parabolic problems.
  17. J. Droniou, R. Eymard, T. Gallouët, and R. Herbin. Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci., 2012. To appear.
  18. R. Eymard, P. Féron, T. Gallouët, R. Herbin, and C. Guichard. Gradient schemes for the stefan problem. 2013. submitted.
  19. R. Eymard, T. Gallouët, and R. Herbin. Cell centred discretisation of non linear elliptic problems on general multidimensional polyhedral grids. J. Numer. Math., 17(3):173-193, 2009.
  20. R. Eymard, C. Guichard, and R. Herbin. Small-stencil 3D schemes for diffusive flows in porous media. ESAIM Math. Model. Numer. Anal., 46(2):265-290, 2012.
  21. R. Eymard, A. Handlovičová, R. Herbin, K. Mikula, and O. Stašová. Gradient schemes for image processing. In Finite volumes for complex applications. VI. Problems & perspectives. Volume 1, 2, volume 4 of Springer Proc. Math., pages 429-437. Springer, Heidelberg, 2011.
  22. R. Eymard and R. Herbin. Gradient scheme approximations for diffusion problems. In Finite volumes for complex applications. VI. Problems & perspectives. Volume 1, 2, volume 4 of Springer Proc. Math., pages 439-447. Springer, Heidelberg, 2011.
  23. R. Eymard and R. Herbin. Mixed finite element methods and gradient schemes for under- ground flow simulations. In Proc. of the 5th International Con-ference on Approximation Methods and Numerical Modelling in Environment and Nat-ural Resources, Granada, Spain, 2013. submitted.
  24. R. S. Falk and M. E. Morley. Equivalence of finite element methods for problems in elasticity. SIAM J. Numer. Anal., 27:1486-1505, 1990.
  25. D.P. Flanagan and T. Belytschko. A uniform strain hexahedron and quadrilateral with or- thogonal hourglass control. International Journal for Numerical Methods in Engineering, 17:679-706, 1981.
  26. G.N. Gatica and E.P. Stephan. A mixed-FEM formulation for nonlinear incompressible elas- ticity in the plane. Numerical Methods for Partial Differential Equations, 18:105-128, 2002.
  27. E. P. Kasper and R. L. Taylor. A mixed-enhanced strain method. Part I: geometrically linear problems. Computers and Structures, 75:237-250, 2000.
  28. P. Knobloch. On korn's inequality for nonconforming finite elements. Technical report, 2000. Band 20, Heft 3.
  29. V.A. Kozlov, V.G. Maz'ya, and J. Rossmann. Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs 85. American Mathematical Society, Providence, RI, 2001.
  30. B. P. Lamichhane. Mortar finite elements for coupling compressible and nearly incompressible materials in elasticity. Int. J. Numer. Anal. Model., 6(2):177-192, 2009.
  31. B.P. Lamichhane. From the Hu-Washizu formulation to the average nodal strain formulation. Computer Methods in Applied Mechanics and Engineering, 198:3957-3961, 2009.
  32. B.P. Lamichhane, B.D. Reddy, and B.I. Wohlmuth. Convergence in the incompressible limit of finite element approximations based on the Hu-Washizu formulation. Numerische Mathe- matik, 104:151-175, 2006.
  33. J. Leray and J.-L. Lions. Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France, 93:97-107, 1965.
  34. G.J. Minty. On a "monotonicity" method for the solution of non-linear equations in Banach spaces. Proceedings of the National Academy of Sciences of the United States of America, 50(6):1038, 1963.
  35. J. Nečas. Introduction to the theory of nonlinear elliptic equations. A Wiley-Interscience Publication. John Wiley & Sons Ltd., Chichester, 1986. Reprint of the 1983 edition.
  36. M. A. Puso and J. Solberg. A stabilized nodally integrated tetrahedral. International Journal for Numerical Methods in Engineering, 67:841-867, 2006.
  37. A. Quarteroni and A. Valli. Numerical approximation of partial differential equations. Springer-Verlag, Berlin, 1994.
  38. G. Romano, F. Marrotti de Sciarra, and M. Diaco. Well-posedness and numerical perfor- mances of the strain gap method. Int. J. Numer. Meth. Engrg., 51:103-126, 2001.
  39. J.C. Simo and M.S. Rifai. A class of assumed strain method and the methods of incompatible modes. Int. J. Numer. Meths. Engrg., 29:1595-1638, 1990.

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