Maio 2014 vol. 1 num. 1 - 10th World Congress on Computational Mechanics
Full Article - Open Access.
A STABILIZED HYBRID FINITE ELEMENT METHOD FOR THE LINEAR ELASTICITY PROBLEMS
Hybrid formulations have been widely used in computational mechanics associated with primal or mixed finite element methods. Recently, hybrid formulations have been developed associated with Discontinuous Galerkin methods. In this work we propose a new primal hybrid finite element method for linear elasticity. Using a stabilization strategy typical of Discontinuous Galerkin methods, we choose as multiplier the displacement field itself and add stabilization and symmetrization terms to generate a stable and adjoint consistent formulation allowing greater flexibility in the choice of basis functions of approximation spaces for the displacement field and the Lagrange multiplier. The local problems, in the displacement field, can always be solved at the element in favor of the Lagrange multiplier defined on each edge of the elements. The global system is assembled involving only the degrees of freedom associated with the Lagrange multipliers, as usual in a hybrid method, where the continuity on the element edges is imposed weakly. Polynomial bases are adopted to approximate both the displacement field and the Lagrange multipliers considering Lagrangian polynomial base. Result of some numerical experiments are presented to illustrate the potential of the proposed formulation.
Palavras-chave: Linear Elasticity, Discontinuous Galerkin, Hybridization, Stabilization,
-  R. Adams. Sobolev Spaces. Academic Press, New York, 1975.
-  D. N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19(4):742–760, 198
-  D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779, 2001/02.
-  F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, 1991.
-  Y. Chen, J. Huang, X. Huang, and Y. Xu. On the Local Discontinuous Galerkin method for linear elasticity. Mathematical Problems in Engineering, 2010:20 pages, 2010.
-  B. Cockburn, B. Dong, and J. Guzmán. A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comp., 77(264):1887–1916, 2008.
-  B. Cockburn, B. Dong, J. Guzmán, M. Restelli, and R. Sacco. A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM J. Sci. Comput., 31(5):3827–3846, 2009.
-  B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of discontinuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems. 47(2):1319–1365, 2009.
-  B. Cockburn, J. Gopalakrishnan, and F.-J. Sayas. A projection-based error analysis of HDG methods. Math. Comp., 79(271):1351–1367, 2010.
-  L. P. Franca, T. J. R. Hughes, A. F. D. Loula, and I. Miranda. A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation. Numerische Mathematik, 53:123–141, 1988.
-  L. P. Franca and R. Stenberg. Error analysis of some Galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal., 28(6):1680–1697, 1991.
-  M. E. Gurtin. An Introductory to Continuum Mechanics. Academic Press, New York, NY, 1981.
-  P. Hansbo and M. G. Larson. Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Engrg., 191:1895–1908, 2002.
-  R. Kouhia and R. Stenberg. A linear nonconforming finite element method for nearly incompressible elasticity and stokes flow. Comput. Methods Appl. Mech. Engrg., 124(3):195–212, 1995.
-  W. M. Lai, D. Rubin, and E. Krempl. Introductory to Continuum Mechanics. Butterworth Heinemann, 3rd Edition, Woburn, MA, 1999.
-  B. Rivière. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: theory and implementation. SIAM, 2008.
-  B. Rivière and M. F. Wheeler. Optimal error estimates for discontiuous Galerkin methods applied to linear elasticity problems. Comput. Math. Appl, 46:141–163, 2000.
-  M. Vogelius. An analysis of the p-version of the finite element method for nearly incompressible materials. Numer. Math., 41:39–53, 1983.
-  T. P. Wihler. Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems. Mathematics of Computation, 75(255):1087–1102, 2006.
Faria, C. O.; Boness, A. J.; Loula, A. F. D.; "A STABILIZED HYBRID FINITE ELEMENT METHOD FOR THE LINEAR ELASTICITY PROBLEMS", p. 2817-2829 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1].
São Paulo: Blucher,
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-19037
últimos 30 dias | último ano | desde a publicação