Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-23T22:14:45.077Z Has data issue: false hasContentIssue false
  • English
  • Français

A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗

Published online by Cambridge University Press:  13 February 2012

Yongxing Shen  and
Adrian J. Lew
Yongxing Shen
Affiliation:
Laboratori de Càlcul Numèric, Universitat Politècnica de Catalunya (UPC BarcelonaTech), Barcelona, Spain. yongxing.shen@upc.edu
Adrian J. Lew
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, 94305-4040 California, USA; lewa@stanford.edu
Get access

Abstract

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order k ≥ 1 for the approximation of the displacement field, and of order k or k − 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields in both cases, with error estimates that are independent of the value of the Poisson’s ratio. These estimates demonstrate that these methods are locking-free. To this end, we prove the corresponding inf-sup condition, which for the equal-order case, requires a construction to establish the surjectivity of the space of discrete divergences on the pressure space. In the particular case of near incompressibility and equal-order approximation of the displacement and pressure fields, the mixed method is equivalent to a displacement method proposed earlier by Lew et al. [Appel. Math. Res. express 3 (2004) 73–106]. The absence of locking of this displacement method then follows directly from that of the mixed method, including the uniform error estimate for the stress with respect to the Poisson’s ratio. We showcase the performance of these methods through numerical examples, which show that locking may appear if Dirichlet boundary conditions are imposed strongly rather than weakly, as we do here.

Keywords

Discontinuous Galerkinlockingmixed methodinf-sup condition
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1003 - 1028
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, D.N., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equations. Calcolo 21 (1984) 337–344. Google Scholar
Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. Google Scholar
Bassi, F. and Rebay, S., A High-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267–279. Google Scholar
Becker, R., Burman, E. and Hansbo, P., A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3352–3360. Google Scholar
Bercovier, M. and Pironneau, O.A., Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1977) 211–224. Google Scholar
Brenner, S.C., Korn’s inequalities for piecewise H 1 vector fields. Math. Comp. 73 (2003) 1067–1087. Google Scholar
Brenner, S.C., Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41 (2003) 306–324. Google Scholar
S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3th edition, Springer (2008).
Brenner, S.C. and Sung, L.-Y., Linear finite element methods for planar linear elasticity. Math. Comp. 59 (1992) 321–338. Google Scholar
Brezzi, F., On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal. Numér. 8 (1974) 129–151. Google Scholar
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, Springer-Verlag, New York (1991).
Brezzi, F., Douglas, J. Jr., and Marini, L.D., Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. Google Scholar
Brezzi, F., Douglas, J. Jr., Durán, R. and Fortin, M., Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237–250. Google Scholar
F. Brezzi, G. Manzini, D. Marini, P. Pietra and A. Russo, Discontinuous galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations (2000) 365–378.
Brezzi, F., Hughes, T.J.R., Marini, L.D. and Masud, A., Mixed discontinuous Galerkin methods for Darcy flow. J. Sci. Comput. 22, 23 (2005) 119–145. Google Scholar
Carrero, J., Cockburn, B. and Schötzau, D., Hybridized globally divergence-free LDG methods. Part I : the Stokes problem. Math. Comp. 75 (2005) 533–563. Google Scholar
P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978).
Cockburn, B., Kanschat, G., Schötzau, D. and Schwab, C., Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40 (2002) 319–343. Google Scholar
Cockburn, B., Schötzau, D. and Wang, J., Discontinuous Galerkin methods for incompressible elastic materials. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3184–3204. Google Scholar
Crouzeix, M. and Raviart, P.-A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Sér. Rouge 7 (1973) 33–75. Google Scholar
Fortin, M., An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11 (1977) 341–354. Google Scholar
Girault, V., Rivière, B. and Wheeler, M.F., A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comp. 74 (2005) 53–84. Google Scholar
Hansbo, P. and Larson, M.G., Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1895–1908. Google Scholar
Hansbo, P. and Larson, M.G., Discontinuous Galerkin and the Crouzeix-Raviart element : Application to elasticity. ESAIM : M2AN 37 (2003) 63–72. Google Scholar
Hansbo, P. and Larson, M.G., Piecewise divergence-free discontinuous Galerkin methods for Stokes flow. Comm. Num. Methods Engrg. 24 (2008) 355–366. Google Scholar
Hecht, F., Construction d’une base de fonctions P 1 non conforme à divergence nulle dans R 3. RAIRO Anal. Numér. 15 (1981) 119–150. Google Scholar
Hood, P. and Taylor, C., Numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1 (1973) 1–28. Google Scholar
P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation. Finite Element Methods in Flow Problems, edited by J.T. Oden. UAH Press, Huntsville, Alabama (1974).
Kouhia, R. and Stenberg, R., A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. 124 (1995) 195–212. Google Scholar
Lew, A., Neff, P., Sulsky, D. and Ortiz, M., Optimal BV estimates for a discontinuous Galerkin method for linear elasticity. Appl. Math. Res. express 3 (2004) 73–106. Google Scholar
Nguyen, N.C., Peraire, J. and Cockburn, B., A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Engrg. 199 (2010) 582–597. Google Scholar
Rivière, B. and Girault, V., Discontinuous finite element methods for incompressible flows on subdomains with non-matching interfaces. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3274–3292. Google Scholar
Schötzau, D., Schwab, C. and Toselli, A., Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40 (2003) 2171–2194. Google Scholar
Scott, L.R. and Vogelius, M., Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modélisation Mathématique et Analyse Numérique 19 (1985) 111–143. Google Scholar
Soon, S.-C., Cockburn, B. and Stolarski, H.K., A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Engrg. 80 (2009), 1058–1092. Google Scholar
Ten Eyck, A., and Lew, A., Discontinuous Galerkin methods for non-linear elasticity. Int. J. Numer. Methods Engrg. 67 (2006) 1204–1243. Google Scholar
Ten Eyck, A., Celiker, F. and Lew, A., Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity : Analytical estimates. Comput. Methods Appl. Mech. Engrg. 197 (2008) 2989–3000. Google Scholar
F. Thomasset, Implementation of finite element methods for Navier-Stokes equations. Springer-Verlag, New York (1981).
Whiteley, J.P., Discontinuous Galerkin finite element methods for incompressible non-linear elasticity, Comput. Methods Appl. Mech. Engrg. 198 (2009) 3464–3478. Google Scholar
Wihler, T.P., Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal. 24 (2004) 45–75. Google Scholar