A Stabilized Mixed Finite Element Method for Finite Elasticity Formulation for Linear Displacement and Pressure Interpolation

A stabilized mixed finite element method for finite elasticity is presented. The method circumvents the fulfillment of the Ladyzenskaya-Babuska-Brezzi condition by adding mesh-dependent terms, which are functions of the residuals of the Euler-Lagrange equations, to the usual Galerkin method. The weak form and the linearized weak form are presented in terms of the reference and current configuration. Numerical experiments using a tetrahedral element with linear shape functions for the displacements and for the pressure show that the method successfully yields a stabilized element. Introduction Galerkin Methods applied to almost incompressible or fully incompressible finite elasticity in the setting of a mixed Finite Element Method have to fulfill the Ladyzenskaya-Babuska-Brezzi condition to achieve unique solvability, convergence and robustness (see e.g. Brezzi and Fortin [1991]). This imposes severe restrictions on the choice of the solution spaces for the unknowns. Without balancing them properly the solution will show significant oscillations rendering it useless. This prohibits the use of convenient elements that employ equal order shape functions for both, the displacements and the pressure. Furthermore, it makes use of p-adaptive methods impossible since the number of different displacement/pressure combinations for the shape functions is quite general in a p-adaptive environment. For about 15 years stabilized methods have been used in fluid flows and linear incompressible and nearly incompressible elastic media to overcome most of the limitations in the Galerkin method. Stabilized finite element methods consist of adding mesh-dependent terms to the usual Galerkin methods. Those terms are functions of the residuals of the Euler-Lagrange equations evaluated elementwise. From the construction, it follows that consistency is not affected since the exact solution satisfies both the Galerkin term and the additional term. See e.g. Franca, Hughes, Loula, Miranda [1988] for an application to nearly incompressible linear elasticity, and Hughes, Franca, Balestra [1985] for an application to the Stokes flow, which is a form identical to the incompressible linear elasticity. Douglas and Wang [1989] propose a slightly different stabilization, which they call absolutely stabilized finite element method. Another family of stabilizing techniques consists of constructing LBB stable elements by adding bubble functions to equal-order continuous interpolation elements. Baiocchi, Brezzi, Franca [1993] show that, under certain assumptions, those methods are identical to stabilized methods. Recently Hughes [1995] demonstrated the relationship between stabilized methods and bubble function methods by identifying the origins of those methods in a particular class of subgrid models.