We propose a new numerical method, namely, the staggered cell-centered finite element method for compressible and nearly incompressible linear elasticity problems. By building a dual mesh and its triangular submesh, the scheme can be constructed from a general mesh in which the displacement is approximated by piecewise linear (P1) functions on the dual submesh and, in the case of nearly incompressible problems, the pressure is approximated by piecewise constant (P0) functions on the dual mesh. The scheme is cell centered in the sense that the solution can be computed by cell unknowns of the primal mesh (for the displacement) and of the dual mesh (for the pressure). The method is presented within a rigorous theoretical framework to show stability and convergence. In particular, for the nearly incompressible case, stability is proved by using the macroelement technique. Numerical results show that the method, compared with other methods, is effective in terms of accuracy and computational cost.
Introduction.Many important physical problems involve analysis of the deformation of compressible and nearly incompressible linear elastic solids. In the biomechanical field, nearly incompressible or incompressible materials at large strains were popularly used. It is well known that low-order finite element methods exhibit a good performance when applied to compressible linear elasticity problems [21,36]. However, for the nearly incompressible elastic case, where the Lamé parameter λ tends to infinity, the accuracy deteriorates. This phenomenon is known as the "locking effect," meaning that the approximate solutions do not converge uniformly. Many methods have been proposed to overcome this problems, for example, the h-version of finite elements [5,6], the B-bar method [21], mixed formulations [3,11,12], enhanced assumed strain modes [15,32,33], the reduced integration stabilization [22,20] and the two-field mixed stress elements [31], the stream function approach [4], and the mimetic finite difference method [7]. In addition, there are several publications investigating an average nodal pressure formulation in which a constant pressure field is