A Geometric Structure-Preserving Discretization Scheme for Incompressible Linearized Elasticity

In this paper, we present a geometric discretization scheme for incompressible linearized elasticity. We use ideas from discrete exterior calculus (DEC) to write the action for a discretized elastic body modeled by a simplicial complex. After characterizing the configuration manifold of volume-preserving discrete deformations, we use Hamilton's principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field. We explicitly derive the governing equations for the two-dimensional case, where the discrete spaces for the displacement field are constructed by P1 polynomials over primal meshes for incompressibility constraint, P0 polynomials over dual meshes for the kinetic energy, and P1 polynomials over support volumes for the elastic energy, and the discrete space of the pressure field is constructed by P0 polynomials over primal meshes. We test the efficiency and robustness of this geometric scheme using some numerical examples. In particular, we do not see any volume locking and/or checkerboarding of pressure in our numerical examples. This suggests that our choice of discrete solution spaces is compatible.