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# Simone Rossi (UNC-CH), Applied Mathematics Colloquium

## October 28, 2016 @ 4:00 pm - 5:00 pm

**Tea at 3:30 in Phillips Hall 330**

**Title: ** Implicit Finite Incompressible Elastodynamics with Linear Finite Elements

**Abstract: **In this talk we will show a new stabilization method for low order tetrahedral elements suitable for incompressible nonlinear implicit elastodynamics. Casting the momentum equation into a first order system, the method is based on a two-steps algorithm where we first solve the mixed P1/P1 velocity/pressure system, with the incompressibility constraint enforced through the divergence of the velocity field, and then we update the displacement field. The residual based stabilization acts only on the divergence-free equation and it constitutes a simple effective modification of the original problem leading to stable solutions. The method is derived from a variational multiscale analysis, where only the stabilizing term is maintained, and combined with several implicit time integrators. As we will show, dissipative time integrators must be used in order to get rid of unphysical high frequency pressure modes. Such modes can be appreciated only in the incompressible limit and become clear when the timestep is taken sufficiently small. The proposed algorithm can be applied to infinitesimal and finite strain deformations. In particular, we consider different classes of isotropic hyperelastic models for incompressible and nearly incompressible motion of rubber-like materials. We report several numerical examples establishing the performances and robustness of the proposed algorithm.