Variational Methods for Nonsmooth Mechanics

In this thesis we investigate nonsmooth classical and continuum mechanics and its discretizations by means of variational numerical and geometric methods. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Next, we develop the foundations of a multisymplectic treatment of nonsmooth classical and continuum mechanics. This work may be regarded as a PDE generalization of the previous formulation of a variational approach to collision problems. The multisymplectic formulation includes a wide collection of nonsmooth dynamical models such as rigid-body collisions, material interfaces, elastic collisions, fluid-solid interactions and lays the groundwork for a treatment of shocks. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplecticmomentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically and the longtime stable energy behavior typical of variational methods is demonstrated.