Time integrators based on approximate discontinuous Hamiltonians

Abstract

We introduce a class of time integration algorithms for finite dimensional mechanical systems whose Hamiltonians are separable. By partitioning the system's configuration space to construct an approximate potential energy, we define an approximate discontinuous Hamiltonian (ADH) whose resulting equations of motion can be solved exactly. The resulting integrators are symplectic and precisely conserve the approximate energy, which by design is always close to the exact one. We then propose two ADH algorithms for finite element discretizations of nonlinear elastic bodies. These result in two classes of explicit asynchronous time integrators that are scalable and, because they conserve the approximate Hamiltonian, could be considered to be unconditionally stable in some circumstances. In addition, these integrators can naturally incorporate frictionless contact conditions. We discuss the momentum conservation properties of the resulting methods and demonstrate their performance with several problems, such as rotating bodies and multiple collisions of bodies with rigid boundaries. © 2011 John Wiley & Sons, Ltd.