Abstract

The theory of energetic rate-independent systems is an elegant way to describe nonlinear systems in mechanics and other fields. One particular advantage is that it yields a natural time discretization that consists of a sequence of minimization problems. Unfortunately, in many interesting cases the objective functional is only given implicitly as the solution of a second minimization problem for a curve length in the state space. Therefore, its evaluation and obtaining derivatives can be very costly. Instead, we present a transformation based on the Finsler exponential map that turns the second minimization problem into an initial-value-problem for a second-order ODE. Solutions of this can be found much cheaper numerically, or may even be available in closed form. We show examples of this construction, and how to use it to obtain fast and robust Proximal Newton solvers for finite-strain elastoplasticity.