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This paper introduces Finite Elements with Switch Detection (FESD), anumerical discretization method for nonsmooth differential equations. We regardthe Filippov convexification of these systems and a transformation into dynamiccomplementarity systems introduced by Stewart. FESD is based on solving ofnonlinear complementarity problems and able to automatically detect thenonsmooth events in time. If standard Runge-Kutta (RK) methods are naivelyapplied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, welet the integrator step-size to be a degree of freedom. Additionalcomplementarity conditions, that we call cross complementarities, enable exactswitch detection, hence FESD is able to recover the high order accuracy whichthe RK methods enjoy for smooth ODE. Additional conditions called stepequilibration allow the step-size to change only when switches occur and thusavoid spurious degrees of freedom. Convergence results for the FESD method arederived, local uniqueness of the solution and convergence of numericalsensitivities are proven. The efficacy of FESD in demonstrated on severalsimulation and optimal control examples. In an optimal control problembenchmark with FESD we achieve up to five orders of magnitude more accuratesolutions than a standard approach for the same computational time.

Finite Elements with Switch Detection for Direct Optimal Control of Nonsmooth Systems