Timestepping schemes for nonsmooth dynamics based on discontinuous Galerkin methods: definition and outlook

Thorsten Schindler 1, * Vincent Acary 2
* Corresponding author
2 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : The contribution deals with timestepping schemes for nonsmooth dynamical systems. Traditionally, these schemes are locally of integration order one, both in smooth and nonsmooth periods. This is inefficient for applications with infinite events but large smooth phases like circuit breakers, valve trains or slider-crank mechanisms. To improve the behavior during smooth episodes, we start activities twofold. First, we include the classic schemes in time discontinuous Galerkin methods. Second, we split smooth and nonsmooth force propagation. The correct mathematical setting is established with mollifier functions, Clenshaw-Curtis quadrature rules and appropriate impact representation. The result is a Petrov-Galerkin distributional differential inclusion. It defines two Runge-Kutta collocation families and enables higher integration order during smooth transition phases. As the framework contains the classic Moreau-Jean timestepping schemes for constant ansatz and test functions on velocity level, it can be considered as a consistent enhancement. An experimental convergence analysis with the bouncing ball example illustrates the capabilities.
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Thorsten Schindler, Vincent Acary. Timestepping schemes for nonsmooth dynamics based on discontinuous Galerkin methods: definition and outlook. [Research Report] RR-7625, INRIA. 2011. ⟨inria-00595460v2⟩

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