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

Thorsten Schindler 1 Vincent Acary 1, *
* Corresponding author
1 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 non-impulsive and impulsive periods. This is inefficient for applications with infinitely many events but large non-impulsive phases like circuit breakers, valve trains or slider-crank mechanisms. To improve the behaviour during non-impulsive episodes, we start activities twofold. First, we include the classic schemes in time discontinuous Galerkin methods. Second, we split non-impulsive and impulsive force propagation. The correct mathematical setting is established with mollifier functions, Clenshaw-Curtis quadrature rules and an 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 non-impulsive 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. Mathematics and Computers in Simulation, Elsevier, 2014, 95, pp.180-199. ⟨10.1016/j.matcom.2012.04.012⟩. ⟨hal-00762850⟩

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