Timestepping schemes for nonsmooth dynamics based on discontinuous Galerkin methods: Definition and outlook
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- 作者：Thorsten Schindler ; Vincent Acary
- 关键词：Timestepping scheme ; High order ; Nonsmooth dynamics ; Time discontinuous Galerkin methods ; Impact
- 刊名：Mathematics and Computers in Simulation
- 出版年：January, 2014
- 年：2014
- 卷：95
- 期：Complete
- 页码：180-199
- 全文大小：628 K
文摘
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.