SIAM Journal on Numerical Analysis
Scientific Computing with Ordinary Differential Equations
Scientific Computing with Ordinary Differential Equations
A Numerical Scheme for Impact Problems I: The One-Dimensional Case
SIAM Journal on Numerical Analysis
A Numerical Scheme for Impact Problems II: The Multidimensional Case
SIAM Journal on Numerical Analysis
Is Gauss Quadrature Better than Clenshaw-Curtis?
SIAM Review
Stability and Convergence of Mechanical Systems with Unilateral Constraints
Stability and Convergence of Mechanical Systems with Unilateral Constraints
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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.