Publication Date:
2014-01-25
Description:
We study the heterogeneous multiscale method (HMM) applied to mechanical systems with solution-dependent high frequencies. Using the example of a stiff spring double pendulum, we formulate a symmetric HMM. It is shown that, for our approach, a correct initialization of the microscale simulation depends crucially on the adiabatic invariance of the actions of the system. Moreover, this almost-invariant property guarantees the existence of an underlying effective system of differential-algebraic equations, which we derive. As we explain, the class of systems under consideration can still be transformed into a setting where an effective system is explicitly available. The analysis is done using canonical transformations proposed by K. Lorenz and Ch. Lubich. It provides general insight into the numerical behaviour and into the development of integrators for the systems under consideration. Finally, we show numerically how other integrators, namely FLAVORS and the impulse method, fail to resolve the correct dynamics.
Print ISSN:
0272-4979
Electronic ISSN:
1464-3642
Topics:
Mathematics
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