# A new conservative/dissipative time integration scheme for nonlinear mechanical systems

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## Abstract

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples.

## Keywords

Conservative/dissipative time integration scheme Nonlinear mechanical systems Linearly constrained quadratic programs Optimality conditions Unconditional energy stability## Notes

### Acknowledgements

Cristian Guillermo Gebhardt and Raimund Rolfes acknowledge the financial support of the Lower Saxony Ministry of Science and Culture (research project *ventus efficiens*, FKZ ZN3024) and the German Federal Ministry for Economic Affairs and Energy (research project *Deutsche Forschungsplattform für Windenergie*, FKZ 0325936E) that enabled this work.

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