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

  • Cristian Guillermo GebhardtEmail author
  • Ignacio Romero
  • Raimund Rolfes
Original Paper


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.


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



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Structural Analysis and ForWind HannoverLeibniz Universität HannoverHannoverGermany
  2. 2.IMDEA Materials InstituteMadridSpain
  3. 3.Escuela Técnica Superior de Ingenieros IndustrialesUniversidad Politécnica de MadridMadridSpain

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