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Variational formulations for dissipative systems

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GCS - Geometry, compatibility and structure preservation in computational differential equations

Variational principles are powerful tools for the modelling and simulation of conservative mechanical and electrical systems. As it is well-known, the fulfilment of a variational principle leads to the Euler-Lagrange equations of motion describing the dynamics of such systems. Furthermore, a variational discretisation directly yields unified numerical schemes with powerful structure-preserving properties. Since many years there have been several attempts to provide a variational description also for dissipative mechanical systems, a task that is addressed in the talk in order to construct both Lagrangian and Hamiltonian pictures of their dynamics. One way doing this is to use fractional terms in the Lagrangian or Hamiltonian function which allows for a purely variational derivation of dissipative systems. Another approach followed in this talk is to embed the non-conservative systems in larger conservative systems. These concepts are used to develop variational integrators for which superior qualitative numerical properties such as the correct energy dissipation rate are demonstrated.




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