Simulations of dynamical systems are intended to reproduce the dynamic behavior in a realistic way. Using structure-preserving integration schemes for the simulation of mechanical systems, certain properties of the real system are conserved in the numerical solution. Examples are the conservation of energy or momentum induced by symmetries in the system (e.g., conservation of the angular momentum in case of rotational symmetry). A special class of structure-preserving integrators are variational integrators that are derived based on discrete variational principles. As it is well-known, the fulfilment of a variational principle leads to the Euler-Lagrange equations of motion describing the dynamics of mechanical or electrical systems. Variational integrators are constructed by directly discretizing the variational principle rather than the Euler-Lagrange equations. The resulting time-stepping schemes are symplectic by construction which leads to excellent long time energy behavior. Furthermore, the fulfilment of the discrete Noether theorem guarantees preservation of momentum maps induced by symmetries.

In our research group we work on higher and mixed order constructions and the analysis of variational integrators as well as their extension to various system classes. The construction of numerical methods of higher order yields so called Galerkin variational integrators whose numerical properties are analyzed in depth by means of particularly tailored variational methods. System classes we consider are, for example, dynamical systems on different time scales, fractional Lagrangian systems, electric circuits and flexible beam dynamics.

Read more about our work on

• Higher and mixed order Galerkin variational integrators
• Variational multirate integration for dynamics on different time scales
• Variational integrators for fractional Lagrangians
• Variational Lie-group integrators for flexible beams
• Variational integrators for electric circuits
• List of publications

For the design of variational integrators of higher order, Galerkin variational integrators are developed and analyzed. Their construction relies on the approximation of the action based on a choice of a finite-dimensional function space and a numerical quadrature formula. To analyze the numerical properties of the Galerkin integrators, various investigations are performed. For particular variational integrators their equivalence to other well-known integration methods can be shown, e.g. the equivalence to the Störmer-Verlet method, to the Newmark algorithm or, generally, to symplectic partitioned and so-called modified symplectic partitioned Runge-Kutta methods. On the one hand, the knowledge that an integrator is equivalent to a variational integrator provides an elegant proof of symplecticity since the classes of variational and symplectic integrators are identical. On the other hand, since Runge-Kutta methods have been analyzed quite extensively during the last decades, the equivalence relation can be exploited to analyze, e.g., stability and convergence properties of variational integrators. Furthermore, novel stability concepts and error analysis are performed exploiting the variational structure of the integrators. In particular, superconvergence of Galerkin variational integrators can be proven based on a variational error analysis. The integrators and their analysis are extended for constrained and optimal control systems as well. Furthermore, mixed order constructions are developed for the efficient simulation of dynamics on different time scales.

Contact: Sina Ober-Blöbaum

Mechanical and electric systems with dynamics on varying time scales, in particular those including highly oscillatory motion, impose challenging questions for numerical integration schemes. Tiny step sizes are required to guarantee a stable integration of the fast frequencies. However, for the simulation of the slow dynamics, integration with a larger time step is accurate enough. Here, small time steps increase integration times unnecessarily, especially for costly function evaluations.

For the efficient simulation of systems comprising fast and slow dynamics, variational multirate integrators are developed or which the slow part of the system is integrated with a relatively large step size while the fast part is integrated with a small time step. A closed variational derivation on a time grid consisting of macro and micro time nodes leads to symplectic, momentum-preserving integration schemes with good long-time energy behavior. Stability, convergence and preservation properties of the integrators are investigated exploiting their variational structure. The variational multirate integrator is extendable to systems with constraints for, e.g., the efficient simulation of multibody systems with dynamics on different time scales. The integrators are adapted for the use in direct optimal control methods to allow an efficient numerical treatment of different time scales in optimal control problems.

Contact: Yana Lishkova, Sina Ober-Blöbaum

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. The use of symplectic or variational integrator for the simulation leads to excellent long-time behavior which is well investigated for conservative systems using techniques of backward error analysis. While excellent long-time energy behavior for dissipative systems (i.e., correct rates of energy decay) is observed numerically, there are no general analytic proofs to explain this phenomenon for non-conservative systems. The key issue is that the description of dissipative systems typically does not rely on purely variational principles and thus the Hamiltonian is no conserved quantity anymore.
Goal of this research project is to develop purely variational principles that are able to model dissipative systems. 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 [Rie97, JOB18]. More concretely, employing a phase space which includes the (Caputo) fractional derivative of curves evolving on real space, a variational principle for Lagrangian systems is developed yielding the so-called restricted fractional Euler-Lagrange equations. This variational principle relies on a particular restriction upon the admissible variation of the curves. In the case of the half-derivative and mechanical Lagrangians, i.e. kinetic minus potential energy, the restricted fractional Euler-Lagrange equations model a dissipative system in both directions of time, summing up to a set of equations that is invariant under time reversal. A discrete fractional Lagrangian version is derived and used to construct fractional variational integrators for the structure-preserving integration of dissipative systems.

Contact: Sina Ober-Blöbaum

For dynamical systems defined on Lie-groups, the corresponding group action can be used to update the group element during the simulation such that the group structure is preserved in a natural way. These integrators are denoted as Lie-group methods. Lie-group formulations are in particular appropriate to describe the orientation of rigid bodies or cross sections of flexible beams. Efficient structure-preserving integrators can be used for the system defined on the Lie- algebra (a linear space) to simulate the dynamics on the Lie-group (typically a nonlinear space). Exploiting the variational structure of beam dynamics variational Lie-group integrators for the simulation of the dynamics of flexible beams are developed and implemented. To realize the simulation of different beam elements with different time steps, so-called asynchronous variational integrators are used.

Contact: Sina Ober-Blöbaum

Variational integrators have been mainly developed and used for a wide variety of mechanical systems. However, considering real-life systems, these are in general not of purely mechanical character. In fact, more and more systems become multidisciplinary in the sense, that not only mechanical parts, but also electric and software subsystems are involved, resulting into mechatronic systems. Since the integration of these systems with a unified simulation tool is desirable, the aim is to extend the applicability of variational integrators to electric systems. The Lagrangian formulation of the dynamics of electric
circuits leads to a degenerate Lagrangian (and thus to a degenerate symplectic form). Reduction techniques are applied to develop variational integrators for the resulting differential-algebraic systems. It can be shown that for the developed integrators the energy rate as well as the current frequencies are better preserved in comparison to a simulation with e.g. Runge-Kutta methods. The developed algorithms are also applied to stochastic circuits. The approach is extended to the treatment of general nonlinear
circuits as well as to hybrid systems, e.g., in the presence of transistors and optimal control problems.

Contact: Sina Ober-Blöbaum