Consistent Integration of Non-Smooth Dynamical Systems
The aim of this PhD project is to develop
numerical methods for the consistent integration of the dynamics of non-smooth mechanical systems.
Non-smooth mechanical models with
set-valued force laws of normal cone type are very well suited for the analysis
of the dynamics of mechanical systems with unilateral contacts and Coulomb
friction. For the numerical integration of the dynamics of these models
time-stepping schemes based on Moreau's midpoint rule in combination with a
reformulation of the normal cone inclusions as proximal point problems have been
proven to be very robust. Usually these time-stepping schemes use a fully
implicit discretisation of all non-smooth forces, while integrating all
classical and smooth forces as well as displacements related to unilateral
contacts with an explicit scheme. If the time steps cannot be chosen small
enough (e.g. for performance reasons), this can lead to an unstable integration,
drift in the energy balance or drift in the unilateral constraints.
Using integrators which discretise all terms with an implicit scheme, one can
achieve an energetically consistent integration. This means, if the total energy
was preserved or strictly decreasing in the model, then the difference scheme
used for integration only allows approximations which have this property as
well. Similarly drift problems with unilateral contacts can be addressed. Both
properties are very useful for increasing the overall robustness of the
integration. When formulating energetically consistent integration schemes one
has to be careful not to increase the number of equations too drastically, or
otherwise performance will suffer. This is the reason why DAE-formulations for
rigid bodies based on quaternions are advantageous in this context.
Normal cones and proximal points.
Fully implicit schemes give rise to new problems, e.g. solving the resulting
normal cone inclusion can become difficult due to additional equations,
especially if they are nonlinear and unstructured. Solutions to this problem can
be obtained by using time splitting or specialised proximal point iteration
methods. Other problems can arise from the model itself in cases where the
impact laws are not energetically consistent.
Beside their direct use for the analysis of non-smooth mechanical applications,
consistent integration schemes are interesting as well for the interpretation
and understanding of the properties of time-stepping schemes closer to Moreau's
Simulation of the dynamics of a Tippe-Top.
Möller, M., Leine, R.I. and
Glocker, Ch., An efficient approximation of set-valued force laws of normal
cone type, Proceedings of the 7th EUROMECH Solid Mechanics Conference
(ESMC2009), Lisbon, Portugal, 2009