Differential-Geometric Methods in Computational Multibody System Dynamics

September 16, 2013 — September 20, 2013


  • Zdravko Terze (University of Zagreb, Zagreb, Croatia)
  • Andreas Müller (Technical University Chemnitz, Chemnitz, Germany)

Multibody system (MBS) dynamics, a branch of computational mechanics dealing with modeling principles and computational methods for dynamic analysis, simulation and control of mechanical systems, requires efficient and reliable formulations and computational methods.
Within research of novel computational concepts, geometric aspects of kinematical and dynamical modeling of MBS are increasingly recognized to play a significant role. By operating on manifolds, and Lie-groups in particular, instead of linear vector spaces, geometric algorithms respect the geometric structure underlying many technical systems and hence offer attractive features such as numerical robustness and efficiency as well as avoidance of the kinematical singularities.
Also, it is well-known that differential-geometric methods are the key concepts in contemporary mechanism design, and control theory. As such, geometric methods can provide the unifying mathematical framework that allows for successful studying of multidisciplinary interactions within complex environments.
Objective and Audience:
The aim of the School is to deliver a panoramic overview of the mathematical concepts underlying modern geometric approaches to modeling, time integration, and control of MBS, followed by an in-depth introduction to the relevant computational algorithms and numerical methods. By merging geometric methods in MBS dynamics, non-linear control and mechanism theory, the School provides unique educational platform that will deliver novel modeling concepts as well as theoretical and computational insights into dynamics and control of mechanical systems. The lectures take an application-driven approach, and numerous case-studies from many fields of engineering are presented and documented.
The School is primarily aimed for the audience of doctoral students and young researchers (post-docs) in engineering, mathematics and applied physics, but will be valuable also for senior researchers and practicing engineers who are interested in the field.
Lectures Outline:
The lectures provide a hands-on introduction to differential-geometric foundations and the audience will make acquaintance with these topics in a natural and application-driven way. A central topic of the school is efficient formulations using Lie-group concepts and screw theory, giving rise to numerically efficient and stable algorithms for MBS comprising rigid and flexible members. Special focus is given to energy and structure preserving numerical integration methods on manifolds for discrete and continuous systems. Natural coupling between mathematical modeling, numerical integration and control issues are covered by the lectures on variational integrators and optimal control with structure preserving integrators.
Specifically, lectures will include:
• introduction to mathematical concepts and differential-geometric modeling (manifolds, Lie-groups, Lie-algebras, exponential maps, screw theory etc.);
• modeling of complex MBS using compact Lie- group formalisms;
• time integration on Lie-groups;
• geometrically exact formulations for beams and shells;
• energy-consistent time integration procedures for MBS with flexible components;
• numerical treatment of holonomic and non-holonomic constraints, constraint stabilization;
• variational integrators, discrete mechanics and optimal control using structure-preserving integration schemes applied to high degree-of-freedom systems;
• Lie-group/screw theoretic framework for design of MBS and articulated mechanisms;
• multi-physics coupling procedures: aero-servo-elastic multidisciplinary models and applications.
A treatment of many numerical case-studies in the domain of robotics, wind energy systems, rotorcraft dynamics, aeronautical and mechatronical systems will highlight compact formulations, relevance and computational advantages of the geometric approach in the modern computational mechanics.
The unifying lecturing approach that combines computational procedures, control algorithms and design aspects, and provides new insights into the coupled modeling procedures, makes this School unique.

Multibody systems dynamics, Geometric modeling, Differential algebraic equations, Robotic applications, Wind turbine applications.


See also