Speaker
Description
Geometrically exact beam models provide a powerful way to accurately model slender elastic elements undergoing large deformations, which is essential in many applications such as medical or soft robotics, cable and pipe modeling, or even biophysics. Due to the underlying kinematic description based on a Cosserat continuum, these models possess an inherent geometric structure, as their configuration space has the structure of a Lie group – more precisely, it is an infinite-dimensional product of the Special Euclidean group SE(3).
Preserving this structure during space and time discretization brings several advantages: It allows the definition of discrete strain measures that are inherently objective, which simultaneously enables interpolation schemes directly on the Lie group, it gives discrete numerical methods highly favorable properties (such as stability and accuracy at coarse discretizations), and it simplifies the integration of the discrete model into rigid-flexible multibody systems.
In this talk, we present a structure-preserving approach for the space and time discretization of geometrically exact beam models in the framework of variational integrators, which provide an elegant way to obtain highly stable, accurate, and efficient fully discrete models. The core of the approach is to define discrete velocity and deformation variables in the Lie algebra of SE(3), which are used to compute the group updates between space and time nodes using the retraction map [1], a generalization of the Lie group exponential map. We further illustrate how switching to a relative-kinematic description of the discretized beam – i.e., using discrete deformation variables as states – brings important practical advantages [2]: The resulting discrete model has a minimum number of states, can be integrated into rigid-flexible multibody systems in a remarkably elegant way, and highly stiff deformation modes can be excluded without algebraic constraints, which is essential for numerical performance.
[1] F. Demoures, F. Gay-Balmaz, S. Leyendecker, S. Ober-Blöbaum, T. S. Ratiu, and Y. Weinand, "Discrete variational Lie group formulation of geometrically exact beam dynamics," Numer. Math., vol. 130, no. 1, pp. 73–123, 2015,
doi: 10.1007/s00211-014-0659-4.
[2] M. Herrmann and P. Kotyczka, "Relative-kinematic formulation of geometrically exact beam dynamics based on Lie group variational integrators," Computer Methods in Applied Mechanics and Engineering, vol. 432, p. 117367, 2024,
doi: 10.1016/j.cma.2024.117367.
