Implementation Details of a Generalized-α Differential-Algebraic Equation Lie Group Method

Abstract

Configuration spaces with Lie group structure display kinematical nonlinearities of mechanical systems. In Lie group time integration, this nonlinear structure is also considered at the time-discrete level using nonlinear updates of the configuration variables. For practical implementation purposes, these update formulae have to be adapted to each specific Lie group setting that may be characterized from the algorithmic viewpoint by group operation, exponential map, tilde, and tangent operator. In this paper, we discuss these practical aspects for the time integration of a geometrically exact Cosserat rod model with rotational degrees-of-freedom being represented by unit quaternions. Shearing and longitudinal extension of the Cosserat rod may be neglected using suitable constraints that result in a differential-algebraic equation (DAE) formulation of the beam structure. The specific structure of unconstrained systems and constrained systems is exploited by tailored algorithms for the corrector iteration of the generalized-α Lie group integrator.

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

“…The map p = [p 0 , p ] → p −1 = [p 0 , −p ] for arbitrary quaternions p is called conjugation and it coincides with the inversion on S 3 . 2 The Lie group (S 3 , * ) is isomorphic to the complex-valued matrix Lie group (SU(2), •) and the map p → R(p) = [R(p) e 1 , R(p) e 2 , R(p) e 3 ], which is called the Euler map, provides a double cover of the matrix Lie group of rotation matrices SO (3). is called the Lie algebra of G. Instead of handling elements v(t) ∈ g of an abstract tangent space, we will express v(t) as the velocity vector v(t) ∈ R n , which bears a physical meaning [4,37].…”

A modified HHT method for the numerical simulation of rigid body rotations with Euler parameters

“…This paper discusses on two mathematical aspects in order to solve also highly complex multibody problems arising in practice. One aspect concentrates on the rotational parametrization in order to get a simplified structure of the equations of motion and the second aspect focuses on the numerical B W. Steiner wolfgang.steiner@fh-wels.at 1 University of Applied Sciences Upper Austria, Josef Ressel Center for Advanced Multibody Dynamics, Stelzhamerstraße 23, Wels 4600, Austria time integration procedure in order to solve the system of equations in reasonable time also for industrial applications.…”

“…Since numerical damping cannot be introduced with the Newmark family integrators unless the order of accuracy is decreased, Hilber, Hughes and Taylor [13] proposed the so-called HHT α-method which can still maintain second order accuracy and stability. A variable damping depending on the value of the parameter α in the interval [− 1 3 , 0] is involved. The smaller the value of α the larger the artificial damping can be chosen in order to damp out high frequency oscillations that are of no interest.…”

“…The group around Arnold, Cardona and Brüls [2,3] suggests a Lie group time integration of constrained systems and presents a stabilized index-2 formulation in terms of Euler parameters; see e.g. [1]. Terze et al [26] propose a Lie group integration method for an index-1 formulation for constrained multibody systems by a reformulated Munthe-Kaas algorithm based on the work by Müller et al [14] in which the importance of constraint satisfaction in the Lie group configuration is pointed out.…”

Improved HHT integration method for index-2 over-determined equations of motion in multibody system dynamics

“…丁洁玉和潘振宽 [20] 基于约束投影策略, 提出了求 解多体系统指标-1超定运动方程的广义-α投影方法. 针对多体系统指标-2超定运动方程, 基于GGL约束 稳定方法 [21] , Negrut和Jay [13,22,23] 提出了HHT-ADD方法, HHT-SI2方法和广义-α-SOI2方法等, Lunk和Simeon [24] 提出了α-RATTLE方法, Arnold等人 [25,26]…”