A Lie Algebra Approach to Lie Group Time Integration of Constrained Systems

Arnold, Martin; Cardona, Alberto; Brüls, Olivier

doi:10.1007/978-3-319-31879-0_3

Cited by 20 publications

(63 citation statements)

References 41 publications

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“…(6-8), we adapt the Lie group generalized-α method proposed in [3] for unified local velocity coordinates. (6-8), we adapt the Lie group generalized-α method proposed in [3] for unified local velocity coordinates.…”

Section: Time Integration Schemementioning

confidence: 99%

“…(3)(4)(5), the symbol • indicates the representation of a vector as skew symmetric matrix, see [3]. In Eqs.…”

Section: Introductionmentioning

confidence: 99%

“…In Eqs. (3)(4)(5), the symbol • indicates the representation of a vector as skew symmetric matrix, see [3]. For the incorporation of constraints, we consider constrained systems with linearly independent holonomic constraints Φ(H) = 0 on the Lie group G = H ∈ SE(3).…”

Section: Introductionmentioning

confidence: 99%

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A formulation for the dynamic analysis of multibody systems using non‐redundant unified local velocity coordinates

Holzinger

Gerstmayr

2019

Proc Appl Math and Mech

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In this paper, we present a formulation for the dynamic analysis of multibody systems using non-redundant unified local velocity coordinates. The motivation for the present work is the result of an earlier investigation of unified local velocity coordinates of Holzinger et al. [1]. The equations of motion presented by the authors allow the description of the rigid body motion in space by local, purely translational velocity coordinates. However, the earlier investigations are limited to the description of the spatial motion of a free rigid body. Therefore, constraints are incorporated in the present paper.

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Section: Time Integration Schemementioning

confidence: 99%

“…(3)(4)(5), the symbol • indicates the representation of a vector as skew symmetric matrix, see [3]. In Eqs.…”

Section: Introductionmentioning

confidence: 99%

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A formulation for the dynamic analysis of multibody systems using non‐redundant unified local velocity coordinates

Holzinger

Gerstmayr

2019

Proc Appl Math and Mech

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“…Euler parameters as a special choice of quaternions are frequently used to define the orientation of the rigid bodies of a multibody system since the rotation matrix becomes quite simple in this case and the equations of motion are simplified. Compared to other parametrizations of SO (3), Euler parameters are singularity-free and avoid the introduction of trigonometric functions. These preferences contribute to reduce the computational time in a multibody simulation program, although an inner constraint has to be observed as four parameters e 0 , e 1 , e 2 , e 3 are used for describing the three rotational degrees of freedom of a rigid rotation.…”

Section: Introductionmentioning

confidence: 99%

“…Another approach concerning the stabilization of constrained mechanical systems has been presented by García Orden [10], in which the artificial energy introduced in the system can be controlled from various points of view. 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].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2019

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In multibody dynamics, the Euler parameters are often used for the numerical simulation of rigid body rotations because they lead to a relatively simple form of the rotation matrix which avoids the evaluation of trigonometric functions and can thus save computational time. The Newmark method and the closely related Hilber-Hughes-Taylor (HHT) method are widely employed for solving the equations of motion of mechanical systems. They can also be applied to constrained systems described by differential algebraic equations. However, in the classical versions, the use of these integration schemes have a very unfavorable impact on the Euler parameter description of rotational motions. In this paper, we show analytically that the angular velocity for a rotation about a single axis under a constant moment will not increase linearly but grows slower. This effect, which does not appear for Euler angles, can be even observed if the numerical damping parameter α in the HHT method is set to zero. To circumvent this problem without losing the advantage of Euler parameters, we present a modified HHT method which reduces the damping effect on the angular velocity significantly and eliminates it completely for α = 0.

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A family of total Lagrangian Petrov–Galerkin Cosserat rod finite element formulations

Eugster

Harsch

2023

GAMM-Mitteilungen

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The standard in rod finite element formulations is the Bubnov-Galerkin projection method, where the test functions arise from a consistent variation of the ansatz functions. This approach becomes increasingly complex when highly nonlinear ansatz functions are chosen to approximate the rod's centerline and cross-section orientations. Using a Petrov-Galerkin projection method, we propose a whole family of rod finite element formulations where the nodal generalized virtual displacements and generalized velocities are interpolated instead of using the consistent variations and time derivatives of the ansatz functions. This approach leads to a significant simplification of the expressions in the discrete virtual work functionals. In addition, independent strategies can be chosen for interpolating the nodal centerline points and cross-section orientations. We discuss three objective interpolation strategies and give an in-depth analysis concerning locking and convergence behavior for the whole family of rod finite element formulations.

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A Lie Algebra Approach to Lie Group Time Integration of Constrained Systems

Cited by 20 publications

References 41 publications

A formulation for the dynamic analysis of multibody systems using non‐redundant unified local velocity coordinates

A formulation for the dynamic analysis of multibody systems using non‐redundant unified local velocity coordinates

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

A family of total Lagrangian Petrov–Galerkin Cosserat rod finite element formulations

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