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See detailError analysis of generalized-alpha Lie group time integration methods for constrained mechanical systems
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

in Numerische Mathematik (in press)

Generalized-alpha methods are very popular in structural dynamics. They are methods of Newmark type and combine favourable stability properties with second order convergence for unconstrained second order ... [more ▼]

Generalized-alpha methods are very popular in structural dynamics. They are methods of Newmark type and combine favourable stability properties with second order convergence for unconstrained second order systems in linear spaces. Recently, they were extended to constrained systems in flexible multibody dynamics that have a configuration space with Lie group structure. In the present paper, the convergence of these Lie group methods is analysed by a coupled one-step error recursion for differential and algebraic solution components. It is shown that spurious oscillations in the transient phase result from order reduction that may be avoided by a perturbation of starting values or by index reduction. Numerical tests for a benchmark problem from the literature illustrate the results of the theoretical investigations. [less ▲]

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See detailOrder reduction in time integration caused by velocity projection
Arnold, Martin; Cardona, Alberto; Bruls, Olivier ULg

in Proceedings of the 3rd Joint International Conference on Multibody System Dynamics and the 7th Asian Conference on Multibody Dynamics (2014, July)

Holonomic constraints restrict the configuration of a multibody system to a subset of the configuration space. They imply so called hidden constraints at the level of velocity coordinates that may ... [more ▼]

Holonomic constraints restrict the configuration of a multibody system to a subset of the configuration space. They imply so called hidden constraints at the level of velocity coordinates that may formally be obtained from time derivatives of the original holonomic constraints. A numerical solution that satisfies hidden constraints as well as the original constraint equations may be obtained considering both types of constraints simultaneously in each time step (stabilized index-2 formulation) or using projection techniques. Both approaches are well established in the time integration of differential-algebraic equations. Recently, we have introduced a generalized- alpha Lie group time integration method for the stabilized index-2 formulation that achieves second order convergence for all solution components. In the present paper, we show that a separate velocity projection would be less favourable since it may result in an order reduction and in large transient errors after each projection step. This undesired numerical behaviour is analysed by a one-step error recursion that considers the coupled error propagation in differential and algebraic solution components. This one-step error recursion has been used before to prove second order convergence for the application of generalized-alpha methods to constrained systems. As a technical detail, we discuss the extension of these results from symmetric, positive definite mass matrices to the rank deficient case. [less ▲]

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See detailSpurious oscillations in generalized-alpha time integration methods
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

Conference (2012, March)

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See detailLie group generalized-alpha time integration of constrained flexible multibody systems
Bruls, Olivier ULg; Cardona, Alberto; Arnold, Martin

in Mechanism & Machine Theory (2012), 48

This paper studies a Lie group extension of the generalized-alpha time integration method for the simulation of flexible multibody systems. The equations of motion are formulated as an index-3 ... [more ▼]

This paper studies a Lie group extension of the generalized-alpha time integration method for the simulation of flexible multibody systems. The equations of motion are formulated as an index-3 differential-algebraic equation (DAE) on a Lie group, with the advantage that rotation variables can be taken into account without the need of introducing any parameterization. The proposed integrator is designed to solve this equation directly on the Lie group without index reduction. The convergence of the method for DAEs is studied in detail and global second-order accuracy is proven for all solution components, i.e. for nodal translations, rotations and Lagrange multipliers. The convergence properties are confirmed by three benchmarks of rigid and flexible systems with large rotation amplitudes. The Lie group method is compared with a more classical updated Lagrangian method which is also formulated in a Lie group setting. The remarkable simplicity of the new algorithm opens interesting perspectives for real-time applications, model-based control and optimization of multibody systems. [less ▲]

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See detailImproved stability and transient behaviour of generalized-alpha time integrators for constrained flexible systems
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

Conference (2011, November)

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See detailTwo Lie Group Formulations for Dynamic Multibody Systems with Large Rotations
Bruls, Olivier ULg; Arnold, Martin; Cardona, Alberto

in Proceedings of the ASME 2011 International Design Engineering Technical Conferences (2011, August)

This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations ... [more ▼]

This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations can then be solved using a Lie group time integration method proposed in a previous work. The general structure of the equations of motion are derived from Hamilton principle in a general and unifying framework. Then, in the case of rigid body dynamics, two particular formulations are developed and compared from the viewpoint of the structure of the equations of motion, of the accuracy of the numerical solution obtained by time integration, and of the computational cost of the iteration matrix involved in the Newton iterations at each time step. In the first formulation, the equations of motion are described on a Lie group defined as the Cartesian product of the group of translations R^3 (the Euclidean space) and the group of rotations SO(3) (the special group of 3 by 3 proper orthogonal transformations). In the second formulation, the equations of motion are described on the group of Euclidean transformations SE(3) (the group of 4 by 4 homogeneous transformations). Both formulations lead to a second-order accurate numerical solution. For an academic example, we show that the formulation on SE(3) offers the advantage of an almost constant iteration matrix. [less ▲]

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See detailConvergence analysis of generalized-alpha Lie group integrators for constrained systems
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

in Proceedings of Multibody Dynamics ECCOMAS Conference (2011, July)

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See detailConvergence analysis for a generalized-alpha Lie group time integrator
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

Conference (2011, January)

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See detailLie group integrators for the numerical solution of DAE’s in flexible multibody dynamics
Cardona, Alberto; Bruls, Olivier ULg; Arnold, Martin

Conference (2010, November)

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See detailNumerical solution of DAEs in flexible multibody dynamics using Lie group time integrators
Bruls, Olivier ULg; Cardona, Alberto; Arnold, Martin

in Proceedings of the First Joint International Conference on Multibody System Dynamics (2010, May)

This paper studies a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parameterization problem and, as an ... [more ▼]

This paper studies a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parameterization problem and, as an extension of the classical generalized-alpha method for dynamic systems, it can deal with constrained equations of motion. Here, second-order accuracy of the Lie group method is demonstrated for constrained problems. The convergence analysis explicitly accounts for the nonlinear geometric structure of the Lie group. The performance is illustrated on two critical benchmarks of rigid and flexible systems with large rotation amplitudes. Second-order accuracy is evidenced in both of them. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control and optimization of multibody systems. [less ▲]

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See detailVariable step-size solvers for coupled DAEs in mechatronic applications
Bruls, Olivier ULg; Arnold, Martin

in 4th International Conference on Advanced Computational Methods in Engineering (ACOMEN) – Book of Abstracts (2008, May)

This work addresses a variable step-size formulation of the generalized- time integration scheme for mechanical and mechatronic systems represented by coupled differential-algebraic equations (DAEs). In ... [more ▼]

This work addresses a variable step-size formulation of the generalized- time integration scheme for mechanical and mechatronic systems represented by coupled differential-algebraic equations (DAEs). In previous publications, a variant of the generalized- alpha algorithm has been proposed, which is able to deal with a non-constant mass matrix, controller dynamics and kinematic constraints in a consistent way. We have shown that this fixed step-size method can be used to solve efficiently industrial problems. The present work focuses on variable step-size schemes. It is well-known that classical formulations of the generalized-alpha method are no more second-order accurate in this case. We argue that second-order accuracy can be recovered provided a modification of the coefficients of the method. Actually, the value of the coefficients should be modified at each time step according to a simple update formula. This approach can thus be implemented very easily in an existing code. We also report practical experience on the implementation of the method. A strategy for the selection of the step-size is described and the importance of a scaling of the equations and unknowns is highlighted. A number of examples and applications are presented in order to illustrate those results. [less ▲]

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See detailThe generalized-alpha scheme as a linear multistep integrator: Toward a general mechatronic simulator
Bruls, Olivier ULg; Arnold, Martin

in Journal Of Computational And Nonlinear Dynamics (2008), 3(4), 041007

This paper presents a consistent formulation of the generalized-alpha time integration scheme for mechanical and mechatronic systems. The algorithm can deal with a nonconstant mass matrix, controller ... [more ▼]

This paper presents a consistent formulation of the generalized-alpha time integration scheme for mechanical and mechatronic systems. The algorithm can deal with a nonconstant mass matrix, controller dynamics; and kinematic constraints. The theoretical background relies on the analogy with linear multistep formulas, which leads to elegant results related to consistency, order conditions for constant and variable step-size methods, as well as global convergence. Those results are illustrated for a controlled spring-mass system, and the method is also applied for the simulation of a vehicle semi-active suspension. [less ▲]

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See detailConvergence of the generalized-alpha scheme for constrained mechanical systems
Arnold, Martin; Bruls, Olivier ULg

in Multibody System Dynamics (2007), 18(2), 185-202

A variant of the generalized-alpha scheme is proposed for constrained mechanical systems represented by index-3 DAEs. Based on the analogy with linear multistep methods, an elegant convergence analysis is ... [more ▼]

A variant of the generalized-alpha scheme is proposed for constrained mechanical systems represented by index-3 DAEs. Based on the analogy with linear multistep methods, an elegant convergence analysis is developed for this algorithm. Second-order convergence is demonstrated both for the generalized coordinates and the Lagrange multipliers, and those theoretical results are illustrated by numerical tests. [less ▲]

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See detailConvergence of the generalized-alpha scheme for constrained mechanical systems
Arnold, Martin; Bruls, Olivier ULg

Report (2007)

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See detailThe generalized-alpha scheme as a linear multistep integrator: Towards a general mechatronic simulator
Bruls, Olivier ULg; Arnold, Martin

in Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (2007)

This paper presents a consistent formulation of the generalized-alpha time integration scheme for mechanical and mechatronic systems. The algorithm can deal with a nonconstant mass matrix, controller ... [more ▼]

This paper presents a consistent formulation of the generalized-alpha time integration scheme for mechanical and mechatronic systems. The algorithm can deal with a nonconstant mass matrix, controller dynamics, and kinematic constraints. The theoretical background relies on the analogy with linear multistep formulae, which leads to elegant results related with consistency, order conditions for constant and variable stepsize methods, as well as global convergence. The algorithm is applied for the simulation of a vehicle semi-active suspension. [less ▲]

Detailed reference viewed: 48 (1 ULg)