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Cuadrado, Javier; Cardenal, Jesús; Bayo, Eduardo

doi:10.1023/a:1009754006096

Cited by 82 publications

(20 citation statements)

References 22 publications

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“…Bayo and Ledesman [28] formulated a "mass-orthogonal" projection method to improve the situation where the augmented Lagrangian formulation [13,14] satisfies the weighted constraint to machine accuracy but doesn't satisfy to the same level of accuracy for individual constraints at position, velocity and acceleration levels. Cuadrado et al [29] developed a more efficient implementation of the mass-orthogonal projection which requires only successive forward reductions and back-substitutions, and then Blajer [30] gave a correcting formulation which doesn't need to update the Lagrange multipliers. The energy consideration with velocity projection was studied by Orden et al [31,32], providing an alternative interpretation of its effect on the stability and a practical criterion for the mass-orthogonal projection matrix selection.…”

Section: Introductionmentioning

confidence: 99%

A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

Zhang

Liu

2015

Multibody Syst Dyn

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The constraint violation problem of spatial multibody systems is analyzed in this paper. The mass matrix is singular in the equations of motion when the Euler parameters with the normalization constraints are used to describe the orientation of the spatial rigid body. The constrained and weighted least-squares-based geometrical projection method is implemented to suppress the constraint violation during numerical integration, and the explicit correction formulation can be obtained by the block matrix inversion scheme. The mass matrix weighted correction formulation gives the physically consistent energy norm, but it needs the mass matrix to be positive definite. To extend the physically consistent correction formulation for solving spatial multibody systems' constraint violation problems with a singular mass matrix, a Modified Mass-Orthogonal Projection Method (MMOPM) and a Generalized Physical Orthogonal Projection Method (GPOPM) are proposed. MMOPM modifies the mass matrix directly by adding a penalty factor matrix which appears in the mass-orthogonal projection method and leads to a positive definite weight matrix that satisfies the block matrix inversion scheme condition. GPOPM is a generalization of the physical orthogonal projection method where the constrained least-squares method is weighted by the positive semi-definite mass matrix and the correction formulation is given by using the generalized block matrix inversion scheme. Numerical results show the feasibility and accuracy of the presented MMOPM and GPOPM. The constraints in position and velocity can reach machine precision during numerical integration. The elimination of violation of position constrains may require few iterations, while the violation of velocity constraints is removed in one step, and GPOPM is more accurate in velocity correction than MMOPM.

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Section: Introductionmentioning

confidence: 99%

A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

Zhang

Liu

2015

Multibody Syst Dyn

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“…However, adopting a Newton-Raphson solution scheme instead of the fixed point one can be advantageous in terms of efficiency and stability [10]. This requires the introduction of the numerical integrator formulas in the equations of motion.…”

Section: Implementation Of the Methods Following A Newton-raphson Schemementioning

confidence: 99%

“…It is then possible to assume that the constraints are exactly fulfilled at some of these levels, which enables one to remove the corresponding terms ,˙ , or¨ from the dynamic equations (6a). An index-1 Newton-Raphson implementation with position and velocity projections was described in [10]. To obtain this algorithm, it was assumed that the projections enforced = 0 anḋ = 0 and only the term¨ had to be considered in the equations of motion.…”

Section: Augmented Lagrangian Methodsmentioning

confidence: 99%

“…(8)- (10) is also equivalent to a penalty method if the number of updates of the multipliers in each evaluation ofq is set to one.…”

Section: Formulation Based On Hamilton's Canonical Equationsmentioning

confidence: 99%

“…The adjusted parameters in this case were the penalty factor α and ξ , which determines the relative weight of in Eq. (10). Parameter ω was set to Fig.…”

Section: Fixed Point and Newton-raphson Implementationsmentioning

confidence: 99%

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Behaviour of augmented Lagrangian and Hamiltonian methods for multibody dynamics in the proximity of singular configurations

et al. 2016

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Augmented Lagrangian methods represent an efficient way to carry out the forward-dynamics simulation of mechanical systems. These algorithms introduce the constraint forces in the dynamic equations of the system through a set of multipliers. While most of these formalisms were obtained using Lagrange's equations as starting point, a number of them have been derived from Hamilton's canonical equations. Besides being efficient, they are generally considered to be robust, which makes them especially suitable for the simulation of systems with discontinuities and impacts. In this work, we have focused on the simulation of mechanical assemblies that undergo singular configurations. First, some sources of numerical difficulties in the proximity of singular configurations were identified and discussed. Afterwards, several augmented Lagrangian and Hamiltonian formulations were compared in terms of their robustness during the forward-dynamics simulation of two benchmark problems. Newton-Raphson iterative schemes were developed for these formulations with the Newmark formula as numerical integrator. These outperformed fixed point iteration approaches in terms of robustness and efficiency. The effect of the formulation parameters on simulation performance was also assessed.

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A multi-index variable time step method for the dynamic simulation of multibody systems

Cardenal

Cuadrado

Morer-Camo

et al. 1999

Int. J. Numer. Meth. Engng.

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SUMMARYThe paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. Various numerical problems that arise during the simulation process are described. The paper also describes ways to circumvent problems.The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. It is also shown how the energy stored in the penalty system is suitable to measure the local integration error. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that it becomes more accurate as the time step size decreases.A section is devoted at the end of the paper to present numerical simulations that illustrate the performance of the proposed method.

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Cited by 82 publications

References 22 publications

A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

Behaviour of augmented Lagrangian and Hamiltonian methods for multibody dynamics in the proximity of singular configurations

A multi-index variable time step method for the dynamic simulation of multibody systems

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