Numerical Methods for Differential‐Algebraic Equations with Application to Real‐Times Simulation of Mechanical Systems

Potra, Florian A.

doi:10.1002/zamm.19940740307

Cited by 5 publications

(5 citation statements)

References 49 publications

(14 reference statements)

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“…Hamiltonian systems arise in applications where dissipative forces can be neglected [3], such as in conservative mechanical systems [6], astronomy [80], electrodynamics, molecular dynamics [1,2,7,8,64,78,84], plasma physics, and fluid dynamics [83]. Mechanical systems arise in many applications, e.g., in multibody dynamics of rigid and/or flexible bodies [36,69,70,71], in structural dynamics [14,15,17,20,37,38,39,40,55,56], in real-time vehicle-systems simulation [57], in aerospace applications [9], in biomechanics [48], and in robotics [43]. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler-Lagrange equations is presented.…”

Section: Introductionmentioning

confidence: 99%

Structure Preservation for Constrained Dynamics with Super Partitioned Additive Runge--Kutta Methods

Jay¹

1998

SIAM J. Sci. Comput.

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A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler-Lagrange equations is presented. A new class of integrators is defined: the super partitioned additive Runge-Kutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structure-preservation, the class of s-stage Lobatto IIIA-B-CC * SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.

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Section: Introductionmentioning

confidence: 99%

Structure Preservation for Constrained Dynamics with Super Partitioned Additive Runge--Kutta Methods

Jay¹

1998

SIAM J. Sci. Comput.

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“…at time 1 n t  , using an explicit integration formula can be made and used to predict the independent velocity ( ) 6. The independent accelerations can be identified and used to define the state space equations which can be integrated forward in time using lower order formulas such as the HHT, BDF, Park, or the Trapezoidal method to obtain ( 1) ,…”

Section: A Prediction Of the Independent Coordinates ( )mentioning

confidence: 99%

“…The independent accelerations can be identified and used to define the state space equations which can be integrated forward in time using the Jacobian-free Newton-Krylov algorithm described before. If convergence is achieved, go to Step 7, otherwise update to obtain ( 1) ,…”

Section: A Prediction Of the Independent Coordinates ( )mentioning

confidence: 99%

“…1-Define the system coordinates 0 3-Using simple iteration algorithm(TLSMNI), or the Jacobian-Free Newton-Krylov an implicit integration formula can be used to obtain 1 1 n q , and 1 1 n q  .…”

Section: Convergence Criteriamentioning

confidence: 99%

“…The numerical solution of constrained MBS problems requires the numerical integration of differential and algebraic equation (DAE) systems. Potra [1] and Negrut [2] discussed several techniques for the numerical solution of the DAE system in MBS dynamics. Large scale and complex MBS applications that include flexible bodies and contact/impact elements lead to stiff problems.…”

Section: Introductionmentioning

confidence: 99%

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Efficient and robust implementation of the TLISMNI method

Aboubakr

Shabana

2015

Journal of Sound and Vibration

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The dynamics of large scale and complex multibody systems (MBS) that include flexible bodies and contact/impact pairs is governed by stiff equations. Because explicit integration methods can be very inefficient and often fail in the case of stiff problems,the use of implicit numerical integration methods is recommended in this case. This paper presents a new and efficient implementation of the two-loop implicit sparse matrix numerical integration (TLISMNI) method proposed for the solution of constrained rigid and flexible MBS differential and algebraic equations. The TLISMNI method has desirable features that include avoiding numerical differentiation of the forces, allowing for an efficient sparse matrix implementation, and ensuring that the kinematic constraint equations are satisfied at the position, velocity and acceleration levels. In this method, a sparse Lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is used to solve for all the accelerations and Lagrange multipliers. The generalized coordinate partitioning or recursive methods can be used to satisfy the constraint equations at the position and velocity levels. In order to improve the efficiency and robustness of the TLISMNI method, the simple iteration and the Jacobian-Free Newton-Krylov approaches are used in this investigation. The new implementation is tested using several low order formulas that include Hilber-Hughes-Taylor (HHT), L-stable Park, A-stable Trapezoidal, and A-stable BDF methods. The HHT method allow for including numerical damping. Discussion on which method is more appropriate to use for a certain application is provided. The paper also discusses TLISMNI implementation issues including the step size selection, the convergence criteria, the error control, and the effect of the numerical damping. The use of the computer algorithm described in this paper is demonstrated by solving complex rigid and flexible tracked vehicle models, railroad vehicle models, and very stiff structure problems. The results, obtained using these low order formulas, are compared with the results obtained using the explicit Adams-Bashforth predictor-corrector method. Using the TLISMNI method, which does not require numerical differentiation of the forces and allows for an efficient sparse matrix implementation, for solving complex and stiff structure problems leads to significant computational cost saving as demonstrated in this paper. In some problems, it was found that the new TLISMNI implementation is 35 times faster than the explicit Adams-Bashforth method.

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