Extrapolation integrators for constrained multibody systems

doi:10.1016/0899-8248(91)90008-I

IMPACT of Computing in Science and Engineering

Volume 3, Issue 3, September 1991, Pages 213-234

IMPACT of Computing ...

https://doi.org/10.1016/0899-8248(91)90008-I Get rights and content

Abstract

Extrapolation methods using the structure in the equations of motion of multibody systems are given in this article. The methods are explicit in the differential part and implicit in the nonlinear constraints. They admit a robust formulation in which only linear systems of equations are solved most of the time. Related methods, which are linearly implicit also in the differential part, are developed for stiff mechanical systems. Numerical results for the extrapolation code MEXX are included.

References (44)

C.W. Gear
Differential-algebraic equations, indices, and integral algebraic equations
SIAM J. Numer. Anal.
(1990)
W.B. Gragg
On extrapolation algorithms for ordinary initial value problems
SIAM J. Numer. Anal.
(1965)
R. März
Some new results concerning index-3 differential-algebraic equations
J. Math. Anal. Appl.
(1989)
A. Ostermann
A half-explicit extrapolation method for differential-algebraic systems of index 3
IMA J. Numer. Anal.
(1990)
W. Auzinger et al.
Asymptotic error expansions for stiff equations: The implicit Euler scheme
SIAM J. Numer. Anal.
(1990)
D.S. Bae et al.
A recursive formulation for constrained mechanical system dynamics I and II
Mech. Structures Mach.
(1987)
D.S. Bae et al.
A recursive formulation for constrained mechanical system dynamics I and II
Mech. Structures Mach.
(1987)
H.G. Bock et al.
Numerical Solution of Constrained Least Squares Boundary Value Problems in Differential-Algebraic Equations
H. Brandl et al.
A very efficient algorithm for the simulation of robots and similar multibody systems without inversion of the mass matrix
H. Brandl et al.
An algorithm for the simulation of multibody systems with kinematic loops
V. Brasey et al.
Half-Explicit Runge-Kutta Methods for Differential-Algebraic Systems of Index 2
(1991)

K.E. Brenan et al.

Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations

(1989)

R. Bulirsch et al.

Numerical treatment of ordinary differential equations by extrapolation methods

Numer. Math.

(1966)

A. Daberkow et al.

Test Examples for Multibody Systems

(1989)

P. Deuflhard

Order and stepsize control in extrapolation methods

Numer. Math.

(1983)

P. Deuflhard

Recent progress in extrapolation methods for ordinary differential equations

SIAM Rev.

(1985)

P. Deuflhard et al.

One-step and extrapolation methods for differential-algebraic systems

Numer. Math.

(1987)

E. Eich et al.

Stabilization and Projection Methods for Multibody Dynamics

(1990)

C. Führer et al.

Formulation and Numerical Solution of the Equations of Constrained Mechanical Motion

(1989)

C. Führer et al.

Generation and solution of multibody system equations

Internat. J. Non-Linear Mech.

(1990)

C.W. Gear et al.

Automatic integration of Euler-Lagrange equations with constraints

J. Comput. Appl. Math.

(1985)

E. Griepentrog

The index of differential-algebraic equations and its significance for the circuit simulation

E. Griepentrog et al.

Differential-Algebraic Equations and Their Numerical Treatment

(1986)

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View full text

Extrapolation integrators for constrained multibody systems

Abstract

SIAM J. Numer. Anal.

SIAM J. Numer. Anal.

J. Math. Anal. Appl.

IMA J. Numer. Anal.

Asymptotic error expansions for stiff equations: The implicit Euler scheme

SIAM J. Numer. Anal.

A recursive formulation for constrained mechanical system dynamics I and II

Mech. Structures Mach.

A recursive formulation for constrained mechanical system dynamics I and II

Mech. Structures Mach.

Numerical Solution of Constrained Least Squares Boundary Value Problems in Differential-Algebraic Equations

A very efficient algorithm for the simulation of robots and similar multibody systems without inversion of the mass matrix

An algorithm for the simulation of multibody systems with kinematic loops

Half-Explicit Runge-Kutta Methods for Differential-Algebraic Systems of Index 2

Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations

Numerical treatment of ordinary differential equations by extrapolation methods

Numer. Math.

Test Examples for Multibody Systems

Order and stepsize control in extrapolation methods

Numer. Math.

Recent progress in extrapolation methods for ordinary differential equations

SIAM Rev.

One-step and extrapolation methods for differential-algebraic systems

Numer. Math.

Stabilization and Projection Methods for Multibody Dynamics

Formulation and Numerical Solution of the Equations of Constrained Mechanical Motion

Generation and solution of multibody system equations

Internat. J. Non-Linear Mech.

Automatic integration of Euler-Lagrange equations with constraints

J. Comput. Appl. Math.

The index of differential-algebraic equations and its significance for the circuit simulation

Differential-Algebraic Equations and Their Numerical Treatment