Order reduction of stiff solvers at elastic multibody systems

Elastic multibody systems arise in the simulation of vehicles, robots, air- and spacecrafts. After semidiscretization in space, a partitioned differential-algebraic system of index 3 with large stiffness terms has to be solved. We investigate the behavior of numerical methods for stiff ODEs and DAEs at this problem class and show that strong order reductions may occur. Examples from structural dynamics and multibody systems illustrate the results.

[1]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[2]  K. Strehmel,et al.  Linear-implizite Runge-Kutta-Methoden und ihre Anwendung , 1992 .

[3]  Linda,et al.  A Time Integration Algorithm forFlexible Mechanism Dynamics : TheDAE-Method , 1996 .

[4]  A. Prothero,et al.  On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations , 1974 .

[5]  Ben P. Sommeijer,et al.  Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions , 1987 .

[6]  C. Lubich.  Integration of stiff mechanical systems by Runge-Kutta methods , 1993 .

[7]  Werner Schiehlen,et al.  Advanced Multibody System Dynamics , 1899 .

[8]  Carmen Arévalo,et al.  Stabilized multistep methods for index 2 Euler-Lagrange DAEs , 1996 .

[9]  B. Owren,et al.  Alternative integration methods for problems in structural dynamics , 1995 .

[11]  R. Courant,et al.  Methoden der mathematischen Physik , .

[12]  Christian Lubich,et al.  Extrapolation integrators for constrained multibody systems , 1991, IMPACT Comput. Sci. Eng..

[13]  Wieslaw Marszalek,et al.  The Index of an Infinite Dimensional Implicit System , 1999 .

[14]  Alexander Ostermann,et al.  Runge-Kutta methods for partial differential equations and fractional orders of convergence , 1992 .

[15]  M. Arnold.  Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2 , 1998 .

[16]  Ernst Hairer,et al.  The numerical solution of differential-algebraic systems by Runge-Kutta methods , 1989 .

[17]  B. L. Ehle.  A-Stable Methods and Padé Approximations to the Exponential , 1973 .

[18]  W. Rheinboldt,et al.  On the numerical solution of the Euler-Lagrange equations , 1995 .

[19]  J. Z. Zhu,et al.  The finite element method , 1977 .

[20]  Bernd Simeon,et al.  Modelling a flexible slider crank mechanism by a mixed system of DAEs and PDEs , 1996 .