On an Implementation of the Hilber-Hughes-Taylor Method in the Context of Index 3 Differential-Algebraic Equations of Multibody Dynamics (DETC2005-85096)

The paper presents theoretical and implementation aspects related to a numerical integrator used for the simulation of large mechanical systems with flexible bodies and contact/impact. The proposed algorithm is based on the Hilber-Hughes-Taylor (HHT) implicit method and is tailored to answer the challenges posed by the numerical solution of index 3 differential-algebraic equations that govern the time evolution of a multibody system. One of the salient attributes of the algorithm is the good conditioning of the Jacobian matrix associated with the implicit integrator. Error estimation, integration step-size control, and nonlinear system stopping criteria are discussed in detail. Simulations using the proposed algorithm of an engine model, a model with contacts, and a model with flexible bodies indicate a 2 to 3 speedup factor when compared against benchmark MSC.ADAMS runs. The proposed HHT-based algorithm has been released in the 2005 version of the MSC.ADAMS/Solver.

[1]  Per Lötstedt,et al.  Numerical solution of nonlinear differential equations with algebraic constraints I: Convergence results for backward differentiation formulas , 1986 .

[2]  J. M. Watt.  Numerical Initial Value Problems in Ordinary Differential Equations , 1972 .

[3]  D. Dopico,et al.  Penalty, Semi-Recursive and Hybrid Methods for MBS Real-Time Dynamics in the Context of Structural Integrators , 2004 .

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

[5]  Javier García de Jalón,et al.  Kinematic and Dynamic Simulation of Multibody Systems , 1994 .

[6]  E. J. Haug,et al.  Computer aided kinematics and dynamics of mechanical systems. Vol. 1: basic methods , 1989 .

[7]  Ch . Engstler,et al.  MEXX - Numerical Software for the Integration of Constrained Mechanical Multibody Systems , 1992 .

[8]  Dena Hendriana,et al.  On a parabolic quadrilateral finite element for compressible flows , 2000 .

[9]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[10]  Javier García de Jalón,et al.  Kinematic and Dynamic Simulation of Multibody Systems: The Real Time Challenge , 1994 .

[11]  Volker Mehrmann,et al.  Differential-Algebraic Equations: Analysis and Numerical Solution , 2006 .

[12]  Laurent O. Jay,et al.  Structure Preservation for Constrained Dynamics with Super Partitioned Additive Runge-Kutta Methods , 1998, SIAM J. Sci. Comput..

[13]  M. Bampton,et al.  Coupling of substructures for dynamic analyses. , 1968 .

[14]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[15]  Leopold Alexander Pars,et al.  A Treatise on Analytical Dynamics , 1981 .

[16]  L. Petzold.  Differential/Algebraic Equations are not ODE's , 1982 .

[17]  Manuel S. Pereira,et al.  Computer-Aided Analysis of Rigid and Flexible Mechanical Systems , 1994 .

[18]  Yi Min Xie,et al.  A simple error estimator and adaptive time stepping procedure for dynamic analysis , 1991 .

[19]  C. W. Gear,et al.  Automatic integration of Euler-Lagrange equations with constraints , 1985 .

[20]  F. Potra.  Implementation of linear multistep methods for solving constrained equations of motion , 1993 .

[21]  A. Faruqui.  The real-time challenge , 2001 .

[22]  G. L. Vey.  Differential algebraic equations : a new look at the index , 1994 .

[23]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[24]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[25]  Bernd Simeon,et al.  Automatic h-Scaling for the Efficient Time Integration of Stiff Mechanical Systems , 2002 .

[26]  U. Nowak,et al.  Numerical Integration of Constrained Mechanical Systems Using MEXX , 1995 .

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

[28]  J. Baumgarte.  Stabilization of constraints and integrals of motion in dynamical systems , 1972 .

[29]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[30]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[31]  Dan Negrut,et al.  The Newmark Integration Method for Simulation of Multibody Systems: Analytical Considerations , 2005 .

[32]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[33]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[34]  Thomas J. R. Hughes,et al.  Improved numerical dissipation for time integration algorithms in structural dynamics , 1977 .

[35]  Björn Engquist,et al.  Backward differentiation approximations of nonlinear differential/algebraic systems , 1988 .

[36]  Kurt S. Anderson,et al.  Dynamic Simulation of Multibody Systems Using a New State-Time Methodology , 2005 .

[37]  M. Géradin,et al.  Numerical Integration of Second Order Differential—Algebraic Systems in Flexible Mechanism Dynamics , 1994 .

[38]  M. Iqbal,et al.  On photon correlation measurements of colloidal size distributions using Bayesian strategies , 2000 .

[39]  Olivier Bruls,et al.  A unified finite element framework for the dynamic analysis of controlled flexible mechanisms , 2005 .

[40]  Olivier A. Bauchau,et al.  Robust integration schemes for flexible multibody systems , 2003 .