Efficient and robust implementation of the TLISMNI method

Abstract

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 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 allows 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.