Extensions of single-step method for equations of motion from multibody dynamics

Abstract

• Improved single-step methods are extended to DAEs from multibody dynamics . • Optimal preconditioning strategies are employed in Newton-Raphson iteration. • All the new methods do not suffer from small time-step-size restriction. • Each new method has second-order accuracy and controllable numerical dissipation. Each second-order, unconditionally stable, linear multi-step (LMS) method has its equivalent single-step (SS) method. For the linear two-step method, LMS2, the corresponding SS method was developed as SS2. In this paper, SS2 for first-order ordinary differential equations (ODEs) is designated as SS2-1 and for second-order ODEs is designated as SS2-2. Then SS2-1 and SS2-2 methods are extended to the numerical integration of index 3 differential-algebraic equations (DAEs), index 2 DAEs, and GGL-stabilized index 2 DAEs from multibody dynamics. Optimal preconditioning strategies are employed in the new methods for DAEs. The right and left preconditioners cure the sensitivity to the solution perturbation and the conditioning of the Jacobian matrix for Newton-Raphson iteration. The extended methods are validated by numerical experiments. The effectiveness of the optimal preconditioning strategy and the violation characteristics of constraint equations for the new methods are also verified. The new methods have the advantages of second-order accuracy, and controllable numerical dissipation and they do not suffer from the small time-step-size restriction. Further, there is no acceleration accuracy reduction for equations of motion (EoM) with a non-constant mass matrix. The efficiency of the new methods and the existing generalized-α method for index 3 DAEs are also compared.