Constructing time integration with controllable errors for constrained mechanical systems

Abstract

This paper investigates the numerical features of equivalent Lagrangian formulations for general mechanical systems and gives new insight into the error reduction of the corresponding numerical integrations. To this end, the modified Lagrange’s equations are derived, which give a unified form of equivalent Lagrangian formulations, featuring different generalized constraint forces. By transforming the modified Lagrange’s equations into their discrete versions, the errors of generalized constraint forces are estimated in strong and weak forms. An error analysis is employed to clarify the error generation of different formulations. This helps to identify whether the generalized constraint force contributes considerable errors compared to other terms in the system. Finally, a mechanism of error reduction is explained for the discrete system, providing a way to reduce total errors by adjusting the error generated from the generalized constraint forces. Inspired by this mechanism, a methodology for constructing synthetic integrations is developed. The synthetic integration scheme consists of the main scheme and the counter scheme. In particular, the counter scheme, discretized from the generalized constraint forces, plays an error conditioner that offsets the total error of the integration. Numerical examples demonstrate the significantly higher numerical accuracy and some other possible computational advantages of the proposed method.