Constructing time integration with controllable errors for constrained mechanical systems

This paper investigates the numerical features of equivalent Lagrangian formulations for general mechanical systems and gives new insight into the error reduction of the corre-sponding numerical integrations. To this end, the modified Lagrange's equations are de-rived, which give a unified form of equivalent Lagrangian formulations, featuring differ-ent 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 er-ror 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 syn-thetic integration scheme consists of the main scheme and the counter scheme. In partic-ular, 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 advan-tages of the proposed method.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

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. The modified Lagrange's equations are derived, which give a unified form of equivalent Lagrangian formulations. An error analysis is employed to clarify the error generation of different formulations and a mechanism of error reduction is explained for the discrete system. A methodology for constructing synthetic integrations is developed, which demonstrates significantly higher numerical accuracy and possible computational advantages.