ON THE IMPROVEMENT OF DISCRETE MECHANICS TIME INTEGRATION SCHEMES UTILIZING INTEGRAL FORMS OF THE RELATIONS OF BALANCE

Abstract eng:
So-called discrete mechanics numerical time integration schemes for dynamic problems became popular in the seventies of the last century, see e.g. Greenspan [1]. For some subsequent contributions on energymomentum conserving time integration schemes in structural dynamics, see also, e.g., Liu et. al. [3] and Hughes et al. [4]. More recently, such formulations have been successfully extended to multi-body dynamic systems, including deformability, see e.g. [5], as well as to coupled multi-physics problems, non-smooth motions and dissipation, etc. The hypothesis followed and exemplarily substantiated in the present contribution is that considerable improvements of such discrete mechanics formulations can be achieved with respect to their computationally efficiency, when they are substituted (as a first guess) into the integral form of the fundamental relations of balance. For a discussion of the integral form of the relations of balance, see e.g. [6]. In contrast to their differential counterparts, to which usual discrete mechanics formulations refer, the integral forms obey less continuity requirements, and thus may be considered as being more fundamental from a physics point of view. In the present contribution, our hypothesis is checked for the example of the plane motion of a pendulum. For the linear solution, it is shown that, repeating the procedure n times in the same time-step, an improvement of the numerical solution is obtaines, which reproduces 2(n+1) terms in the Taylor series representation of the harmonic functions exactly. For the non-linear solution, the same is true for the Jacobian elliptic functions, where numerically beneficial additional correction terms of higher order do appear also. These facts give evidence for allowing the use of substantially larger time-steps in numerical time-stepping procedures, when compared to the usual discrete mechanics formulations. Besides the demonstration of a high accuracy of the proposed strategy, numerical stability issues are discussed, as well as the question of conservation of invariants of the motion.

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