Inherently energy conserving time finite elements for classical mechanics

In this paper, we develop a finite element method for the temporal discreti zation of the equations of motion. The continuous Galerkin method is bused upon a weighted-residual statement of Hamilton's canonical equations. We sh ow that the proposed finite element formulation is energy conserving in a n atural sense. A family of implicit one-step algorithms is generated by spec ifying the polynomial approximation in conjunction with the quadrature form ula used for the evaluation of time integrals. The numerical implementation of linear, quadratic, and cubic time finite elements is treated in detail for the model problem of a circular pendulum. In addition to that, concerni ng dynamical systems with several degrees of freedom, we address the design of nonstandard quadrature rules which retain the energy conservation prope rty. Our numerical investigations assess the effect of numerical quadrature in time on the accuracy and energy conservation property of the time-stepp ing schemes. (C) 2000 Academic Press.