A Taylor basis for kinematic nonlinear real-time simulations. Part I: The complete modal derivatives

摘要

Despite todays computational power, only small nonlinear numerical substructures may be simulated in real time. The size restriction on the substructures in nonlinear finite element analysis is primarily due to the time-consuming evaluation of the internal restoring forces, which is performed element-by-element in every iteration step. The present work constitutes the first of two papers presenting a method to simulate kinematic nonlinear structures more efficiently. It involves applying a reduced basis with modal derivatives representing the nonlinearities of the system in an efficient way. Previously, the modal derivatives have been determined from a set of approximate governing equations. In the present paper, a novel set of equations governing the complete modal derivatives is derived. This is done by introducing a Taylor series into the free undamped kinematic nonlinear equations of motion. Also, the approximate governing equations are improved by introducing a novel geometric restriction. By way of an example, it is shown that only the modal derivatives determined from the complete set of equations are consistent with the Taylor series. In the second paper, it is shown that the novel modal derivatives may be used in a so-called Taylor basis and that they improve the computational time and stability significantly.