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

Despite todays computational power, only small nonlinear numerical substruc-tures may be simulated in real time. The size restriction on the substructures innonlinear finite element analysis is primarily due to the time-consuming eval-uation of the internal restoring forces, which is performed element-by-elementin every iteration step. The present work constitutes the first of two papers pre-senting a method to simulate kinematic nonlinear structures more efficiently.It involves applying a reduced basis with modal derivatives representing thenonlinearities of the system in an efficient way. Previously, the modal deriva-tives have been determined from a set of approximate governing equations.In the present paper, a novel set of equations governing the complete modalderivatives is derived. This is done by introducing a Taylor series into the freeundamped kinematic nonlinear equations of motion. Also, the approximate gov-erning equations are improved by introducing a novel geometric restriction.By way of an example, it is shown that only the modal derivatives determinedfrom the complete set of equations are consistent with the Taylor series. In thesecond paper, it is shown that the novel modal derivatives may be used in aso-called Taylor basis and that they improve the computational time and stabilitysignificantly.