Energy decaying scheme with adaptive time step methodology for nonlinear beam models

摘要

This paper is concerned with the time integration of the equations of motion describing the nonlinear dynamic response of beams. Desirable characteristics of integration schemes for structural dynamics problems include unconditional stability, accuracy, and high frequency numerical dissipation. Several schemes exist that present all these features when applied to linear problems. Though the application of those schemes to nonlinear problems is often sucessful, proofs of unconditional stability are rarely derived. A finite difference integration scheme is derived in this paper for the nonlinear dynamic response of beams. Though a finite difference nature, the proposed scheme mimics the integration scheme obtained by applying the time discontinuous Galerkin methodology to a single degree of freedom linear oscillator. A formal proof of unconditional stability for the nonlinear problem is derived based on an energy decay argument. Numerical examples using the proposed scheme are given, and the results are compared with the predictions of other available schemes. An adaptive time stepping methodology is proposed, and numerical experimentation demonstrate its efficacy.