Characterization of the dynamic response of continuous system discretized using finite element methods

Abstract

Nonlinear dynamic physical systems exhibit a rich variety of behaviors. In many cases, the system response is unstable, and the behavior may become unpredictable. Since an unstable or unpredictable response is usually undesirable in engineering practice, the stability characterization of a system's behavior becomes essential. In this work, a numerical procedure to characterize the dynamic stability of continuous solid media, discretized using finite element methods, is proposed. The procedure is based on the calculation of the maximum Lyapunov characteristic exponent (LCE), which provides information about the asymptotic stability of the system response. The LCE is a measure of the average divergence or convergence of nearby trajectories in the system phase space, and a positive LCE indicates that the system asymptotic behavior is chaotic, or, in other words, asymptotically dynamically unstable. In addition, a local temporal stability indicator is proposed to reveal the presence of local dynamic instabilities in the response. Using the local stability indicator, dynamic instabilities can be captured shortly after they occur in a numerical calculation. The indicator can be obtained from the successive approximations of the response LCE calculated at each discretized time step. Both procedures can also be applied to fluid-structure interaction problems in which the analysis focuses on the behavior of the structural part.

(cont.) The response of illustrative structural systems and fluid flow-structure interaction systems, in which the fluid is modeled using the Navier-Stokes equations, was calculated. The systems considered present both stable and unstable behaviors, and their LCEs and local stability indicators were computed using the proposed procedures. The stability of the complex behaviors exhibited by the problems considered was properly captured by both approaches, confirming the validity of the procedures proposed in this work.