Semi-Implicit Integration and Data-Driven Model Order Reduction in Structural Dynamics With Hysteresis

Abstract

Structural damping is often empirically rate-independent wherein the dissipative part of the stress depends on the history of deformation but not its rate of change. Hysteresis models are popular for rate-independent dissipation; and a popular hysteresis model is the Bouc-Wen model. If such hysteretic dissipation is incorporated in a refined finite element model, then the model involves the usual structural dynamics equations along with nonlinear nonsmooth ordinary differential equations for a large number of internal hysteretic states at Gauss points used within the virtual work calculation. For such systems, numerical integration is difficult due to both the distributed nonanalytic nonlinearity of hysteresis as well as large natural frequencies in the finite element model. Here, we offer two contributions. First, we present a simple semi-implicit integration approach where the structural part is handled implicitly based on the work of Piché, while the hysteretic part is handled explicitly. A cantilever beam example is solved in detail using high mesh refinement. Convergence is good for lower damping and a smoother hysteresis loop. For a less smooth hysteresis loop and/or higher damping, convergence is noted to be roughly linear on average. Encouragingly, the time-step needed for stability is much larger than the time period of the highest natural frequency of the structural model. Subsequently, data from several simulations conducted using the above semi-implicit method are used to construct reduced order models of the system, where the structural dynamics is projected onto a few modes and the number of hysteretic states is reduced significantly as well. Convergence studies of error against the number of retained hysteretic states show very good results.