Structural damping is known to be approximately rate-independent in many cases. Popular models for rate-independent dissipation are hysteresis models; and a highly popular hysteresis model is the Bouc-Wen model. If such hysteretic dissipation is incorporated in a refined finite element model, then the mathematical model includes the usual structural dynamics equations along with nonlinear nonsmooth ordinary differential equations for a large number of internal hysteretic states at Gauss points, to be used within the virtual work calculation for dissipation. For such systems, numerical integration becomes difficult due to both the distributed non-analytic nonlinearity of hysteresis as well as the very high 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 Pich\'e, and where 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 observed 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 small number of 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.

翻译：结构阻尼在许多情况下近似率无关。流行的率无关耗散模型为迟滞模型，而广受应用的迟滞模型是Bouc-Wen模型。当此类迟滞耗散被纳入精细化有限元模型时，数学模型包含常规结构动力学方程以及大量高斯点处内部迟滞状态的非线性非光滑常微分方程，用于耗散的虚功计算。对于此类系统，由于迟滞的非解析非线性分布特性及有限元模型中极高固有频率的存在，数值积分变得困难。本文提出两项贡献：首先，基于Piché工作引入一种简单的半隐式积分方法，其中结构部分采用隐式处理，而迟滞部分采用显式处理。通过高网格细化的悬臂梁算例进行详细求解。在较低阻尼和较平滑迟滞回线情况下收敛性良好；对于较不平滑迟滞回线和/或较高阻尼，观察到平均近似线性收敛。令人振奋的是，稳定性所需时间步长远大于结构模型最高固有频率对应的周期。随后，利用上述半隐式方法生成的多个仿真数据构建系统降阶模型，将结构动力学投影至少量模态，同时显著减少迟滞状态数量。针对保留迟滞状态数量与误差的收敛性研究表明结果非常理想。