Optimized Equivalent Linearization for Random Vibration

Abstract

A fundamental limitation of various Equivalent Linearization Methods (ELMs) in nonlinear random vibration analysis is that they are approximate by their nature. A quantity of interest estimated from an ELM has no guarantee to be the same as the solution of the original nonlinear system. In this study, we tackle this fundamental limitation. We sequentially address the following two questions: i) given an equivalent linear system obtained from any ELM, how do we construct an estimator so that, as the linear system simulations are guided by a limited number of nonlinear system simulations, the estimator converges on the nonlinear system solution? ii) how to construct an optimized equivalent linear system such that the estimator approaches the nonlinear system solution as quickly as possible? The first question is theoretically straightforward since classical Monte Carlo techniques such as the control variates and importance sampling can improve upon the solution of any surrogate model. We adapt the well-known Monte Carlo theories into the specific context of equivalent linearization. The second question is challenging, especially when rare event probabilities are of interest. We develop specialized methods to construct and optimize linear systems. In the context of uncertainty quantification (UQ), the proposed optimized ELM can be viewed as a physical surrogate model-based UQ method. The embedded physical equations endow the surrogate model with the capability to handle high-dimensional random vectors in stochastic dynamics analysis.

Figures (16)

• Figure 1: General framework of the optimized equivalent linearization method. The proposed optimized equivalent linearization method can be reworded as a linear dynamic system-based surrogate modeling approach for nonlinear stochastic dynamics analysis. The linear system is represented by a parametric impulse/frequency response function with physical parameters such as damping, stiffness, modal participation factor, degree-of-freedom, etc. This is a parsimonious surrogate model for a high-dimensional uncertainty quantification problem. The computational flows (arrows) are bidirectional since the modules are coupled, e.g., the quantity of interest evaluations can guide computational efforts in the surrogate model training and UQ solver.
• Figure 2: Simulation result of the cubic oscillator for the mean peak absolute response. To achieve the same coefficient of variation of $1\%$, the ACV-ELM requires $50$(for optimization)+$78=128$ nonlinear model runs, and the estimate is $0.35\,\rm{m}$; the AIS-ELM requires $50$(for optimization)$+74=124$ nonlinear model runs, and the estimate is $0.35\,\rm{m}$; the direct Monte Carlo simulation requires $1124$ nonlinear model runs, and the estimate is $0.35\,\rm{m}$. The shaded areas represent approximate confidence intervals obtained from the mean plus and minus $1.96$ standard deviations.
• Figure 3: Impulse and frequency response functions of the optimized linear system for the mean peak response of the cubic oscillator. The impulse/frequency response functions obtained from both methods exhibit dominant low-frequency components.
• Figure 4: Simulation result for the first-passage probability estimation of the cubic oscillator. To achieve the same coefficient of variation of $10\%$, the AIS-ELM based on relaxation requires $653$(for optimization)$+84=737$ nonlinear model runs, and the probability estimate is $8.84\times10^{-4}$; the AIS-ELM based on conditioning requires $653$(for optimization)$+500=1153$ nonlinear model runs, and the probability estimate is $8.28\times10^{-4}$; the direct Monte Carlo simulation requires $1.18\times10^5$ nonlinear model runs, and the probability estimate is $8.50\times10^{-4}$. The AIS-ELM based on conditioning involves "hidden" nonlinear model runs embedded in the MCMC sampler, which cannot be shown in the figure (but are already included in the reported computational cost). The shaded areas represent approximate confidence intervals obtained from the mean plus and minus $1.96$ standard deviations.
• Figure 5: Response trajectories and distributions of the optimized linear system. The linear system could imitate the actual trajectory of the cubic oscillator (although the optimization objective is to match the mean peak response); their peak response distributions slightly differ. The mean peak responses (dashed lines) of the two systems match, because the linear system is optimized to estimate the mean peak response.