[Paper Review] Optimized Equivalent Linearization for Random Vibration

Ziqi Wang|arXiv (Cornell University)|Aug 18, 2022

Probabilistic and Robust Engineering Design43 references4 citations

TL;DR

This paper proposes an optimized equivalent linearization method (ELM) for nonlinear random vibration by enhancing surrogate linear models with Monte Carlo variance reduction techniques—control variates and importance sampling—to correct linear system responses toward the true nonlinear system solution. The method ensures convergence to the correct response statistics and rare event probabilities, even under strong nonlinearity, by optimizing the linear system for maximum correlation with the nonlinear response in critical regions.

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.

Motivation & Objective

• To overcome the fundamental limitation of equivalent linearization methods (ELMs), which are inherently approximate and lack guarantees of convergence to the true nonlinear system solution.
• To develop a framework that uses limited nonlinear system simulations to guide and correct linear system simulations, ensuring the estimator converges on the true solution.
• To optimize the parametric linear system such that the response estimator approaches the nonlinear system solution as quickly as possible, especially for rare event probabilities.
• To integrate physical interpretability of linear systems with statistical variance reduction techniques, enabling efficient uncertainty quantification in stochastic dynamics.

Proposed method

• Adapts classical Monte Carlo techniques—control variates and importance sampling—within the context of equivalent linearization to reduce variance in response estimators.
• Optimizes the parameters of a parametric linear system to maximize correlation between its response and the nonlinear system response in critical regions, particularly the tail regions for rare events.
• Uses a control variate-enhanced ELM for estimating response statistics like mean peak responses, leveraging Pearson correlation for variance reduction.
• Employs an importance sampling-enhanced ELM for rare event probability estimation, minimizing sampling variance by focusing simulations on critical response regions.
• Integrates the optimized linear system with limited nonlinear system simulations to iteratively correct the linear model toward the true nonlinear solution.
• Employs mathematical formulations for optimization that balance correlation and variance reduction, ensuring convergence under limited sampling.

Experimental results

Research questions

• RQ1 How can an estimator based on an equivalent linear system be constructed to converge on the true solution of a nonlinear stochastic system when guided by a limited number of nonlinear simulations?
• RQ2 What optimization strategy can be applied to a parametric linear system so that its response estimator approaches the true nonlinear system solution as quickly as possible?
• RQ3 How can control variates and importance sampling be adapted to the equivalent linearization framework to improve convergence and efficiency?
• RQ4 What are the performance limits of linearization in capturing non-Gaussian and rare event responses, and how can they be mitigated?
• RQ5 Can the proposed method be generalized to other response quantities beyond mean peak responses and first-passage probabilities?

Key findings

• The control variate-enhanced ELM achieves significant variance reduction for response statistics, particularly when the correlation between linear and nonlinear responses is high.
• The importance sampling-enhanced ELM is especially effective for estimating rare event probabilities, where standard ELMs often fail due to low sampling efficiency.
• In highly nonlinear hysteretic systems, the maximum achievable correlation between linear and nonlinear responses is capped around 0.8, indicating a fundamental limit of linearization.
• The proposed method ensures convergence of the linear system estimator toward the true nonlinear system solution as the number of nonlinear simulations increases, overcoming the core limitation of conventional ELMs.
• The optimized linear system encodes more than just a point estimate—it retains physical information such as power spectral density, enabling further use of linear system theories.
• The method is robust even under strong nonlinearity, as demonstrated across three benchmark nonlinear random vibration problems.