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.

翻译：各类等效线性化方法（ELMs）在非线性随机振动分析中的一个根本局限性在于其本质上的近似性。通过ELM计算得到的关注量无法保证与原非线性系统的解一致。本研究致力于解决这一根本性挑战。我们依次探讨以下两个问题：（i）对于任意ELM方法获得的等效线性系统，如何在有限数量非线性系统模拟的引导下构建估计器，使其收敛于原非线性系统的解？（ii）如何构建优化等效线性系统，使估计器能够以最快速度逼近非线性系统的解？第一个问题在理论上较为直接——经典蒙特卡洛技术（如控制变量法和重要性抽样法）可以改进任意代理模型的求解精度。我们将成熟的蒙特卡洛理论创新性地应用于等效线性化特定场景。第二个问题则具有挑战性，特别是在涉及稀有事件概率分析时。我们开发了专门的方法来构建和优化线性系统。在不确定性量化（UQ）框架下，本文提出的优化等效线性化方法可视为基于物理代理模型的UQ方法。其中嵌入的物理方程使代理模型具备处理随机动力学分析中高维随机向量的能力。