Computationally Efficient Bayesian Model Selection for Locally Nonlinear Structural Dynamic Systems

Abstract

Models, typically given by systems of mathematical equations, are built to help represent, understand, and further characterize physical phenomena. The choice of a model for a particular phenomenon is made based on user judgment, evidence from measurement data, and/or the ease of its use. Generally, many linear and nonlinear models are available to describe a particular structural dynamical system. Bayesian model selection is a probabilistic tool to help select suitable mathematical model(s) among a possible set of models using Bayes’ theorem. To simplify the analysis, linear structural dynamical models are often used, regardless of whether the dynamical system behaves linearly or not. However, linear models are not always adequate to accurately compute structural responses. When the models also involve some nonlinearity, the required computation for Bayesian model selection increases significantly. An important class of nonlinear problems consists of models that are mostly linear except for some spatially localized nonlinearities. For example, in a building with base isolation, the superstructure is designed to behave essentially linearly in an earthquake excitation and only the isolation layer behaves nonlinearly. Similarly, spacecraft may be modeled with linear components that are connected by spatially localized nonlinear joints. To lessen this increased computational burden, a method is proposed in this paper, combining the senior authors’ previously developed efficient dynamic response algorithm for locally nonlinear systems and an intelligent sampling algorithm, to calculate the evidence for, or marginal likelihood of, a model. The efficient dynamic response algorithm helps achieve significant gains in computational efficiency by exactly transforming the potentially high-dimensional state-space equation of the structural dynamical system into a low-order nonlinear Volterra integral equation. This algorithm is embedded into a nested sampling algorithm, which samples parameters more frequently from the high likelihood region (even if the region of higher likelihood is very different from the region where the prior parameter density function is large), resulting in a computationally efficient framework for Bayesian model selection with locally nonlinear systems; a (moderately) alternate derivation of nested sampling is developed. The approach is demonstrated using three numerical examples. The first two examples consider building models mounted on a hysteretic isolation layer that are subjected to an earthquake excitation. Both single and 99-degree-of-freedom superstructure models are investigated. Different candidate linear and nonlinear models representing the hysteretic behavior of the isolation layer are used. The third example consists of a three-dimensional wind-excited 1,623-degree-of-freedom building structure with tuned mass dampers (TMDs) attached to its roof, where different linear and nonlinear candidate damping models are considered in the TMDs. All three examples include cases in which the true model is not among the models evaluated. The application of the proposed synergistic approach, consisting of an intelligent sampling and an efficient dynamic response algorithm, demonstrates gains in Bayesian model selection computational efficiency of up to three orders of magnitude relative to conventional approaches with comparable accuracy, reducing days of computation to minutes or hours.