MS20.1 - UQ and Probabilistic Learning in Computational Dynamics

The process of Finite Element (FE) modelling is often associated with uncertainties which necessitates its updating using measured data. Updating dynamical model has a considerable importance in Structural Health Monitoring (SHM), estimating the structural responses, and structural control applications. Among the various presently available approaches for performing model updating, the probabilistic updating approach such as Bayesian model updating has gained a great deal of attention in the last two decades. It can aggregate all the types of uncertainties, e.g., prediction errors, modelling errors, measurement errors, etc., and deal with them as epistemic uncertainties about the system. This paper introduces a two-stage Bayesian model updating procedure of FE dynamical model. The targeted systems encounter a non-classical damping yielding complex modes of vibration. The original FE model is reduced using the method of dynamic condensation, where the reduced FE model is related to the observed Degrees-of-Freedoms (DOFs). Due to the usual availability of a limited number of sensors during real-world applications, modal data used to update the FE model is obtained from multiple setups. Finally, to simulate samples that approximate the posterior Probability Density Function (PDF), a new MWG sampling algorithm is used and compared with Transitional Markov Chain Monte Carlo (TMCMC) algorithm. The proposed approach has been applied to a numerical dynamical system with synthetic modal data acquired from multiple setups. Results revealed that the proposed approach could update the system's uncertain parameters efficiently with a visible reduction in the uncertainty in both undamaged and damaged system.

Structural optimization is the process of identifying the optimal set of design parameters for a structural system. Optimization technique provides an effective approach for rationally improving engineering structural design, both for structural system with deterministic and uncertain parameters. Moreover, it is unanimously agreed that in engineering design applications, knowledge about a planned system is never comprehensive. Therefore, these uncertainties resulting from incomplete information are often assessed probabilistically. In this probabilistic framework, the system design process is referred to as stochastic system design, and the concomitant design optimization problem is alluded to as stochastic optimization. The stochastic optimization process has been widely used in civil, mechanical, and aeronautical engineering designs. Available stochastic system optimization approaches include two stage approaches where initially the performance measure is approximated either by Taylor series approximation or the metamodels and in second stage the approximated performance function is optimized by implementing the available optimization techniques such as nonlinear programming methods, gradient based methods, metaheuristic methods, etc. These aforementioned approaches can effectively take into account the uncertainties but still, achieving higher accuracy and lower computational cost remains a challenging task for designing complex and realistic structural systems. This is also true for the complex deterministic structural systems. To obliviate these aforementioned limitations, an iterative simulation-based approach know as Stochastic Subset Optimization (SSO) is adopted and improved to more effectively determine the optimal design parameter of the system in the present study. The basic principle in original SSO is the formulation of an augmented problem where the design variables are artificially considered as uncertain. Then the design space size is reduced by iteratively identifying a subset of the original design space that has high plausibility of containing the optimal design variables. However, the success of the approach depends on the shape selection of the design space while implementing the SSO. Therefore, in the present study the dependency of the original SSO over the selection of the shape of the design subset is obviated by implementing the novel idea of Voronoi Tessellation for the exploration of the design space rather than presuming its shape as Hyper-Rectangle or Hyper-Ellipse. This improves the applicability of the SSO to the complex realistic problem where the later fails to identify the global minima. The improved SSO can be employed to perform the optimal design of structures in a probabilistic framework: Reliability-Based Design Optimization (RBDO) and Robust Design Optimization (RDO) as well as in the deterministic framework. To demonstrate the efficiency of the proposed approach, three optimization problems: quadratic function and Sinc function for deterministic optimization and 10-bar truss structure for RDO. The results obtained illustrates the efficiency and accuracy of the proposed approach.

Surrogate models enable the efficient propagation of uncertainties in computationally demanding models of physical systems. We employ surrogate models that draw upon polynomial bases to model the stochastic response of structural dynamics systems. In linear structural dynamics problems, the system response can be described by the frequency response function. In [1] we proposed a rational approximation that expresses the system frequency response as a rational of two polynomials with complex coefficients. We showed that the proposed model is able to capture accurately the highly nonlinear nature of the frequency response function, even for structures with low damping. To estimate the coefficients of the approximation, a non-intrusive regression approach was employed that can be coupled easily with existing deterministic solvers. Recently, a sparse Bayesian learning approach was proposed in [2] in which only the polynomial terms that significantly contribute to the predictability of the surrogate are retained. The proposed approaches are based on global, static experimental designs. Especially in applications, where the relevant region in the input space is not known a-priori, e.g., Bayesian parameter updating, reliability analysis or optimization, using a static experimental design can lead to significant errors. In such cases it is beneficial to resort to active learning approaches, in which the experimental design is adaptively enriched during the training process. A so-called learning or acquisition function is employed as a means to choose further experimental design points. Often, the formulation of the acquisition function is based on the availability of the uncertainty in the surrogate model prediction. In the context of linear, Gaussian surrogate models, such as Gaussian processes, the predictive covariance can be obtained analytically. For non-linear models, such as the rational approximation, the predictive covariance is seldom available in closed-form. In this contribution we assess simulation- and approximation-based approaches to active learning using rational surrogate models. We specifically focus on Bayesian updating problems in which we are interested in learning the posterior distribution of a set of system parameters based on measured frequency response data. Furthermore, we demonstrate the applicability of the presented methods on different dynamic models. [1] F. Schneider, I. Papaioannou, M. Ehre and D. Straub, Polynomial chaos based rational approximation in linear structural dynamics with parameter uncertainties. Computers & Structures 233 (2020): 106223. [2] F. Schneider, I. Papaioannou and G. Müller, Sparse Bayesian Learning for Complex-Valued Rational Approximations, https://arxiv.org/abs/2206.02523, 2022