MS20.5 - UQ and Probabilistic Learning in Computational Dynamics

In this paper surrogate models of the performance metrics of early-stage design of aerostructures are created to optimize a subset of design parameters based on some prescribed limits of the intended real system response. The designed system is that of an aircraft wing that is subjected to a design of experiments to collect data for training of the surrogate models. There are two quantities of interest for this problem related to the load distribution on the wing and its aerodynamic performance for which surrogate models have been built covering the entire design space. Priors have been chosen for the high-dimensional design space which includes both uncertain design parameters and stochastic covariates with which the inverse design problem of optimizing a subset of design parameters is defined. Gaussian processes have been used for surrogate models wherein the uncertainty due to lack of data and model form error can be quantified and propagated. The approach taken requires novel formulation of the likelihood function to tackle the following issues: (1) the output of the surrogate model is dependent on the remaining unoptimized design parameters and therefore their influence must be marginalized out each time the likelihood function is evaluated in the Markov chain; (2) ensuring that the posterior distribution over the quantities of interest is within the 95% confidence interval of the constraints; (3) ensuring that the optimization process does not favor one quantity of interest over the other or fail completely due to compromise. Latin hyper cube sampling of the remaining unoptimized parameters is used to solve (1). As Gaussian process surrogate models are used it is possible to solve (2) given the predictions are in the form of the first two moments of a normal distribution. Finally, as the likelihood is the joint likelihood of both systems remaining within their constraints given the same set of parameters, their marginal likelihoods take on similar forms based around the CDF of a standard normal distribution. Markov Chain Monte Carlo is used to obtain the posterior distribution over the parameters, and subsequently the posterior distribution over the quantities of interest, given the constraints. Several case studies are presented that study how the likelihood function can be manipulated to address issue (3), it is found that the optimization is sensitive to decreases and increases of the variances of the marginal standard normal CDF’s such that it can be used as a weight to direct the optimization towards a quantity of interest, therefore adjustment of this parameter is used to balance the optimization.

Finding the response of a large dynamical system demands considerable computational resources. This demand further amplifies where repeated solutions are sought, such as uncertainty quantification, sensitivity analysis, optimization. Besides this, for a complex problem such as soil-structure interaction, access to the source code is not readily available. To address these issues, recently, several non-intrusive reduced order models (ROMs) have been developed that significantly bring down the computational cost and do not require access to the source code. However, all the existing non-intrusive ROMs address uncertainty in system parameters but not in the excitation. The key challenge in developing a non-intrusive ROM for random excitation is performing interpolation or regression in a very high dimension. This step is practically infeasible in the state-of-the-art methods. This high dimension corresponds to the large number of parameters needed to represent or approximate random excitations. This paper addresses this issue. To the authors’ knowledge, this is the first reported study on the non- intrusive ROM for random excitation. A novel proper orthogonal decomposition-based non-intrusive ROM is proposed here using a neural network. In various applications neural networks have been found to be very effective in high dimensional regression. In the proposed method, first, a principal component analysis (PCA) is performed on a set of random excitations generated from a given power spectral den- sity spectrum. The PCA extracts the dominant features of the random excitation, and thus helps in reducing the number of parameters. Then, these dominant features are used to train a fully connected deep feed-forward neural network. Upon completion of the training phase, vibration response for a new excitation is found from this network. The accuracy of the proposed method is tested numerically for a soil-structure interaction problem modeled as a beam on Winkler foundation. A stationary and Gaussian excitation, generated from the Clough-Penzien spectrum, is chosen here for the numerical studies. The results show that the proposed method is accurate. The accuracy depends on the number of training points used for training the network, which, in turn, depends on the range of the frequency content. A wider frequency range demands a larger training sample size. Finally, the efficiency of the proposed method in performing uncertainty quantification is tested by comparing it with the direct Monte Carlo simulation. The results show a significant speed-up. Similar to most machine learning applications, the architecture of the network is problem specific. However, numerical experiments on various vibration problems showed a generic nature of this network and the training scheme. Applicability of the proposed method for non-stationary excitations remains to be explored.

Informal construction comprises 50% to 90% of residential construction in low and middle-income countries. A significant amount of these buildings is constructed in masonry infilled reinforced concrete (RC) frames, without engineering oversight and the license required for the construction. They are often built by residents or builders who may not have formal construction training and thus do not comply with regulations or design codes. This type of construction is prone to high levels of variability in the material properties, posing a significant challenge in their seismic performance evaluation. This study presents a framework to evaluate the seismic fragility of informally constructed masonry infilled RC frame buildings that accounts for the variability in the materials. The framework consists of four stages. The first one is a statistical characterization of material properties typically used in the construction process by builders. The second stage involved inelastic modeling of the structure capable of capturing the performance of concrete elements and masonry walls. A parametric analysis is proposed in the third stage using Monte Carlo simulations that integrate the first and the second stages to capture the variability of material properties. The fourth stage consists of a performance assessment represented in a probability distribution for each engineering demand parameter (EDP) considered in the study. This framework is demonstrated using an informal house representative of the informal construction in Sincelejo. This is a municipality located on the north coast of Colombia, and according to the national census, 35% of the dwellings in its urban area are built with inappropriate materials. Fieldwork conducted as part of this study suggests that this percentage may be higher. The results showed that the framework allows capturing the uncertainty in the materials construction variability of informal houses. The results also showed that using mean values of material properties may lead to significant errors in the seismic performance assessment of informally constructed masonry infilled RC frames, with estimations of the probabilities of exceeding damage limits that may deviate more than 80% of the median values.

Sequential Bayesian estimators are often employed in structural dynamics to estimate unknown probability density functions using incoming measurements and a mathematical process model. In the simplest case, where all variables of interest are normally distributed and the transitions are linear, the Bayesian estimator becomes equal to the Kalman filter. For more complex cases, a wide variety of nonlinear alternatives and algorithms able to deal with non-Gaussian distributions has been proposed. These sequential estimators rely on (possibly erroneous) mathematical state and observation equations, and by far the majority of them assume that the errors between the true and modelled states and observations can be modelled as Gaussian white noise. It is well-known, however, that for structural systems this assumption is questionable in the least, and mostly false. This could for instance be due to biases stemming from wrongfully modelled dynamic properties, or specific noise colour generated by the measurement equipment. In this contribution, we systematically characterize the process noise typically found in state-space formulations of structural systems, based on commonly encountered modelling discrepancies: damping errors, offsets in natural frequencies, and mode shape errors. The characterization is a first step towards more focused attempts at identifying or developing algorithms able to deal with different types and distributions of noise.