F.A. DiazDelaO

Bayesian updating and model class selection with Subset Simulation

Identifying the parameters of a model and rating competitive models based on measured data has been among the most important and challenging topics in modern science and engineering, with great potential of application in structural system identification, updating and development of high fidelity models. These problems in principle can be tackled using a Bayesian probabilistic approach, where the parameters to be identified are treated as uncertain and their inference information are given in terms of their posterior probability distribution. For complex models encountered in applications, efficient computational tools robust to the number of uncertain parameters in the problem are required for computing the posterior statistics, which can generally be formulated as a multi-dimensional integral over the space of the uncertain parameters. Subset Simulation has been developed for solving reliability problems involving complex systems and it is found to be robust to the number of uncertain parameters. An analogy has been recently established between a Bayesian updating problem and a reliability problem, which opens up the possibility of efficient solution by Subset Simulation. The formulation, called BUS (Bayesian Updating with Structural reliability methods), is based the standard rejection principle. Its theoretical correctness and efficiency requires the prudent choice of a multiplier, which has remained an open question. This paper presents a fundamental study of the multiplier and investigates its bias effect when it is not properly chosen. A revised formulation of BUS is proposed, which fundamentally resolves the problem such that Subset Simulation can be implemented without knowing the multiplier a priori. An automatic stopping condition is also provided. Examples are presented to illustrate the theory and applications.

Gaussian process hyper-parameter estimation using Parallel Asymptotically Independent Markov Sampling

Gaussian process emulators of computationally expensive computer codes provide fast statistical approximations to model physical processes. The training of these surrogates depends on the set of design points chosen to run the simulator. Due to computational cost, such training set is bound to be limited and quantifying the resulting uncertainty in the hyper-parameters of the emulator by uni-modal distributions is likely to induce bias. In order to quantify this uncertainty, this paper proposes a computationally efficient sampler based on an extension of Asymptotically Independent Markov Sampling, a recently developed algorithm for Bayesian inference. Structural uncertainty of the emulator is obtained as a by-product of the Bayesian treatment of the hyper-parameters. Additionally, the user can choose to perform stochastic optimisation to sample from a neighbourhood of the Maximum a Posteriori estimate, even in the presence of multimodality. Model uncertainty is also acknowledged through numerical stabilisation measures by including a nugget term in the formulation of the probability model. The efficiency of the proposed sampler is illustrated in examples where multi-modal distributions are encountered. For the purpose of reproducibility, further development, and use in other applications the code used to generate the examples is freely available for download at https://github.com/agarbuno/paims_codes.

Transitional annealed adaptive slice sampling for Gaussian process hyper-parameter estimation

Surrogate models have become ubiquitous in science and engineering for their capability of emulating expensive computer codes, necessary to model and investigate complex phenomena. Bayesian emulators based on Gaussian processes adequately quantify the uncertainty that results from the cost of the original simulator, and thus the inability to evaluate it on the whole input space. However, it is common in the literature that only a partial Bayesian analysis is carried out, whereby the underlying hyper-parameters are estimated via gradient-free optimisation or genetic algorithms, to name a few methods. On the other hand, maximum a posteriori (MAP) estimation could discard important regions of the hyper-parameter space. In this paper, we carry out a more complete Bayesian inference, that combines Slice Sampling with some recently developed Sequential Monte Carlo samplers. The resulting algorithm improves the mixing in the sampling through delayed-rejection, the inclusion of an annealing scheme akin to Asymptotically Independent Markov Sampling and parallelisation via Transitional Markov Chain Monte Carlo. Examples related to the estimation of Gaussian process hyper-parameters are presented. For the purpose of reproducibility, further development, and use in other applications, the code to generate the examples in this paper is freely available for download at this http URL

Inferring Structural Variability Using Modal Analysis in a Bayesian Framework

Dynamic systems with different geometric configurations may present remarkable distinct dynamic behaviour. However, variability is identifiable in the measured modal shapes and modal frequencies. This paper explores a Bayesian framework in order to infer structural variability based on modal parameters. This is relevant in cases of difficult access for inspection in finished products/structures. An approach using a radial basis neural network benchmarked by a Gaussian process metamodel is developed and then followed by a test case with experimental data. It is concluded that the proposed methodology shows promise in solving this kind of problems.

An experimental study on the manufacture and characterization of in-plane fibre-waviness defects in composites

A new method has been developed for creating localized in-plane fibre waviness in composite coupons and used to create a large batch of specimens. This method could be used by manufacturers to experimentally explore the effect of fibre waviness on composite structures both directly and indirectly to develop and validate computational models. The specimens were assessed using ultrasound, digital image correlation and a novel inspection technique capable of measuring residual strain fields. To explore how the defect affects the performance of composite structures, the specimens were then loaded to failure. Predictions of remnant strength were made using a simple ultrasound damage metric and a new residual strain-based damage metric. The predictions made using residual strain measurements were found to be substantially more effective at characterizing ultimate strength than ultrasound measurements. This suggests that residual strains have a significant effect on the failure of laminates containing fibre waviness and that these strains could be incorporated into computational models to improve their ability to simulate the defect.

PROBABILISTIC SENSITIVITY ANALYSIS OF CORRUGATED SKINS WITH RANDOM ELASTIC PARAMETERS AND SURFACE TOPOLOGY

The distinction between epistemic and aleatory uncertainty can sometimes be useful for practical purposes. In principle, epistemic uncertainty is reducible by obtaining more or better information. However, the boundaries between both types of uncertainties are often blurred. Fortunately, it is often the case that not all sources of uncertainty have the same impact on the predictive performance of numerical models, so that there is considerable computational benefit from investigating potential simplification in the number of parameters that should be modelled as being stochastic. In order to determine which parameters are the most important in terms of their contribution to the output’s uncertainty, probabilistic sensitivity analysis can be performed. We apply this idea to a numerical model of corrugated skins. Corrugated skins are particularly suitable for morphing applications in aerospace structures, largely due to the very high compliance they offer along the corrugation direction.

Approximate Bayesian Computation for Finite Element Model Updating

In recent years, there has been a growing interest in Bayesian model updating methods. The learning process is characterised by estimating the probability distribution of a random parameter within an ensemble of data and prior information. A crucial component of these methods is a marginal likelihood term. However, for most models, an analytical expression cannot be found or can be computationally intractable. A possible solution is to perform likelihood-free inference. Recently, there has been a development of techniques known as Approximate Bayesian Computation (ABC) methods. This work explores the coupling between finite element model updating and ABC, its potential and its limitations.

Uncertainty analysis of corrugated skin with random elastic parameters and surface topology

Uncertainty analysis of corrugated skins has been performed for random perturbations in geometric and elastic parameters of a chosen baseline model. The corrugated skins are particularly suitable for morphing applications in aerospace structures and their sensitivity to the various input uncertainties is a major concern in their design. The various sources of uncertainty include random perturbations of the geometrical parameters of the corrugation units, surface roughness and parametric uncertainty of the elastic parameters. These uncertainties are described here within the probabilistic framework and have been incorporated into the discretized stochastic finite element model used for their analysis. The propagation of these uncertainties to the dynamic response of the structure is a computationally intensive exercise especially for high dimensional stochastic models. Such high dimensional models have been resolved with statistical methods such as Gaussian Process Emulation and polynomial interpolation based sparse grid collocation techniques. The brute force Monte Carlo simulation technique results have been used as benchmark solutions. A global sensitivity analysis has been performed to identify the key uncertainty sources which affect the system response and the equivalent models using Sobol’s importance measure.