Model selection and parameter estimation in structural dynamics using approximate Bayesian computation

This paper will introduce the use of the approximate Bayesian computation (ABC) algorithm for model selection and parameter estimation in structural dynamics. ABC is a likelihood-free method typically used when the likelihood function is either intractable or cannot be approached in a closed form. To circumvent the evaluation of the likelihood function, simulation from a forward model is at the core of the ABC algorithm. The algorithm offers the possibility to use different metrics and summary statistics representative of the data to carry out Bayesian inference. The efficacy of the algorithm in structural dynamics is demonstrated through three different illustrative examples of nonlinear system identification: cubic and cubic-quintic models, the Bouc-Wen model and the Duffing oscillator. The obtained results suggest that ABC is a promising alternative to deal with model selection and parameter estimation issues, specifically for systems with complex behaviours

Model selection, updating and prediction of fatigue crack propagation using nested sampling algorithm

Mathematical models are often used to interpret experimental data, estimate the parameters and then predictions can be made. In practice, and in several applications, it is common that often more than one model could be used to describe the dynamics of a given phenomenon. Modelling and prediction of fatigue crack growth (FCG) is one of the engineering problems where a number of models with different levels of complexities exist and the selection of the most suitable one is always a challenging task. In this study, model selection, updating and prediction of fatigue crack propagation is carried out under a Bayesian framework. The nested sampling algorithm is selected to estimate the evidence of each competing model using an experimental data set of Aluminum 2024-T3. The obtained results are very encouraging and show the efficiency of the proposed approach when dealing with model selection, updating and prediction issues

A probabilistic approach for optimising hydroformed structures using local surrogate models to control failures

A probabilistic approach is proposed to optimise hydroformed structures by taking into account the potential variabilities. An efficient implementation requires an appropriate strategy for uncertainty representation and propagation. Moreover, the probability of failure associated to each failure mode must be accurately estimated. To this end, the failure modes are controlled locally only at the highly strained regions which reduces the problem complexity and increases the precision of the generated surrogate models. In this study, finite element simulations with material formability diagrams are used to predict the critical zones in which failure modes may initiate. The predicted zones agree well with the experimental and numerical simulations. By this simplification, the optimisation problem is formulated differently while retaining the relevant physical features of the process. To illustrate this strategy, tee-shaped tube hydroforming process is proposed due to its complexity to demonstrate the benefits of the probabilistic approach. The optimisation problem is formulated within deterministic and probabilistic frameworks to determine the optimal loading paths. It will be shown that probabilistic optimum allows better process mechanics and improved thickness distribution in the hydroformed tube. This approach can be extended to other metal forming processes and easily implemented for industrial products within reasonable computational time

Optimization of Tube Hydroforming Process Using Probabilistic Constraints on Failure Modes

In metal forming processes, different parameters (Material constants, geometric dimensions, loads …) exhibits unavoidable scatter that lead the process unreliable and unstable. In this paper, we interest particularly in tube hydroforming process (THP). This process consists to apply an inner pressure combined to an axial displacement to manufacture the part. During the manufacturing phase, inappropriate choice of the loading paths can lead to failure. Deterministic approaches are unable to optimize the process with taking into account to the uncertainty. In this work, we introduce the Reliability-Based Design Optimization (RBDO) to optimize the process under probabilistic considerations to ensure a high reliability level and stability during the manufacturing phase and avoid the occurrence of such plastic instability. Taking account of the uncertainty offer to the process a high stability associated with a low probability of failure. The definition of the objective function and the probabilistic constraints takes advantages from the Forming Limit Diagram (FLD) and the Forming Limit Stress Diagram (FLSD) used as a failure criterion to detect the occurrence of wrinkling, severe thinning, and necking. A THP is then introduced as an example to illustrate the proposed approach. The results show the robustness and efficiency of RBDO to improve thickness distribution and minimize the risk of potential failure modes

Global sensitivity analysis and multi-objective optimisation of loading path in tube hydroforming process based on metamodelling techniques

Tube hydroforming process is widely used in various industrial applications which consists of combining internal pressure and axial displacement to manufacture tubular parts. Inappropriate choice as small changes in such variables may affect the process stability and, in some cases, lead to failure. Consequently, loading path should be optimised to better control the process and to guarantee hydroformed parts with desired specifications. However, optimisation procedure requires several evaluations of the real models which induces a huge computational time. To cope with this limitation, we propose to compare two metamodelling techniques to solve the problem efficiently: the response surface method and the least squares support vector regression. To enhance the metamodels precision, optimal latin hypercube design is used to generate sampled points. It is obtained through iterative optimisation procedure based on a modified version of the simulated annealing algorithm by minimising simultaneously two optimality criterions. Then, multi-objective optimisation problem is formulated to search for the Pareto optimal solutions. Fuzzy classification is then applied to rank the non-dominated solutions which helps designers in the decision-making phase. Before optimising the process, a global sensitivity analysis is carried out using the variance-based method by coupling metamodels and Monte Carlo simulations in order to identify the relative importance of the design variables in terms of internal pressure and axial displacement on the variance of the responses of interest defined to control the process

Increasing the stability of T-shape tube hydroforming process under stochastic framework

Metal forming processes present several sources of uncertainties coming from material properties, geometric characteristics, and loading paths. During the manufacturing phase, such parameters may vary affecting the process stability and increasing the defect parts. Stochastic framework seems more pertinent than classical deterministic approaches to treat such problems since it is intended to include variabilities at the early design stage. In the present work, tube hydroforming process widely used in various industry applications is investigated. To ensure the process stability, loading paths should be optimized with taking into account randomness associated to the input parameters. To control the potential failure modes, the Forming Limit Stress Diagram is implemented in the finite element code to avoid necking while a simple geometrical criterion is defined for wrinkling. A global sensitivity analysis using the variance-based method is done which shows that the selected random parameters impact considerably the variance of failure indicators. Then, a numerical example of T-shape tube hydroforming process is proposed to show the efficiency of the stochastic framework. Statistical and probabilistic observations of the optimum solution show that the stochastic approach yields to an optimum less sensitive to such fluctuations which improves the process stability and minimizes considerably the percentage of defect parts in a mass production environment

ABC-NS: a new computational inference method applied to parameter estimation and model selection in structural dynamics

The inference of dynamical systems is a challenging issue, particularly when the dynamics include complex phenomena such as the existence of bifurcations and/or chaos. In this situation, the likelihood function formulated based on time-series data may be complex with several local minima and as a result not suitable for parameter inference. In the most challenging scenarios, the likelihood function may not be available in an analytical form, so a standard statistical inference is impossible to carry out. To overcome this problem, the inclusion of new features/invariants less sensitive to small variations from either the time or frequency domains seems to be potentially a very useful way to make Bayesian inference. The use of approximate Bayesian computation (ABC) or likelihood-free algorithms is an appropriate option as they offer the flexibility to use different metrics for parameter inference. However, most variants of the ABC algorithm are inefficient due to the low acceptance rate. In this contribution, a new ABC algorithm based on an ellipsoidal nested sampling technique is proposed to overcome this issue. It will be shown that the new algorithm performs perfectly well and maintains a relatively high acceptance rate through the iterative inference process. In addition to parameter estimation, the new algorithm allows one to deal with the model selection issue. To demonstrate its efficiency and robustness, a numerical example is presented

Stochastic modelling and prediction of fatigue crack propagation using piecewise-deterministic Markov processes

Fatigue crack propagation is a stochastic phenomenon due to the inherent uncertainties originating from material properties, environmental conditions and cyclic mechanical loads. Stochastic processes thus offer an appropriate framework for modelling and predicting crack propagation. In this paper, fatigue crack growth is modelled and predicted by a piecewise-deterministic Markov process associated with deterministic crack laws. First, a regime-switching model is used to express the transition between the Paris regime and rapid propagation that occurs before failure. Both regimes of propagation are governed by a deterministic equation whose parameters are randomly selected in a finite state space. This one has been adjusted from real data available in the literature. The crack growth behaviour is well-captured and the transition between both regimes is well-estimated by a critical stress intensity factor range. The second purpose of our investigation deals with the prediction of the fatigue crack path and its variability based on measurements taken at the beginning of the propagation. The results show that our method based on this class of stochastic models associated with an updating method provides a reliable prediction and can be an efficient tool for safety analysis of structures in a large variety of engineering applications. In addition, the proposed strategy requires only little information to be effective and is not time-consuming

Identification of nonlinear dynamical systems using approximate Bayesian computation based on a sequential Monte Carlo sampler

The Bayesian approach is well recognised in the structural dynamics community as an attractive approach to deal with parameter estimation and model selection in nonlinear dynamical systems. In the present paper, one investigates the potential of approximate Bayesian computation employing sequential Monte Carlo (ABC-SMC) sampling [1] to solve this challenging problem. In contrast to the classical Bayesian inference algorithms which are based essentially on the evaluation of a likelihood function, the ABC-SMC uses different metrics based mainly on the level of agreement between observed and simulated data. This alternative is very attractive especially when the likelihood function is complex and cannot be approximated in a closed form. Moreover, this flexibility allows one to use new features from either the temporal or the frequency domains for system identification. To demonstrate the practical applicability of the ABC-SMC algorithm, two illustrative examples are considered in this paper