Broggi, matteo

Patelli, edoardo

Rocchetta, Roberto

University of Liverpool Repository

Bayesian Updating for Real-Time Crack Detection  Roberto Rocchetta robertor@liverpool.ac.uk  Supervisors:  Dr. Edoardo Patelli (Engineering) Dr. Sven  Schewe (Computer science)  Dr. Broggi Matteo (Engineering) Academic partner: Professor Enrico Zio (Ecole central de  Paris, and Politecnico di Milano)  Fatigue is the most dangerous failure mode for mechanical components subject to alternating loads. Due to repeated loading and unloading, one or several cracks can be initiated and propagated through the cross section of the structure. Once a critical crack length is exceeded, fracture may occur and the structure will catastrophically fail even for stress level much lower than the design stress limit. Non-destructive inspections may be performed at predetermined time intervals in order to detect the cracks (e.g. Magnetic particle inspections, ultrasonic inspection). Continuous monitoring of the dynamic response of the structure can allow real-time cracks detection and corrective maintenance procedures might be taken in case the monitoring procedure identifies a crack. In the proposed work, Bayesian model updating procedures is adopted for the detection of crack location and length on a suspension arm, normally used by automotive industry and subject to fatigue stress. Experimental data of the damaged structure (frequency response function) are simulated using a high-fidelity numerical finite element (FE) model of the arm. Surrogate models have been calibrated and selected based on their capability of represent the FE results. The idea underlining the approach is to identify the most probable model consistent with the observa-tions. The likelihood is the key mathematical formulation to include the experimental knowledge in the updating of the probabilistic model. Three empirical likelihood functions have considered and results compared to verify the capability of Bayesian procedure in monitoring the system health state.    Industrial partner: European Centre for Soft Computing  (Oviedo, Spain) 1) Objective:   Framework for real-time identification of cracks in mechanical components subject to fatigue;     Evaluation of the posterior distribution given the observed data, exploration of different empirical likelihood functions;   Computational tractability to meet Real-Time detection requirement;   Account for aleatory and epistemic uncertainty 4) Case Study and Preliminary Results: 3) Proposed Methodology:  Bayesian Model Updating and to estimate proba-bility distribution function of the crack parame-ters;   Effective Transitional Markov Chain Monte Car-lo (TMCMC) ;   Extended Finite Element Model (XFEM) for crack modelling and solver parallelization on a computer cluster;   Surrogate Model selection to best mimic the XFEM results and further reduce computational time, which is a strong constraint for real time application.    Ad-hoc Likelihoods Expression, to account dif-ferent frequency domain of the Frequency Re-sponse Function (FRF); …….. 6) Synergic Research directions: Diagram of the Parallelization and Meta-Model Strategy for computational time reduction: Advanced uncertainty quantification frameworks: Conference paper: ‘’A Computational Framework for Classical and Generalized Uncertainty Quantification: Solution to the NAFEMS Challenge Problem’’; R. Rocchetta, M. Broggi, E. Patelli, NAFEMS Conference San Diego 2015   Risk assessment framework of complex systems:  Conference paper: ‘’Simulation-Based Risk Assessment Framework for Electric Vehicles Strategies Compar-ison Accounting Renewable Energy Sources’’; R. Rocchetta, E. Patelli  IPW conference November 2015  References: Goller, B., M. Broggi, A. Calvi, & G. Schu¨eller (2011). A stochastic model updating technique for complex aerospace structures. Finite Elements in Analysis and Design 47(7), 739–752.  Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London 53, 370–418.  Patelli, E., M. Broggi, M. de Angelis, & M. Beer (2014). Opencossan: an efficient open tool for dealing with epistemic and aleatory uncertainties. Vulnerability, Uncertainty, and Risk: Analysis, Modelling, and Management, 2564–2573.  Ching, J. & Y.-C. Chen (2007). Transitional Markov chain Monte Carlo method for bayesian model updating, model class selection, and model averaging. Journal of engineering mechanics133(7), 816–832.  Abdelaziz, Y. & A. Hamouine (2008). A survey of the extended finite element. Computers & Structures 86(11-12), 1141–1151. Surrogate Model Training and Selection Frequency response function(Ҩ1)  Parallelized Ex-tended finite ele-ment Model  (XFEM) Frequency response function(Ҩ2)  Frequency response function(Ҩn)  Frequency response function(Ҩexperiment)  Experiment Meas-urement Indicator of the FRF  Comparison Mean Square Error (MSE) Likelihood The analysed mechanical component is a suspension arm, similar to those used in the automotive indus-try.  It can freely rotate along the axis indicated by the dashed line; the suspension spring and the wheel structure are connected at the location indicated by “S”. The stress concentration points, and candidate crack locations, are indicated in the figure by the numbers 1 to 6. A crack with fixed length is inserted in one of the candidate position, and the reference FRF is computed at the position indicated by “O”. The crack lengths and positions are considered as uncertain parameters and updated in order to minimize the differences respect to the synthetic measurement of the system to be monitored.  5)Conclusions and Future developments: A Bayesian model updating procedure for cracks detection has been applied to detect cracks in a suspension arm. Reference dynamic data from vibration anal-ysis (FRF) was used as target for the updating. The effects of different numeri-cal likelihood expressions and different experimental data on the crack detec-tion strategy have been analysed. The procedure has been tested first to detect a single crack with unknown length but known position, the result comparison did not suggest major differences between the likelihood formulas and the de-tection procedure appear promisingly precise. Nevertheless, the second case point out the limitations. This is possibly due to similarity in the FRF for differ-ent cracks or shortcoming in the computational accuracy. In both the analysed cases, the crack was detect correctly around the true length and position and the computational time have been dramatically reduced. Future development and additional research will be taken by using real experimental data to further vali-date and expand the proposed approach, furthermore uncertainty in the device parameters will be accounted to increase robustness.   Microscopic visualization of  cracks A Mechanical system subject to fatigue Example of  Dye Penetrant Non Destructive Inspection  Frequency response function(Ҩexperiment)  Synthetic Experi-mental data Surrogate Model Training and Selection Frequency response function(Ҩ1)  Model (XFEM) Artificial neural network (ANN) Calibration  Polyharmonic spline (PHS) Calibration  Example of Frequency response functions. A single crack of 5mm length is inserted in each of the stress concentration points. On top: re mashing around crack length; On bottom: the suspension arm FE model, the six possible crack positions and measurement point O;  2) Challenges: Numerical Likelihood functions Comparison Bayesian Updating , TMCMC and Result Visualization Posterior distributions obtained with two different likelihood functions, synthetic experimental data of crack in position 6 of length 8.04 mm, Empirical likelihood functions computed considering three different frequency regions  Example of  frequency domain selected to compute  Mean Square Errors Likelihood 1:  All the considered frequencies  Likelihood 2:  Low frequency range  Likelihood 3:  Around the main resonance peak  An illustrative example of the Bayesian Model Updating Framework Abstract: 

Bayesian Model Updating for Damage Identification

http://livrepository.liverpool.ac.uk/3034572/1/ON%20BAEYSIAN%20UPDATING%20FOR%20CRACK%20DETECTION.pdf

Bayesian Model Updating for Damage Identification

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