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In the Kennedy and O'Hagan framework for Bayesian calibration of physics models, selection of an appropriate prior form for the model discrepancy function is a challenging issue due to the lack of physics knowledge regarding model inadequacy. Aiming to address the uncertainty arising from the selection of a particular prior, this paper first conducts a study on possible formulations of the model discrepancy function. A first-order Taylor series expansion-based method is developed to investigate the potential redundancy caused by adding a discrepancy function to the original physics model. Further, we propose a three-step (calibration, validation, and combination) approach in order to inform the decision on the construction of model discrepancy priors. In the validation step, a reliability-based metric is used to evaluate posterior model predictions in the validation domain. The validation metric serves as a quantitative measure of how well the discrepancy formulation captures the missing physics in the model. In the combination step, the posterior distributions of model parameters and discrepancy corresponding to different priors are combined into a single distribution based on the probabilistic weights derived from the validation step. The combined distribution acknowledges the uncertainty in the prior formulation of model discrepancy function. [ABSTRACT FROM AUTHOR]

Abstract: This paper develops new insights into quantitative methods for the validation of computational model prediction. Four types of methods are investigated, namely classical and Bayesian hypothesis testing, a reliability-based method, and an area metric-based method. Traditional Bayesian hypothesis testing is extended based on interval hypotheses on distribution parameters and equality hypotheses on probability distributions, in order to validate models with deterministic/stochastic output for given inputs. Formulations and implementation details are outlined for both equality and interval hypotheses. Two types of validation experiments are considered—fully characterized (all the model/experimental inputs are measured and reported as point values) and partially characterized (some of the model/experimental inputs are not measured or are reported as intervals). Bayesian hypothesis testing can minimize the risk in model selection by properly choosing the model acceptance threshold, and its results can be used in model averaging to avoid Type I /II errors. It is shown that Bayesian interval hypothesis testing, the reliability-based method, and the area metric-based method can account for the existence of directional bias, where the mean predictions of a numerical model may be consistently below or above the corresponding experimental observations. It is also found that under some specific conditions, the Bayes factor metric in Bayesian equality hypothesis testing and the reliability-based metric can both be mathematically related to the p-value metric in classical hypothesis testing. Numerical studies are conducted to apply the above validation methods to gas damping prediction for radio frequency (RF) micro-electro-mechanical-system (MEMS) switches. The model of interest is a general polynomial chaos (gPC) surrogate model constructed based on expensive runs of a physics-based simulation model, and validation data are collected from fully characterized experiments. [Copyright &y& Elsevier]

Abstract: This paper presents a Bayesian probabilistic methodology to integrate model-based fatigue damage prognosis (FDP) with online and offline structural health monitoring (SHM) data. The prognosis uses fracture mechanics-based fatigue crack growth modeling, along with quantification of various sources of uncertainty, including natural variability, data uncertainty and model errors. These uncertainty sources are connected using a Bayesian network and a probabilistic sensitivity analysis is performed to assess the uncertainty contributions from these sources. The cycle-by-cycle simulation of fatigue crack growth is expedited via the use of a surrogate modeling technique (Gaussian process model) to replace computationally expensive finite element analysis. Real-time monitoring data of external variable amplitude loading history is used to construct a Bayesian autoregressive integrated moving average (ARIMA) model to predict and update the loading. On-ground crack inspection data is used to quantify the uncertainty in the initial and current size of an existing crack, using the Bayesian approach. Three possible cases of inspection results are considered: (1) crack is not detected; (2) crack is detected but not measured; (3) crack is detected and measured. Different scenarios of data availability (load monitoring data and inspection data) are considered for the prognosis of an individual component in a fleet. A numerical example, surface cracking in a rotorcraft mast under service loading, is implemented to illustrate the proposed methodology. The results of prognosis are validated using Bayesian hypothesis testing. [Copyright &y& Elsevier]

Abstract: Accurate characterization and prediction of loading, while properly accounting for uncertainty, are essential for probabilistic fatigue damage prognosis. Three different techniques – rainflow counting, the Markov chain method, and autoregressive moving average (ARMA) modeling – are investigated for stochastic characterization and reconstruction of the fatigue load history. The ARMA method is extended in this paper by introducing random coefficients and probabilistic weights, to account for the uncertainty in the selection of the model, inherent variability in loading, and uncertainty due to sparse data. A continuous model updating approach based on real-time monitoring data is developed and applied to all the three techniques mentioned above. The relation between prediction accuracy and updating interval is evaluated quantitatively. A quantitative model validation approach using Bayesian hypothesis testing is proposed to assess the confidence in load prediction from all the three methods. [Copyright &y& Elsevier]

Abstract: This paper presents a methodology for uncertainty quantification and model validation in fatigue crack growth analysis. Several models – finite element model, crack growth model, surrogate model, etc. – are connected through a Bayes network that aids in model calibration, uncertainty quantification, and model validation. Three types of uncertainty are included in both uncertainty quantification and model validation: (1) natural variability in loading and material properties; (2) data uncertainty due to measurement errors, sparse data, and different inspection results (crack not detected, crack detected but size not measured, and crack detected with size measurement); and (3) modeling uncertainty and errors during crack growth analysis, numerical approximations, and finite element discretization. Global sensitivity analysis is used to quantify the contribution of each source of uncertainty to the overall prediction uncertainty and to identify the important parameters that need to be calibrated. Bayesian hypothesis testing is used for model validation and the Bayes factor metric is used to quantify the confidence in the model prediction. The proposed methodology is illustrated using a numerical example of surface cracking in a cylindrical component. [Copyright &y& Elsevier]

Abstract: This paper presents several efficient statistical inference techniques to calibrate the equivalent initial flaw size (EIFS) of fatigue cracks for mechanical components with complicated geometry and multi-axial, variable amplitude loading. Finite element analysis is used to address the complicated geometry and calculate the stress intensity factors. Multi-modal stress intensity factors due to multi-axial loading are combined to calculate an equivalent stress intensity factor using a characteristic plane approach. During cycle-by-cycle integration of the crack growth law, a Gaussian process surrogate model is used to replace the expensive finite element analysis, resulting in rapid computation. Experimental data (crack size after a particular number of loading cycles) and statistical methods are used to calibrate the EIFS. The methods of least squares and maximum likelihood method are extended to evaluate the entire probability distribution of EIFS. Bayesian techniques are also implemented for this purpose. A fast numerical integration technique is developed as an efficient alternative to the expensive Markov Chain Monte Carlo sampling approach in the Bayesian analysis. An application problem of cracking in a cylindrical structure is used to illustrate the proposed methods. [Copyright &y& Elsevier]