A Quantitative Validation Method of Kriging Metamodel for Injection Mechanism Based on Bayesian Statistical Inference

metals Article A Quantitative Validation Method of Kriging Metamodel for Injection Mechanism Based on Bayesian Statistical Inference Dongdong You 1,2,3,* , Xiaocheng Shen 1,3, Yanghui Zhu 1,3, Jianxin Deng 2,* and Fenglei Li 1,3 1 National Engineering Research Center of Near-Net-Shape Forming for Metallic Materials, South China University of Technology, Guangzhou 510640, China; [email protected] (X.S.); [email protected] (Y.Z.); fl[email protected] (F.L.) 2 Guangxi Key Lab of Manufacturing System and Advanced Manufacturing Technology, Guangxi University, Nanning 530003, China 3 Guangdong Key Laboratory for Advanced Metallic Materials processing, South China University of Technology, Guangzhou 510640, China * Correspondence: [email protected] (D.Y.); [email protected] (J.D.); Tel.: +86-20-8711-2933 (D.Y.); +86-137-0788-9023 (J.D.) Received: 2 April 2019; Accepted: 24 April 2019; Published: 27 April 2019 Abstract: A Bayesian framework-based approach is proposed for the quantitative validation and calibration of the kriging metamodel established by simulation and experimental training samples of the injection mechanism in squeeze casting. The temperature data uncertainty and non-normal distribution are considered in the approach. The normality of the sample data is tested by the Anderson–Darling method. The test results show that the original difference data require transformation for Bayesian testing due to the non-normal distribution. The Box–Cox method is employed for the non-normal transformation. The hypothesis test results of the calibrated kriging model are more reliable after data transformation. The reliability of the kriging metamodel is quantitatively assessed by the calculated Bayes factor and confidence. The Bayesian factor and the confidence level results indicate that the kriging model demonstrates improved accuracy and is acceptable after data transformation. The influence of the threshold " on both the non-normally and normally distributed data in the model is quantitatively evaluated. The threshold " has a greater influence and higher sensitivity when applied to the normal data results, based on the rapid increase within a small range of the Bayes factors and confidence levels. Keywords: squeeze casting; bayesian inference; uncertainty; kriging metamodel 1. Introduction For the injection mechanism, which is a significant component of a squeeze casting machine, uncertainty analysis is very important for the optimal design of mechanical structures and injection processes. The metamodel method is effective and feasible for data analysis and prediction. In our previous study [1,2], an uncertainty analysis based on a kriging metamodel of the injection mechanism in squeeze casting and a literature review were performed. To evaluate and improve the prediction accuracy with uncertainty, the proposed kriging model needs quantitative validation and calibration. This study focuses on the quantitative method that considers data non-normality and uncertainty for the validation and calibration of kriging metamodels. Model validation employs a reasonable validation criterion established by probability and statistics theory, fuzzy mathematics, interval estimation methods, and other information theories to evaluate the prediction model and experimental model, considering data uncertainty, to obtain Metals 2019, 9, 493; doi:10.3390/met9050493 www.mdpi.com/journal/metals Metals 2019, 9, 493 2 of 15 a reliable prediction model and quantitatively infer the confidence of the predicted results of the total structural response. During the past ten years, model validation and calibration methods that consider uncertainties have increasingly attracted attention. Commonly used approaches, such as parameter estimation, model parameter correction, and system identification, update computer models to improve the consistency of model output. Oberkampf and Trucano [3] investigated model calibration and validation benchmarks. The design and use method of the code and validation metrics based on experimental data uncertainty was recommended. Chen et al. [4] presented a generic model validation methodology that replaced parts of the experimental evaluation with the model uncertainty analysis and substantially reduced the number of physical tests. Beek et al. [5] proposed a heuristics-enhanced model fusion method that updates the estimated input parameters by reducing information about available points to calibrate a kriging metamodel and accurately obtain the system responses. Recent studies have progressed to the use of statistical analysis methods, such as a hypothesis testing and Bayesian statistical inference, to measure and verify the consistency of the probability distribution between model prediction output and reference data. Oberkampf and Barone [6] conducted a quantitative model validation with uncertainty by developing a statistical hypothesis testing-based method. Hypothesis testing includes the classic method and the Bayesian method. Previous studies have discussed the differences between the two approaches (e.g., [7]). Compared with the classical hypothesis test, the Bayesian approach seems relatively simple and focuses on model acceptance. Selecting test statistics and determining the sampling distribution of statistics is not required. Furthermore, the significance level is not needed in advance, and its rejection domain does not have to be determined. Thus, accurate and stable test results can be obtained. A remarkable feature of the Bayesian approach is the introduction of prior information for Bayesian utilization. In decision-making, prior distribution, which is formed by the quantification of prior information, total information, and samples are integrated by a Bayesian formula to obtain the posterior distribution. Then, the statistical inference based on a posterior distribution may be used to validate the results. Rebba and Mahadevan [8] developed aggregate validation metrics that are based on classical and Bayesian hypothesis testing by multivariate analysis and applied experimental data to validate computational models with uncertainties. Jiang and Mahadevan [9] derived a formula to calculate the Bayes factor and confidence for interval hypothesis testing to improve the total reliability evaluation of the model. A method based on Bayesian inference and factor analysis is applied for the quantitative evaluation of the model confidence in dynamic systems at various levels [10]. They employ the proposed method and a power density pseudospectrum approach for system identification and neural network model validation [11]. In recent years, the Bayesian method has been demonstrated to be effective and feasible in validating and calibrating the kriging model. Arendt et al. [12,13] developed the modular Bayesian approach, which is a form of preposterior analysis using a Gaussian process, and calculated the preposterior covariance matrix of the calibration parameters to predict the degree of identifiability of the systems to be calibrated. Han et al. [14] introduced a statistical methodology that is based on a hierarchical Bayesian model and its application in a biomechanical engineering example, where tuning parameters and calibration parameters were simultaneously determined. Pronzato and Rendas [15] applied a Bayesian local kriging metamodel approach to replace the computationally expensive simulation codes with localized covariance functions and construct interpolators/predictors of random fields. Bachoc et al. [16] proposed a global statistical approach based on a universal kriging model that was calibrated by classical statistical inference to improve the predictions of the underlying physical system, including an uncertainty quantification, and determine the calibration parameters in a Bayesian framework. Jensen et al. [17] explored the feasibility of integrating an adaptive kriging metamodel into a finite element model updating formulation, in which a Bayesian model updating approach based on stochastic simulation was adopted using dynamic response data. De Baar et al. [18] derived gradient-enhanced kriging by the Bayesian theorem for uncertainty quantification, and in the Bayesian framework, improved the kriging process by increasing the posterior covariance when Metals 2019, 9, 493 3 of 15 making maximum likelihood estimates of the hyperparameters. Boojari et al. [19] developed a fixed rank kriging model to simultaneously handle skewness and nonhomogeneity and adopted a Bayesian framework for posterior distribution sampling and spatial prediction. Zhang et al. [20] adopted a kriging model to approximate the inaccessible point in one dimension and combined multiple predictions using Bayesian inference to improve the precision of function estimation. Angelikopoulos et al. [21] drastically reduced the computational cost of uni-modal posterior probability distribution functions in large-scale structural dynamics simulation by combining adaptive kriging with the Bayesian inference of models. Plumlee [22] presented a Bayesian calibration method that considers the orthogonality between the kriging model gradient and the prior distribution on the model bias to mitigate the suboptimally broad problem of the posterior of the calibration parameter and model bias. In statistical analysis and prediction, the interval hypothesis test method requires that the analysis and calculation data follow a normal distribution. An error in normal distribution assumption is usually made when employing Bayesian inference for computational model validation (e.g., [8]).

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