Kriging-based Method Research Articles

A Kriging-based method for calibrating the bonded-particle model parameters of iron ore

In order to successfully simulate the crushing process of iron ore in a gyratory crusher, it is crucial to precisely model the iron ore based on the bonded-particle model, this requires reasonably calibrated micromechanical parameters in the model. In this study, a calibration method for iron ore BPM bonding parameters was proposed. Firstly the combination of the design of experimental method and discrete element simulation tests was used to investigate the effects of normal stiffness per unit area, shear stiffness per unit area, critical normal stress, and critical shear stress on the compressive strength and critical strain properties of the BPM. Furthermore, a fitted mathematical model between micro-mechanical parameters and macro-mechanical property parameters of iron ore was established. According to this, a bonding parameter calibration method based on Kriging surrogate model for the iron ore bonded-particle model is proposed. Finally, the micromechanical parameters of the calibrated actual iron ore bonded-particle model were verified by simulation test, which confirmed the accuracy and reliability of the proposed method and provided a new idea for the subsequent related research.

Adaptive Kriging-based method with learning function allocation scheme and hybrid convergence criterion for efficient structural reliability analysis

Adaptive Kriging-based method with learning function allocation scheme and hybrid convergence criterion for efficient structural reliability analysis

AK-SYS-IE: A novel adaptive Kriging-based method for system reliability assessment combining information entropy

Structural reliability assessment is a popular topic in engineering problems, particularly for the larger and more complex systems with implicit performance functions, also called black-box problems. The reliability assessment for a black-box problem must continuously be computed using simulation models, such as the finite element model, a highly time-consuming process with a high computational cost. The adaptive Kriging has gained considerable attention over the past decade. The Kriging-based reliability assessment method reduces the computational cost to a great extent, on the premise of ensuring the accuracy of reliability assessment. However, many of the currently published system reliability assessment methods, construct adaptive Kriging models by reducing the probability of incorrect prediction of the system state, and do not make full use of the uncertainty of the system state prediction information. To this end, a new Kriging-based method for structural system reliability assessment is proposed in this study. First, the probabilities of incorrect or correct system state predictions were understood from the perspective of information entropy. Second, an active learning strategy is proposed based on information entropy theory. Finally, the advantages of the proposed method are demonstrated and highlighted through several numerical examples. The results show that the proposed method achieves a good balance between the accuracy and computational cost, and the numerical magnitude effect does not affect the computational cost. Moreover, this is an effective method for assessing the reliability of complex systems.

A Kriging-based method for the efficient computation of debris impact zones

To prevent or assess launch risk, evaluation of launchers impact zones is a key element. Several methods are currently used to predict impact zones at the French space agency (CNES), but the highest-fidelity method uses a series of computationally costly Monte Carlo simulations. This process can be very time consuming and the computation time can become prohibitive. A machine learning method called Kriging or Gaussian Process Regression is studied as a potential avenue to speed up the impact zones evaluation. This Kriging-based method, is tested in this paper in different flight phases and its potential for estimating debris impact zones is evaluated in terms of processing time, accuracy and genericity.

A novel enhanced Kriging-based method for reliability analysis integrating Bayesian optimization with ensemble strategy

A novel enhanced Kriging-based method for reliability analysis integrating Bayesian optimization with ensemble strategy

Optomechanical integrated performance analysis and optimization of a greenhouse gas absorption spectrometer

Design of a space-borne opomechanical system with high spatial resolution, spectral resolution and quality imagery demands a tightly integrated performance optimization method to expand the design space and improve the optical performance of the system. In this paper, a Kriging-based method is proposed for optomechanical integrated performance optimization of a greenhouse gas absorption spectrometer (GGAS). The proposed method combines the thermal-structural-optical (TSO) integrated analysis method and the Kriging meta-modeling technique, in which the latter one is used to replace the former analysis process and approximates the I/O functions implied in the underlying TSO simulation model. Structural deformation characteristics and optical performance of GGAS are evaluated firstly. Then, optomechanical structural optimization of GGAS is carried out to achieve lightweight of three sub-benches. Results show that the proposed method is able to reduce the weight of three sub-benches by 11.85 % compared to the common opomechanical design method while satisfying almost the same constraints.

Reliability-based design optimization using adaptive Kriging-A single-loop strategy and a double-loop one

With the development of the surrogate-assisted Reliability-Based Design Optimization (RBDO) methods in recent decade, efficiency has been continuously improved by various state-of-the-art learning schemes. However, ensuring sufficient accuracy and feasibility at the optimum is still challenging, especially for real scenarios with complex probabilistic constraints and high fidelity. In order to achieve a good balance between efficiency and accuracy and feasibility at the optimum, both a single-loop and a double-loop adaptive Kriging-based RBDO methods are proposed. By directly approximating the probabilistic constraints using Kriging models in the single-loop method, the original RBDO problem is converted into a trivial deterministic optimization, which can be solved by any type of optimization method. The real values of failure probabilities are estimated by Generalized Subset Simulation (GSS) at the initial construction or during each update of the Kriging models. To further raise efficiency in the double-loop adaptive Kriging-based method, GSS is substituted by an inner-loop Kriging-based failure probability estimation, which includes each local enrichment at the limit states corresponding to the active probabilistic constraints based on an initial global enrichment within the augmented reliability space. Once a certain Kriging model meets the requirement during the refining of all, only the others are kept updating until the accuracy of all are accepted. Four examples are used to demonstrate the performance of the proposed two methods.

A new active-learning estimation method for the failure probability of structural reliability based on Kriging model and simple penalty function

Since the Kriging model can provide the mean value of the performance function at a sample point and the corresponding variance among the various surrogate models, many Kriging-based reliability analysis methods have been developed to estimate accurate failure probabilities of engineering structure problems. Various active learning functions have been proposed to reduce the computation burden of Kriging-based reliability analysis for engineering problems. This paper proposes a new active learning Kriging-based method for estimating the failure probability. A penalty learning function is proposed to improve the efficiency and accuracy of the failure probability estimate based on a simple penalty function. Also, a distance constraint term is considered to avoid a cluster of the selected sample points. Additionally, by an error-based stopping criterion, convergence conditions of the proposed new active learning reliability analysis method are investigated. Finally, five numerical examples are provided to verify the accuracy and effectiveness of the proposed method and convergence strategy. Numerical results demonstrated that the novel proposed method significantly improves the computational efficiency of reliability analysis with high estimating accuracy of the failure probability.

A dimension-wise analysis driven active learning paired-Kriging (DWA-ALK) method for the hybrid reliability analysis

Hybrid reliability analysis (HRA) has been extensively explored and mainly performed based on two categories of models. Mathematical properties on these models are given and proven in a rigorous manner, providing insights into HRA problems. Potential obstacles of the most appealing Kriging-based method available in literature for HRA include: metamodeling in an augmented-dimensional space spanned by both random and interval parameters resulting in substantial inversion of a large correlation matrix, an absence of distance metric in learning functions causing sample clustering, repeated interval analysis mainly completed by either the sampling-based or optimization-based algorithm leading to excessive computational cost and memory requirements especially for high-dimensional interval inputs. In this sense, a dimension-wise analysis driven active learning paired-Kriging method is proposed, where a paired-Kriging metamodel is constructed solely in the random input space and, at each training point, the interval analysis is conducted by a dimension-wise analysis method. A new learning function and termination criterion are conceived for refining the paired-Kriging model. It is concluded that the proposed method outperforms the common-in-use ones for HRA with high-dimensional interval inputs while its accuracy and efficiency are comparable in the case of low-dimensional interval inputs after validating its numerical performance by typical issues.

Method for predicting tool condition using kriging analysis of normalised frequency shifts of spindle acceleration signature and validation in trimming operations of CFRPs

ABSTRACT This work presents a method that deals with the problem of determining tool wear, surface roughness, and cutter temperate in the machining of a carbon fibre-reinforced polymer (CFRP) workpiece without the need for removing the tool for inspection. The spindle acceleration signature is dominated by the cutter passing frequency and its first harmonic. Cutting parameters and tool conditions strongly correlate with shifts in the latter two frequencies. The kriging method is utilised to construct prediction models of tool conditions using cutting parameters and spectral shifts. The presented Kriging-based method is unique in its ability to generate acceptable predictions of tool conditions using a small set of experimental data to construct the prediction models. Validation experiments have shown that the proposed method predicted tool wear, surface roughness, and cutter temperature with accuracies of 91%, 86%, and 84%, respectively. Benchmarking of the proposed method against another prediction method has shown acceptable performance.

Soft Tissue Deformation Modeling in the Procedure of Needle Insertion: A Kriging-Based Method

The simulation and planning system (SPS) requires accurate and real-time feedback regarding the deformation of soft tissues during the needle insertion procedure. Traditional mechanical-based models such as the finite element method (FEM) are widely used to compute the deformations of soft tissue. However, it is difficult for the FEM or other methods to find a balance between an acceptable image fidelity and real-time deformation feedback due to their complex material properties, geometries and interaction mechanisms. In this paper, a Kriging-based method is applied to model the soft tissue deformation to strike a balance between the accuracy and efficiency of deformation feedback. Four combinations of regression and correlation functions are compared regarding their ability to predict the maximum deformations of ten characteristic markers at a fixed insertion depth. The results suggest that a first order regression function with Gaussian correlation functions can best fit the results of the ground truth. The functional response of the Kriging-based method is utilized to model the dynamic deformations of markers at a series of needle insertion depths. The feasibility of the method is verified by investigating the adaptation to step variations. Compared with the ground truth of the finite element (FE) results, the maximum residual is less than 0.92 mm in the Y direction and 0.31 mm in the X direction. The results suggest that the Kriging metamodel provides real-time deformation feedback for a target and an obstacle to a SPS.

An active learning Kriging-based method combining the weight information entropy function and the adaptive candidate sample pool

The reliability analysis methods based on the Kriging model have been explored to update the design of experiments (DoE), but the number of calls to the performance function is high. In this paper, a new active learning method combining the weight information entropy function (WH) and the adaptive candidate sample pool is proposed to improve the reliability analysis efficiency. Based on the information entropy function (H), the learning function WH is constructed to consider Kriging variance and the joint probability density function (PDF). In this proposed method, the sample points can be assigned different weight by the learning function WH according to the important degrees of sample points. The point which not only nears the limit state function (LSF), but also has a high probability density function value and large Kriging variance is assigned more weight than others. To select the sample points with lower confidence level, the adaptive candidate sample pool is generated by Markov Chain Monte Carlo (MCMC) simulation. The Kriging model can be updated efficiently by the proposed method. Four numerical examples and an engineering example with implicit performance function are used to verify the efficiency and accuracy of the proposed method. The results show that the proposed method can significantly improve the computational efficiency of the reliability analysis without losing accuracy.

Locking-free Kriging-based Timoshenko Beam Elements using an Improved Implementation of the Discrete Shear Gap Technique

Kriging-based finite element method (K-FEM) is an enhancement of the conventional finite element method using a Kriging interpolation as the trial solution in place of a polynomial function. In the application of the K-FEM to the Timoshenko beam model, the discrete shear gap (DSG) technique has been employed to overcome the shear locking difficulty. However, the applied DSG was only effective for the Kriging-based beam element with a cubic basis and three element-layer domain of influencing nodes. Therefore, this research examines a modified implementation of the DSG by changing the substitute DSG field from the Kriging-based interpolation to linear interpolation of the shear gaps at the element nodes. The results show that the improved elements of any polynomial degree are free from shear locking. Furthermore, the results of beam deflection, cross-section rotation, and bending moment are very accurate, while the shear force field is piecewise constant.

Uncertainty quantification for initial geometric imperfections of cylindrical shells: A novel bi-stage random field parameter estimation method

Thin-walled cylindrical shells are widely used as main load-carrying structures in the aerospace industry. Considering the high sensitivity of the buckling load capacity to initial geometric imperfections (GIs), uncertainty quantification (UQ) of GIs is crucial to both structural safety as well as lightweight design. However, the inherent strong random characteristics of GIs and limited experimental data exacerbate the difficulty. Therefore, a novel bi-stage random field parameter estimation method with limited samples is proposed for the UQ of GIs. In stage I, a new optimization formula based on average coefficient of determination (AR) is established to overcome the complexity caused by both parameter coupling and limited samples and then, Kriging-based method via chunking and selecting mechanism (KM-CMS) is proposed to find the optimal correlation length and number of truncated terms. In stage II, a maximum likelihood (ML) based method for estimating standard deviation (SD) of random field is established, benefitted from the SD-independence of AR. Numerical results show that the proposed random field parameter estimation method achieves the best performance in terms of accuracy, efficiency and stability. Finally, a high-fidelity UQ model of measured GIs of cylindrical shells is obtained by proposed method in a rational manner.

AKSE: A novel adaptive Kriging method combining sampling region scheme and error-based stopping criterion for structural reliability analysis

The reliability analysis of complex structures usually involves implicit performance function and expensive-to-evaluate computational models, which pose a great challenge for the estimation of failure probability. In this paper, an adaptive Kriging-based method is proposed for the efficient estimation of failure probability with high accuracy. Starting from a small set of initial design of experiments (DoE), a Kriging model is constructed and iteratively refined by adding judiciously selected sample points to the DoE. To effectively select more sample points for updating the Kriging model, a new learning function is proposed for the selection of representative samples following the idea of the penalty function method in optimization. Besides, the inclusion of the term that measures the distance between the candidate samples and those in DoE controls the density of samples, and thus enables efficient exploration of the regions of interest. To improve the efficiency of the algorithm, this learning function is integrated with a sampling region scheme to filter out sample points in regions with rather low probability density from the candidate sampling pool. Moreover, a convergence criterion based on the error-based stopping criterion is developed to terminate the learning process at an appropriate stage. Hence, the proposed method is referred to as the Adaptive Kriging method combining Sampling region scheme and Error-based stopping criterion (AKSE). To further explore the possible schemes for the improvement of the proposed method, the combinations of AKSE with a dynamic Kriging method and the bootstrap variance are also investigated. For rare failure events, the Monte Carlo simulation (MCS) in AKSE is replaced by the importance sampling (IS) to establish an improved version of AKSE, namely the AKSE-IS. The performance of the proposed algorithm is evaluated through five numerical examples and the results demonstrate the superior performance of the proposed method both in terms of accuracy and efficiency.

An efficient and robust Kriging-based method for system reliability analysis

System reliability analysis involving multiple failure modes is challenging when performance functions are associated with time-consuming codes. This paper aims to enhance the efficiency of system reliability analysis by reducing the number of evaluations of time-consuming models. To achieve that, an adaptive Kriging-based method is proposed. In order to develop the method, a quantificational error measure of Kriging models (i.e. surrogate models of performance functions associated with each failure mode) is first derived. The stepwise accuracy-improvement strategy (SAIS) is then modified to solve system reliability problems, and the modified SAIS is called SAIS-SYS. The method for system reliability analysis is finally developed based on the derived error measure and SAIS-SYS. In the proposed method, Kriging models, i.e. the surrogate models of original performance functions, are adaptively refreshed according to SAIS-SYS until the associated error measure is smaller than a prescribed threshold. After Kriging models meet with accuracy requirement, the system failure probability can be obtained through a random simulation method and no additional evaluations of original performance functions is needed. The accuracy, efficiency and robustness of the proposed method are validated by four examples.

Uncertainty quantification for impact location and force estimation in composite structures

Structural health monitoring of impact location and severity using Lamb waves has been proven to be a reliable method under laboratory conditions. However, real-life operational and environmental conditions (vibration noise, temperature changes, different impact scenarios, etc.) and measurement errors are known to generate variation in Lamb wave features which may significantly affect the accuracy of these estimates. Therefore, these uncertainties should be considered, as a deterministic approach may lead to erroneous decisions. In this article, a novel data-driven stochastic Kriging-based method for impact location and maximum force estimation, that is able to reliably quantify the output uncertainty is presented. The method utilises a novel modification of the kriging technique (normally used for spatial interpolation of geostatistical data) for statistical pattern matching and uncertainty quantification using Lamb wave features to estimate the location and maximum force of impacts. The data was experimentally obtained from a composite panel equipped with piezoelectric sensors. Comparison with a deterministic benchmark method developed in prior studies shows that the proposed method gives a more reliable estimate for experimental impacts under various simulated environmental and operational conditions by estimating the uncertainty. The developed method highlights the suitability of data-driven methods for uncertainty quantification, by taking advantage of the relationship between data points in the reference database that is a mandatory component of these methods (and is often seen as a disadvantage). By quantifying the uncertainty, there is more information for operators to reliably locate impacts and estimate the severity, leading to robust maintenance decisions.

Uncertainty analysis of motion error for mechanisms and Kriging-based solutions

In this work, the issue of uncertainty analysis of motion errors for mechanisms is studied. First, two definitions of motion errors, that is, maximum error and accumulative error, are introduced to measure the motion accuracy in a comprehensive way. Then, the method for performing an uncertainty analysis of the two errors is discussed. For the maximum error, an analysis of the failure probability and the corresponding sensitivity analysis are carried out. For the accumulative error, the variance-based sensitivity is studied to quantify the contributions of input variables to the mechanism performance. A Kriging-based method is proposed to reduce the computational cost of the uncertainty analysis of motion errors. Finally, four examples including the classical four-bar mechanism and the landing gear mechanism for aeronautical purposes are studied with the proposed method. Discussions on the results show that the uncertainty analysis of motion errors can provide helpful information for mechanisms.

Improved active learning probabilistic approach for the computation of failure probability

This paper presents a cost-effective probabilistic approach to be used in engineering applications. The proposed approach consists of an improved Kriging-based method aiming at reducing to a minimum the number of evaluations of the true performance function when computing a failure probability. It is a kind of variant of the classical Active learning method combining Kriging and Monte Carlo Simulation (AK-MCS) developed by Echard et al. (2011) [1], where some improvements are introduced to enhance the learning process. Some illustrative and practical examples are presented and discussed. The proposed approach has shown a great efficiency as compared to the classical AK-MCS approach.

Kriging-based unconstrained global optimization through multi-point sampling

ABSTRACTThe generalized efficient global optimization (GEGO) method is able to solve expensive black-box problems. However, selecting one sampling point per cycle may result in large time consumption and loss of convergence accuracy. To this end, a kriging-based multi-point unconstrained global optimization (KMUGO) method is proposed. It extends the GEGO method with the improved constant liar (CL) strategy. For each cycle, the kriging model is first constructed or updated by the existing sampled data. Then, the enhanced alternative CL strategy is used to find multiple points, which will be further screened to identify the final expensive-evaluation points. Test results of numerical problems and an engineering simulation case show that KMUGO can deliver better convergence than GEGO, multi-point sampling method-based kriging (MPSK), hybrid and adaptive meta-model-based global optimization (HAM) and the kriging-based global optimization method using multi-point infill search criterion (KGOMISC).