Kriging Model Research Articles

Scenario-driven distributionally robust optimization model for a rural virtual power plant considering flexible energy-carbon-green certificate trading

With the increased coupling of agriculture and energy, there is a trend to aggregate and manage distributed energy resources in agricultural parks using rural virtual power plants (RVPP). This paper investigates the impact of uncertainties in renewable energy generation and energy usage, as well as the flexibility of energy‑carbon-green certificate (GC) trading, on the planning and operation of RVPP. Firstly, the basic architecture of RVPP is constructed, and a joint trading mechanism for the carbon emission allowance (CEA) and GC is designed. On this basis, a two-stage deterministic optimization model is developed considering capacity configuration in the planning stage and the Stackelberg game in the operation stage of RVPP. Then, several typical scenarios considering the correlation of uncertainties are generated, and the deterministic model is transformed into a distributionally robust optimization (DRO) model in a scenario-driven manner. The confidence intervals of the scenario probability distributions are constrained by a combination of 1-norm and infinity-norm. Finally, the DRO model is decomposed into two problems, solved iteratively using a revised Kriging model and a column-and-constraint generation (C&CG) algorithm. Several cases covering different transaction forms and solution methods are analyzed comparatively to validate the effectiveness of the DRO model. The simulation results indicate that, compared to the energy trading with a fixed price, flexible trading based on the Stackelberg game can reduce the total planning and operating costs by 22.49 %. Compared to the separate trading of GC and CEA, the trading volume of CEA decreases by 44.21 % under the joint trading mechanism, with the increased configuration of renewable energy resources.

A Force-Sensor-Less Approach for Rapid Young's Modulus Identification of Heterogeneous Soft Tissue.

Due to individual differences, accurate identification of tissue elastic parameters is essential for biomechanical modeling in surgical guidance for hepatic venous injections. This paper aims to acquire the absolute Young's modulus of heterogeneous soft tissues during endoscopic surgery with 2D ultrasound images. First, we introduced a force-sensor-less approach that utilizes a pre-calibrated soft patch with a known Young's modulus and its ultrasound images to calculate the external forces exerted by the probe on the tissue. Second, we introduced a Kriging-based inverse algorithm to identify the relative Young's modulus (RYM) between the inclusion and the background tissue. The RYM was estimated based on 2D plane strain approximation and mapped to the RYM of 3D soft tissue through a trained Kriging model. Finally, we developed a direct method to identify the background Young's modulus (BYM) based on calculated external forces and RYM. The simulation results demonstrate the high efficiency and robustness of the Kriging-based inverse algorithm in identifying RYM. Physical experiments on the three phantoms show that the errors of the identified BYM and RYM are all below 15%. The proposed methodology for Young's modulus identification is feasible and achieves satisfactory accuracy and computational efficiency in both simulations and physical experiments.

Global sensitivity analysis of high-speed train dynamics system with high-dimensional inputs and multiple outputs

A high-speed train dynamics system involves numerous uncertainty parameters, and there are multiple evaluation indices for its dynamic performance. In this paper, we conduct high-dimensional and multi-output global sensitivity analysis for the dynamic performance of high-speed trains by comprehensively considering these dynamics parameters and indices. Firstly, the dynamics simulation model of the high-speed train is established and six dynamics indices for evaluating the safety and comfort of the train are obtained under operating conditions. Subsequently, considering the multi-output and high-dimensional characteristics of train sensitivity analysis, we propose a multi-output global sensitivity analysis method based on the correlation-analysis-guided Kriging model (CA-K). Finally, global sensitivity analysis of 96 uncertainty parameters on six dynamics indices is conducted under three scenarios utilising the proposed method. The results show that (1) CA-K achieves higher accuracy in approximating high-dimensional train dynamics indices than existing conventional surrogate models. (2) The multi-output global sensitivity analysis method can simultaneously identify all important dynamics parameters and exclude unimportant parameters.

Rare reliability analysis by beta-surfaces and Kriging model

Rare reliability analysis by beta-surfaces and Kriging model

Multifidelity co-kriging metamodeling based on multivariate data fusion for dynamic fit improvement of injection mechanism in squeeze casting

To improve the dynamic fit the injection mechanism in a squeeze casting machine, this paper proposes a novel multifidelity co-kriging (MFCK) metamodeling method, which fuses high-fidelity measured data with low-fidelity simulated data and considers data uncertainty and multivariate correlation influence to accurately predict response values when experimental sample data are insufficient. An MFCK model was established to predict the deformation and dynamic fit clearance, by selecting experimental and simulated values of deformation and temperature as the principal and covariates for correlation testing. The results indicate that the proposed MFCK model significantly improved the prediction accuracy by 34.18 %, 73.53 %, 41.57 % and 37.93 %, respectively, compared with the ordinary kriging model and finite element method. This method was applied to the multicycle injection process of a 2,500-kN squeeze casting machine, revealing the variation law of the fit clearance. The MFCK model improved the prediction accuracy of the fit clearance by 72.7 %, which is beneficial for process control. The accuracy and industrial applicability of the proposed MFCK method was thus verified.

AN INVESTIGATION OF CRIME IN NIGER REPUBLIC USING REGRESSION KRIGING MODELS

This study investigates crime patterns using regression kriging. The methodology involves determining variogram models for various crime types and fitting multiple regression kriging models. Most variogram models were Gaussian, with some exhibiting smooth transition estimates or spherical structures. The analysis revealed that the majority of crime types exhibit spatial correlation, with range values greater than zero. In the regression kriging models, population size and literacy rate emerged as the most significant predictors. Additionally, the variograms of the kriging residuals indicated trends for certain crime types. However, for most crimes, the residuals displayed spatial autocorrelation, suggesting that the models are suitable for predictive purposes.

Metamodeling of turbofan engine compressor characteristics using SVM-based improved Kriging method

PurposeThe purpose of this paper is to overcome the inherent lack of precision in commonly used interpolation procedures when solving the mathematical model of turbofan engines, as well as to address the issue that the theoretical variogram model in traditional Kriging models is prone to subjective selection bias, which makes it impossible to accurately capture the inherent fluctuation patterns in compressor data.Design/methodology/approachTo mitigate this challenge, based on the spatial distribution characteristics of the compressor characteristic data of a certain type of turbofan engine, the input and output dimensions of the model are defined. By determining the stable operating region from the original component data, the authors use the proposed Kriging method improved with a support vector machine model to reconstruct the characteristics at unknown speeds within this region. The effectiveness of the proposed method is evaluated using the established assessment metrics.FindingsExperimental results demonstrate that the proposed method exhibits significant advantages over the conventional Kriging approach. Specifically, it leads to a substantial reduction in root mean square error and mean absolute error by 0.0153/0.0118 (low speed), 0.1306/0.0362 (medium speed) and −0.0066/0.2366 (high speed).Originality/valueThis refined approach not only offers notable engineering applicability but also contributes significantly to the enhancement of aerospace engine model solutions’ precision.

A multi-layer Kriging surrogate model for the reliability analysis of variable stator vanes in aero engines

Abstract The variable stator vanes (VSV) are a set of typical spatial linkage mechanisms widely used in the variable cycle engine compressor. Various factors influence the angle adjustment precision of the VSV, leading to the failure of the mechanism. The reliability analysis of VSV is a complex task due to the involvement of multiple components, high dimensionality input and computational inefficiency. Considering the hierarchical characteristics of VSV structure, we propose a novel multi-layer Kriging surrogate (MLKG) for the reliability analysis of VSV. The MLKG combines multiple Kriging surrogate models arranged in a hierarchical structure. By breaking the problem down into more minor problems, MLKG works by presenting each small problem as a Kriging model and reducing the input dimension of the sub-layer Kriging model. In this way, the MLKG can capture the complex interactions between the inputs and outputs of the problem while maintaining a high degree of accuracy and efficiency. This study proves the error propagation process of MLKG. To evaluate MLKG’s accuracy, we test it on two typical high-dimensional non-linearity functions (Rosenbrock and Michalewicz function). We compared MLKG with some contemporary KG surrogate modeling techniques using mean squared error (MSE) and R square ( ${R^2}$ ). Results show that MLKG achieves an excellent level of accuracy for reliability analysis in high-dimensional problems with a small number of sample points.

The Influence of Spatial Scale Effect on Rock Spectral Reflectance: A Case Study of Huangshan Copper–Nickel Ore District

The spectral reflectance measured in situ is often regarded as the “truth”. However, its limited coverage and large spatial heterogeneity often make the ground-based reflectance unable to represent the remote sensing images. Since the spatial scale mismatch between ground-based, airborne, and spaceborne measurements, the applications of geological exploration, metallogenic prognosis and mine monitoring are facing severe challenges. In order to explore the influence of spatial scale effect on rock spectra, spectral reflectance with uncertainty caused by differences in illumination view geometry and spatial heterogeneity is introduced into the Bayesian Maximum Entropy (BME) method. Then, the rock spectra are upscaled from the point-scale to meter-scale and to 10 m-scale, respectively. Finally, the influence of spatial scale effect is evaluated based on the reflectance value, spectral shape, and spectral characteristic parameters. The results indicate that the BME model shows better upscaling accuracy and stability than Ordinary Kriging and Ordinary Least Squares model. The maximum Euclidean Distance of rock spectra caused by spatial resolution change is 6.271, and the Spectral Angle Mapper can reach 0.370. The spectral absorption position, absorption depth, and spectral absorption index are less affected by scale effect. For the area with similar spatial heterogeneity to the Huangshan Copper–Nickel Ore District, when the spatial resolution of the image is greater than 10 m, the rock’s spectrum is less influenced by the change in spatial resolution. Otherwise, the influence of spatial scale effect should be considered in applications. In addition, this work puts forward a set of processes to evaluate the influence of spatial scale effect in the study area and carry out the upscaling.

Efficient Hierarchical Kriging Modeling Method for High-dimension Multi-fidelity Problems

The multi-fidelity Kriging model is a promising technique in surrogate-based design, balancing model accuracy and the cost of sample generation by combining low- and high-fidelity data. However, the cost of building a multi-fidelity Kriging model increases significantly as problem complexity grows. To address this issue, we propose an efficient Hierarchical Kriging modeling method. In building the low-fidelity model, distance correlation is used to determine the relative value of the hyperparameter. This transforms the maximum likelihood estimation problem into a one-dimensional optimization task, which can be solved efficiently, significantly improving modeling efficiency. The high-fidelity model is built similarly, with the low-fidelity model's hyperparameter used as the relative value for the high-fidelity model's hyperparameter. The proposed method's effectiveness is evaluated through analytical problems and a real-world engineering problem involving modeling the isentropic efficiency of a compressor rotor. Experimental results show that the proposed method reduces modeling time significantly without compromising accuracy. For the compressor rotor isentropic efficiency model, the proposed method yields over 99% cost savings compared to conventional approaches, while also achieving higher accuracy.

Bayesian updating using accelerated Hamiltonian Monte Carlo with gradient-enhanced Kriging model

Bayesian methods have been widely used to improve the accuracy of finite element model in civil engineering. However, Bayesian methods generally suffer from the computational complexity involved in accurately identifying the posterior distribution. To address this issue, this paper proposes a novel method by combining the Hamiltonian Monte Carlo (HMC) algorithm with the gradient-enhanced Kriging (GEK) model, termed HMC-GEK, for more efficient model updating. The proposed method uses the potential function and gradient information generated during the burn-in phase of the HMC to train the GEK model. By replacing high-cost potential function with the GEK model, the original HMC sampling process is accelerated. An eight-story frame structure and a Y-shaped arch bridge are used to validate the accuracy and efficiency of the proposed method. Furthermore, the HMC-GEK method has been employed to identify damage of a real eight-story steel frame structure. Compared with the HMC method with the traditional Kriging model, HMC-GEK makes more full use of the gradient information of the potential function and significantly improves the sample acceptance rate and computational efficiency. In addition, the successful application of the method in damage identification of the real structure demonstrates its value for engineering applications.

Research on Fast Prediction Methods of Ice Shape in Ice-tolerant Airfoil Design

Abstract Ice accretion damages the aerodynamic performance of aircraft during flight, so the airfoil design based on the concept of ice-tolerant has become a brand-new research direction. In order to shorten the Computational Fluid Dynamics (CFD) time of ice shape in the design of ice-tolerant airfoil, this paper studies a fast prediction method of ice shape based on the Class function/Shape function Transformation (CST) method and Kriging model. Firstly, airfoils are parameterized by the CST method to establish the airfoil database. Secondly, the ice shape database is obtained by CFD calculations, and then the ice shapes mapping on airfoils are generated by the Kriging model. Finally, the sample number, CST order, POD energy ratio and other relative parameters are studied based on the NACA0012 airfoil. Results show that the CFD time can be extensively saved in ice-tolerant design for airfoils, and further, for three-dimensional optimization problems with variable icing conditions.

Research on multi-objective optimization method of Z-shaped pipeline structure based on Kriging model

Slurry pipeline transportation is widely used in dredging and serves as an essential method for conveying solid materials. However, accurately describing the interaction between slurry and particles through numerical simulations, while optimizing the pipeline structure to improve the performance of slurry pipelines, poses a significant engineering challenge. In this study, a Z-shaped continuous pipeline, designed using B-spline curves, is implemented in the slurry circulation system. The validity of the CFD–DEM method is confirmed through slurry circulation experiments, and the effects of various structural parameters on transportation efficiency and flow characteristics are investigated. By integrating the Kriging surrogate model with the NSGA-II algorithm, a multi-objective optimization method is proposed to reduce computational complexity and improve the sediment-carrying capacity of the Z-shaped pipeline. Using a slurry shield tunneling machine with a diameter of 6.24 m as an example, this study optimizes a Z-shaped pipeline with an inner diameter of 0.3 m and an axial height of 1.5 m. The optimized structural parameters increase the average particle velocity by 8.9% and reduce particle accumulation by 21.3%. Additionally, the interaction between the pipeline’s inclination angle \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta$$\\end{document}, the B-spline control parameter \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_{4}$$\\end{document}, and the sediment-carrying capacity is analyzed.

Surrogate model-based modelling and optimization of the adsorption performance of a vacuum adsorption system under leakage conditions

The leakage phenomenon has become a major challenge in engineering applications for vacuum adsorption technology. To achieve efficient design optimization of the vacuum adsorption system based on the finite element model, an optimization design method based on residual adaptive fitting of the surrogate model is proposed. The surrogate model for the adsorption performance and influencing parameters of the vacuum adsorption system is established under leakage conditions by the response surface method (RSM). The residuals generated by the RSM are fitted using the kriging model and incorporated into the construction of the RSM. The established surrogate models are updated by learning functions, combining genetic algorithms to optimize the parameters of the vacuum adsorption system. The optimization results indicate that appropriate parameter configurations can enhance adsorption performance and reduce energy consumption, and also validate the superiority of the proposed method.

A novel multi-objective robust optimization method based on improved Gray Wolf optimizer and Kriging model

A novel multi-objective robust optimization method for mechanical structure products is introduced. Firstly, the method introduces Bernoulli Dynamic Adaptive Gray Wolf Optimizer (BDAGWO) to address inherent limitations within the original Gray Wolf Optimizer (GWO), such as falling into local optimal solutions and low convergence rate. Subsequently, BDAGWO is utilized to search optimal correlation coefficients of Kriging. The proposed surrogate model exhibits high fitting accuracy on different test cases, with coefficients of determination reaching above 0.99 on the test set, and mean relative error, mean absolute error, mean absolute percentage error and root mean squared error are all close to 0. To solve multi-objective optimization problems, an improved NSGA-II is introduced to amplify search capabilities. Then based on robust optimization method, combine BDAGWO-Kriging and improved NSGA-II to establish a multi-objective robust optimization framework with high accuracy and high solution efficiency. The proposed method is implemented to an EMU bogie frame, demonstrating that the robust optimization scheme reduces the maximum equivalent stress and mass, and exhibits lower fluctuation compared to deterministic optimization. This substantiates the method’s effectiveness and provides an optimization approach of large and complex mechanical products.

An efficient sequential Kriging model for structure safety lifetime analysis considering uncertain degradation

Safety lifetime analysis performs a crucial role in ensuring structural safety in service and developing effective maintenance strategies, which also places higher demands on calculation. However, existing safety lifetime analysis methods generally suffer from inefficiency, which is more prominent for complex engineering structures. In this paper, a novel sequential single-loop Kriging (SSK) surrogate modeling approach is proposed to calculate the safety lifetime in an efficient and accurate manner. To reduce the computational cost, a single-loop safety lifetime analysis framework is proposed. In this framework, there is no need to accurately calculate the time-dependent failure probability (TDFP) in any sub-time interval. By searching the safety lifetime in the process of time-dependent reliability analysis (TRA) and dynamically adjusting the interest time interval, the safety lifetime can be quickly determined by constructing only one Kriging model. To maximize the utilization of sample information, SSK employs a modified learning function that allows most of the training points to be added before the safety lifetime. For accuracy, a convergence criterion that includes two Kriging models is proposed. Mathematical engineering examples are used to illustrate the accuracy and efficiency of SSK. The proposed method offers a promising approach for efficient safety lifetime analysis of engineering problems.

A new adaptive analysis method based on the Kriging model for structural reliability analysis

Widespread uncertainty in engineering problems makes it necessary to carry out structural reliability analysis. The crude Monte Carlo simulation (MCS) method can obtain accurate results, but it requires a large number of model evaluations. The Kriging-based method is a feasible way to reduce the computational cost. This study proposes a novel adaptive analysis method. Firstly, the convergence condition based on estimation accuracy is introduced. This condition focuses on the precision of the failure probability rather than the state of the points in the candidate sample pool. Then three extended U learning strategies are proposed. Sequence strategy (#1) focuses on evenly selecting samples by exploiting information on both sides of the limit state function. Strategy (#2) adopts the parallel adaptive learning technique to simultaneously select samples in both the safe and failure domains. Strategy (#3) pays attention to low-precision domains and can adaptively choose between sequential and parallel analysis modes. The choice of the three strategies can be based on the parallel computing resources available to researchers. Finally, three numerical cases and one engineering case are presented. This study provides an efficient tool for reliability evaluation of practical engineering problems.

Research on Optimal Design of Ultra-High-Speed Motors Based on Multi-Physical Field Coupling Under Mechanical Boundary Constraints

This study investigates the impact of rotor structure, material selection, and cooling methods on ultra-high-speed motor performance, revealing performance variation laws under multi-physical field coupling. Considering mechanical boundary constraints, we propose an optimization design method based on a multi-physical field coupling model. Using a MaxPro experimental design, initial samples are obtained and fitted using a Kriging surrogate model. The NSGA-2 algorithm is then applied for optimization, with Relative Maximum Absolute Error (RMAE) and Relative Average Absolute Error (RAAE) employed for accuracy evaluation. The Kriging model is iteratively updated based on evaluation results until the optimal design is achieved. This method enhances motor performance, ensures mechanical boundary conditions, and reduces computational load. Experimental results show significant improvements in efficiency and power density. This study provides theoretical support and technical guidance for ultra-high-speed motor design and offers new ideas for related motor research and development. Future work will explore more efficient and intelligent optimization algorithms to continuously advance ultra-high-speed motor technology.

Optimization of Thermal Management for the Environmental Worthiness Design of Aviation Equipment Using Phase Change Materials

A phase change materials (PCM)-based heat sink is an effective way to cool intermittent high-power electronic devices in aerospace applications such as airborne electronics and aircraft external carry. Optimizing the heat sink is significant in designing a compact and efficient system. This paper proposes an optimization procedure for the PCM/expanded graphite (EG)-based heat sink to minimize the temperature of the heat source. The numerical model is built to estimate the thermal response, and a surrogate model is fitted using the Kriging model. An optimization algorithm is constructed to predict the optimum parameters of the heat sink, and the effects of heat sink volume, heat flux, and working time on the optimal parameters of the heat sink are investigated. This shows that the numerical results agree well with the experimental data, and the proposed optimization method effectively obtains the optimal EG mass fraction and the geometric dimensions of the PCM enclosure. The optimal EG mass fraction increases with the rise in heat sink volume and decreases with the increase in heat flux and working time. The optimal ratio of the height to the length of the heat sink is 0.43. This study provides practical guidance for the optimal design of a PCM/EG-based heat sink.

Numerical study on the hydrothermal characteristics of a wire-wrapped rod bundle with nonuniform wire pitches

In this study, thermal hydraulic behaviors in a 19-pin bundle fuel assembly with nonuniform wire pitches is investigated by combing CFD with the Kriging method. To optimize the design, two geometric variables—the ratio of inner pitch to reference pitch (Pi/P) and the ratio of outer pitch to reference pitch (Po/P)—are selected, and the design space is sampled using Latin Hypercube Sampling (LHS). The sampled points are then subjected to CFD analysis. Convergence is considered achieved when the residuals of all variables are below 1e-5. The optimization problem aims to minimize the objective function, which is a linear combination of the cross-sectional temperature difference and friction factor. Sequential Quadratic Programming (SQP) is employed to search for the optimal point using a constructed meta-model. When compared to the reference shape, the optimal shape exhibits higher axial velocity in the inner channel, higher average temperature, smaller temperature difference at the outlet section, and reduced pressure drop in the fuel assembly. The Kriging model accurately predicts the cross-sectional temperature difference and friction coefficient for the optimal shape, consistent with the CFD calculation results. This confirms the accuracy and feasibility of the Kriging model in fuel assembly optimization.