Kriging Model Method Research Articles

Reliability-Based Robust Design Optimization with Fourth-Moment Method for Ball Bearing Wear

Ball bearings operating at low speeds and under heavy loads are susceptible to wear failure, leading to significant economic losses. The existing reliability-based robust design optimization method of the fourth-moment method has high accuracy and does not need to determine the random distribution of the input variables, but it is not possible to apply it to ball bearing wear due to the complexity of the bearing wear state function that cannot be characterized as an explicit form. To address this issue, this paper proposes a novel design method for ball bearing wear. Firstly, a surrogate model is constructed using the Kriging model method to establish a relationship between the bearing design parameters and the mechanical response. Subsequently, a wear reliability model is developed on the basis of the fourth-moment method, and reliability sensitivity analysis is conducted. Finally, the ball bearing wear reliability-based robust design optimization is accomplished through the use of a genetic algorithm. The results of the case calculations demonstrate that the proposed method effectively calculates the ball bearing wear reliability and analyzes the impact of design parameter randomness on reliability. Furthermore, optimizing the design parameters reduces the sensitivity of wear reliability to parameter randomness.

Active on-orbit surface accuracy adjustment for spaceborne antennas based on multi-fidelity adaptive migration learning strategy

For the large spaceborne synthetic aperture radar antenna, its surface accuracy has a decisive impact on the on-orbit electrical performance, which is affected by manufacturing errors, assembly misalignments, and thermal deformations. Therefore, ensuring the surface accuracy on orbit and reducing its fluctuation are highly concerned issues for the large spaceborne antenna in practice. To this end, this study proposes an active adjustment method based on a multi-fidelity adaptive migration learning strategy to guarantee on-orbit surface accuracy. First, the theoretical surface accuracy model considering space thermal deformation is completely established. Then, the surrogate model of surface accuracy is developed to realize fast prediction and optimization, in which a new dynamic Kriging modeling method is proposed. Furthermore, the surrogate prediction model at other sites on the orbit is efficiently obtained by an innovative multi-fidelity migration learning strategy. On this basis, the active adjustment optimization model is constructed based on the surrogate prediction model, and the corresponding adjustment amounts can be obtained accordingly by solving this optimization problem. Finally, the effectiveness and advantages of the proposed method are comprehensively proven through case studies and experimental verification.

A dimension reduction-based Kriging modeling method for high-dimensional time-variant uncertainty propagation and global sensitivity analysis

Surrogate models have been widely used in the uncertainty propagation and global sensitivity analysis of complex evaluation-expensive engineering problems. However, the construction of high-accuracy surrogate models for time-variant problems with high-dimensional inputs and outputs using a small number of training samples remains a challenge. To address this challenge, we propose a dimension reduction-based Kriging modeling (KMDR) method for high-dimensional time-variant uncertainty propagation and global sensitivity analysis. Singular value decomposition is performed on the original time-variant response to extract principal components from high-dimensional outputs. And the improved sufficient dimension reduction (ISDR) is performed on high-dimensional inputs to identify the latent input space with respect to each principal component of outputs. A ladle estimator with a rigorous mathematical definition is then employed to determine the number of principal components of the outputs and the dimensionalities of the latent input spaces. The ladle estimator considers variabilities in both eigenvalues and eigenvectors of a matrix and can determine the latent dimensionality more accurately and efficiently than existing approaches. Subsequently, Kriging models between high-dimensional inputs and each principal component of the outputs are constructed based on a newly devised Kriging kernel, which embeds the information of the ISDR into the kernel function and can achieve higher accuracy than directly constructing Kriging models between latent inputs and outputs. In addition, a generalized variance-based sensitivity index, which can quantify the effects of the inputs on the overall time-variant response, is defined and computed directly from Sobol’ sensitivity indices of the inputs at each time node. Finally, the surrogate model of the time-variant system constructed by the KMDR is directly adopted for efficient time-variant uncertainty propagation and global sensitivity analysis. Several examples demonstrate that the proposed approach can construct more accurate surrogate models and obtain more accurate time-variant uncertainty propagation and global sensitivity analysis results than existing methods with a small training set.

Importance sampling enhanced by adaptive two-stage Kriging model and active subspace for analyzing rare probability with high dimensional input vector

For the high dimensional and small failure probability events widely existing in engineering, structural reliability analysis method still faces challenges at present. Since active subspace (AS) method can effectively reduce the dimension of input vector space, and metamodel-based importance sampling (IS) can efficiently estimate small failure probability, this paper combines AS with IS to propose an adaptive two-stage Kriging model method (AS-IS-AK2) for rare failure event with high dimensional problem. The proposed AS-IS-AK2 includes two stages. In the first stage, a new learning function is proposed to construct the Kriging model for roughly estimating the performance function values of all IS samples, and the gradient information is provided as a by-product for AS. In the second stage, an adaptive Kriging method is constructed in the reduced dimension space to accurately judge the states of all IS samples. Since AS reduces the dimension of input vector space, and IS reduces the required size of candidate sample pool for obtaining the convergent failure probability estimation, the computational complexity is reduced in reconstructing the Kriging model to judge the states of IS samples. The results of three examples verify the advantage of the proposed method for high dimensional problem with small failure probability.

An improved high-dimensional Kriging modeling method utilizing maximal information coefficient

PurposeKriging surrogate model has demonstrated a powerful ability to be applied to a variety of engineering challenges by emulating time-consuming simulations. However, when it comes to problems with high-dimensional input variables, it may be difficult to obtain a model with high accuracy and efficiency due to the curse of dimensionality. To meet this challenge, an improved high-dimensional Kriging modeling method based on maximal information coefficient (MIC) is developed in this work.Design/methodology/approachThe hyperparameter domain is first derived and the dataset of hyperparameter and likelihood function is collected by Latin Hypercube Sampling. MIC values are innovatively calculated from the dataset and used as prior knowledge for optimizing hyperparameters. Then, an auxiliary parameter is introduced to establish the relationship between MIC values and hyperparameters. Next, the hyperparameters are obtained by transforming the optimized auxiliary parameter. Finally, to further improve the modeling accuracy, a novel local optimization step is performed to discover more suitable hyperparameters.FindingsThe proposed method is then applied to five representative mathematical functions with dimensions ranging from 20 to 100 and an engineering case with 30 design variables.Originality/valueThe results show that the proposed high-dimensional Kriging modeling method can obtain more accurate results than the other three methods, and it has an acceptable modeling efficiency. Moreover, the proposed method is also suitable for high-dimensional problems with limited sample points.

An improved sufficient dimension reduction-based Kriging modeling method for high-dimensional evaluation-expensive problems

Kriging is a powerful surrogate modeling method for the analysis and optimization of computationally expensive problems. However, the efficient construction of high-precision models for high-dimensional expensive problems is an important challenge for Kriging method and has received much attention recently. To address this challenge, we propose an improved sufficient dimension reduction-based Kriging modeling method (KISDR) for high-dimensional evaluation-expensive problems. First, the martingale difference divergence is introduced into the sufficient dimension reduction to obtain a more accurate and stable estimate of the projection matrix, which can project the high-dimensional inputs into a low-dimensional latent space while maintaining sufficient information for response prediction. Second, we propose to utilize the ladle estimator to identify the dimension of latent space. The ladle estimator combines both eigenvalues and eigenvectors of the matrix and can identify the latent dimensionality more precisely. After that, a new Kriging correlation function is constructed by integrating the information of dimension reduction into the correlation structure, which significantly reduces the number of hyperparameters to be estimated. Finally, we devise a local optimization approach to fine-tune the Kriging hyperparameters to further improve the modeling accuracy. In this study, six mathematical examples and three engineering examples with dimensions varying from 30-D to 100-D are employed for performance analysis and comparison. The results indicate that the proposed KISDR can precisely identify and exploit the low-dimensional linear structure in the data to improve modeling accuracy and efficiency for high-dimensional evaluation-expensive problems.

An efficient gradient-enhanced kriging modeling method assisted by fast kriging for high-dimension problems

PurposeThe modeling cost of the gradient-enhanced kriging (GEK) method is prohibitive for high-dimensional problems. This study aims to develop an efficient modeling strategy for the GEK method.Design/methodology/approachA two-step tuning strategy is proposed for the construction of the GEK model. First, an auxiliary kriging is built efficiently. Then, the hyperparameter of the kriging model is served as a good initial guess to that of the GEK model, and a local optimal search is further used to explore the search space of hyperparameter to guarantee the accuracy of the GEK model. In the construction of the auxiliary kriging, the maximal information coefficient is adopted to estimate the relative magnitude of the hyperparameter, which is used to transform the high-dimension maximum likelihood estimation problem into a one-dimensional optimization. The tuning problem of the auxiliary kriging becomes independent of the dimension. Therefore, the modeling efficiency can be improved significantly.FindingsThe performance of the proposed method is studied with analytic problems ranging from 10D to 50D and an 18D aerodynamic airfoil example. It is further compared with two efficient GEK modeling methods. The empirical experiments show that the proposed model can significantly improve the modeling efficiency without sacrificing accuracy compared with other efficient modeling methods.Originality/valueThis paper developed an efficient modeling strategy for GEK and demonstrated the effectiveness of the proposed method in modeling high-dimension problems.

A Gray-Box Surrogate Vehicle Energy Consumption Model Capable of Real-Time Updating

Vehicle energy consumption model, as a function of its operational environment, plays a significant role in real-time optimization of vehicle route and speed for a given pair of origin and destination with a desired arrival time for minimal energy consumption. In this article, a gray-box vehicle energy consumption model is developed based on a high-fidelity vehicle dynamic model with environmental influence based on the kriging modeling method, which includes rolling resistance, aerodynamics, gravity, and energy consumption of air conditioning and heater (HVAC), along with environmental conditions, such as temperature, wind speed, etc. The data-driven model, trained based on Gaussian process assumption, ensures the accuracy of the resulting model with a modeling error below 2.5%. The real-time model updating is based on recursive least-squares optimization using current driving data so that the model used for route and speed optimization represents the current vehicle status. The proposed gray-box model is validated in computer-in-the-loop simulations using SUMO and MATLAB with less than 2% error of energy consumption, which is a significant improvement over the vehicle dynamic model with up to 35% error in certain cases. A case study also indicates energy consumption reduction for vehicle route-speed optimization.

An efficient variable selection-based Kriging model method for the reliability analysis of slopes with spatially variable soils

The random finite element method (RFEM) is an efficient tool to demonstrate the spatial variability of soil properties during reliability analysis of slopes, but it requires remarkable model evaluations and computational efforts. In this paper, an efficient variable selection-based Kriging model method is proposed to approximate the finite element analysis model in reliability analysis of slopes. The variable selection technique successfully remedies the “curse of dimensionality” within Kriging model induced by the numerous random variables in random field discretization. The implementation procedure of this method for the reliability analysis of slopes is introduced in detail. Two typical examples of soil slopes, as well as a real complex slope are subsequently analyzed to illustrate the validity of the proposed method. The results show that the local loss of variability of random field due to the variable selection method has little impact on the safety factor of slopes. The proposed method can significantly reduce the number of finite element analysis and obtain accurate results in reliability analysis of slopes considering the spatial variability of soil properties.

Study on the degradation law of harmonic gear drive backlash with wear and assembly errors

Harmonic gear drive mechanism has been widely used in space mechanism, robot and other precision actuators. Based on the theory of probability and statistics, this paper establishes a mathematical model of harmonic gear transmission error considering the assembly and wear among components, and conducts sensitivity analysis and normalization of factors affecting harmonic gear transmission error model. Through Kriging model and Monte Carlo method, reliability degradation law of transmission accuracy is obtained by transmission error function and reliability analysis. Some suggestions are given in the design of harmonic gear drive in the future.

An envelope-function-based algorithm for time-dependent reliability analysis of structures with hybrid uncertainties

This paper proposes a novel envelope-function-based algorithm for time-dependent reliability analysis considering random and interval uncertainties. First, the envelope function of the bound of the limit-state function is approximated with assistance of the first-order reliability method, thereby transforming time-dependent reliability problem into a time-independent one. Here, the Kriging modeling method is utilized to construct the expansion point determination function for reducing the most probable point search, which significantly accelerates the expansion point determination in constructing the envelope function. After the envelope function is found, the time-dependent reliability in terms of random and interval uncertainties is efficiently calculated by a multivariate Gaussian integral. Finally, five case studies are used to demonstrate the effectiveness of the proposed algorithm. The results indicate it can provide accurate and efficient time-dependent reliability analysis for structures where the random and interval uncertainties coexist.

The intelligent design of reflow soldering based on an improved high-dimensional surrogate assisted optimization method

The Efficient Global Optimization (EGO) optimizes the temperature curve of the reflow soldering process by constructing the Kriging model. Compared with the traditional Surrogate Assisted Optimization (SAO), it can meet two different requirements: exploitation of optima and exploration of uncertain areas. However, due to the inversion of the covariance correlation matrix and the solving of Kriging-related parameters, the Kriging approximation process for high-dimensional problems is time consuming and even impossible to construct. For this reason, a High-Dimensional Kriging Modeling method through Linear Discriminant Analysis (HDKRG-LDA) method is proposed by considering the correlation parameters and the design variables of a Kriging model. It uses LDA to transform a high-dimensional correlation parameter vector in Kriging into low-dimensional one, which is used to reconstruct a new correlation function. In this study, the HDKRG-LDA method assisted EGO algorithm is applied to optimize the temperature curve of the reflow soldering process. In the optimization results of the reflow soldering, the method can effectively predict the temperature curve, and provide guidance for the production process of the reflow soldering.

An Efficient Kriging Modeling Method Based on Multidimensional Scaling for High-Dimensional Problems

Kriging-based modeling has been widely used in computationally intensive simulations. However, the Kriging modeling of high-dimensional problems not only takes more time, but also leads to the failure of model construction. To this end, a Kriging modeling method based on multidimensional scaling (KMDS) is presented to avoid the “dimensional disaster”. Under the condition of keeping the distance between the sample points before and after the dimensionality reduction unchanged, the KMDS method, which mainly calculates each element in the inner product matrix due to the mapping relationship between the distance matrix and the inner product matrix, completes the conversion of design data from high dimensional to low dimensional. For three benchmark functions with different dimensions and the aviation field problem of aircraft longitudinal flight control, the proposed method is compared with other dimensionality reduction methods. The KMDS method has better modeling efficiency while meeting certain accuracy requirements.

Time-varying reliability analysis of compressor blisk based on particle swash optimization extreme Kriging model

In order to effectively realize the dynamic time-varying reliability analysis of the radial deformation of the aero-engine low pressure compressor, based on Kriging model combined with particle swarm optimization (PSO) and extreme value idea, a particle swarm optimization extremum Kriging model (PSOEKM) method was proposed. Firstly, the analysis principle of PSOEKM method is expounded. Secondly, the modeling idea of PSOEKM method is discussed. Then, the implementation approach of time-varying reliability analysis based on PSOEKM is explored. Finally, taking the aero-engine low pressure compressor blisk as an example, the dynamic reliability analysis is carried out by using PSOEKM. The analysis results show that the reliability is 99.76% when the allowable value of radial deformation of the compressor blisk is 1.594×10-3 m. Compared with the traditional Kriging model and the extreme response surface model, the PSOEKM method has high analysis accuracy and calculation efficiency. The method presented in this paper provides a new research idea for time-varying reliability analysis of complex structures.

An analytical framework for local and global system kinematic reliability sensitivity of robotic manipulators

This paper develops a novel analytical framework for system kinematic reliability sensitivity analysis of robotic manipulators, which can provide the analytical results of local and global reliability sensitivity defined on the pose and position errors. First, uncertainty analysis of the pose error of the end-effector is accomplished by virtue of the second-order closed-form error propagation formula on motion groups. Then, the system kinematic reliability, namely the probability of the system kinematic error located within the prescribed safe boundary, is analytically calculated. Specifically, the non-central chi-square approximation and expectation propagation technique are employed to compute the system kinematic reliability defined on the position and pose errors, respectively. On this basis, the local and global system kinematic reliability sensitivity of robotic manipulators are analytically obtained. Finally, the effectiveness of the proposed method is validated by comparison with the Monte Carlo simulation and Kriging modeling method. The results indicate the proposed method has good efficacy and efficiency in system kinematic reliability sensitivity analysis for robotic manipulators.

An Improved High-Dimensional Kriging Surrogate Modeling Method through Principal Component Dimension Reduction

The Kriging surrogate model in complex simulation problems uses as few expensive objectives as possible to establish a global or local approximate interpolation. However, due to the inversion of the covariance correlation matrix and the solving of Kriging-related parameters, the Kriging approximation process for high-dimensional problems is time consuming and even impossible to construct. For this reason, a high-dimensional Kriging modeling method through principal component dimension reduction (HDKM-PCDR) is proposed by considering the correlation parameters and the design variables of a Kriging model. It uses PCDR to transform a high-dimensional correlation parameter vector in Kriging into low-dimensional one, which is used to reconstruct a new correlation function. In this way, time consumption of correlation parameter optimization and correlation function matrix construction in the Kriging modeling process is greatly reduced. Compared with the original Kriging method and the high-dimensional Kriging modeling method based on partial least squares, the proposed method can achieve faster modeling efficiency under the premise of meeting certain accuracy requirements.

A novel Nested Stochastic Kriging model for response noise quantification and reliability analysis

Surrogate models and adaptive methods can release the huge computational burden of structural reliability analysis. However, it is very difficult to guarantee the accuracy of uncertainty quantification, especially when noises are contained in samples, which may greatly reduce the confidence of reliability analysis. In this study, we propose a novel Nested Stochastic Kriging (NSK) model method with response noise parameters decoupled from other surrogate model parameters, which can significantly improve the accuracy of uncertainty quantification and modeling efficiency. The proposed NSK is for deterministic data with response noise, aiming at reliability analysis. Various types of uncertainty can be identified by the NSK, including the conditions of known and unknown, constant and variable variances. Moreover, a local sample verification method is established to improve the accuracy of uncertainty quantification. To further improve the accuracy of reliability analysis, an adaptive NSK framework is established with new samples added, and then reliability analysis can be conducted. Through several numerical examples, it can be seen that the NSK always provides the most accurate results with the fewest analysis calls.

Direct Calculation of Updating Parameters Based on Kriging Model for Bridge Finite Element Model Updating

To improve the calculation efficiency and to avoid the updating results being affected by optimization algorithms, a direct calculation of updating parameters method based on Kriging model for model updating is proposed in this paper. The proposed method first generates the data samples of updating parameters by using Latin hypercube Design (LHD). The corresponding structural response is calculated by initial finite element model. Then, the explicit function between updating parameters and structural responses is obtained based on kriging theory. Finally, the best design parameters are directly solved using the experimentally obtained structural response as the objective value based on the established Kriging model. Model updating of Caiyuanba Yangtze River Bridge is employed to test the proposed method. The results show that the model updating based on direct calculation method avoids the iterative calculation and improves the computational efficiency. The proposed method can provide the approximate accuracy level compared with the traditional Kriging model method.

High-Precision Kriging Modeling Method Based on Hybrid Sampling Criteria

Finding new valuable sampling points and making these points better distributed in the design space is the key to determining the approximate effect of Kriging. To this end, a high-precision Kriging modeling method based on hybrid sampling criteria (HKM-HS) is proposed to solve this problem. In the HKM-HS method, two infilling sampling strategies based on MSE (Mean Square Error) are optimized to obtain new candidate points. By maximizing MSE (MMSE) of Kriging model, it can generate the first candidate point that is likely to appear in a sparse area. To avoid the ill-conditioned correlation matrix caused by the too close distance between any two sampling points, the MC (MSE and Correlation function) criterion formed by combining the MSE and the correlation function through multiplication and division is minimized to generate the second candidate point. Furthermore, a new screening method is used to select the final expensive evaluation point from the two candidate points. Finally, the test results of sixteen benchmark functions and a house heating case show that the HKM-HS method can effectively enhance the modeling accuracy and stability of Kriging in contrast with other approximate modeling methods.

Multi-objective optimization of CFRP engine hood considering dynamic performance based on surrogate model

This paper uses a steel engine hood as a prototype to design a carbon fiber reinforced plastic (CFRP) engine hood, analyzes the dynamic performance of the engine hood and optimizes the dynamic stiffness index while considering lightweight of the structure. Taking the layer thickness of each ply orientation of the CFRP engine hood as the design variable, the optimal Latin hypercube design (Opt-LHD) method is applied for uniform sampling. Moreover, the Kriging model method and response surface model (RSM) method are used to construct the surrogate model of the total mass, equivalent dynamic stiffness of critical points, and the first-order modal frequency. Then the Non-dominated Sorting Genetic Algorithm NSGA-II is used for multi-objective optimization with the equivalent dynamic stiffness and mass of the CFRP engine hood as the optimization objective. Finally, the optimized CFRP engine hood was trial-produced and compared with the original steel engine hood. The results indicate that the optimized CFRP engine hood has better dynamic stiffness performance; furthermore, the lightweight effect reaches 24.4 %.