An improved interval model updating method via adaptive Kriging models

Interval model updating(IMU)methods have been widely used in uncertain model updating due to their low requirements for sample data.However,the surrogate model in IMU methods mostly adopts the one-time construction method.This makes the accuracy of the surrogate model highly dependent on the experience of users and affects the accuracy of IMU methods.Therefore,an improved IMU method via the adaptive Kriging models is proposed.This method transforms the objective function of the IMU problem into two deterministic global optimization problems about the upper bound and the interval diameter through universal grey numbers.These optimization problems are addressed through the adaptive Kriging models and the particle swarm optimization(PSO)method to quantify the uncertain parameters,and the IMU is accomplished.During the construction of these adaptive Kriging models,the sample space is gridded according to sensitivity information.Local sampling is then performed in key subspaces based on the maximum mean square error(MMSE)criterion.The interval division coefficient and random sampling coefficient are adaptively adjusted without human interference until the model meets accuracy requirements.The effectiveness of the proposed method is demonstrated by a numerical example of a three-degree-of-freedom mass-spring system and an experimental example of a butted cylindrical shell.The results show that the updated results of the interval model are in good agreement with the experimental results.

Tunnel face reliability analysis using active learning Kriging model——Case of a two-layer soils

This paper is devoted to the probabilistic stability analysis of a tunnel face excavated in a two-layer soil. The interface of the soil layers is assumed to be positioned above the tunnel roof. In the framework of limit analysis, a rotational failure mechanism is adopted to describe the face failure considering different shear strength parameters in the two layers. The surrogate Kriging model is introduced to replace the actual performance function to perform a Monte Carlo simulation. An active learning function is used to train the Kriging model which can ensure an efficient tunnel face failure probability prediction without loss of accuracy. The deterministic stability analysis is given to validate the proposed tunnel face failure model. Subsequently, the number of initial sampling points, the correlation coefficient, the distribution type and the coefficient of variability of random variables are discussed to show their influences on the failure probability. The proposed approach is an advisable alternative for the tunnel face stability assessment and can provide guidance for tunnel design.

Dynamic characteristic analysis of whole machine tools based on Kriging model

In order to study the variation of machine tools’dynamic characteristics in the manufacturing space,a Kriging approximate model is proposed.Finite element method(FEM)is employed on the platform of ANSYS to establish finite element(FE)model with the dynamic characteristic of combined interface for a milling machine,which is newly designed for producing aero engine blades by a certain enterprise group in China.The stiffness and damping of combined interfaces are adjusted by using adaptive simulated annealing algorithm with the optimizing software of iSIGHT in the process of FE model update according to experimental modal analysis(EMA)results.The Kriging approximate model is established according to the finite element analysis results utilizing orthogonal design samples by taking into account of the range of configuration parameters.On the basis of the Kriging approximate model,the response surfaces between key response parameter and configuration parameters are obtained.The results indicate that configuration parameters have great effects on dynamic characteristics of machine tools,and the Kriging approximate model is an effective and rapid method for estimating dynamic characteristics of machine tools in the manufacturing space.

Moment-independence global sensitivity analysis for the system with fuzzy failure state and its Kriging method

For the system with the fuzzy failure state, the effects of the input random variables and the fuzzy failure state on the fuzzy probability of failure for the structural system are studied, and the moment-independence global sensitivity analysis（GSA） model is proposed to quantitatively measure these effects. According to the fuzzy random theory, the fuzzy failure state is transformed into an equivalent new random variable for the system, and the complementary function of the membership function of the fuzzy failure state is defined as the cumulative distribution function（CDF） of the new random variable. After introducing the new random variable, the equivalent performance function of the original problem is built. The difference between the unconditional fuzzy probability of failure and conditional fuzzy probability of failure is defined as the moment-independent GSA index. In order to solve the proposed GSA index efficiently, the Kriging-based algorithm is developed to estimate the defined moment-independence GSA index. Two engineering examples are employed to verify the feasibility and rationality of the presented GSA model, and the advantages of the developed Kriging method are also illustrated.

Global optimization for ducted coaxial-rotors aircraft based on Kriging model and improved particle swarm optimization algorithm

To improve the operational efficiency of global optimization in engineering, Kriging model was established to simplify the mathematical model for calculations. Ducted coaxial-rotors aircraft was taken as an example and Fluent software was applied to the virtual prototype simulations. Through simulation sample points, the total lift of the ducted coaxial-rotors aircraft was obtained. The Kriging model was then constructed, and the function was fitted. Improved particle swarm optimization(PSO) was also utilized for the global optimization of the Kriging model of the ducted coaxial-rotors aircraft for the determination of optimized global coordinates. Finally, the optimized results were simulated by Fluent. The results show that the Kriging model and the improved PSO algorithm significantly improve the lift performance of ducted coaxial-rotors aircraft and computer operational efficiency.

Probabilistic Load Flow Calculation of Power System Integrated with Wind Farm Based on Kriging Model

Because of the randomness and uncertainty,integration of large-scale wind farms in a power system will exert significant influences on the distribution of power flow.This paper uses polynomial normal transformation method to deal with non-normal random variable correlation,and solves probabilistic load flow based on Kriging method.This method is a kind of smallest unbiased variance estimation method which estimates unknown information via employing a point within the confidence scope of weighted linear combination.Compared with traditional approaches which need a greater number of calculation times,long simulation time,and large memory space,Kriging method can rapidly estimate node state variables and branch current power distribution situation.As one of the generator nodes in the western Yunnan power grid,a certain wind farm is chosen for empirical analysis.Results are used to compare with those by Monte Carlo-based accurate solution,which proves the validity and veracity of the model in wind farm power modeling as output of the actual turbine through PSD-BPA.

Identification of material parameters from punch stretch test

To accurately describe the mechanical properties of aluminium alloy sheet during deformation, an inverse identification was presented to deal with material parameters from the popular punch stretch test. In the identification procedure, the optimization strategy combines finite element method （FEM）, Latin hypercube sampling （LHS）, Kriging model and multi-island genetic algorithm （MIGA）. The proposed approach is used on material parameter identification of aluminium alloy sheet 2D12. The anisotropic yield criterion Hill’90 is discussed. The results show that the Hill’90 anisotropic yield criterion with identified anisotropic material parameters has a good potential in describing the anisotropic behaviours. It provides a way to obtain the material parameters for FE simulations of sheet metal forming.

EFFICIENT METHOD FOR MULTIDISCIPLINARY DESIGN OPTIMIZATION BY CONSIDERING UNCERTAINTY

A new reliability-based multidisciplinary design optimization （RBMDO） framework is proposed by combining the single-loop-based reliability analysis （SLBRA） method with multidisciplinary feasible （MDF） method. The Kriging approximate model with updating is introduced to reduce the computational cost of MDF caused by the complex structure. The computational efficiency is remarkably improved as the lack of iterative process during reliability analysis. Special attention is paid to a turbine blade design optimization by adopting the proposed method. Results show that the method is much more efficient than the commonly used double-loop based RBMDO method. It is feasible and efficient to apply the method to the engineering design.

A study of groundwater irrigation water quality in south-central Bangladesh: a geo-statistical model approach using GIS and multivariate statistics

Southern Bangladesh＇s irrigation and drinking water is threatened by saline intrusion. This study aimed to establish an irrigation water quality index （IWQI） using a geostatistical model and multivariate indices in Gopalganj district, south-central Bangladesh. Groundwater samples were taken randomly （different depths） in two seasons （wet-monsoon and dry-monsoon）. Hydrochemical analysis revealed groundwater in this area was neutral to slightly alkaline and dominating cations were Na^＋, Mg^2＋, and Ca^2＋ along with major anions Cl^- and HCO3^-. Principal component analysis and Gibbs plot helped explain possible geochemical processes in the aquifer. The irrigation water evaluation indices showed： electrical conductivity （EC） 〉750 μS/cm, moderate to extreme saline; sodium adsorption ratio （SAR）, excellent to doubtful; total hardness （TH）, moderate to very hard; residual sodium bicarbonate, safe to marginal; Kelly＇s ratio 〉1; soluble sodium percentage （SSP）, fair to poor; magnesium adsorption ratio, harmful for soil; and IWQI, moderate to suitable. In addition, the best fitted semivariogram for IWQI, EC, SAR, SSP, and TH confirmed that most parameters had strong spatial dependence and others had moderate to weak spatial dependence. This variation might be due to the different origin/sources of major contributing ions along with the influence of variable river flow and small anthropogenic contributions. Furthermore, the spatial distribution maps for IWQI, EC, SSP, and TH during both seasons confirmed the influence of salinity from the sea; low-flow in the major river system was the driving factor of overall groundwater quality in the study area. These findings may contribute to management of irrigation and/or drinking water in regions with similar groundwater problems.

Multi-objectives nonlinear structure optimization for actuator in trajectory correction fuze subject to high impact loadings

This paper presents an actuator used for the trajectory correction fuze,which is subject to high impact loadings during launch.A simulation method is carried out to obtain the peak-peak stress value of each component,from which the ball bearings are possible failures according to the results.Subsequently,three schemes against impact loadings,full-element deep groove ball bearing and integrated raceway,needle roller thrust bearing assembly,and gaskets are utilized for redesigning the actuator to effectively reduce the bearings’stress.However,multi-objectives optimization still needs to be conducted for the gaskets to decrease the stress value further to the yield stress.Four gasket’s structure parameters and three bearings’peak-peak stress are served as the four optimization variables and three objectives,respectively.Optimized Latin hypercube design is used for generating sample points,and Kriging model selected according to estimation result can establish the relationship between the variables and objectives,representing the simulation which is time-consuming.Accordingly,two optimization algorithms work out the Pareto solutions,from which the best solutions are selected,and verified by the simulation to determine the gaskets optimized structure parameters.It can be concluded that the simulation and optimization method based on these components is effective and efficient.

Rotary-scaling fine-tuning (RSFT) method for optimizing railway wheel profiles and its application to a locomotive

The existing multi-objective wheel profile optimization methods mainly consist of three sub-modules:(1)wheel profile generation,(2)multi-body dynamics simulation,and(3)an optimization algorithm.For the first module,a comparably conservative rotary-scaling finetuning(RSFT)method,which introduces two design variables and an empirical formula,is proposed to fine-tune the traditional wheel profiles for improving their engineering applicability.For the second module,for the TRAXX locomotives serving on the Blankenburg–Rubeland line,an optimization function representing the relationship between the wheel profile and the wheel–rail wear number is established based on Kriging surrogate model(KSM).For the third module,a method combining the regression capability of KSM with the iterative computing power of particle swarm optimization(PSO)is proposed to quickly and reliably implement the task of optimizing wheel profiles.Finally,with the RSFT–KSM–PSO method,we propose two wear-resistant wheel profiles for the TRAXX locomotives serving on the Blankenburg–Rubeland line,namely S1002-S and S1002-M.The S1002-S profile minimizes the total wear number by 30%,while the S1002-M profile makes the wear distribution more uniform through a proper sacrifice of the tread wear number,and the total wear number is reduced by 21%.The quasi-static and hunting stability tests further demonstrate that the profile designed by the RSFT–KSM–PSO method is promising for practical engineering applications.

Adaptive sampling approach based on Jensen-Shannon divergence for efficient reliability analysis

Extensive studies have been carried out for reliability studies on the basis of the surrogate model,which has the advantage of guaranteeing evaluation accuracy while minimizing the need of calling the real yet complicated performance function.Here,one novel adaptive sampling approach is developed for efficiently estimating the failure probability.First,one innovative active learning function integrating with Jensen-Shannon divergence(JSD)is derived to update the Kriging model by selecting the most suitable sampling point.For improving the efficient property,one trust-region method receives the development for reducing computational burden about the evaluation of active learning function without compromising the accuracy.Furthermore,a termination criterion based on uncertainty function is introduced to achieve better robustness in different situations of failure probability.The developed approach shows two main merits:the newly selected sampling points approach to the area of limit state boundary,and these sampling points have large discreteness.Finally,three case analyses receive the conduction for demonstrating the developed approach s feasibility and performance.Compared with Monte Carlo simulation or other active learning functions,the developed approach has advantages in terms of efficiency,convergence,and accurate when dealing with complex problems.

Bi-objective path optimization of flapping airfoils based on a surrogate model

A bi-objective optimization problem for flapping airfoils is solved to maximize the time-averaged thrust coefficient and the propulsive efficiency. Design variables include the plunging amplitude, the pitching amplitude and the phase shift angle. A well defined Kriging model is used to substitute the time-consuming high fidelity model, and a multi-objective genetic algorithm is employed as the search algorithm. The optimization results show that the propulsive efficiency can be improved by reducing the plunging amplitude and the phase shift angle in a proper way. The results of global sensitivity analysis using the Sobol’s method show that both of the time-averaged thrust coefficient and the propulsive efficiency are most sensitive to the plunging amplitude, and second most sensitive to the pitching amplitude. It is also observed that the phase shift angle has an un-negligible influence on the propulsive efficiency, and has little effect on the time-averaged thrust coefficient.

Performance Comparison of Two Meta-Model for the Application to Finite Element Model Updating of Structures

To investigate the application of meta-model for finite element( FE) model updating of structures,the performance of two popular meta-model,i. e.,Kriging model and response surface model( RSM),were compared in detail. Firstly,above two kinds of meta-model were introduced briefly. Secondly,some key issues of the application of meta-model to FE model updating of structures were proposed and discussed,and then some advices were presented in order to select a reasonable meta-model for the purpose of updating the FE model of structures. Finally,the procedure of FE model updating based on meta-model was implemented by updating the FE model of a truss bridge model with the measured modal parameters. The results showed that the Kriging model was more proper for FE model updating of complex structures.

Contrastive analysis and crashworthiness optimization of two composite thin-walled structures

For the safety protection of passengers when train crashes occur, special structures are crucially needed as a kind of indispensable energy absorbing device. With the help of the structures, crash kinetic-energy can be completely absorbed or dissipated for the aim of safety. Two composite structures(circumscribed circle structure and inscribed circle structure) were constructed. In addition, comparison and optimization of the crashworthy characteristic of the two structures were carried out based on the method of explicit finite element analysis(FEA) and Kriging surrogate model. According to the result of Kriging surrogate model, conclusions can be safely drawn that the specific energy absorption(SEA) and ratio of specific energy absorption to initial peak force(REAF) of circumscribed circle structure are lager than those of inscribed circle structure under the same design parameters. In other words, circumscribed circle structure has better performances with higher energy-absorbing ability and lower initial peak force. Besides, error analysis was adopted and the result of which indicates that the Kriging surrogate model has high nonlinear fitting precision. What is more, the SEA and REAF optimum values of the two structures have been obtained through analysis, and the crushing results have been illustrated when the two structures reach optimum SEA and REAF.

Variable Importance Measure System Based on Advanced Random Forest

The variable importance measure(VIM)can be implemented to rank or select important variables,which can effectively reduce the variable dimension and shorten the computational time.Random forest(RF)is an ensemble learning method by constructing multiple decision trees.In order to improve the prediction accuracy of random forest,advanced random forest is presented by using Kriging models as the models of leaf nodes in all the decision trees.Referring to the Mean Decrease Accuracy(MDA)index based on Out-of-Bag(OOB)data,the single variable,group variables and correlated variables importance measures are proposed to establish a complete VIM system on the basis of advanced random forest.The link of MDA and variance-based sensitivity total index is explored,and then the corresponding relationship of proposed VIM indices and variance-based global sensitivity indices are constructed,which gives a novel way to solve variance-based global sensitivity.Finally,several numerical and engineering examples are given to verify the effectiveness of proposed VIM system and the validity of the established relationship.

Uncertainty Quantification of Store-Separation Simulation Due to Ejector Modeling using a Monte Carlo Approach with Kriging Model

Precise calculation of the trajectory of store separation is critical in assess-ing whether the store can be released safely.Store ejection is the initial stage of the releasing process and any uncertainty introduced at this stage will propagate through the whole trajectory.In this work,the impact of the uncertainties in ejector modeling on the simulation of a generic store separation is investigated by using a Monte-Carlo-based approach.To reduce the extremely large computation cost resulted from the direct use CFD in Monte Carlo simulation,the CFD solutions are represented by a time-dependent Kriging model,which is constructed at each time step by using the samples from the URANS simulations.The stochastic outputs,including the distri-bution of probability density function,expected value and 95%confidence interval of store separation trajectory,are obtained by the Monte Carlo simulations.The sensitiv-ity analysis is also carried out by using the Monte-Carlo-based method to determine the most significant variables in ejector modeling,which affect the output uncertainty.Our results show that ejector modeling is one of the main uncertainty sources of store separation simulation and the approximation in ejector modeling can cause a signifi-cant deviation,especially in the angular displacement.

Kriging Model Averaging Based on Leave-One-Out Cross-Validation Method

In recent years,Kriging model has gained wide popularity in various fields such as space geology,econometrics,and computer experiments.As a result,research on this model has proliferated.In this paper,the authors propose a model averaging estimation based on the best linear unbiased prediction of Kriging model and the leave-one-out cross-validation method,with consideration for the model uncertainty.The authors present a weight selection criterion for the model averaging estimation and provide two theoretical justifications for the proposed method.First,the estimated weight based on the proposed criterion is asymptotically optimal in achieving the lowest possible prediction risk.Second,the proposed method asymptotically assigns all weights to the correctly specified models when the candidate model set includes these models.The effectiveness of the proposed method is verified through numerical analyses.

Optimal Model Average Prediction in Orthogonal Kriging Models

The main objective of this paper is to consider model averaging methods for kriging models.This paper proposes a Mallows model averaging procedure for the orthogonal kriging model and demonstrate the asymptotic optimality of the model averaging estimators in terms of mean square error.Simulation studies are conducted to evaluate the performance of the proposed method and compare it with the competitors to demonstrate its superiority.The authors also analyse a real dataset for an illustration.

Evaluation of Different Topographic Parameters Extracted from the Digital Elevation Measurements with the Use of Geostatistical Interpolation Methods

Gradient slope, aspect slope, profiling and contourlines are important topographic parameters that can be derived from digital elevation data obtained from different sources with exploitation of different interpolation techniques. Geostatistical interpolation methods such as ordinary kriging models constitute reliable alternatives to deterministic approaches in creation of continuous surface models from discrete elevation data. This research aimed at extraction, analysis, and evaluation of different terrain parameters elevation measurements with the use of different ordinary kriging models including the linear model, the circular model, the spherical model, the exponential models, and the Gaussian model. Different ordinary kriging models under ESRI ArcView 3.3 package along with its 3D analyst and Spatial analysis extensions have been exploited in extraction of gradient slope maps, aspect slope maps, and hillshade maps in addition to contourline maps from a sample of elevation data. Visual analysis of the gradient slope maps shows great similarities between the slope maps from the linear, circular, spherical, and exponential OK models, however, that from OK Gaussian models look very different as different sizes and arrangements of the colour patches, referring to different tones and different textures where smooth tones and smooth textures dominate the gradient slope map from the OK Gaussian model. Thus, gradient slope degradation and smoothing are considerably high in the gradient slope map from Gaussian model compared to the slope maps from the other four OK models. Also, the mean slope in the Gaussian model records the lowest value with the lowest value of the standard deviation of slopes in the same map reflecting less structured and highly smoothed gradient slope map compared to the slope maps from the other OK models. Thus, similar sizes of the colour patches and similar tones and similar texture dominate the different aspect slope maps. This is not the case in Figure 2(e) which depicts the aspect slope map extracted with the use of the Gaussian OK model where the smooth colour patches, smooth tones and smooth textures can be observed. Also, the Aspect map, hillshade map and the contourline map from Gaussian OK model are visually and statistically different from their corresponding maps created with the other four OK models. Finally, analysis of extracted two groups of profiles shows that the profiles extracted with the use of linear, circular, spherical, and exponential OK models run close and show highly corrugated and varied terrain. This is different from the profiles with the use of the Gaussian model which are less corrugated and tend to smooth and approximate different parts of the terrains.