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.

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.

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.

Multi-infill strategy for kriging models used in variable fidelity optimization

In this paper, a computationally efficient optimization method for aerodynamic design has been developed. The low-fidelity model and the multi-infill strategy are utilized in this approach.Low-fidelity data is employed to provide a good global trend for model prediction, and multiple sample points chosen by different infill criteria in each updating cycle are used to enhance the exploitation and exploration ability of the optimization approach. Take the advantages of lowfidelity model and the multi-infill strategy, and no initial sample for the high-fidelity model is needed. This approach is applied to an airfoil design case and a high-dimensional wing design case.It saves a large number of high-fidelity function evaluations for initial model construction. What＇s more, faster reduction of an aerodynamic function is achieved, when compared to ordinary kriging using the multi-infill strategy and variable-fidelity model using single infill criterion. The results indicate that the developed approach has a promising application to efficient aerodynamic design when high-fidelity analyses are involved.

Resonance System Reliability and Sensitivity Analysis Method for Axially FGM Pipes Conveying Fluid with Adaptive Kriging Model

This paper aims to solve the resonance failure probability and develop an effective method to estimate the effects of variables and failure modes on failure probability of axially functionally graded material(FGM)pipe conveying fluid.Correspondingly,the natural frequency of axially FGM pipes conveying fluid is calculated using the differential quadrature method(DQM).A variable sensitivity analysis(VSA)is introduced to measure the effect of each random variable,and a mode sensitivity analysis(MSA)is introduced to acquire the importance ranking of failure modes.Then,an active learning Kriging(ALK)method is established to calculate the resonance failure probability and sensitivity indices,which greatly improves the application of resonance reliability analysis for pipelines in engineering practice.Based on the resonance reliability analysis method,the effects of fluid velocity,volume fraction and fluid density of axially FGM pipe conveying fluid on resonance reliability are analyzed.The results demonstrate that the proposed method has great performance in the anti-resonance analysis of pipes conveying fluid.

Method of a New Iteration Scheme Combined with Kriging Model for Structural Reliability Evaluation

The first order reliability method (FORM) is widely adopted for structural reliability evaluation due to its numerical efficiency. Concerning the issue of FORM often failing to converge when the limit state function (LSF) behaves high nonlinearity, a new iteration scheme called "rotated gradient algorithm (RGA)" is proposed and combined with Kriging model to evaluate the reliability of implicit performance function. In this paper, the Kriging model is applied to approximate the real LSF first. Then the scheme of RGA, constructed in terms of gradient information of two adjacent design points obtained during the process of calculation, is used to calculate the reliability index. Numerical examples show the validity in convergence and accuracy of the proposed methodfor arbitrary nonlinear performance function.

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.

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.

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.

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.

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.

Multi-objective structural optimization and degradation model of magnesium alloy ureteral stent

Background:Mg alloys have attractive properties,including biocompatibility,biodegradability,and ideal mechanical properties.Moreover,Mg alloys are regarded as one of the promising candidates for manufacturing ureteral stents.This study proposed a multi-objective optimization method based on the Kriging surrogate model,NSGA-III,and finite element analysis to improve the degradation performance of Mg alloy ureteral stents.Methods:The finite element model for the degradation of Mg alloy ureteral stents has been established to compare the degradation performance of the stents under different parameters.Latin hypercube sampling was adopted to generate train sample points in the design space.Meanwhile,the Kriging surrogate model was constructed between strut parameters and stent degradation behavior.The NSGA-III was utilized to determine the optimal solution in the global design space.Results:The optimized stent achieved 5.52degradation uniformity(M),10degradation time(DT),and 4work time(FT).The errors between the Kriging surrogate model and the finite element calculation results were less than 6%.Conclusion:The optimized stent achieved better degradation performance.The degradation behavior of stents was dependent on the design parameters.The multi-objective optimization method based on the Kriging surrogate model and finite element analysis was effective in stent design optimization 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.

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.

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.

Multi-Objective Optimization for Structure Crashworthiness Based on Kriging Surrogate Model and Simulated Annealing Algorithm

Multi-objective optimization of crashworthiness in automobile front-end structure was performed,and finite element model(FEM)was validated by experimental results to ensure that FEM can predict the response value with sufficient accuracy.Seven design variables and four crashworthiness indicators were defined.Through orthogonal design method,18 FEMs were established,and the response values of crashworthiness indicators were extracted.By using the variable-response specimen matrix,Kriging surrogate model(KSM)was constructed to replace FEM to refect the function correlation between variables and responses.The accuracy of KSM was also validated.Finally,the simulated annealing optimization algorithm was implemented in KSM to seek optimal and reliable solutions.Based on the optimal results and comparison analysis,the 9096-th iteration point was the optimal solution.Although the intrusion of firewall and the mass of optimal structure increased slightly,the vehicle acceleration of the optimal solution decreased by 6.9%,which fectively reduced the risk of occupant injury.

Reliability and reliability sensitivity analysis of structure by combining adaptive linked importance sampling and Kriging reliability method

The application of reliability analysis and reliability sensitivity analysis methods to complicated structures faces two main challenges:small failure probability(typical less than 10-5)and time-demanding mechanical models.This paper proposes an improved active learning surrogate model method,which combines the advantages of the classical Active Kriging–Monte Carlo Simulation(AK-MCS)procedure and the Adaptive Linked Importance Sampling(ALIS)procedure.The proposed procedure can,on the one hand,adaptively produce a series of intermediate sampling density approaching the quasi-optimal Importance Sampling(IS)density,on the other hand,adaptively generate a set of intermediate surrogate models approaching the true failure surface of the rare failure event.Then,the small failure probability and the corresponding reliability sensitivity indices are efficiently estimated by their IS estimators based on the quasi-optimal IS density and the surrogate models.Compared with the classical AK-MCS and Active Kriging–Importance Sampling(AK-IS)procedure,the proposed method neither need to build very large sample pool even when the failure probability is extremely small,nor need to estimate the Most Probable Points(MPPs),thus it is computationally more efficient and more applicable especially for problems with multiple MPPs.The effectiveness and engineering applicability of the proposed method are demonstrated by one numerical test example and two engineering applications.

Rotorcraft flight endurance estimation based on a new battery discharge model

To avoid the numerical complexities of the battery discharge law of electric-powered rotorcrafts,this study uses the Kriging method to model the discharge characteristics of Li-Po batteries under standard conditions.A linear current compensation term and an ambient temperature compensation term based on radial basis functions are then applied to the trained Kriging model,leading to the complete discharged capacity-terminal voltage model.Using an orthogonal experimental design and a sequential method,the coefficients of the current and ambient temperature compensation terms are determined through robust optimization.An endurance calculation model for electric-powered rotorcrafts is then established,based on the battery discharge model,through numerical integration.Laboratory tests show that the maximum relative error of the proposed discharged capacity-terminal voltage model at detection points is 0.0086,and that of the rotorcraft endurance calculation model is 0.0195,thus verifying their accuracy.A flight test further demonstrates the applicability of the proposed endurance model to general electric-powered rotorcrafts.

Efficient adaptive Kriging for system reliability analysis with multiple failure modes under random and interval hybrid uncertainty

In the field of the system reliability analysis with multiple failure modes,the advances mainly involve only random uncertainty.The upper bound of the system failure probability with multiple failure modes is usually employed to quantify the safety level under Random and Interval Hybrid Uncertainty(RI-HU).At present,there is a lack of an efficient and accurate method for estimating the upper bound of the system failure probability.This paper proposed an efficient Kriging model based on numerical simulation algorithm to solve the system reliability analysis under RI-HU.This method proposes a system learning function to train the system Kriging models of the system limit state surface.The convergent Kriging models are used to replace the limit state functions of the system multi-mode for identifying the state of the random sample.The proposed system learning function can adaptively select the failure mode contributing most to the system failure probability from the system and update its Kriging model.Thus,the efficiency of the Kriging training process can be improved by avoiding updating the Kriging models contributing less to estimating the system failure probability.The presented examples illustrate the superiority of the proposed method.

基于自适应近似法的复合材料先进格栅加筋结构优化设计(英文)

An adaptive approximation-based optimization （AABO） procedure is developed for the optimum design of a composite advanced grid-stiffened （AGS） cylinder subject to post-buckling. The design taking account of post-buckling under ultimate load will be able to promote the structural efficiency compared to the conventional design in which only the linear buckling is allowed. The beam-shell offsets technique is utilized for modeling the stiffener-skin connection, and the Newton-Raphson method is employed for the post-buckling analysis. A few structural analysis efforts are carried out for establishing the Kriging model of the collapse load of the AGS cylinder for optimization to significantly increase the optimization efficiency. The multi-island genetic algorithm （MIGA） is utilized for global optimum search. An adaptive approximation framework is proposed to resolve the computational burden caused by the large domain of design variables, and it is demonstrated that much less computational expense than that of the traditional approximation-based optimization method can be achieved. The utility of making use of commercial optimization package iSIGHT in conjunction with the finite element （FE） code MSC.MARC to develop the preliminary design tool of the composite AGS cylinder is evaluated as well.