Kriging Surrogate Research Articles

Uncertainty quantization and reliability analysis for rotor/stator rub-impact using advanced Kriging surrogate model

This study provides new insights into the uncertainty quantification and reliability analysis of a flexible rotor system undergoing rub-impact. A probabilistic nonlinear formulation is proposed by considering the parameter uncertainties of different random distribution characteristics. The harmonic balance and alternating frequency/time (HB-AFT) scheme are employed to analyze the nonlinear behaviors of the system. To avoid the computationally intensive nonlinear calculations, the efficient Kriging surrogate strategy is adopted for the propagation of uncertainties in rotor dynamics, which enables rapid estimation of the statistics of nonlinear responses. Then, the probability density function (PDF) is evaluated for reliability analysis to determine the conditions for the occurrence of the rub-impact fault. Numerical results are compared with experimental results and those obtained by Monte Carlo simulation (MCS) to verify the effectiveness of the new modeling. Effects of stochastic parameters on vibrational behaviors and reliability of the rub-impact rotor system are finally performed, which demonstrates the proposed method is high-efficiency and high-precision for predicting the high-nonlinearity between output response and input variables. This contribution will further enrich the theory and application of probabilistic statistical analysis and reliability evaluation for complex rotating machinery.

Bayesian inversion using adaptive Polynomial Chaos Kriging within Subset Simulation

In this paper, we propose a Bayesian inversion approach combining adaptive Polynomial Chaos Kriging (PCK) surrogate models and a rare event estimation method called Subset Simulation (SuS). It is based on the recently introduced Bayesian Updating with Structural reliability (BUS) framework that enables to reformulate the classical Bayesian inference into a rare event estimation problem. In this context, the SuS method aims at drawing samples from the posterior distribution as well as estimating the model evidence, which is usually computationally intractable when considering classical MCMC approaches. The proposed approach involves the construction of a PCK surrogate model which provides both global and local approximations of the likelihood function, through the combination of Polynomial Chaos and Kriging surrogates. Furthermore, we propose an adaptive scheme for enriching the PCK surrogate throughout the SuS sampling procedure, in order to improve its accuracy near informative regions. The applicability and the efficiency of the proposed approach are assessed through several cases studies with increasing complexity. Results highlight that the proposed approach enables to accurately approximating posteriors with a limited amount of full model calls, even in the case of multi-modal posteriors, which are usually difficult to sample when using classical MCMC algorithms.

Bayesian Model Updating Based on Kriging Surrogate Model and Simulated Annealing Algorithm

Aiming at the problem of difficulty in selecting the proposal distribution and low computational efficiency in the traditional Markov chain Monte Carlo algorithm, a Bayesian model updating method using surrogate model technology and simulated annealing algorithm is proposed. Firstly, the Kriging surrogate model is used to mine the implicit relationship between the structural parameters to be updated and the corresponding dynamic responses, and the Kriging model that meets the accuracy requirement is used to replace the complex finite element model to participate in the iterative calculation to improve the model updating efficiency. Then, the simulated annealing algorithm is introduced to reorganize the Markov chains from different proposal distributions to obtain high-quality posterior samples, which are used to estimate the parameters posterior distributions. Finally, a space truss structure is used to verify the effectiveness of the proposed method.

Optimized biodiesel production from C. innophyllum bio-oil using Kriging and Ann predicitive models

This work aimed at optimizing the two-stage transesterification efficiency of the production of C. inophyllum biodiesel using artificial neural network and Kriging predictive models. Response surface methodology was used to develop the central rotatable composite design of 27 trial experimental runs with variations in the input process parameters like methanol to oil molar ratio, potassium hydroxide catalyst loading, and reaction time. A multi-layered non-linear regressive artificial neural network model with feed-forward propagation and a numerical surrogate Kriging model was used to predict the C. inophyllum biodiesel yield. The efficacy of the developed model was verified using analysis of variance by com-paring its coefficient of determination and the mean relative percentage deviation values. The optimized C. inophyllum biodiesel as 98.1% is derived with 0.94 v/v of methanol to oil molar ratio, 0.98 wt.% of potassium hydroxide catalyst loading, and 80 minutes reaction time with 70?C constant reaction temperature as predicted by Kriging model. The optimized parameters were also verified experimentally.

Framework for Embedding Process Simulator in GAMS via Kriging Surrogate Model Applied to C3MR Natural Gas Liquefaction Optimization

Rigorous black-box simulations are useful to describe complex systems. However, it cannot be directly integrated into mathematical programming models in some algebraic modeling environments because of the lack of symbolic formulation. In the present paper, a framework is proposed to embed the Aspen HYSYS process simulator in GAMS using kriging surrogate models to replace the simulator-dependent, black-box objective, and constraints functions. The approach is applied to the energy-efficient C3MR natural gas liquefaction process simulation optimization using multi-start nonlinear programming and the local solver CONOPT in GAMS. Results were compared with two other meta-heuristic approaches, Particle Swarm Optimization (PSO) and Genetic Algorithm (GA), and with the literature. In a small simulation evaluation budget of 20 times the number of decision variables, the proposed optimization approach resulted in 0.2538 kW of compression work per kg of natural gas and surpassed those of the PSO and GA and the previous literature from 2.45 to 15.3 %.

Stochastic Crashworthiness Optimization Accounting for Simulation Noise

Abstract Finite element-based crashworthiness optimization is extensively used to improve the safety of motor vehicles. However, the responses of crash simulations are characterized by a high level of numerical noise, which can hamper the blind use of surrogate-based design optimization methods. It is therefore essential to account for the noise-induced uncertainty when performing optimization. For this purpose, a surrogate, referred to as Non-Deterministic Kriging (NDK), can be used. It models the noise as a non-stationary stochastic process, which is added to a traditional deterministic kriging surrogate. Based on the NDK surrogate, this study proposes an optimization algorithm tailored to account for both epistemic uncertainty, due to the lack of data, and irreducible aleatory uncertainty, due to the simulation noise. The variances are included within an extension of the well-known expected improvement infill criterion referred to as Modified Augmented Expected Improvement (MAEI). Because the proposed optimization scheme requires an estimate of the aleatory variance, it is approximated through a regression kriging, which is iteratively refined. The proposed algorithm is tested on a set of analytical functions and applied to the optimization of an Occupant Restraint System (ORS) during a crash.

Kriging and dimension reduction techniques for delamination detection in composites using electrical resistance tomography

The electrical resistance tomography (ERT) is a non-destructive evaluation (NDE) technique that uses the inherent conductivity variations of composites to characterize internal damage. This article presents a study on the use of Kriging surrogate models for inverse identification of delamination in composites using ERT. Kriging surrogates are used to predict the change in electrical response of composites with delamination and matrix cracking damage. The results show that Kriging surrogates can successfully identify delamination even in the presence of matrix cracking. The technique demonstrated here that combines model order reduction with Kriging surrogates provides a computationally efficient and accurate method for damage identification using ERT.

Optimal Design of IPMSM for EV Using Subdivided Kriging Multi-Objective Optimization

In this paper, subdivided kriging multi-objective optimization (SKMOO) is proposed for the optimal design of interior permanent magnet synchronous motor (IPMSM). The SKMOO with surrogate kriging model can obtain a uniform and accurate pareto front set with a reduced computation cost compared to conventional algorithms which directly adds the solution in the objective function area. In other words, the proposed algorithm uses a kriging surrogate model, so it is possible to know which design variables have the value of the objective function on the blank space. Therefore, the solution can be added directly in the objective function area. In the SKMOO algorithm, a non-dominated sorting method is used to find the pareto front set and the fill blank method is applied to prevent premature convergence. In addition, the subdivided kriging grid is proposed to make a well-distributed and more precise pareto front set. Superior performance of the SKMOO is confirmed by compared conventional multi objective optimization (MOO) algorithms with test functions and are applied to the optimal design of IPMSM for electric vehicle.

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.

Bayesian Optimization of a Low-Boom Supersonic Wing Planform

A surrogate-assisted methodology to accelerate the design space exploration process and multi-objective optimization of a notional low-drag, low-boom supersonic transport planform using Euler computational fluid dynamics with an empirical parasite drag addition and an augmented Burgers equation solver is demonstrated. The use of Kriging-based surrogate models, coupled with expected hypervolume improvement in a Kriging believer framework, allows efficient global optimization of the selected wing design parameters. The use of effective nondominated sampling is used to further improve the solutions found. The effects of parameterizing the planform of a baseline model (NASA’s 69 deg wing–body) with 6- and 11-variables are explored. Genetic and local optimizers are used to search the Kriging surrogates. The results show the tradeoff between planform shapes that efficiently use compression lift to increase the lift-to-drag ratio and a smooth undertrack pressure signature to reduce ground-level noise. Furthermore, correctly positioned wingtips may also be used to smooth the undertrack signature, reducing the sonic boom.

Uncertainty Quantification of Aero-Thermal Performance of a Blade Endwall Considering Slot Geometry Deviation and Mainstream Fluctuation

Abstract With the ever-increasing aerodynamic and thermal loads, the endwalls of modern gas turbines have become critical areas that are susceptible to manufacturing and operational uncertainties, making them highly prone to thermal failures. Therefore, it is of vital importance to quantify the impacts of input uncertainties on the aero-thermal performance of endwalls. First, based on the Kriging surrogate, an efficient uncertainty quantification (UQ) method suitable for expensive computational fluid dynamics (CFD) problems is proposed. By using this method, the impacts of slot geometry deviations (slot width, endwall misalignment) and mainstream condition fluctuations (turbulence intensity, inlet flow angle) on the aero-thermal performance of endwalls are quantified. Results show that the actual performance of endwalls has a high probability of deviating from its nominal value. The maximum deviations of aerodynamic loss, area-averaged film cooling effectiveness, and area-averaged Nusselt number reach 0.33%, 45%, and 5.0%, respectively. The critical regions that are most sensitive to the input uncertainties are also identified. Second, a global sensitivity analysis method is also performed to pick out the significant uncertain parameters and explore the relationship between input uncertainties and performance output. The inlet flow angle is proved to be the most significant parameter among the four input uncertain parameters. Besides, a positive incidence angle is found to be detrimental to both the aerodynamic performance and the thermal management of endwalls. Finally, the influence mechanisms of the inlet flow angle on endwall aero-thermal performance are clarified by a fundamental analysis of flow and thermal fields.

Wind-Resistant Capacity Modeling for Electric Transmission Line Towers Using Kriging Surrogates and Its Application to Structural Fragility

Wind loading on a transmission tower structure is jointly influenced by the wind field, structural parameters, and the geo-spatial configuration of the transmission line. Considering the multi-parametric effect, this paper aims at developing a limit capacity model for transmission towers under strong winds. To this end, the limit capacity of the tower is expressed via two equivalent means: one is the limit wind speed as a function of the wind angle of attack and the span of transmission line; the other is a limit capacity surface with three fundamental wind load components as the principal axes. An adaptive kriging surrogate modeling is constructed to approximate the function/surface with structural uncertainties considered. The performance of the surrogate model is improved by adding support points and then evaluated by the overall accuracy validation and local error check. A numerical example demonstrating the feasibility of the surrogate modeling for the limit capacity of the transmission tower under winds is presented. Finally, a fragility assessment concerning a practical transmission line and towers subjected to typhoons is accomplished using the established limit capacity model of the tower.

Aerodynamic optimization design of single-layer spherical domes using kriging surrogate model

This paper aims to provide an aerodynamic optimization procedure to improve the aerodynamic performance of single-layer spherical domes, by coupling the kriging surrogate model with computational fluid dynamics (CFD) and finite element analysis (FEA). Firstly, a series of wind tunnel tests on the mean pressures and wind-induced behavior of a single-layer spherical latticed shell, were carried out to investigate the effect of dome geometric parameters. Then, the Reynolds-averaged Navier-Stokes equations and RSM turbulence model were utilized for simulating the wind loads on spherical domes, and the numerical results are validated with experimental data. On this basis, the single-objective aerodynamic optimization of spherical domes based on ordinary kriging surrogates has been carried out to find out the optimal geometric parameters (rise/span and wall-height/span ratios). The objectives were minimizing the highest mean suction and the maximum vertical displacement, respectively. The optimization results showed that the optimal design of spherical domes exhibits a reasonable aerodynamic performance improvement compared with the near optimal solutions. In addition, the highest mean suction and the maximum vertical displacement can be reduced by decreasing the wall-height of the dome, and a good trade-off between the two objectives can be achieved by selecting suitable dome geometric parameters.

Dual-polarized nanocrescent antenna designed using efficient optimization techniques.

A novel nanocrescent antenna with polarization diversity is introduced. It is formed from a crescent-shaped patch fed with a coupled strip transmission line. The antenna is located on top of a SiO2 thin film with a shielding ground layer underneath. The structure is supported by an arbitrary substrate. Polarization of the radiated field can be adjusted to be along either one of the two orthogonal polarizations based on which one of the two crescent patch modes is going to be excited. The excitation of either one of these two modes of the patch is achieved by switching between the two propagating modes of the feeding coupled strip transmission line. Using a dual-polarized antenna allows doubling the optical communication system's capacity via frequency reuse. The new crescent antenna dimensions are optimized to satisfy several goals, such as minimizing the losses, the deviation of the main beam direction away from broadside, and maximizing the radiation efficiency and axial ratio. Through the optimization process, simple surrogate kriging models replace the detailed electromagnetic simulation. The optimal response is achieved by applying two different optimizers. The first optimizer employs the design-centering technique using normed distances. The multiobjective particle swarm with the preference ranking organization method for enrichment evaluations is used by the second optimizer. In order to identify the critical dimensions to which the nanoantenna is most sensitive, a sensitivity analysis is used. The optimized antenna is capable of switching its radiation between two orthogonal pure linear polarizations with maximum radiation along the broadside direction. The size of the proposed antenna is about 500nm×500nm. Its impedance-matching bandwidth is higher than 30 THz centered around 193 THz (1550 nm). Its gain and radiation efficiency are higher than 5.2 dBi and 85%, respectively, all over the working frequency band.

New Adaptive Surrogate-Based Approach Combined Swarm Optimizer Assisted Less Tuning Cost of Dynamic Production-Inventory Control System

Today, most manufacturing control systems are complex and expensive, so they are limited to employ a small number of function evaluations for optimal design. Yet, looking for optimization methods with the less-computational cost is an open issue in engineering control systems. This paper aims to propose an effective adaptive optimization approach by integrating Kriging surrogate and Particle Swarm Optimization (PSO). In this method, a novel iterative adaptive approach is utilized using two sets of training samples including initial training and adaptive sample points. The initial training points are designed by space-filling design, while the adaptive points are generated using a new jackknife resampling approach. The proposed approach can effectively convergence towards the global optimal point using a small number of function evaluations. The efficiency and applicability of the proposed algorithm are evaluated using the optimal design of the fractional-order PID (FOPID) controller for some benchmark transfer functions. Then, the introduced approach is applied for tuning the parameters and the sensitivity analysis of the FOPID controller for a dynamic production-inventory control system. The results are in good agreement with the results reported in the literature, while the proposed approach is executed with a lower computational burden.

Structural reliability assessment through surrogate based importance sampling with dimension reduction

We present a method for reliability assessment in extreme conditions from a numerical simulator through surrogate based importance sampling. As proposed in recent works in the literature, a Kriging surrogate is used to build an approximation of the limit state function and the optimal importance density. Our contribution is then the use of a sufficient dimension reduction method which enables the construction of the limit state function metamodel in lower dimension. The so called augmented failure probability and correction factor are recast in this dimension reduction framework. Simple strategies for metamodel refinement in the dimension reduction subspace are described and, in the case of Gaussian inputs, a computationally efficient MCMC scheme aimed at sampling the quasi-optimal importance density is presented. The case of non-Gaussian inputs is also laid out and it is argued and demonstrated through simulations that this approach can reduce the number of calls to the computer model, which is a crucial factor in reliability analysis. Advantages of this method are also supported by numerical simulations carried on an industrial case study concerned with the extreme response prediction of a wind turbine under wind loading.

A single-loop approach with adaptive sampling and surrogate Kriging for reliability-based design optimization

Surrogate models have been widely used for Reliability-Based Design Optimization (RBDO) to solve complex engineering problems. However, the accuracy and efficiency of surrogate-based RBDO largely rely on the sample size and sampling methods. For this reason, successive sampling methods that update the surrogate successively are more promising. Nowadays, several Kriging-based RBDO approaches have been proposed with different successive sampling techniques. However, these approaches are based on Monte Carlo simulations and double-loop approaches such that most of them would be time consuming for high target reliability levels or high dimensional problems. To improve the efficiency of surrogate-based RBDO, this article proposes a Single-Loop Approach (SLA) combined with the Kriging surrogate. This Kriging model is updated efficiently by using the Most Probable Points (MPPs) from the last SLA iteration. A very simple and effective stopping criterion is proposed. Compared with other sampling methods, the initial Kriging can be started with very few training points and converges to the right optimum very efficiently. Three mathematical examples and a practical engineering problem are used to demonstrate the effectiveness, the advantages and also the limitations of this method.

Robust optimization of an organic Rankine cycle for geothermal application

A robust design optimization (RDO) methodology for Organic Rankine Cycles (ORC) is presented, allowing to ensure an improved, stable performance over a large range of operating conditions. In contrast with classical ORC design methods, whereby all modeling hypotheses and operating conditions are considered as perfectly known, i.e. deterministic, the RDO approach allows to account for the manifold sources of uncertainty affecting the system. For geothermal ORC, the latter are related on one hand with the ill-known properties of the geothermal source and, on the other, with intrinsically random parameters, such as the condensation temperature. The proposed RDO approach selects values of the design parameters that maximize the expected (average) performance while minimizing its variance under uncertain nominal operating conditions. The optimal design delivered by the proposed strategy outperforms the one derived from the standard deterministic approach: specifically, the expected power output is increased by 1.5%, while its standard deviation is reduced by 8.5% and the surface of the heat exchangers by 34%. • Geothermal Organic Rankine cycle performance is subject to many uncertainties. • A robust optimization strategy is devised for cycle design under uncertainty. • The design maximizes average performance while minimizing the variance. • Nested Kriging surrogates are used to efficiently speed-up the design loop. • Robust optimum has +1.5% power output and smaller variance wrt standard design.

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 reflect 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 effectively reduced the risk of occupant injury.

An Adaptive Searching Kriging Surrogate Model for Aerodynamic Optimization

The Kriging-based genetic algorithm applied to aerodynamic optimization design encounters a problem of unexpected sample size. In this paper, an adaptive method is proposed that the search space moves with the local optimum. The automatic division and hierarchical approximation of search space are realized by taking the selection of refinement samples into account. The typical function optimization and transonic supercritical airfoil drag reduction design are performed using this method. Results show that the number of samples required is greatly reduced, and the aerodynamic performance of the airfoil is efficiently improved.