Kriging Surrogate Model Research Articles

Multi-Fidelity Sparse Polynomial Chaos and Kriging Surrogate Models Applied to Analytical Benchmark Problems

In this article, multi-fidelity kriging and sparse polynomial chaos expansion (SPCE) surrogate models are constructed. In addition, a novel combination of the two surrogate approaches into a multi-fidelity SPCE-Kriging model will be presented. Accurate surrogate models, once obtained, can be employed for evaluating a large number of designs for uncertainty quantification, optimization, or design space exploration. Analytical benchmark problems are used to show that accurate multi-fidelity surrogate models can be obtained at lower computational cost than high-fidelity models. The benchmarks include non-polynomial and polynomial functions of various input dimensions, lower dimensional heterogeneous non-polynomial functions, as well as a coupled spring-mass-system. Overall, multi-fidelity models are more accurate than high-fidelity ones for the same cost, especially when only a few high-fidelity training points are employed. Full-order PCEs tend to be a factor of two or so worse than SPCES in terms of overall accuracy. The combination of the two approaches into the SPCE-Kriging model leads to a more accurate and flexible method overall.

Multifidelity Surrogate Models for Efficient Uncertainty Propagation Analysis in Salars Systems

Salars are complex hydrogeological systems where the high-density contrasts require advanced numerical models to simulate groundwater flow and brine transport. Applying those models over large spatial and temporal scales is important to understand the various subsurface processes in salars, but the associated computational cost hinders an analysis based on repetitive numerical simulations. Single fidelity surrogate modeling is a common approach to alleviate computational burden with computationally expensive physics-based models of high-fidelity. However, due to the complexity in salars modeling it might not be affordable to run high-fidelity simulations many times until we build a surrogate model of acceptable accuracy. Here, we investigate if multifidelity surrogate methods, that exploit information from inexpensive lower fidelity models, can show promise for computationally demanding tasks for salars systems. Additive, multiplicative and co-Kriging multifidelity surrogates are developed based on the combination of training data from low fidelity sharp interface models and a higher fidelity variable-density flow and solute transport model. Their performance is compared against a single fidelity Kriging surrogate model, and they are all employed to conduct a Monte-Carlo-based uncertainty propagation analysis where recharge, hydraulic conductivity and density differences between freshwater and brine are considered uncertain model inputs. Results showed that multifidelity methods are a promising alternative for time-intensive numerical models of salars under limited high-fidelity samples. In addition, sharp interface models, despite commonly used in coastal aquifer problems, can also be applied in salars modeling as cheap lower fidelity models for interface calculations via a multifidelity framework. The Monte-Carlo outputs based on the surrogate models, resulted in estimated probability density functions characterized by long tails, thus, highlighting the need to reduce parametric uncertainty in real world models of salars.

Hydraulic Capture Optimization and Risk Assessment of Polluted Groundwater Based on Kriging Surrogate Model

Hydraulic Capture Optimization and Risk Assessment of Polluted Groundwater Based on Kriging Surrogate Model

Structural reliability analysis of redundant systems considering global structural responses and using Kriging surrogate models

Structural reliability analysis of redundant systems considering global structural responses and using Kriging surrogate models

Investigation on thermal model updating of Alpha Magnetic Spectrometer in orbit based on Kriging meta-modeling

The purpose of the update of the space instrument thermal model is to make adjustments to the model parameters to maintain the consistency between the temperature simulation consequences and the experimental data, which is essential for thermal control design. This study proposed an approach of combining Kriging meta-modeling and Genetic Algorithm (GA) instead of direct iteration to update the Alpha Magnetic Spectrometer (AMS) thermal model, which significantly boosts the updating efficiency. Firstly, the importance ranking of the thermal model parameters affecting its temperature response is determined by the parameter sensitivity analysis. The Kriging surrogate model of the AMS simulation temperature is then established by training samples from sensitive parameters using Latin Hypercube Sampling (LHS) method, and its reliability is demonstrated by the maximum error of 0.86 °C in contrast to thermal model results at other specific model parameter values. Based on above information, a set of optimal parameters are eventually generated to update the original AMS thermal model by GA with the improvement of the temperature prediction accuracy up to 13.96%. The current work provides an efficient and accurate approach for thermal model optimization and thermal control design of the space-borne instruments.

A two-step identification method for crack parameter identification of train axles

Abstract In this paper, a crack parameter identification method based on the coordinate of 1x intersections and Kriging surrogate model with enhanced samples is proposed to solve the crack parameter identification problem of train axles in start-stop condition. The location of the crack is identified in the first step by the x-coordinate of the linear intersection of 1x frequency components. Then, the intersecting coordinates and rigid constraint points were used to solve response curves of equal crack depth, and numerous enhanced samples of same depth at different positions were obtained to establish a Kriging model. Finally, the crack depth was directly obtained by the Kriging model. In this paper, response curve s of equal crack depth is proposed according to the symmetry of the train structure, which greatly reduces the number of samples needed to construct the Kriging model. Numerical simulation and experimental results demonstrate the effectiveness and robustness of the proposed method.

Geometric design of friction ring dampers in blisks using nonlinear modal analysis and Kriging surrogate model

Integrally bladed disks (blisk) have been widely used in the turbo-machinery industry due to its high aerodynamic performance and structural efficiency. A friction ring damper (FRD) is usually integrated in the system to improve its low damping. However, the design of the geometry of this FRD become complex and computationally expensive due to the strong nonlinearities from friction interfaces. In this work, we propose an efficient modelling strategy based on advanced nonlinear modal analysis and Kriging surrogate models to design and optimize the geometry of a 3D FRD attached to a high fidelity full-scale blisk. The 3D ring damper is parametrised with a few key geometrical parameters. The impact of each geometric parameter and their sensitivities to nonlinear dynamic response can be efficiently assessed using Kriging meta-modelling based on a few damped nonlinear normal modes. Results demonstrate that the damping performances of ring dampers can be substantially optimized through the proposed modelling strategy whilst key insights for the design of the rings are given. It is also demonstrated that the distribution of the contact normal load on the contact interfaces has a strong influence on the damping performances and can be effectively tuned via the upper surface geometry of the ring dampers.

An effective method to optimise plasma immersion ion implantation: Sensitivity analysis and design based on low‐density polyethylene

AbstractThe performance of polymer surface treatment using plasma immersion ion implantation (PIII) depends on many operating parameters, such as treatment duration, radiofrequency power, pulsed bias voltage and pulse repetition rate, and the ion fluence applied on the polymer surface. Currently, the identification of optimal operating parameters to achieve specific performance targets is heavily based on trail and error with extensive experimental testing. Herein, we present an optimisation method based on sensitivity analysis using polynomial chaos expansion and experimental design with Kriging surrogate model to greatly reduce the amount of experiments. The combined effects of PIII operating parameters on low‐density polyethylene surface modifications are investigated, demonstrating the validation and effectiveness of the method and design. The new approach offers highly accurate and computationally efficient way for achieving optimum radical density, wettability and optical transmittance that are important for biomedical applications.

Reliability analysis of autonomous underwater vehicle aft pressure shell for optimal design and strength

The lightweight design of the autonomous underwater vehicle (AUV) is a means to improve its resistance. In order to meet reliability requirements of AUV, the lightweight optimization design of aft pressure shell (APS) with sandwich structure is studied. A multi-objective optimization mathematical model is established for the mass and maximum stress as objectives and the strength as constraint condition, which is efficiently adopted by NSGA-II based on Kriging surrogate model. In the reliability analysis, the statistical distributions of the yield strength and the maximum stress is obtained according to the random variable sample and finite element method and then the stress-strength interference model is established. The result shows that the APS with sandwich structure meets the strength requirements and the maximum stress appears near outer skin. The thickness of the sandwich structure has a great influence on the mass and maximum stress. The reliability index of the optimized APS is 5.02, which satisfies three sigma level of random variables and ensures safety and reliability of AUV. The analysis provides reference and technical support for lightweight of pressure shell.

Crushing analysis and optimization for bio-inspired hierarchical 3D cellular structure

Three-dimensional (3D) cellular structures have been widely used in many aspects, e.g., protective equipment, aerospace, vehicles and so on, due to its lightweight and excellent energy absorption characteristics. Triply periodic minimal surface (TPMS) has more uniform energy absorption and mechanical properties than foam due to its structural characteristics. Therefore, a novel 3D hierarchical structure based on TPMS inspired by natural hierarchical biological structure is proposed in this study. Then, the numerical simulation is carried out using the nonlinear finite element code through LS-DYNA to study the crushing behavior of the hierarchical 3D cellular structure. And, the accuracy of finite element model is validated by quasi-static compression experiments on hierarchical structures manufactured by 3D printing. According to the numerical simulation results, the bio-inspired hierarchical 3D lattice is found to have excellent energy absorption characteristics than the non-hierarchical 3D lattice. In addition, the energy absorption characteristics of hierarchical structure is found to be affected by the design parameters of density and constant C. In order to obtain the optimal design of the bio-inspired hierarchical 3D lattice, Kriging (KRG) surrogate models and non-dominated sorting genetic algorithm II (NSGA-II) are jointly used to optimize the hierarchical 3D cellular structure. Moreover, the approximate expression of the specific energy absorption (SEA) of the hierarchical 3D cellular structure is obtained by fitting Gibson and Ashby empirical formula for prediction. Based on our study, the novel hierarchical 3D cellular structure has extremely excellent energy absorption characteristics and has very good application prospect in impact engineering.

A new Bayesian finite element model updating method based on information fusion of multi-source Markov chains

In this paper, we present a new Bayesian model updating method that could overcome the problem of low sampling efficiency and over-reliance on single-chain proposal distribution of traditional Markov chain Monte Carlo algorithm. The delayed rejection and adaptive strategies are introduced in sampling process to obtain a certain number of Markov chains from different proposal distributions, which can independently adjust the variances of proposal distributions and improve the acceptance rate of candidate samples. The abnormal chain detection criterion is adopted to eliminate abnormal Markov chains. Then, the initial variances of different proposal distributions are analogous to the accuracy index of multi-source sensors in the signal domain. And the multi-source sensors grouping weighted fusion algorithm is introduced to fuse the screened Markov chains to approach the posterior probability distribution with high accuracy. The implicit relationship between the parameters to be updated and the responses of the finite element model is fitted by the Kriging surrogate model to improve the computational efficiency. The results of study cases demonstrate that the proposed method has good updating efficiency, excellent updating accuracy, and a higher acceptance rate of samples, which provides a new idea for solving the stochastic model updating.

Safety lifetime analysis method for multi-mode time-dependent structural system

It is important to determine the safety lifetime of Multi-mode Time-Dependent Structural System (MTDSS). However, there is still a lack of corresponding analysis methods. Therefore, this paper establishes MTDSS safety lifetime model firstly, and then proposes a Kriging surrogate model based method to estimate safety lifetime. The first step of proposed method is to construct the Kriging model of MTDSS performance function by using extremum learning function. By identifying possible extremum mode of MTDSS, the performance function of MTDSS can be equivalently transformed into the one of Single-mode Time-Dependent Structure (STDS). The second step is to use the Advanced First Failure Instant Learning Function (AFFILF) to train the Kriging model constructed in the first step, so that the convergent Kriging model can identify the possible First Failure Instant (FFI) of STDS. Then safety lifetime can be searched quickly by dichotomy search. By using AFFILF, the minimum instant that the state is not accurately identified by the current Kriging model is selected as the training point, which avoids the unnecessary calculation which may be introduced into the existing First Failure Instant Learning Function (FFILF). In addition, the Candidate Sample Pool (CSP) reduction strategy is also adopted. By adaptively deleting the random candidate sample points whose FFI have been accurately identified by the current Kriging model, the training efficiency is further improved. Three cases show that the proposed method is accurate and efficient.

Time-dependent reliability analysis method based on ARBIS and Kriging surrogate model

Time-dependent reliability analysis method based on ARBIS and Kriging surrogate model

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.

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.

A construction strategy for conservative adaptive Kriging surrogate model with application in the optimal design of contaminated groundwater extraction-treatment.

When the simulation-optimization model to optimize the groundwater extraction-treatment schemes is used, the construction of a surrogate model for the numerical simulation model has become an effective means to overcome the large calculation load of repeatedly calling the numerical model. However, there are still some problems in using the surrogate model, such as large training sample size, low accuracy, and poor optimization results. In this paper, a conservative adaptive Kriging surrogate model (CAKSM) was proposed by coupling the Kriging surrogate model, optimal solution adaptive sampling method (OSAS), and conservative prediction idea. Firstly, an initial Kriging surrogate model (IKSM) was built for the numerical simulation model of groundwater flow and solute transport. Then, the IKSM was coupled with the optimization model to construct the adaptive Kriging surrogate model (AKSM) by using OSAS. A safety margin was added to the AKSM to build the CAKSM. Finally, the simulation-optimization models based on IKSM, AKSM, and CAKSM were solved by the genetic algorithm, respectively. The results showed that the IKSM could well substitute for the simulation model. The AKSM significantly improved the approximation degree between the surrogate model and the simulation model at the optimal solution by supplementing a small number of new samples. CAKSM could effectively constrain the pollutant mass concentrations within the controlled value, improving the reliability of the optimization scheme. The optimal extraction wells based on different surrogate models were all well 5, well 6, and well 9. They were concentrated in the middle and lower reaches of the contaminated plume's central axis. The sequence for the remediation effects by different surrogate models from high to low was as follows: CAKSM, AKSM, and IKSM. The risk rate of the optimal remediation scheme from the hydraulic conductivity random fields was as high as 12.12%, and the risks were mainly located upstream of the pollution plume's central axis.

Aerodynamic-Aeroacoustic Optimization of a Baseline Wing and Flap Configuration

Optimization design was widely used in the high-lift device design process, and the aeroacoustic reduction characteristic is an important objective of the optimization. The aerodynamic and aeroacoustic study on the baseline wing and flap configuration was performed numerically. In the current study, the three-dimensional Large Eddy Simulation (LES) equations coupled with dynamic Smagorinsky subgrid model and Ffowcs–William and Hawkings (FW–H) equation are employed to simulate the flow fields and carry out acoustic analogy. The numerical results show reasonable agreement with the experimental data. Further, the particle swarm optimization algorithm coupled with the Kriging surrogate model was employed to determine optimum location of the flap deposition. The Latin hypercube method is used for the generation of initial samples for optimization. In addition, the relationship between the design variables and the objective functions are obtained using the optimization sample points. The optimized maximum overall sound pressure level (OASPL) of far-field noise decreases by 3.99 dB with a loss of lift-drag ratio (L/D) of less than 1%. Meanwhile, the optimized performances are in good and reasonable agreement with the numerical predictions. The findings provide suggestions for the low-noise and high-lift configuration design and application in high-lift devices.

Comparison of parallel infill sampling criteria based on Kriging surrogate model

One of the key issues that affect the optimization effect of the efficient global optimization (EGO) algorithm is to determine the infill sampling criterion. Therefore, this paper compares the common efficient parallel infill sampling criterion. In addition, the pseudo-expected improvement (EI) criterion is introduced to minimizing the predicted (MP) criterion and the probability of improvement (PI) criterion, which helps to improve the problem of MP criterion that is easy to fall into local optimum. An adaptive distance function is proposed, which is used to avoid the concentration problem of update points and also improves the global search ability of the infill sampling criterion. Seven test problems were used to evaluate these criteria to verify the effectiveness of these methods. The results show that the pseudo method is also applicable to PI and MP criteria. The DMP and PEI criteria are the most efficient and robust. The actual engineering optimization problems can more directly show the effects of these methods. So these criteria are applied to the inverse design of RAE2822 airfoil. The results show the criterion including the MP has higher optimization efficiency.

Selection of Existing Sail Designs for Multi-Fidelity Surrogate Models

Velocity Prediction Programs (VPPs) are commonly used to help predict and compare the performance of different sail designs. A VPP requires an aerodynamic input force matrix which can be computationally expensive to calculate, limiting its application in industrial sail design projects. The use of multi-fidelity kriging surrogate models has previously been presented by the authors to reduce this cost, with high-fidelity data for a new sail being modelled and the low-fidelity data provided by data from existing, but different, sail designs. The difference in fidelity is not due to the simulation method used to obtain the data, but instead how similar the sail’s geometry is to the new sail design. An important consideration for the construction of these models is the choice of low-fidelity data points, which provide information about the trend of the model curve between the high-fidelity data. A method is required to select the best existing sail design to use for the low-fidelity data when constructing a multi-fidelity model. The suitability of an existing sail design as a low fidelity model could be evaluated based on the similarity of its geometric parameters with the new sail. It is shown here that for upwind jib sails, the similarity of the broadseam between the two sails best indicates the ability of a design to be used as low-fidelity data for a lift coefficient surrogate model. The lift coefficient surrogate model error predicted by the regression is shown to be close to 1% of the lift coefficient surrogate error for most points. Larger discrepancies are observed for a drag coefficient surrogate error regression.

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.