Kriging Metamodel Research Articles

Practical metamodel-assisted multi-objective design optimization for improved sustainability and buildability of wind turbine foundations

In this work, we study the potential of using kriging metamodelling to perform multi-objective structural design optimization using finite element analysis software and design standards while keeping the computational efforts low. A method is proposed, which includes sustainability and buildability objectives, and it is applied to a case study of reinforced concrete foundations for wind turbines based on data from a large Swedish wind farm project. Sensitivity analyses are conducted to investigate the influence of the penalty factor applied to unfeasible solutions and the size of the initial sample generated by Latin hypercube sampling. A multi-objective optimization is then performed to obtain the optimum designs for different weight combinations for the four objectives considered. Results show that the kriging-obtained designs from samples of 20 designs outperform the best designs in the samples of 1000 designs. The optimum designs obtained by the proposed method have a sustainability impact 8–15% lower than the designs developed by traditional methods.

Proposing an Uncertainty Management Framework to Implement the Evidence Theory for Vehicle Crash Applications

Abstract The purpose of this work is to enable the use of the Dempster–Shafer evidence theory (ET) for uncertainty propagation on computationally expensive automotive crash simulations. This is necessary as the results of these simulations are influenced by multiple possibly uncertain aspects. To avoid negative effects, it is important to detect these factors and their consequences. The challenge when pursuing this effort is the prohibitively high computational cost of the ET. To this end, we present a framework of existing methods that is specifically designed to reduce the necessary number of full model evaluations and parameters. An initial screening removes clearly irrelevant parameters to mitigate the curse of dimensionality. Next, we approximate the full-scale simulation using metamodels to accelerate output generation and thus enable the calculation of global sensitivity indices. These indicate effects of the parameters on the considered output and more profoundly sort out irrelevant parameters. After these steps, the ET can be performed rapidly and feasibly due to fast-responding metamodel and reduced input dimension. It yields bounds for the cumulative distribution function of the considered quantity of interest. We apply the proposed framework to a simplified crash test dummy model. The elementary effects method is used for screening, a kriging metamodel emulates the finite element simulation, and Sobol' sensitivity indices are determined before the ET is applied. The outcome of the framework provides engineers with information about the uncertainties they may face in hardware testing and that should be addressed in future vehicle design.

Kriging-assisted design of functionally graded cellular structures with smoothly-varying lattice unit cells

Cellular structures in nature have attracted great attention in the field of structural optimization. This paper proposes a Kriging-assisted topology optimization method for design of functionally graded cellular structures (FGCS), which are infilled by smoothly-varying lattice unit cells (LUCs). Specifically, LUCs are depicted by level set functions and the shape interpolation method is employed to generate sample LUCs. Then, one Kriging metamodel is constructed to predict the mechanical properties of LUCs within FGCS, so as to reduce the computational expense involved in finite element analysis of LUCs. Meanwhile, the other Kriging metamodel is created to predict the values of shape interpolation function, and a Kriging-assisted morphological post-process method is put forward to achieve the smooth transition between adjacent graded LUCs. In the proposed method, the effective densities, mechanical properties, and geometrical configurations of LUCs are coupled by Kriging metamodels, so that the multiscale design of FGCS can be realized at a low computational burden by optimizing the distribution of elemental densities, and this also paves the way for design of FGCS with irregular geometries by morphological post-process and geometry reconstruction. Numerical examples are presented to validate the accuracy and effectiveness of the proposed method for design of FGCS. What is more, the design of a pillow bracket with a slightly complex geometry is provided to illustrate the engineering application of the proposed method. The results indicate that the proposed method is effective and universal for designing FGCS with smoothly-varying LUCs.

Steam boiler mixing channel optimization with a surrogate based multi objective genetic algorithm

Nowadays, thanks to ever-increasing computational resources, a viable path to a robust and fast design strategy for both thermal machines and turbomachinery is the coupling of Computational Fluid Dynamics (CFD) and shape optimization algorithms. In general, numerical optimization approaches require less time than the trial-and-error procedure traditionally employed, where the designer produces only a tentative initial geometry. This work assesses the capability of a shape optimization algorithm to enhance the design of a steam boiler mixing channel to guarantee negligible NOx production, avoid combustion instabilities especially at lower thermal powers, due to a bad mixing quality of the mixture, and thermal deformation on the burner surface mesh, due to a non uniform distribution of the flame. In particular, the effect of the mixing quality, flow uniformity and the pressure losses at the outlet section of the mixing channel are investigated. The shape optimization approach is here based on a Surrogate Based Optimization (SBO) with the Multi Objective Genetic Algorithm (MOGA), where response surfaces based on the Kriging meta-model are adopted to decrease the computational cost of the proposed approach.

Robust Optimization Design of Supply Chain Based on Correlated Multi-Performance Responses

Aiming at the problem of supply chain robust optimization with multiple performance responses, a robust optimization method was proposed by combining principal component analysis with double response surface method based on Kriging meta-model. Firstly, through principal component analysis, the location characteristics and divergence characteristics of multiple related performance responses were transformed into the principal component comprehensive score of supply chain performance; secondly, the Kriging metamodels of location characteristics and divergence characteristics of multiple performance responses were constructed respectively, and the principal component comprehensive score model of comprehensive performance based on Kriging meta-model was constructed. Thus the optimal operating conditions by optimizing the constructed robust optimization strategy were obtained in order to realize the robust optimization of supply chain system with correlated multiple responses. Finally, a supply chain simulation case was given to illustrate the effectiveness of our proposed optimization method, and the robustness of different methods was also discussed. The comparison results showed that our proposed was more robust.

A dynamic sample augmentation Kriging metamodeling method for industrial residue hydrogenation process

AbstractResidue hydrogenation process (RHP) plays an important role in efficient utilization of heavy oil resources. The high‐fidelity model of RHP is so complex that its optimization cost is expensive and even intractable. Furthermore, in the actual industrial processes, due to the low sampling frequencies of some sensors, only a few sampling points can be obtained with some missing values. The insufficient samples can directly affect the performance of the final model. Therefore, a new adaptive dynamic sampling (ADS) method for Kriging metamodeling is proposed. Based on a small number of valid industrial sample points, the proposed method adaptively obtains key information sampling points and selectively adds key information sampling points to update the model. First, the difference maximization strategy and the outlier distance strategy are proposed to obtain new sampling points with both global and regionalized information. Then, the new key points are used to iteratively update the Kriging predictor to obtain satisfactory accuracy. Finally, the calculation result of benchmark cases and two actual experiments validate the effectiveness of the proposed method for constructing surrogate models of complex industrial processes.

Optimization based on electro-thermo-mechanical modeling of the high electron mobility transistor (HEMT)

The electro-thermomechanical modeling study of the High Electron Mobility Transistor (HEMT) has been presented, all the necessary equations are detailed and coupled. This proposed modeling by the finite element method using the Comsol multiphysics software, allowed to study the multiphysics behaviour of the transistor and to observe the different degradations in the structure of the component. Then, an optimization study is necessary to avoid failures in the transistor. In this work, we have used the Covariance Matrix Adaptation-Evolution Strategy (CMA-ES) method to solve the optimization problem, but it requires a very important computing time. Therefore, we proposed the kriging assisted CMA-ES method (KA-CMA-ES), it is an integration of the kriging metamodel in the CMA-ES method, it allows us to solve the problem of optimization and overcome the constraint of calculation time. All these methods are well detailed in this paper. The coupling of the finite element model developed on Comsol Multiphysics and the KA-CMA-ES method on Matlab software, allowed to optimize the multiphysics behaviour of the transistors. We made a comparison between the results of the numerical simulations of the initial state and the optimal state of the component. It was found that the proposed KA-CMA-ES method is efficient in solving optimization problems.

Validation of a multi-physics simulation platform for engine emissions modelling

This paper introduces a comprehensive and systematic Design of Experiments based methodology deployed in conjunction with a multi-physics engine air-path and combustion co-simulation, leading to the development of a global transient simulation capability for engine out NOx emissions. The proposed multi-physics engine simulation framework couples a real-time one-dimensional air flow model with a Probability Density Function based Stochastic Reactor Model that accounts for detailed in-cylinder combustion chemistry to predict combustion emissions. The integration challenge stemming from the different computation complexities and time scales required to ensure adequate fidelity levels across multi-physics simulations was addressed through a comprehensive Design of Experiments methodology to develop a reduction of the slower Stochastic Reactor Model simulation to enable a transient simulation focussed on NOx emissions. The Design of Experiments methodology, based on Optimal Latin Hypercube design experiments, was deployed on the multi-physics engine co-simulation platform and systematically validated against both steady state and transient light-duty Diesel engine test data. The surrogate selection process included the evaluation of a range of metamodels, with Kriging metamodels selected based on both the statistical performance criteria and consideration of physical phenomena trends. The transient validation was carried out on a simulated New European Drive Cycle against the experimental data available, showing good capability to capture transient NOx emission behaviour in terms of trends and values. The significance of the results is that it proves the transient and drive cycle capability of the multi-physics simulation platform, suggesting a promising potential applicability for early powertrain development work focussed on drive cycle emissions.

An improved screen force model based on CFD simulations of the hydrodynamic loads on knotless net panels

Knotless net panels are widely-adopted for traditional fish cages and offshore fish farms. A new screen force model based on numerical simulations of a portion of knotless net panels is proposed in this paper. At first, the hydrodynamic loads on knotless smooth and textile twisted net panels are studied using Improved Delayed Detached Eddy Simulations while varying the incoming velocity as well as the diameter and length of the net twines. The accuracy of the simulations is validated through a computational grid and domain independence study for the mean hydrodynamic loads on full-size net panels. The comparison of the hydrodynamic loads on a circular cylinder, cruciform and net panel reveals the unique hydrodynamics around net panels which emphasises the importance of studying complete panels. Then, a Kriging metamodel is established from the simulation-based data to generate response surfaces for the hydrodynamic load coefficients for a large range of flow and geometrical properties. Hence, this combination of high-fidelity CFD and Kriging metamodelling provides a more flexible approach than the common theoretical or experimental variations of screen force models. The metamodel predicts well the non-linear relation between drag coefficient, twine diameter and length of the twines. A second-order regression fitting is applied to generate a closed polynomial which is compared to existing approaches and physical measurements for validation. The proposed approach outperforms previous screen force models for the prediction of the drag and lift coefficients for a wide range of nets and inflow velocities due to the inclusion of different roughnesses and more geometrical properties of the net in the derivation. Finally, the proposed formula for the drag force coefficients is applied to a two-way coupled CFD method which is based on screen force models to determine the hydrodynamic loads. It can be shown that the new approach is superior to previously used screen force models for the simulation of the drag forces on and the velocity reduction behind net panels.

An improved Morison hydrodynamics model for knotless nets based on CFD and metamodelling methods

The hydrodynamic loadings on the knotless nets are studied using Improved Delayed Detached Eddy Simulations (IDDES) with Kriging metamodelling, then an improved Morison model for predicting hydrodynamic loadings on net twines is derived for the first time. The high-fidelity IDDES turbulence model is capable of simulating the separated flows from cylindrical objects. The 6 × 6 portions of smooth and fabric nets with roughness features are considered in flow models. The accuracy of the numerical method is validated through a typical cruciform case in fluid, wherein the pressure fields and turbulent intensity are compared with physical measurements. Initially, a comparative study of flowing over a circular cylinder, cruciform, and net panel is conducted, illustrating the necessity of the net twines utilisation to analyse the flow fields and hydrodynamic loadings accurately. Based on the review of existed Morison model for net structures, the design point cases varying the normal incoming velocities, diameters and lengths are simulated numerically, then Kriging metamodels of hydrodynamic loading coefficients are trained. It is addressed that the over-estimation of Reynolds number effects, the overlook of net solidity and tangential forces on the twine, or the constant-adoption in the previous models contribute to the deviations of hydrodynamic modelling. The polynomial-type, as well as power-type formulae of normal and tangential force coefficients, are fitted non-linearly using the metamodelling results. Finally, the applications of newly-proposed Morison models embedded in finite element modellings are presented to emphasise its improving abilities. The hydrodynamic loadings with new formulae are validated and compared with existed models through net panels, fish cage and trawl net cases.

Efficient surrogate strategy for investigating stick‐slip instability

AbstractAn innovative sampling strategy called MiVor coupled with kriging metamodeling is employed for detecting stick‐slip instabilities within a parametric domain based on very few simulations. The interest of the approach is here exposed on an oscillator of Duffing's type in combination with an elasto‐plastic friction force model, more details can be found in [3].

Stochastic optimization of high-altitude airship envelopes based on kriging method

High-altitude airships can be used to transport substantial payloads to the stratosphere and remain there over long periods of time. In this paper, an algorithm for the design of high-altitude airship envelopes, accounting for uncertainties, is developed and applied. The algorithm is based on the non-intrusive polynomial chaos expansion scheme, which is employed to build a stochastic kriging metamodel. Two uncertainties are examined and characterized: 1) the stratospheric wind fluctuations using reanalysis datasets and 2) the variability in the turbulence levels. The method results are discussed to address the relevancy of the uncertainties. It is found that the drag coefficient of stratospheric envelopes can vary by as much as 30 percent. As a case of study, an ideal stratospheric airship is considered, operating at an altitude of 20 km, at a latitude of 30cN and carrying a payload of 250 kg. The baseline design follows the shape of the ZHIYUAN-1 envelope and the cost function to be minimized is the average mission drag coefficient. Due to the new method, a significant reduction (4%) of the average drag of the aircraft is achieved.

Optimization of semi-interlocking heat sinks for hotspot thermal management using multi-objective genetic algorithm

A semi-interlocking heat sink that operates based on the effect of flow acceleration in a curved channel is proposed to mitigate local heat fluxes from power electronics. The proposed heat sink offers easy manufacturability owing to its simple structure. A computational fluid dynamics simulation is developed and validated experimentally. The simulation is conducted based on periodic boundary conditions. The design of experiment method and the Kriging meta-modeling are used to optimize the heat sink. Furthermore, a multi-objective genetic algorithm is used to minimize thermal resistance and pressure drop. Based on the optimization results, the thermal and hydraulic characteristics are analyzed according to geometric changes. The analysis shows that the proposed heat sink affords a 30.4%–34.7% lower thermal resistance than plate-finned heat sinks at the same pumping power.

Optimal reliable design of brake disk using a Kriging surrogate model

In this article, structural analysis is performed on the disk due to the brake pad actuating load. Thermal analysis is addressed on the disk to observe the maximum temperature rise after the braking operation. A reliability-based design optimization using Kriging meta-model is performed to determine the optimal reliable design. Kriging meta-model provides a more accurate approximation of the original model with 50 LHS points. The objective is to minimize the brake disk volume, whereas the constraints are to minimize its stress and temperature. The optimal design gives a minimum volume with a weight reduction of 32.67% compared to the initial model.

A note on the effect of material uncertainty on acoustic source localization error in anisotropic plates

The uncertainty in material properties of an anisotropic plate may influence the acoustic source localization process undertaken for the plate. To study this effect of material uncertainty, the two moduli of elasticity of an orthotropic plate material are considered in this note as independent random variables and the propagation of this material uncertainty through the wave front shape-based acoustic source localization approach is investigated. Assuming lognormal probability distributions for the two random variables, several design points in lognormal spaces are picked using Latin Hypercube Sampling. Finite element analysis is performed for each design point to simulate the elastic wave propagation due to an acoustic event and wave front shape-based approach is applied to estimate the source location. The time-of-arrivals and source localization errors obtained for each design point are considered as separate response functions at that design point and regression kriging metamodels through the responses at the design points are constructed. Monte Carlo simulations are carried out using these metamodels to obtain the distribution parameters (i.e., ranges, means and standard deviations) of the time-of-arrivals and localization errors. A global sensitivity analysis is performed to estimate the effect of each random variable on the localization errors. It is observed that for lognormally distributed moduli of elasticity with same coefficients of variation, uncertainty in the modulus of elasticity in the major direction affects the source localization accuracy more compared to the uncertainty in the modulus of elasticity in the minor direction, particularly when the ellipse-based technique is used.

AK-SESC: a novel reliability procedure based on the integration of active learning kriging and sequential space conversion method

To deal with evaluating small failure probabilities, AK–SESC: a novel approach integrating an active learning Kriging meta-model (AK-MCS) and the SESC, a sequential space conversion method, is suggested. The efficiency of the proposed approach relies on the advantages of the AK-MCS and its updating feature to evaluate the actual performance function and the superiority of SESC in estimating small failure probabilities. Although there are effective methods for small probabilities, the beauty of this approach is that it is derived from the probability integral with no simplifications while providing results of high accuracy.Different problems were solved to study the AK–SESC applicability. The main effort of this method is reducing the function call numbers of the original SESC while reaching the same accuracy as Monte Carlo Simulation (MCS). The reliability analysis results were compared with the main reliability methods of the Importance Sampling (IS), Subset Simulation (SubSim), Line Sampling (LS), First and also Second-Order Reliability Method (FORM and SORM). The solved problems indicate that the proposed approach provides accurate answers with much fewer function calls than SESC. So, it can be a promising method for reliability analyses involving nonlinear or high-dimensional performance functions with small failure probabilities.

Metamodeling and On-Line Clustering for Loss-of-Flow Accident Precursors Identification in a Superconducting Magnet Cryogenic Cooling Circuit

In the International Thermonuclear Experimental Reactor, plasma is magnetically confined with Superconductive Magnets (SMs) that must be maintained at the cryogenic temperature of 4.5 K by one or more Superconducting Magnet Cryogenic Cooling Circuits (SMCCC). To guarantee cooling, Loss-of-Flow Accidents (LOFAs) in the SMCCC are to be avoided. In this work, we develop a three-step methodology for the prompt detection of LOFA precursors (i.e., those combinations of component failures causing a LOFA). First, we randomly generate accident scenarios by Monte Carlo sampling of the failures of typical SMCCC components and simulate the corresponding transient system response by a deterministic thermal-hydraulic code. In this phase, we also employ quick-running Proper Orthogonal Decomposition (POD)-based Kriging metamodels, adaptively trained to reproduce the output of the long-running code, to decrease the computational time. Second, we group the generated scenarios by a Spectral Clustering (SC) employing the Fuzzy C-Means (FCM), in order to identify the main patterns of system evolution towards abnormal states (e.g., a LOFA). Third, we develop an On-line Supervised Spectral Clustering (OSSC) technique to associate time-varying parameters measured during plant functioning to one of the prototypical groups obtained, which may highlight the related LOFA precursors (in terms of SMCCC components failures). We apply the proposed technique to the simplified model of a cryogenic cooling circuit of a single module of the ITER Central Solenoid Magnet (CSM). The framework developed promptly detects 95% of LOFA events and around 80% of the related precursors.

Performance Enhancement of the Micromixer by the Multiobjective Genetic Algorithm and Surrogate Model Based on a Navier–Stokes Analysis Using Trade-Off Objective Functions

Optimal structure of the micromixer with a two-layer serpentine crossing device was accomplished by a multiobjective genetic algorithm and surrogate modeling based on a Navier–Stokes analysis using the trade-off objective functions behavior. The optimization analysis was conducted with three design parameters, i.e., channel width to the pitch span ( w / P ) ratio, major channel width to the pitch span (H/P) ratio, and channel depth to the pitch span (d/P) ratio. Two objective functions (i.e., mixing index and pressure drop) with trade-off characteristics have been used to solve the multiobjective optimization problem. The design domain was predetermined by a parametric investigation; afterward, the Latin hypercube sampling method was employed to select the appropriate design points surrounded by the design domain. The numerical data of the thirty-two design points were used to create the surrogate model; among the different surrogate models, in this study, the Kriging metamodel has been used. The concave pareto-optimal curve signifies the trade-off characteristics linking the objective functions.

Kriging-based optimization design of deep tunnel in the rheological Burger rock

The principal purpose of this work consists in optimizing the support system of a deep tunnel accounting for the uncertainty of the time-dependent behaviour of the surrounding rock, which is described by the rheological Burger law. The stochastic approach is chosen for this aim. On one hand the Quantile Monte Carlo (QMC) simulation is used to determine the optimal design variables (i.e., the thickness of two liners). On the other hand, the well-known Kriging metamodeling technique is undertaken to approximate the limit state function in the augmented reliability space (i.e., the tensor product between the random variable space and the design variable space). The adopted optimization process allows to derive the optimal tunnel support that verifies two failure modes, namely the support capacity criterion and the maximum tunnel convergence.

A link-to-link segment based metamodel for dynamic network loading

Dynamic network loading (DNL) is a core part of simulation-based dynamic traffic assignment (DTA) models, which describes how traffic propagates on a transportation network and usually maps a set of route departure rates to a set of route travel times. Due to the incorporation of different intra- and inter-link dynamics such as queuing, spillback, etc., DNL becomes complex, resulting in models with undesirable analytical properties (e.g., discontinuity, non-differentiability, and lack of closed-form) and requiring intensive computational efforts. Moreover, as DNL is applied repeatedly in simulation-based DTA algorithms, it often is the computational bottleneck of large-scale traffic assignment and related optimization problems. In this research, we address these problems by proposing a link-to-link segment-based Kriging metamodel for dynamic network loading. The proposed Kriging model calculates route travel time, link travel time, and link flow. It systematically approximates the traditional DNL model while being faster and maintaining some desirable analytical properties. Both temporal and spatial proximity information and the route segment incidence information are incorporated in the metamodel formulation. The models are trained using maximum likelihood estimation with different dataset sizes. In the experiments, the proposed Kriging metamodels performed better than a state-of-the-art neural network model, achieving an average error of 1.64% and 5.32% in the Nguyen Dupuis (ND) network and the Sioux Falls (SF) network, respectively, along with at least 43 times, sometimes as high as over 1300 times, improvement in computational speed. The results, however, suggest a tradeoff between efficiency and accuracy and the size of the training dataset. With balanced route demand, the model’s performance was reasonable even under congested traffic conditions. The MAPE values were at most 3% and 18% in the ND network and the SF network, respectively, for different level of congestion demonstrating the effectiveness of the proposed model.