Surrogate Modeling Strategy Research Articles

Micromechanics-based surrogate models for the response of composites: A critical comparison between a classical mesoscale constitutive model, hyper-reduction and neural networks

Although being a popular approach for the modeling of laminated composites, mesoscale constitutive models often struggle to represent material response for arbitrary load cases. A better alternative in terms of accuracy is to use the FE2 technique to upscale microscopic material behavior without loss of generality, but the associated computational effort can be extreme. It is therefore interesting to explore alternative surrogate modeling strategies that maintain as much of the fidelity of FE2 as possible while still being computationally efficient. In this work, three surrogate modeling approaches are compared in terms of accuracy, efficiency and calibration effort: the state-of-the-art mesoscopic plasticity model by Vogler et al. (Vogler et al., 2013), regularized feed-forward neural networks and hyper-reduced-order models obtained by combining the Proper Orthogonal Decomposition (POD) and Empirical Cubature Method (ECM) techniques. Training datasets are obtained from a Representative Volume Element (RVE) model of the composite microstructure with a number of randomly-distributed linear-elastic fibers surrounded by a matrix with pressure-dependent plasticity. The approaches are evaluated with a comprehensive set of numerical tests comprising pure stress cases and three different stress combinations relevant in the design of laminated composites. The models are assessed on their ability to accurately reproduce the training cases as well as on how well they are able to predict unseen stress combinations. Gains in execution time are compared by using the trained surrogates in the FE2 model of an interlaminar shear test.

Force-field coefficient optimization of coarse-grained molecular dynamics models with a small computational budget

Coarse-grained models in molecular dynamics are low-fidelity models developed to study the properties and behavior of materials. These force-field models are popular due to their simpler implementation and significant computational benefit of their use as compared to complex high-fidelity atomistic models. On the other hand, these advantages often come at the expense of accuracy. One effective way to resolve this issue is tuning their corresponding force-field coefficients such that the estimation error of atomistic properties is minimal with respect to high-fidelity reference data. However, acquisition/computation of these data is often costly, and in most practical cases in materials sciences only a small experimental/computational budget is available to obtain high-fidelity reference data. In this work, a multi-fidelity surrogate modeling strategy for resolving the trade-off between accuracy of a full of set of quantities of interest for an atomistic system and computational expenses is proposed. The proposed procedure takes advantage of the multi-fidelity upper error bound calculated using low-fidelity data by (i) employing an efficient point selection mechanism in the design space and obtaining a set of coarse-grained models, (ii) building stochastic collocation models corresponding to each selected low-fidelity model and obtaining the optimal small group of physical input parameter sets at which a high-fidelity model is evaluated, (iii) constructing a non-intrusive spectral polynomial-based surrogate based on the error of the low-fidelity emulators corresponding to the selected coarse-grained models, and (iv) optimizing force-field coefficients using the resultant surrogate with the objective of error minimization. Moreover, in the context of using a multi-fidelity framework and in order to construct a more accurate emulator, an approach for optimal kernel function selection is implemented in this work. Also, for the purpose of obtaining a set of optimal force-field coefficients using the constructed surrogate model, we used a heuristic population-based optimization approach. For the proof of concept, we applied the proposed approach for the force-field coefficient optimization of the coarse-grained monatomic water (mW) model based on the data obtained from the high-fidelity TIP4P/2005 water model simulations. The outcome of this study is an optimal coarse-grained monatomic water model that along with the TIP4P/2005 water model, provides an accurate emulator for exploration of the properties of water in the designated pressure-temperature input parameter space. Furthermore, this approach provides a more accurate way of sampling important points in the pressure-temperature plane. Finally, the cross-validation results indicate that a certain combination of quantities of interest can produce a more accurate multi-fidelity emulator for the TIP4P/2005 water model.

A Data Compression Strategy for the Efficient Uncertainty Quantification of Time-Domain Circuit Responses

This paper presents an innovative modeling strategy for the construction of efficient and compact surrogate models for the uncertainty quantification of time-domain responses of digital links. The proposed approach relies on a two-step methodology. First, the initial dataset of available training responses is compressed via principal component analysis (PCA). Then, the compressed dataset is used to train compact surrogate models for the reduced PCA variables using advanced techniques for uncertainty quantification and parametric macromodeling. Specifically, in this work sparse polynomial chaos expansion and least-square support-vector machine regression are used, although the proposed methodology is general and applicable to any surrogate modeling strategy. The preliminary compression allows limiting the number and complexity of the surrogate models, thus leading to a substantial improvement in the efficiency. The feasibility and performance of the proposed approach are investigated by means of two digital link designs with 54 and 115 uncertain parameters, respectively.

Principal component analysis and sparse polynomial chaos expansions for global sensitivity analysis and model calibration: Application to urban drainage simulation

This paper presents an efficient surrogate modeling strategy for the uncertainty quantification and Bayesian calibration of a hydrological model. In particular, a process-based dynamical urban drainage simulator that predicts the discharge from a catchment area during a precipitation event is considered. The goal of the case study is to perform a global sensitivity analysis and to identify the unknown model parameters as well as the measurement and prediction errors. These objectives can only be achieved by cheapening the incurred computational costs, that is, lowering the number of necessary model runs. With this in mind, a regularity-exploiting metamodeling technique is proposed that enables fast uncertainty quantification. Principal component analysis is used for output dimensionality reduction and sparse polynomial chaos expansions are used for the emulation of the reduced outputs. Sobol’ sensitivity indices are obtained directly from the expansion coefficients by a mere post-processing. Bayesian inference via Markov chain Monte Carlo posterior sampling is drastically accelerated.

Decomposed-coordinated surrogate modeling strategy for compound function approximation in a turbine-blisk reliability evaluation

Performing probabilistic analyses on a complex structure is challenging with a high computational burden, owing to the many components and multiple disciplines involved, with high nonlinearity and many hyperparameters. Despite the advancements in surrogate models, they are still insufficient to accurately model compound functions with many sub-layers and sub-functions. In this paper, we propose the decomposed-coordinated surrogate model method (DCSMM) to improve the modeling accuracy and efficiency of compound functions. This improvement will enable performing expensive probabilistic analyses, which typically involve thousands of Monte Carlo simulation runs, efficiently. This type of analysis would be too computationally expensive to perform when using full-scaled models. The proposed DCSMM uses the decomposition and coordination strategy, and combines it with surrogate modeling methods. In this work, we establish the mathematical model of the DCSMM with quadratic polynomial (QP) and Kriging model, and develop QP-DCSMM (the DCSMM based on QP), K-DCSMM (the DCSMM based on Kriging) and M-DCSMM (the DCSMM based on the mixture of QP and Kriging). The approximation accuracy and simulation performance (including computational precision and efficiency) of the DCSMM are demonstrated with an analytical model and a turbine blisk multi-failure modes of an aeroengine as an engineering case study. The proposed DCSMM is demonstrated to be effective in modeling the high-nonlinearity between output response and input variables, in addition to being robust. These benefits become even more prominent as we increase the number of Monte Carlo simulation runs. Overall, this study shows a high-efficiency and high-precision approximation method for complex compound functions and complex structures. This contribution will further enrich the theory and application of probabilistic statistical analysis as well. This paper also offers useful insights into engineering optimization and reliability design pertaining to multi-model mechanical systems.

Surrogate modeling of phase equilibrium calculations using adaptive sampling

Equation of state models as the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) model are accurate and reliable prediction models for phase equilibria. But due to their iterative nature, they are difficult to apply in chemical process optimization, because of long computation times. To overcome this issue, surrogate modeling – replacing a complex model by a black-box model – can be used. A novel surrogate modeling strategy for phase equilibria is presented, combining the training of a classifier model with regression models for the phase composition using a mixed adaptive sampling method. We discuss the selection of the parameters of the sampling algorithm and a suitable stop criterion for the example ternary liquid-liquid equilibrium system of n-decane, dimethylformamide and 1-dodecene in detail. The sequential mixed adaptive sampling method is compared to the one-shot Latin hypercube sampling design.

High-Dimensional Reliability-Based Design Optimization Involving Highly Nonlinear Constraints and Computationally Expensive Simulations

Reliability-based design optimization (RBDO) aims at optimizing the design of an engineered system to minimize the design cost while satisfying reliability requirements. However, it is challenging to perform RBDO under high-dimensional uncertainty due to the often prohibitive computational burden. In this paper, we address this challenge by leveraging a recently developed method for reliability analysis under high-dimensional uncertainty. The method is termed high-dimensional reliability analysis (HDRA). The HDRA method optimally combines the strengths of univariate dimension reduction (UDR) and kriging-based reliability analysis to achieve satisfactory accuracy with an affordable computational cost for HDRA problems. In this paper, we improve the computational efficiency of high-dimensional RBDO by pursuing two new strategies: (i) a two-stage surrogate modeling strategy is adopted to first locate a highly probable region of the optimum design and then locally refine the accuracy of the surrogates in this region; and (ii) newly selected samples are updated for all the constraints during the sequential sampling process in HDRA. The results of two mathematical examples and one real-world engineering example suggest that the proposed HDRA-based RBDO (RBDO-HDRA) method is capable of solving high-dimensional RBDO problems with higher accuracy and comparable efficiency than the UDR-based RBDO (RBDO-UDR) and ordinary kriging-based RBDO (RBDO-kriging) methods.

Expensive Black-Box Model Optimization Via a Gold Rush Policy

The optimization of black-box models is a challenging task owing to the lack of analytic gradient information and structural information about the underlying function, and also due often to significant run times. A common approach to tackling such problems is the implementation of Bayesian global optimization techniques. However, these techniques often rely on surrogate modeling strategies that endow the approximation of the underlying expensive function with nonexistent features. Further, these techniques tend to push new queries away from previously queried design points, making it difficult to locate an optimum point that rests near a previous model evaluation. To overcome these issues, we propose a gold rush (GR) policy that relies on purely local information to identify the next best design alternative to query. The method employs a surrogate constructed pointwise, that adds no additional features to the approximation. The result is a policy that performs well in comparison to state of the art Bayesian global optimization methods on several benchmark problems. The policy is also demonstrated on a constrained optimization problem using a penalty method.

Maximum energy conversion from human motion using piezoelectric flex transducer: A multi-level surrogate modeling strategy

Conventional engineering design optimization requires a large amount of expensive experimental tests from prototypes or computer simulations, which may result in an inefficient and unaffordable design process. In order to overcome these disadvantages, a surrogate model may be used to replace the prototype tests. To construct a surrogate model of sufficient accuracy from limited number of tests/simulations, a multi-level surrogate modeling strategy is introduced in this article. First, a chosen number of points determined by optimal Latin Hypercube Design of Experiments are used to generate global-level surrogate models with genetic programming and the fitness landscape can be explored by genetic algorithms for near-optimal solutions. Local-level surrogate models are constructed then from the extended-optimal Latin Hypercube samples in the vicinity of global optimum on the basis of a much smaller number of chosen points. As a result, an improved optimal design is achieved. The efficiency of this strategy is demonstrated by the parametric optimization design of a piezoelectric flex transducer energy harvester. The optimal design is verified by finite element simulations and the results show that the proposed multi-level surrogate modeling strategy has the advantages of faster convergence and more efficiency in comparison with the conventional single-single level surrogate modeling technique.

Strategies for efficient machine learning of surrogate drag models from three-dimensional mesoscale computations of shocked particulate flows

Macroscale simulations of shocked particulate flows rely on closure laws to model momentum transfer between the fluid and dispersed particles phase. Developing closure models from experimental data is expensive. Robust and accurate closures laws can be obtained through surrogate modeling using high-resolution mesoscale simulations. However, development of surrogate models for drag from 3D high-fidelity simulations of shock interaction with clusters of particles can be computationally prohibitive. This paper explores various strategies to efficiently construct surrogate models for drag on particles in the shocked flow. The cost of generating training data is reduced by selecting optimal grid resolutions, particle arrangements in clusters, and size of particle clusters, i.e., by selecting suitable representative volumes (RVEs). Different surrogate modeling strategies such as multi-fidelity and parameter-by-parameter construction approaches are examined. The surrogate models obtained from the different methods are compared to determine the most cost-effective machine learning based surrogate modeling method in the context of shock-particle interactions.

A two-step surrogate modeling strategy for single-objective and multi-objective optimization of radiofrequency circuits

The knowledge-intensive radiofrequency circuit design and the scarce design automation support play against the increasingly stringent time-to-market demands. Optimization algorithms are starting to play a crucial role; however, their effectiveness is dramatically limited by the accuracy of the evaluation functions of objectives and constraints. Accurate performance evaluation of radiofrequency passive elements, e.g., inductors, is provided by electromagnetic simulators, but their computational cost makes their use within iterative optimization loops unaffordable. Surrogate modeling strategies, e.g., Kriging, support vector machines, artificial neural networks, etc., arise as a promising modeling alternative. However, their limited accuracy in this kind of applications has prevented a widespread use. In this paper, inductor performance properties are exploited to develop a two-step surrogate modeling strategy in order to evaluate the behavior of inductors with high efficiency and accuracy. An automated design flow for radiofrequency circuits using this surrogate modeling of passive components is presented. The methodology couples a circuit simulator with evolutionary computation algorithms such as particle swarm optimization, genetic algorithm or non-dominated sorting genetic algorithm (NSGA-II). This methodology ensures optimal performances within short computation times by avoiding electromagnetic simulations of inductors during the entire optimization process and using a surrogate model that has less than 1% error in inductance and quality factor when compared against electromagnetic simulations. Numerous real-life experiments of single-objective and multi-objective low-noise amplifier design demonstrate the accuracy and efficiency of the proposed strategies.

Radio-frequency inductor synthesis using evolutionary computation and Gaussian-process surrogate modeling

In recent years, the application of evolutionary computation techniques to electronic circuit design problems, ranging from digital to analog and radiofrequency circuits, has received increasing attention. The level of maturity runs inversely to the complexity of the design task, less complex in digital circuits, higher in analog ones and still higher in radiofrequency circuits. Radiofrequency inductors are key culprits of such complexity. Their key performance parameters are inductance and quality factors, both a function of the frequency. The inductor optimization requires knowledge of such parameters at a few representative frequencies. Most common approaches for optimization-based radiofrequency circuit design use analytical models for the inductors. Although a lot of effort has been devoted to improve the accuracy of such analytical models, errors in inductance and quality factor in the range of 5%–25% are usual and it may go as high as 200% for some device sizes. When the analytical models are used in optimization-based circuit design approaches, these errors lead to suboptimal results, or, worse, to a disastrous non-fulfilment of specifications. Expert inductor designers rely on iterative evaluations with electromagnetic simulators, which, properly configured, are able to yield a highly accurate performance evaluation. Unfortunately, electromagnetic simulations typically take from some tens of seconds to a few hours, hampering their coupling to evolutionary computation algorithms. Therefore, analytical models and electromagnetic simulation represent extreme cases of the accuracy-efficiency trade-off in performance evaluation of radiofrequency inductors. Surrogate modeling strategies arise as promising candidates to improve such trade-off. However, obtaining the necessary accuracy is not that easy as inductance and quality factor at some representative frequencies must be obtained and both performances change abruptly around the self-resonance frequency, which is particular to each device and may be located above or below the frequencies of interest. Both, offline and online training methods will be considered in this work and a new two-step strategy for inductor modeling is proposed that significantly improves the accuracy of offline methods The new strategy is demonstrated and compared for both, single-objective and multi-objective optimization scenarios. Numerous experimental results show that the proposed two-step approach outperforms simpler application strategies of surrogate modelling techniques, getting comparable performances to approaches based on electromagnetic simulation but with orders of magnitude less computational effort.

Snap-Through Buckling Reliability Analysis Under Spatiotemporal Variability and Epistemic Uncertainty

Surrogate models often provide an effective tradeoff between accuracy and efficiency during reliability analysis with expensive physics models. In snap-through buckling reliability analysis, a surrogate model could be built for the critical buckling load, as a function of loading, material properties, geometry, and boundary conditions. However, in the presence of spatiotemporal variability, the response surface of the critical buckling load is often highly nonlinear and irregular, thus rendering commonly used response surface-type surrogate modeling strategies ineffective. This paper proposes a new buckling reliability analysis method based on support vector machines for structures subjected to spatiotemporal variability and in the presence of epistemic uncertainty regarding model inputs and parameters. Bayesian calibration is first used to quantify the epistemic uncertainty in the modeling of spatiotemporal variability under limited data. Upon the modeling of spatiotemporal variability and epistemic uncertainty, a time-dependent reliability analysis method is developed for the snap-through buckling failure by constructing a nonlinear support vector machine classifier. Considering that the computer simulation is computationally expensive and the support vector machine classifier may not be well trained due to limited computational resources, a method is also developed to quantify the uncertainty in the reliability estimate due to classification uncertainty. A curved beam with an uncertain boundary condition, spatially varying cross-section geometry, and spatiotemporally varying loading is used to demonstrate the effectiveness of the proposed method.

Optimization of an Implicit Large-Eddy Simulation Method for Underresolved Incompressible Flow Simulations

In engineering applications, resolution is often low. In these underresolved regions, the truncation error of the underlying numerical schemes strongly affects the solution. If the truncation error functions as a physically consistent subgrid-scale model (that is, it models the evolution of otherwise resolved scales), resolution may remain low. Thereby, computational efficiency is improved. The sixth-order adaptive central-upwind weighted essentially nonoscillatory scheme with implicit scale separation, denoted as WENO-CU6-M1, potentially allows for physically consistent implicit subgrid-scale modeling, when shaped accordingly. In this work, finding an optimal formulation of WENO-CU6-M1 is considered within a deterministic design optimization framework. Possible surrogate modeling and sampling strategies are considered. Design optimization is based on evaluating the potential of a WENO-CU6-M1 scheme formulation to reproduce Kolmogorov scaling for a Taylor–Green vortex in its quasi-isotropic state. As in the absence of physical viscosity, kinetic energy dissipates exclusively due to the subgrid-scales, the Reynolds number is infinite, and the evolution of the flow is determined by proper subgrid-scale modeling. To complete the work, the effective numerical dissipation rate of the WENO-CU6-M1 model optimized for artificially compressible fluid flows is quantified, and it is compared to the original one. Not only is the zero viscosity limit considered, but the model behavior is benchmarked offdesign, for low to high Reynolds numbers. A comparison to an alternative explicit and implicit subgrid-scale model demonstrates its superior behavior for the chosen test flow.

Isothermal Combustor Prediffuser and Fuel Injector Feed Arm Design Optimization Using the prometheus Design System

The prometheus combustor design system aims to reduce the complexity of evaluating combustor designs by automatically defining preprocessing, simulation, and postprocessing tasks based on the automatic identification of combustor features within the computer-aided design (CAD) environment. This system enables best practice to be codified and topological changes to a combustor's design to be more easily considered within an automated design process. The following paper presents the prometheus combustor design system and its application to the multiobjective isothermal optimization of a combustor prediffuser and the multifidelity isothermal optimization of a fuel injector feed arm in combination with a surrogate modeling strategy accelerated via a high-performance graphical processing unit (GPU).

An Innovative Surrogate-Based Searching Method for Reducing Warpage and Cycle Time in Injection Molding

This paper presents a hybrid multiobjective optimization system for minimizing the warpage and cycle time of the plastic injection molding process. During optimization, an innovative adaptive Kriging surrogate modeling strategy is applied to substitute the computationally intensive numerical simulation of the injection molding process. The system consists of two main stages. In the first stage, the Taguchi optimization method, the analysis of variance, and the signal-to-noise ratio are employed to find the significant process parameters. The most influential factors’ domains will be divided into two subregions equally in the next step. The Kriging surrogate model is developed to obtain the mathematical relationship between the warpage and process parameters based on the fractional factorial design. A parallel efficient global optimization, named subregional efficient global optimization (SEGO), is proposed. SEGO algorithm is continued till the convergence accuracy is satisfied, and the sufficiently accurate Kriging surrogate model is established. In the second stage, the nondominated sorting-based genetic algorithm II is used to search for the pareto-optimal solutions for two objectives. Numerical result shows that the pareto-frontier is well identified with a small number of simulation runs. The proposed approach can help engineers obtain optimal process conditions and achieve competitive advantages of product quality and efficiency.

Comparison of Adaptive Sampling Methods for Generation of Surrogate Aerodynamic Models

A surrogate modeling strategy, using effective interpolation and sampling methods, facilitates a reduction in the number of computational fluid dynamics simulations required to construct an aerodynamic model to a specified accuracy. In this paper, two adaptive sampling strategies are compared for generating surrogate models, based on Kriging and radial basis function interpolation, respectively. The relationships between the two model formulations are discussed, and three test cases are considered, including analytic functions and recovery of aerodynamic coefficients for two example applications: longitudinal flight mechanics analysis for the DLR-F12 aircraft and structural loads analysis of an RAE2822 airfoil. For the airfoil example, models of , , and were constructed with the two sampling strategies using Euler/boundary-layer-coupled computational fluid dynamics and a three-dimensional flight envelope of incidence, Mach, and Reynolds number. The two sampling approaches direct some samples toward exploration of the domain by minimizing model uncertainty and some toward refinement of local nonlinearities, by adapting to model curvature or extrema. The results provide some evidence that, for certain functions, in certain scenarios, each update scheme could be useful. Both methods were at least better than traditional space-filling sampling for all the test cases presented.

Modeling of turbulent, isothermal and cryogenic cavitation under attached conditions

Cavitation is often triggered when the fluid pressure is lower than the vapor pressure at a local thermodynamic state. The present article reviews recent progress made toward developing modeling and computational strategies for cavitation predictions under both isothermal and cryogenic conditions, with an emphasis on the attached cavity. The review considers alternative cavitation models along Reynolds-averaged Navier–Stokes and very lager eddy simulation turbulence approaches to ensure that the computational tools can handle flows of engineering interests. Observing the substantial uncertainties associated with both modeling and experimental information, surrogate modeling strategies are reviewed to assess the implications and relative importance of the various modeling and materials parameters. The exchange between static and dynamic pressures under the influence of the viscous effects can have a noticeable impact on the effective shape of a solid object, which can impact the cavitation structure. The thermal effect with respect to evaporation and condensation dynamics is examined to shed light on the fluid physics associated with cryogenic cavitation. The surrogate modeling techniques are highlighted in the context of modeling sensitivity assessment.

A hybrid of mode-pursuing sampling method and genetic algorithm for minimization of injection molding warpage

This paper presents a hybrid optimization method for minimizing the warpage of injection molded plastic parts. This proposed method combines a mode-pursuing sampling (MPS) method with a conventional global optimization algorithm, i.e. genetic algorithm, to search for the optimal injection molding process parameters. During optimization, Kriging surrogate modeling strategy is also exploited to substitute the computationally intensive Computer-Aided Engineering (CAE) simulation of injection molding process. With the application of genetic algorithm, the “likelihood-global optimums” are identified; and the MPS method generates and chooses new sample points in the neighborhood of the current “likelihood-global optimums”. By integrating the two algorithms, a new sampling guidance function is proposed, which can divert the search process towards the relatively unexplored region resulting in less likelihood of being trapped at the local minima. A case study of a food tray plastic part is presented, with the injection time, mold temperature, melt temperature and packing pressure selected as the design variables. This case study demonstrates that the proposed optimization method can effectively reduce the warpage in a computationally efficient manner.