Kriging-based Surrogate Model Research Articles

Improved Method for Distributed Parameter Model of Solenoid Valve Based on Kriging Basis Function Predictive Identification Program

In this paper, a method for the improvement of the calculation accuracy of the distributed parameter model (DPM) of electromagnetic devices is proposed based on the kriging basis function predictive identification program (PIP). Kriging is mainly an optimal interpolation method which uses spatial self-covariance, and takes a polynomial as the basis function. The accuracy of the kriging-based surrogate model can be improved by adjusting the related functions and hyperparameters. Based on the DPM of a solenoid valve, there exist certain errors in the estimation. They can be summarized as follows: Firstly, the estimation error of magnetic flux leakage (MFL) permeance is caused directly by the deviation of the magnetic flux tube due to the segmented magnetic field line. Secondly, the estimation error of soft magnetic resistance because of the nonlinearity of the permeability of soft magnetic material leads to the change of soft magnetic resistance alongside the magnetic flux. In this paper, an improved kriging error correction method is applied to modify the leak permeance and soft magnetic resistance calculation. The kriging basis function is adjusted to adapt to the data curve of the MFL permeance error data. The calculated MFL permeance data are compared with the error variation data to select the appropriate basis function. To improve the computational efficiency, the PIP is proposed to select the appropriate basis function. The modified MFL permeance data and soft magnetic resistance are substituted into the DPM for improving the computational accuracy and efficiency of the solenoid valve.

DEVELOPMENTS IN STORM SURGE ESTIMATION USING SURROGATE MODELING TECHNIQUES

Numerical advances in storm surge prediction over the past couple of decades have produced high-fidelity simulation models that permit a detailed representation of hydrodynamic processes and therefore support high accuracy forecasting. Unfortunately, the computational burden of such numerical models is large, requiring thousands of CPU hours for each simulation, something that limits their applicability for hurricane risk assessment. Use of Kriging-based surrogate modeling techniques has been examined to address the aforementioned challenge Jia et al. [2016], Zhang et al. [2018]. This approach can provide fast predictions using a database of high-fidelity, synthetic storms, with the goal of maintaining the accuracy of the numerical model utilized to produce this database, while offering computational efficiency. This contribution overviews initially recent research developments for the application of Kriging for storm surge predictions. Topics discussed include: enhancement of the initial database for nodes (i.e., geographical locations) that have remained dry in some of the database storms; adaptive selection of storms forming the initial database; use of different surrogate modeling tuning techniques and their impact on the metamodel predictive capabilities for storm surge estimation; implementation for estimation of impact due to near-shore processes (breaking waves), something that requires coupling of different numerical models.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/vL38Kv3kLDM

From probabilistic back analyses to probabilistic run-out predictions of landslides: A case study of Heifangtai terrace, Gansu Province, China

The input parameters of dynamic numerical models for landslide run-out distance simulations cannot be measured accurately, which results in significant uncertainties in the predictions obtained using these models. To address this issue, a novel framework combining probabilistic back analyses and probabilistic predictions of landslide run-out distance is developed in this study, with the aim of ensuring the reliability of run-out predictions in risk assessments. First, a probabilistic back analysis method is applied to calibrate the input parameters and quantify their uncertainties; the calibrated parameters are then used for a probabilistic forward run-out prediction. Unlike the trial and error calibration approach, the back analysis method is implemented probabilistically by modeling the input parameters as random variables and efficiently improving their distributions through Markov chain Monte Carlo-based Bayesian inference using the observed run-out distance of a previous landslide event. These improved posterior distributions are employed to obtain the run-out distance exceedance probability curve of a potential landslide similar to the previous event; this curve can be used to estimate the probability of the run-out distance being exceeded at different locations along the landslide's path. By selecting different threshold values for the exceedance probabilities, a landslide hazard zoning map can be generated, serving as a valuable tool for landslide risk assessment and management. Moreover, a Kriging-based surrogate model is used to approximate the dynamic numerical model employed in this study, which significantly improves the computational efficiency of the proposed methodology. Finally, two sequential landslides that have occurred in the Heifangtai terrace of Gansu Province, China, are considered as a case study to demonstrate the performance of the proposed methodology.

Biomechanical rupture risk assessment of abdominal aortic aneurysms using clinical data: A patient-specific, probabilistic framework and comparative case-control study.

We present a data-informed, highly personalized, probabilistic approach for the quantification of abdominal aortic aneurysm (AAA) rupture risk. Our novel framework builds upon a comprehensive database of tensile test results that were carried out on 305 AAA tissue samples from 139 patients, as well as corresponding non-invasively and clinically accessible patient-specific data. Based on this, a multivariate regression model is created to obtain a probabilistic description of personalized vessel wall properties associated with a prospective AAA patient. We formulate a probabilistic rupture risk index that consistently incorporates the available statistical information and generalizes existing approaches. For the efficient evaluation of this index, a flexible Kriging-based surrogate model with an active training process is proposed. In a case-control study, the methodology is applied on a total of 36 retrospective, diameter matched asymptomatic (group 1, n = 18) and known symptomatic/ruptured (group 2, n = 18) cohort of AAA patients. Finally, we show its efficacy to discriminate between the two groups and demonstrate competitive performance in comparison to existing deterministic and probabilistic biomechanical indices.

Ordinary kriging - and cokriging - based surrogate model for ionospheric TEC prediction using NavIC/GPS data

The ionospheric total electron content (TEC) plays a major role in the estimation of positional accuracy of satellite-based navigation systems. TEC disturbs the transmission of the signal transmitted from the satellite to receiver, which leads to range error. Hence, the prediction of VTEC can rectify the range errors in advance. Navigation with Indian Constellation (NavIC) is a navigational satellite system that gives position of a user and time information in India. NavIC receiver is installed at ACSCE, Bangalore, and it is utilized to estimate vertical total electron content (VTEC). In this paper, VTEC estimated from ACSCE station and VTEC obtained from IISC station, Bangalore, are compared and also ordinary kriging-based surrogate model (OKSM) and cokriging-based surrogate model (COKSM) are developed for forecasting NavIC/GPS VTEC. In these models, input parameters comprise the time, Bt, SSN, Ap and F10.7. NavIC and GPS VTEC are predicted for quiet days, i.e. from 16/1/2018 to 26/1/2018 and from 7/8/2018 to 13/8/2018, by using previous one week of VTEC data, and input parameters belong to ACSCE (12.8914° N, 77.4657° E) and IISC (13.0219° N, 77.5671° E) stations, Bangalore. The VTEC prediction results from OKSM and COKSM are compared. It shows that OKSM results are comparatively better than COKSM. In order to validate our developed models, OKSM and COKSM prediction results are compared with the SWIF model during storm days. Based on the comparison, it is observed that RMSE of OKSM is 3.5202 TECU, COKSM gives RMSE as 4.7126 TECU, and the SWIF model’s RMSE is 4.6804. From the comparison results, it is evident that OKSM yields better prediction results than COKSM and SWIF. These results indicate that the proposed model will be useful for correcting range measurement data in advance.

Surrogate Assisted Optimization for Low-Voltage Low-Power Circuit Design

Low-voltage low-power (LVLP) circuit design and optimization is a hard and time-consuming task. In this study, we are interested in the application of the newly proposed meta-modelling technique to alleviate such burdens. Kriging-based surrogate models of circuits’ performances were constructed and then used within a metaheuristic-based optimization kernel in order to maximize the circuits’ sizing. The JAYA algorithm was used for this purpose. Three topologies of CMOS current conveyors (CCII) were considered to showcase the proposed approach. The achieved performances were compared to those obtained using conventional LVLP circuit sizing techniques, and we show that our approach offers interesting results.

Robust aerodynamic shape optimization using a novel multi-objective evolutionary algorithm coupled with surrogate model

The fruit fly optimization algorithm (FOA) is a newly developed bio-inspired algorithm that has exhibited enormous values in engineering optimization. In this study, an effective multi-objective fruit fly optimization algorithm (MOFOA) incorporated with the Pareto dominance is investigated and applied to robust aerodynamic shape optimization considering uncertainties in the design process. In the MOFOA, the fruit flies are required to perform solution explorations within an adaptive search scope by flying around the Pareto non-dominated solutions. An enhanced clustering evolution mechanism of combining the cooperative local search and differential evolution operator is elaborately designed to enrich the exploratory capacities for complex optimization. Paired with the search strategy, the non-dominated sorting technique considering the quality as well as uniform distribution of Pareto solutions is adopted to deal with the multiple objectives. To generate high-quality and uniformly spread-out Pareto non-dominated solutions, the roulette wheel selection (RWS) operator based on crowding distance is employed to guide the selection processes of individuals throughout various stages of the MOFOA. The effectiveness and superiority of MOFOA are demonstrated by the comparative studies on several benchmark functions. Finally, a robust aerodynamic design system for combining the MOFOA optimizer with a Kriging-based surrogate model and full Navier–Stokes computation is advantageously implemented to perform shape optimization on a transonic airfoil using the Taguchi robust design methodology that emphasizes the inherent variability while improving engineering productivity. The design results show that the robust design, compared with the single-point design, is capable of producing a set of robust solutions that are significantly less sensitive to input variations by capturing the Pareto-optimal front with respect to the criteria of aerodynamic performance and its stability.

Surrogate modeling for fast uncertainty quantification: Application to 2D population balance models

Surrogate modeling is a useful tool for enabling uncertainty quantification (UQ) tasks that require many expensive model evaluations, as it replaces expensive high-fidelity models with cheap-to-evaluate surrogates. This paper investigates sparse polynomial chaos and Kriging methods for surrogate modeling of first-principles models with probabilistic uncertainty in parameters and initial conditions. The surrogate modeling methods are demonstrated on a 2-dimensional population balance (2D-PB) model for batch cooling crystallization of ibuprofen with 20 uncertain parameters. Our analysis indicates that not only sparse polynomial chaos expansions are powerful for probabilistic UQ, but also the approximation accuracy of Kriging surrogate models can be significantly improved when polynomial chaos expansions are used to describe their trend. A basis-adaptive least-angle-regression strategy is shown to be particularly useful for inducing sparsity in polynomial chaos expansions, allowing for dealing with problems with a relatively large number of uncertain inputs. The utility of sparse polynomial chaos- and Kriging-based surrogate models is illustrated for various forward and inverse UQ problems, including global sensitivity analysis as well as Bayesian and maximum a posteriori parameter estimation of the 2D-PB model, where massive savings in computational cost (up to 30,000-fold) are observed.

A Bayesian Updating Method for Non-Probabilistic Reliability Assessment of Structures with Performance Test Data

For structures that only the predicted bounds of uncertainties are available, this study proposes a Bayesian method to logically evaluate the nonprobabilistic reliability of structures based on multi-ellipsoid convex model and performance test data. According to the given interval ranges of uncertainties, we determine the initial characteristic parameters of a multi-ellipsoid convex set. Moreover, to update the plausibility of characteristic parameters, a Bayesian network for the information fusion of prior uncertainty knowledge and subsequent performance test data is constructed. Then, an updated multi-ellipsoid set with the maximum likelihood of the performance test data can be achieved. The credible non-probabilistic reliability index is calculated based on the Kriging-based surrogate model of the performance function. Several numerical examples are presented to validate the proposed Bayesian updating method.

Reliability-based design optimization of crane bridges using Kriging-based surrogate models

Cranes as indispensable and important hoisting machines of modern manufacturing and logistics systems have been wildly used in factories, mines, and custom ports. For crane designs, the crane bridge is one of the most critical systems, as its mechanical skeleton bearing and transferring the operational load and the weight of the crane itself thus must be designed with sufficient reliability in order to ensure safe crane services. Due to extremely expensive computational costs, current crane bridge design has been primarily focused either on deterministic design based on conventional design formula with empirical parameters from designers’ experiences or on reliability-based design by employing finite-element analysis. To remove this barrier, the paper presents the study of using an advanced surrogate modeling technique for the reliability-based design of the crane bridge system to address the computational challenges and thus enhance design efficiency. The Kriging surrogate models are first developed for the performance functions for the crane system design and used for the reliability-based design optimization. Comparison studies with existing crane design methods indicated that employing the surrogate models could substantially improve the design efficiency while maintaining good accuracy.

Prediction of ionospheric vertical total electron content from GPS data using ordinary kriging-based surrogate model

The total electron content (TEC) is the number of electrons present in a ray path between satellite and receiver. TEC affects the propagation of radio signal from satellite to receiver, which causes a ranging error. In the case of single frequency user receiver, the prediction of TEC helps to correct the range errors. TEC is measured using TEC unit (TECU), where $1~\text{TECU} = 1 \times 10^{16}~\text{electrons}/\text{m}^{2}$ . The vertical total electron content (VTEC) of the L1 band is estimated by using data collected from GPS receiver which is installed at the ACSCE station Bangalore. In this work, an ordinary kriging (OK)-based surrogate model algorithm and Matlab code is developed and used to predict the hourly basis ionospheric VTEC. Six parameters such as time, sun spot number (SSN), the solar flux index at 10.7 cm (F10.7), Kp and Ap and observed TEC are used to build the surrogate model. These parameters are related to the ionospheric diurnal variations, solar cycle and geomagnetic activities. In order to cover all regions of the world during different solar activity periods six input parameters are collected from low-, mid- and high-latitude regions during low, medium and high solar activity periods. The root mean square error (RMSE) of the OK-based surrogate model ranges from 0.6–3.6 TECU, and the correlation coefficient varies between 0.79–0.99 and range error varies from 0.04654–0.3017244 m at three regions. The TEC prediction results from the OK-based surrogate model are compared with results obtained from (i) a genetic algorithm-based neural network (GA-NN), (ii) a back-propagation neural network (BP-NN) and (iii) the IRI-2012 model at the CHAN station. For a typical dataset the RMSE of the OK-based surrogate model is 4.523 TECU and the correlation coefficient is 0.9733. The RMSE values for the GA-NN, BP-NN and IRI-2012 models are 5.3529, 6.2913 and 6.7179 TECU and The correlation coefficients are 0.8343, 0.7869 and 0.7797, respectively. The OK model is also compared with a time series method for the CHAN station; it is observed that, for 3 days (7-1-2008 to 9-1-2008) prediction, the OK model gives a 75% result for $\Delta \textit{TEC} <1~\text{TECU}$ condition, the time series method gives 39% for the same condition. The results indicate that the OK-based surrogate model is suitable for applications in ionospheric TEC predictions.

Simulation and optimization of cooling water systems with closed wet cooling tower based on reduced order model

This study presents a CFD (computational fluid dynamics)-assistant multi-scale modeling and optimization strategy for synthesizing cooling water systems which incorporates multiple closed wet cooling towers (CWCTs) installed with plain tubes. The utilization of CFD is capable of providing accurate and visual results with respect to complex thermal and fluid flow phenomena that can be used to explore the synergistic mechanism of multi-parameter for CWCTs. Based on CFD results, a systematic reduced order model (ROM) for CWCTs is developed by closely approximating the high-fidelity CWCTs models through integrating principal component analysis (PCA) with Kriging-based surrogate model. This ROM is accessible to the desired result through input data without accounting for the complex geometries and detailed thermal-hydraulics property models. The proposed CFD-assistant ROM for the CWCTs is applied to enhance solution convergence and obtain optimal results of cooling water systems. An industrial case study is used to demonstrate the superiority of the proposed CFD-assistant strategy for the cooling water system synthesis and optimization.

Design of a Fractional Order Frequency PID Controller for an Islanded Microgrid: A Multi-Objective Extremal Optimization Method

Fractional order proportional-integral-derivative(FOPID) controllers have attracted increasing attentions recently due to their better control performance than the traditional integer-order proportional-integral-derivative (PID) controllers. However, there are only few studies concerning the fractional order control of microgrids based on evolutionary algorithms. From the perspective of multi-objective optimization, this paper presents an effective FOPID based frequency controller design method called MOEO-FOPID for an islanded microgrid by using a Multi-objective extremal optimization (MOEO) algorithm to minimize frequency deviation and controller output signal simultaneously in order to improve finally the efficient operation of distributed generations and energy storage devices. Its superiority to nondominated sorting genetic algorithm-II (NSGA-II) based FOPID/PID controllers and other recently reported single-objective evolutionary algorithms such as Kriging-based surrogate modeling and real-coded population extremal optimization-based FOPID controllers is demonstrated by the simulation studies on a typical islanded microgrid in terms of the control performance including frequency deviation, deficit grid power, controller output signal and robustness.

Evolutionary Reliable Regional Kriging Surrogate and Soft Outer Array for Robust Engineering Optimization

Robust optimization seeks designs with optimized performance and low sensitivity to possible variations in a product’s life-cycle. As a popular robust design scheme in industry, Taguchi method uses the signal-to-noise ratio (SNR) as a metric of robustness. However, the Taguchi experimental design includes an inner orthogonal arrays (OA) for control factors and an outer OA for noise factors in estimating SNR-based robustness, raising a serious cost concern, especially if expensive samples are involved. Furthermore, rigorous control of noise factors to prespecified levels in the outer OA is impractical in engineering applications. This paper presents a novel approach, robust optimization using evolving reliable Kriging surrogate (ROERKS) that uses an evolving surrogate model to approximate the actual system, and uses a soft outer array to estimate the robustness. Both control variables and noise factors are merged into a combined experimental design served as the training samples to construct a Kriging-based surrogate model. An evolutionary optimizer is applied to search of the subspace of the design variables for a robust optimal solution, and a soft outer array is introduced to estimate the fitness function consisted of the mean and the variance response of evolving individual. To accommodate reduced accuracy of the surrogate model owing to an inadequate sample size, an issue that commonly arises in expensive optimization, the proposed algorithm uses a reliable region to constrain the genetic algorithm search. The verification result of the quasi-optimum is then added to the training samples to refine the evolving surrogate and to adjust the reliable region. A hybrid infilling strategy is then introduced to prevent the early convergence of the quasi-optimum. If the predicted optimum is in close proximity to current samples, the infilling strategy switches to an alternative sample with maximum expected improvement to improve sample efficiency. The iteration process continues until it converges to a robust optimum. The robust optimization of a numerical example for minimization and the robust optimization of a micro accelerometer design with a nominal-the-best sensitivity are presented to demonstrate the effectiveness of the proposed method. ROERKS outperforms Taguchi method and the RSM-based robust optimization, and derives a superior robust optimum using many fewer experiments.

Genetic algorithm optimized Taylor Kriging surrogate model for system reliability analysis of soil slopes

A Kriging-based surrogate model provides a logically strict and efficient tool to evaluate the system reliability of a slope. However, the constant trend function adopted in the ordinary Kriging (OK) cannot always well capture the nonlinear non-smooth properties of a slope stability problem. Although the universal Kriging (UK) with a linear or a quadratic trend function could be an alternative for some cases, a higher order nonlinear trend function is preferable for some more complicated nonlinear non-smooth cases in the slope stability analysis. To address this problem, a genetic algorithm (GA) optimized Taylor Kriging (TK) surrogate model is proposed for the system reliability analysis of soil slopes in this paper. The proposed surrogate model allows a unified framework of the Kriging, considering different extents of nonlinear properties according to the Taylor expansion order (e.g., can be as high as the fourth order). The GA is introduced to search for the optimal correlation parameters, of which the effectiveness is verified by an analytical example. The feasibility of the proposed surrogate model is then validated by two analytical examples before its application to the practical slope reliability analyses. The results show that the UK model can be incorporated into the TK model, and the TK model provides a higher accuracy and efficiency when facing the highly nonlinear slope stability problems. It is also found that the UK model cannot fully capture the potential nonlinear properties existed in a slope stability model as compared with the higher order TK model.

Time-variant reliability assessment through equivalent stochastic process transformation

Time-variant reliability measures the probability that an engineering system successfully performs intended functions over a certain period of time under various sources of uncertainty. In practice, it is computationally prohibitive to propagate uncertainty in time-variant reliability assessment based on expensive or complex numerical models. This paper presents an equivalent stochastic process transformation approach for cost-effective prediction of reliability deterioration over the life cycle of an engineering system. To reduce the high dimensionality, a time-independent reliability model is developed by translating random processes and time parameters into random parameters in order to equivalently cover all potential failures that may occur during the time interval of interest. With the time-independent reliability model, an instantaneous failure surface is attained by using a Kriging-based surrogate model to identify all potential failure events. To enhance the efficacy of failure surface identification, a maximum confidence enhancement method is utilized to update the Kriging model sequentially. Then, the time-variant reliability is approximated using Monte Carlo simulations of the Kriging model where system failures over a time interval are predicted by the instantaneous failure surface. The results of two case studies demonstrate that the proposed approach is able to accurately predict the time evolution of system reliability while requiring much less computational efforts compared with the existing analytical approach.

A Fully Optimized Electrowinning Cell for Achieving a Uniform Current Distribution at Electrodes Utilizing Sampling-Based Sensitivity Approach

In this paper, a zinc electrowinning cell is fully optimized to achieve a uniform current distribution at electrode surfaces. To effectively deal with an electromagnetically coupled problem with multi-dimensional design variables, a sampling-based sensitivity approach is combined with a highly tuned multiphysics simulation model. The model involves the interrelation between electrochemical reactions and electromagnetic phenomena so as to predict accurate current distributions in the electrowinning cell. In the sampling-based sensitivity approach, Kriging-based surrogate models are generated in a local window, and accordingly their sensitivity values are extracted. Such unique design strategy facilitates optimizing very complicated multiphysics and multi-dimensional design problems. Finally, ten design variables deciding the electrolytic cell structure are optimized, and then the uniformity of current distribution in the optimized cell is examined through the comparison with existing cell designs.

Performance study of gradient-enhanced Kriging

The use of surrogate models for approximating computationally expensive simulations has been on the rise for the last two decades. Kriging-based surrogate models are popular for approximating deterministic computer models. In this work, the performance of Kriging is investigated when gradient information is introduced for the approximation of computationally expensive black-box simulations. This approach, known as gradient-enhanced Kriging, is applied to various benchmark functions of varying dimensionality (2D-20D). As expected, results from the benchmark problems show that additional gradient information can significantly enhance the accuracy of Kriging. Gradient-enhanced Kriging provides a better approximation even when gradient information is only partially available. Further comparison between gradient-enhanced Kriging and an alternative formulation of gradient-enhanced Kriging, called indirect gradient-enhanced Kriging, highlights various advantages of directly employing gradient information, such as improved surrogate model accuracy, better conditioning of the correlation matrix, etc. Finally, gradient-enhanced Kriging is used to model 6- and 10-variable fluid---structure interaction problems from bio-mechanics to identify the arterial wall's stiffness.

Multi-fidelity information fusion based on prediction of kriging

In this paper, a novel kriging-based multi-fidelity method is proposed. Firstly, the model uncertainty of low-fidelity and high-fidelity models is quantified. On the other hand, the prediction uncertainty of kriging-based surrogate models(SM) is confirmed by its mean square error. After that, the integral uncertainty is acquired by math modeling. Meanwhile, the SMs are constructed through data from low-fidelity and high-fidelity models. Eventually, the low-fidelity (LF) and high-fidelity (HF) SMs with integral uncertainty are obtained and a proposed fusion algorithm is implemented. The fusion algorithm refers to the Kalman filter's idea of optimal estimation to utilize the independent information from different models synthetically. Through several mathematical examples implemented, the fused SM is certified that its variance is decreased and the fused results tend to the true value. In addition, an engineering example about autonomous underwater vehicles' hull design is provided to prove the feasibility of this proposed multi-fidelity method in practice. In the future, it will be a helpful tool to deal with reliability optimization of black-box problems and potentially applied in multidisciplinary design optimization.

Performance study of multi-fidelity gradient enhanced kriging

Multi-fidelity surrogate modelling offers an efficient way to approximate computationally expensive simulations. In particular, Kriging-based surrogate models are popular for approximating deterministic data. In this work, the performance of Kriging is investigated when multi-fidelity gradient data is introduced along with multi-fidelity function data to approximate computationally expensive black-box simulations. To achieve this, the recursive CoKriging formulation is extended by incorporating multi-fidelity gradient information. This approach, denoted by Gradient-Enhanced recursive CoKriging (GECoK), is initially applied to two analytical problems. As expected, results from the analytical benchmark problems show that additional gradient information of different fidelities can significantly improve the accuracy of the Kriging model. Moreover, GECoK provides a better approximation even when the gradient information is only partially available. Further comparison between CoKriging, Gradient Enhanced Kriging, denoted by GEK, and GECoK highlights various advantages of employing single and multi-fidelity gradient data. Finally, GECoK is further applied to two real-life examples.