To address the efficiency limitations of the Kriging (KG) model caused by single-point infill criteria, which add only one sample per iteration, and the excessive sample size requirement of multi-point infill criteria, which add a fixed number of samples per iteration, this paper proposes an AMSS-QBC-KG framework. This framework allows the sample size to decrease as model accuracy improves in the later stages of KG modeling, thereby enhancing modeling efficiency while controlling sample size. The effectiveness of the AMSS-QBC-KG framework was validated using standard test functions, with its predictive performance compared against existing infill criteria: Expected Improvement (EI), q-EI, Query-by-Committee (QBC), Adaptive Multi-Scale Sampling (AMSS), and Latin Hypercube Sampling (LHS). The results demonstrate that while achieving accuracy comparable to other methods for low-dimensional weakly nonlinear problems, the AMSS-QBC-KG framework exhibits superior predictive accuracy and reduced computational cost across diverse problem types. Furthermore, the framework’s effectiveness was verified through an engineering case study on fatigue life analysis of vortex-induced vibration in submarine pipelines.

The aerodynamic optimization of airfoil shapes remains a critical research area for enhancing aircraft performance under various flight conditions. In this study, the RAE 2822 airfoil was selected as a benchmark case to investigate and compare the effectiveness of surrogate-based methods under an Efficient Global Optimization (EGO) framework and an adjoint-based approach in both single-point and multi-point optimization settings. Prior to optimization, the computational fluid dynamics (CFD) model was validated against experimental data to ensure accuracy. For the surrogate-based methods, Kriging (KRG), Kriging with Partial Least Squares (KPLS), Gradient-Enhanced Kriging (GEK), and Gradient-Enhanced Kriging with Partial Least Squares (GEKPLS) were employed. In the single-point optimization, the GEK method achieved the highest drag reduction, outperforming other approaches, while in the multi-point case, GEKPLS provided the best overall improvement. Detailed comparisons were made against existing literature results, with the proposed methods showing competitive and superior performance, particularly in viscous, transonic conditions. The results underline the importance of incorporating gradient information into surrogate models for achieving high-fidelity aerodynamic optimizations. The study demonstrates that surrogate-based methods, especially those enriched with gradient information, can effectively match or exceed the performance of gradient-based adjoint methods within reasonable computational costs.

Proper Orthogonal Decomposition (POD) is a well-examined order reduction method. In this paper, it is implemented as a non-intrusive method for the cost-effective simulation of a multi-stream heat exchanger. Three-dimensional fluid-solid domains are analyzed conjugately, and the numerical results provide snapshots. The dimensionless vector-valued approach is suggested for the extraction of the base vectors. The Kriging (KRG) method is employed to approximate the coefficients of the bases. Reynolds numbers of the flow streams are variable parameters in the snapshots, and the field is reconstructed at two Reynolds number combinations. The field is reconstructed for two situations: one with full interpolation of the POD weighting coefficients and another with partial extrapolation for them. The deviation of the results is calculated regarding the corresponding CFD solution as the reference. In 96% of the comparative test cases, the estimated total heat exchanges and exit temperatures via POD have deviations less than 20%, and a substantial reduction in computational cost is attained. A deviation accumulation is observed due to the integration in post-processing. The partial extrapolation increases the deviation of heat exchange and temperature all over the domain because of the unified analysis.

A novel machine learning framework for impact force prediction of foam-filled multi-layer lattice composite structures

Aiming to address the issue of the ‘curse of dimensionality’ encountered in approximating high-dimensional problems using surrogate-based methods, high-dimensional model representation (HDMR), decomposing the high-dimensional problem into summands of different-order component functions, has been widely studied. To reduce the computational demands of the current HDMR metamodelling techniques, an improved high-dimensional model representation framework (iHDMR) is presented, taking full advantage of the relationships between the first-order and second-order component functions in the Cut-HDMR theory. then, a novel metamodelling technique, termed kriging (KRG)-iHDMR, is implemented by integrating the kriging technique and the suggested surface-axes alternating sampling strategy into the iHDMR framework. The accuracy and efficiency of KRG-iHDMR are demonstrated by various numerical and engineering examples with different dimensions. Finally, an optimization problem of the multi-stage solid launch vehicle propulsion system is introduced to verify the engineering feasibility of the KRG-iHDMR metamodelling technique.

Estimating intensity−duration−frequency (IDF) curves requires local historical information of precipitation intensity. When such information is unavailable, as in areas without rain gauges, it is necessary to consider other methods to estimate curve parameters. In this study, three methods were explored to estimate IDF curves in ungauged areas: Kriging (KG), Inverse Distance Weighting (IDW), and Storm Index (SI). To test the viability of these methods, historical data collected from 31 rain gauges distributed in central Chile, 35° S to 38° S, are used. As a result of the reduced number of rain gauges to evaluate the performance of each method, we used LOOCV (Leaving One Out Cross Validation). The results indicate that KG was limited due to the sparse distribution of rain gauges in central Chile. SI (a linear scaling method) showed the smallest prediction error in all of the ungauged locations, and outperformed both KG and IDW. However, the SI method does not provide estimates of uncertainty, as is possible with KG. The simplicity of SI renders it a viable method for extrapolating IDF curves to locations without data in the central zone of Chile.

A linear geostatistical model is considered. Properties of a universal kriging are studied when the locations of observations are measured with errors. Alternative prediction procedures are introduced and their least squares errors are analyzed.

Droplet ejection technology is widely used in green and intelligent manufacturing. A stable jetting can be defined as no obvious satellite droplets during the whole ejection process, which is of great importance to ensure the quality and efficiency of the printed products; However, due to the multi-parameter features and the interaction between different physics, using traditional analytical-based approaches to analyze and/or optimize is usually difficult and even unfeasible. Experimental tests using a PZT printhead design-optimization method based on surrogate modeling are proposed in this paper to overcome this challenge, which can synthesize the advantages of numerical simulation. The basic data for surrogate model construction was obtained by the Computational Fluid Dynamics (CFD) numerical-based model, which was developed to predict the flow characteristic under different parameter settings of the printhead. The accuracy of the developed numerical model was validated by performing experimental tests; thereby, the predictive ability of the numerical model in droplet ejection was verified. With the validated numerical model, the Design of Experiments (DoE) was performed to generate the necessary training and validation sample dataset required by the surrogate modeling. Thereafter, four surrogate modeling methods were adopted to construct the relationship between the design parameters and flow features, where the Kriging (KRG) was identified as the optimal modeling method. Based on the developed KRG model, global sensitivity analysis (GSA) of the parameters was carried out with Sobol’s method; thereby, the influence of different parameters can be quantified. Finally, a genetic algorithm (GA) was used to optimize the structure of the droplet printhead. Through validation, the optimized design model increases the droplet ejection speed by 20.84% while keeping no satellite droplet formation, confirming the efficient and stable printhead ejection, and verifying the feasibility and effectiveness of the analysis/optimization method proposed in this paper.

Novel modeling and optimization framework for Navy Blue adsorption onto eco-friendly magnetic geopolymer composite

To address the severe spectrum scarcity problem and achieve efficient green communications, we propose a new and practical scheme to obtain electromagnetic data (ED) and a composite electromagnetic map reconstruction method (CEMRM). The scheme uses a small number of sensing nodes to obtain incomplete sampled ED, then uses CEMRM to reconstruct complete ED according to the incomplete ED and builds realistic electromagnetic maps (EMs) in various propagation scenarios. Specifically, we firstly adopt kriging (KG) method to obtain the geography-based ED (GED) according to geographical correlation of the locations of the sampled ED. Meanwhile, a novel algorithm, named filtered subdistrict sparsity adaptive matching pursuit (FSSAMP), is proposed to obtain the pure ED (PED) according to electromagnetic correlation of the sampled ED. Then, weight factors are mapped into the above two types of data and the fast gradient projection (FGP) method is employed to obtain the highly accurate combined ED (CED). Based on the CED, the accurate and practical EMs can be drawn. Simulation results demonstrate that the proposed scheme can provide more accurate ED and EMs than existing benchmark schemes in various propagation scenarios, and the built EMs can provide accurate information for the assessment of spectrum resources utilization to make spectrum resources efficiently used.

Metamodels are developed and used for aerodynamic optimization of a ducted opening integrated into a high-rise building to maximize the amplification factor within the duct. The duct consists of a nozzle, a throat, and a diffuser. 211 high-resolution 3D RANS CFD simulations are performed to generate training and testing datasets. The space-filling design and Genetic algorithm are used for data sampling and optimization, respectively. The performance of five commonly-used metamodels is systematically investigated: Response Surface Methodology (RSM), Kriging (KG), Neural Network (NN), Support Vector Regression (SVR), and Genetic Aggregation Response Surface (GARS). The investigation is based on (i) detailed in-sample and out-of-sample evaluations of the metamodels, (ii) annual available power in the wind (Pavailable), and (iii) annual energy production (AEP) for a 3-bladed horizontal-axis wind turbine (HAWT) installed in the mid-throat for the optimum designs obtained by the metamodels. The results show that converging-diverging ducted openings can magnify the experienced wind speed by the turbine and enhance the available wind power. In addition, the use of different metamodels can lead to a variation of up to 153% in the estimated Pavailable. For a small dataset, crude yet still acceptable accuracy can be achieved for Genetic Aggregation Response Surface and Kriging at a very low computational time.

Finite element analysis (FEA) is increasingly applied to medicine because it could increase accuracy and rapid outcomes. However, there is a lack of the method to determine Young’s modulus and Poisson’s ratio for fresh femoral bone and the mathematical principle’s optimization for calculating nonuniform configuration. This study aimed to investigate the surrogate model for the optimization method to determine Young’s modulus and Poisson’s ratio of the fresh femoral bone. Young’s modulus and Poisson’s ratio obtained 20 ranked pairs by the Latin hypercube sampling method. The values ​​were calculated in the finite element for root mean square error (RMSE) and were then used for solutions by a quadratic function, radial basis function (RBF), and Kriging (KG). The lowest RMSE value was 0.1518 for the RBF method, with the young’s modulus at 304.4756 and the Poisson’s ratio at 0.3334. The current study identified the RBF technique to determine the properties of the femoral bone. Moreover, the RBF procedure might apply to other long bones because of the comparable nonuniform configuration.

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

In the present study, a new approach by coupling the interpolation method with computation-based technique (data-mining algorithms and an optimization algorithm) is introduced for modeling and optimization removal of Reactive Orange 7 (RO7) dye removal from synthetic wastewater. To this end, four significant factors like pH, electrolyte concentration, current density, and electrolysis time are considered as input variables. Thus, modeling of RO7 removal is implemented using eight data mining algorithms including multi- variate linear regression (MLR), ridge regression (RR), multivariate nonlinear regression (MNLR), artificial neural network (ANN), classification and regression tree (CART), k nearest neighbor (KNN), random forest (RF), and support vector machine (SVM). These al- gorithms require a large data set for creating reliable results. However, creating a large number of experimental data request consuming high cost and time. Hence, the interpolation methods of kriging (KRG) and inverse distance weight (IDW) are applied for generating more data, whereas KRG has more accuracy than IDW by increasing the 47.080, 36.914, and 1.77% in MAE, RMSE, and R values, res- pectively. Then, the data mining algorithms are used for modeling the decolorization efficiency (DE) based on the original data and new data from KRG. It is found that using new data leads to significantly increasing accuracy (94.47, 96.43, 1.52, and 2.77% for MAE, RMSE, R and R2, respectively) of DE modeling. Also, SVM has demonstrated the highest accuracy out of all data mining algorithms (by in- creasing the 97.13, 98.30, and 14.42% in MAE, RMSE, and R2 values, respectively). Another challenge in the removal of RO7 from synthetic wastewater is predicting the maximum removal amount and optimal input variables. For this purpose, the hybrid of SVM and whale optimization algorithm (WOA) is employed. Finally, SVM-WOA has predicted the maximum of DE (91%) by optimal values of 4.2, 1.5 g/L, 4.2 mA/cm2, and 18 min for pH, C, I, and Time, respectively. In light of the high performance of the introduced approach for modeling removal process and predicting optimal conditions of removal process, this approach can be suggested for the removal of other pollutants from wastewater when the number of experimental data set is limited.

This study applies, tests, and compares comprehensive surrogate-based optimization techniques to optimize the performance of polymer electrolyte membrane fuel cells (PEMFCs). Moreover, parametric cases considering four important design variables, i.e., gas diffusion layer thickness (), channel depth (), channel width (), and land width (), are defined using the latin hypercube sampling technique under reasonable constraint conditions. Multidimensional, two-phase PEMFC model simulations are performed to generate the training and test data under these design conditions. Three famous surrogate models, i.e., response surface approximation (RSA), radial basis neural network (RBNN), and kriging (KRG), are employed to construct objective functions for the PEMFC cell voltages operating in the galvanostatic mode, and their accuracies are tested and compared using root mean square error and adjusted R-square. The surrogates are then linked to stochastic optimization algorithms, i.e., genetic algorithm and particle swarm optimization, to determine the optimal design points. A comparative study of these surrogates reveals that the KRG model outperforms the RBNN and RSA models in terms of prediction capability. Furthermore, the PEMFC model simulations at the optimal design points clearly demonstrate that performance improvements of around 56–69 mV at 2.0 A cm−2 are achieved with the optimum design values compared to typical PEMFC design conditions.

Developing air pollution concentration fields for health studies using multiple methods: Cross-comparison and evaluation

The bond between the concrete and bar is a main factor affecting the performance of the reinforced concrete (RC) members, and since the steel corrosion reduces the bond strength, studying the bond behavior of concrete and GFRP bars is quite necessary. In this research, a database including 112 concrete beam test specimens reinforced with spliced GFRP bars in the splitting failure mode has been collected and used to estimate the concrete-GFRP bar bond strength. This paper aims to accurately estimate the bond strength of spliced GFRP bars in concrete beams by applying three soft computing models including multivariate adaptive regression spline (MARS), Kriging, and M5 model tree. Since the selection of regularization parameters greatly affects the fitting of MARS, Kriging, and M5 models, the regularization parameters have been so optimized as to maximize the training data convergence coefficient. Three hybrid model coupling soft computing methods and genetic algorithm is proposed to automatically perform the trial and error process for finding appropriate modeling regularization parameters. Results have shown that proposed models have significantly increased the prediction accuracy compared to previous models. The proposed MARS, Kriging, and M5 models have improved the convergence coefficient by about 65, 63 and 49%, respectively, compared to the best previous model.

In this paper, a data-driven multi-objective optimization approach using optimal Latin hypercube sampling (OLHS), Kriging (KRG) metamodel and the non-dominated sorting genetic algorithm II (NSGA-II) is presented for the laser welding process parameters on 6061-T6 aluminum alloy. The experiments are designed by OLHS and carried out to obtain the data results. The complex relationship between the process parameters and the bead profile geometry is established by KRG using the data results. The accuracy of the established KRG metamodel is validated using experiments. Then, NSGA-II is used to explore the design space and search the Pareto optimal solutions of process parameters. Besides, the validation experiments were carried out to obtain ideal LW bead profile, which shows that the approach can bring dependable guidance for LW experiments.

This article proposes a new approach for determining the optimal parameter (β) in the Inverse Distance Weighted Method (IDW) for spatial interpolation of hydrological data series. This is based on a genetic algorithm (GA) and finds a unique β for the entire study region, while the classical one determines different βs for different interpolated series. The algorithm is proposed in four scenarios crossover/mutation: single-point/uniform, single-point/swap, two-point/uniform, and two-point swap. Its performances are evaluated on data series collected for 41 years at ten observation sites, in terms of mean absolute error (MAE) and mean standard error (MSE). The smallest errors are obtained in the two-point swap scenario. Comparisons of the results with those of the ordinary kriging (KG), classical IDW (with β = 2 and the optimum beta found by our algorithm), and the Optimized IDW with Particle Swarm Optimization (OIDW) for each study data series show that the present approach better performs in 70% (80%) cases.

Satellite altimetry and tide gauges are the two main techniques used to measure sea level. Due to the limitations of satellite altimetry, a high-quality unified sea level model from coast to open ocean has traditionally been difficult to achieve. This study proposes a fusion approach of altimetry and tide gauge data based on a deep belief network (DBN) method. Taking the Mediterranean Sea as the case study area, a progressive three-step experiment was designed to compare the fused sea level anomalies from the DBN method with those from the inverse distance weighted (IDW) method, the kriging (KRG) method and the curvature continuous splines in tension (CCS) method for different cases. The results show that the fusion precision varies with the methods and the input measurements. The precision of the DBN method is better than that of the other three methods in most schemes and is reduced by approximately 20% when the limited altimetry along-track data and in-situ tide gauge data are used. In addition, the distribution of satellite altimetry data and tide gauge data has a large effect on the other three methods but less impact on the DBN model. Furthermore, the sea level anomalies in the Mediterranean Sea with a spatial resolution of 0.25° × 0.25° generated by the DBN model contain more spatial distribution information than others, which means the DBN can be applied as a more feasible and robust way to fuse these two kinds of sea levels.