A hybrid surrogate and pattern search optimization method and application to microelectronics

This paper presents a new method to construct response surface function and a new hybrid optimization method. For the response surface function, the radial basis function is used for a zeroth-order approximation, while new bases is proposed for the moving least squares method for a first-order approximation. For the new hybrid optimization method, the gradient-based algorithm and pattern search algorithm are integrated for robust and efficient optimization process. These methods are based on: (1) multi-point approximations of the objective and constraint functions; (2) a multi-quadric radial basis function for the zeroth-order function representation or radial basis function plus polynomial based moving least squares approximation for the first-order function approximation; and (3) a pattern search algorithm to impose a descent condition. Several numerical examples are presented to illustrate the accuracy and computational efficiency of the proposed method for both function approximation and design optimization. The examples for function approximation indicate that the multi-quadric radial basis function and the proposed radial basis function plus polynomial based moving least squares method can yield accurate estimates of arbitrary multivariate functions. Results also show that the hybrid method developed provides efficient and convergent solutions to both mathematical and structural optimization problems.

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results.

In this study numerical solutions for the compressible fluid flow around obstacles, based on a meshless method with Radial Basis Functions (RBFs) are presented. The numerical solutions are obtained by a hybrid method that uses the first step of the boundary element method, namely a singular boundary integral equation, which is an equivalent mathematical model for the problem of the compressible fluid around an obstacle, and then a meshless method based on RBFs for solving this singular boundary integral equation. Two types of RBFs are used, namely the Multiquadric RBFs and Gauss-type RBFs. An analytical check is made in order to study the numerical solutions' accuracy. Using computer simulations, we compare our results with an analytical solution available for a specific case. We examine how different parameters used to obtain the numerical solutions affect their accuracy, and we compare these numerical solutions to determine which RBF is most suitable for this scenario. We also note that the shape parameters of RBFs have a great influence on numerical solutions accuracy and, through simulations, optimal values are found. For evaluating the integrals of singular kernels which appear, the truncation method is applied. The influence of the parameter used to evaluate these integrals is also highlighted in the paper.

Shale plays with pre-existing natural fractures can yield significant production when operating horizontal wells with multi-stage hydraulic fracturing (HWMHF). This work proposes a general, robust, and integrated framework for estimating optimal HWMHF design parameters in an unconventional naturally fractured oil reservoir. This work considers uncertainty in both the distribution of the natural fractures and uncertainty in three geo-mechanical parameters: the internal friction factor, the cohesion coefficient, and the tensile strength. Because a maximum of five design variables is considered, it is appropriate to apply derivative-free algorithms. This work considers versions of the genetic algorithm (GA), particle swarm optimization (PSO), and general pattern search (GPS) algorithms. The forward model consists of two linked software programs: a geo-mechanical simulator and an unconventional shale oil simulator. The two simulators run sequentially during the optimization process without human intervention. The in-house geo-mechanical simulator model provides sufficient computational efficiency so that it is feasible to solve the robust optimization problem. An embedded discrete fracture model (EDFM) is implemented to model large-scale fractures. Two cases strongly verified the feasibility of the framework for the optimization of HWMHF, and the average comprehensive NPV increases by 35% and 102.4%, respectively. By comparison, the pattern search algorithm is more suitable for HWMHF optimization. In this way, oil and gas scientists are contributing to the energy industry more accurately and resolutely.

This research investigated and optimized the separation of carbon dioxide (CO2) from natural gas in an adsorption column filled with grafted-beam nanofiber adsorbent. The main purpose of using ANN and RSM models in this manuscript is to compare these two methods in predicting the CO2 adsorption capacity. In other words, it was made to find a suitable model that has the highest agreement with the experimental data. Also, the other purpose of using the RSM model is to detect the optimized empirical conditions. Moreover, two common ANN models are applied in this work, including a multilayer perceptron (MLP) and radial basis function (RBF). The novelties of this work are explained as follows: (1) detecting the optimized synthesis factors of NF-PAN/PUGMA sorbent which possess the highest CO2 adsorption capacity with the help of response surface methodology (RSM), (2) studying the simultaneous interaction of synthesis parameters on the CO2 adsorption capacity with the help of both RSM and artificial neural networks (ANNs), (3) testing two types of ANN models including multilayer perceptron (MLP) and radial basis function (RBF) to predict the effect of monomer volume percentage and irradiation dose on the CO2 adsorption capacity. Indeed, finding the best model (ANN or RSM) can help engineers in practical applications predict CO2 adsorption capacity using NF-PAN/PUGMA under different conditions without incurring high-priced chemical materials, electricity, or human resources. The validation results were examined using correlation coefficients (R2) of RSM, RBF, and MLP models. The correlation coefficients for the RSM, RBF, and MLP models were 0.9910, 0.9949, and 0.9968, respectively. Additionally, the average absolute relative deviation (AARD) values for the RBF and MLP models were 0.00046512 and 0.00045511, respectively, indicating that the MLP model is better than the RBF model. To identify the optimal network structure, the trial-and-error method was conducted for MLP and RBF models. The number of neurons was found at 12 and 45 for MLP and RBF, respectively. The optimized effective parameters were obtained using RSM: 25.80% GMA, 66.45% amine, and an irradiation intensity of 28kGy.

This paper discusses the influence of different Radial Basis Function (RBF) in metamodel construction to be applied to 3D aerodynamic wing optimization problems using the Constrained Optimization with Response Surface (CORS) methodology in conjunction with a stochastic Controlled Random Search Algorithm (CRSA). CORS methodology is based on the iterative construction and optimization of response surfaces with a robust search pattern application. In the CORS methodology the response surface may be generate by at least three types of methods: (i) Classical (polynomials and parametric surfaces), (ii) Statistical (K-Nearest, Kriging and Gaussian Processes) and (iii) Advanced (RBF and Neural Network). The response surfaces used in this paper are constructed using six different types of RBF: Gaussian, Hardy’s Multiquadric and Inverse Multiquadric, which depend on a shape parameter, and linear, cubic and thin plate spline, which do not depend on a shape parameter. The RBF’s are directly applied on the response surface construction inside the CORS structure. Thus the choice of a RBF and of a shape parameter (being the case) may influence significantly the methodology efficiency. The CORS methodology is here applied for accelerating the optimization process of wing aerodynamic designs with a solver based on a first order 3D panel method and a 2D boundary layer model. Since the main objective of this paper is of prospective nature, the choice of a relatively low-fidelity flow computation solver is justified. One considers problems of minimizing the aerodynamic coefficient relation (CD/CL) and the inverse of lift coefficient (1/CL). Comparative influence of the RBF choice on the acceleration induced by CORS methodology is investigated taking into account the number of expensive objective function evaluations necessary to find the minimum value in each problem.

This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [1] in solution of coupled heat transfer and fluid flow problems associated with solidification of a binary alloy. The involved temperature, velocity, species and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [2,3]. The solution procedure is demonstrated in simulation of solidification of 8 wt % Pb-Sn alloy and 5 wt % Sn-Pb alloy in rectangular cavity at conditions, defined in [4]. The numerical examples include studies with different uniform discretization and its influence on final macrosegregation map. The solution is assessed by comparison with the reference results of the finite volume and finite element method. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements. The methodology will be use in complementing our heat transfer model [5] of the continuous casting of steel.

Radial basis function (RBF) methods have been actively developed in the last decades. RBF methods are global methods which do not require the use of specialized points and that yield high order accuracy if the function is smooth enough. Like other global approximations, the accuracy of RBF approximations of discontinuous problems deteriorates due to the Gibbs phenomenon, even as more points are added. In this paper we show that it is possible to remove the Gibbs phenomenon from RBF approximations of discontinuous functions as well as from RBF solutions of some hyperbolic partial differential equations. Although the theory for the resolution of the Gibbs phenomenon by reprojection in Gegenbauer polynomials relies on the orthogonality of the basis functions, and the RBF basis is not orthogonal, we observe that the Gegenbauer polynomials recover high order convergence from the RBF approximations of discontinuous problems in a variety of numerical examples including the linear and nonlinear hyperbolic partial differential equations. Our numerical examples using multi-quadric RBFs suggest that the Gegenbauer polynomials are Gibbs complementary to the RBF multi-quadric basis.

The design optimization of acoustic metamaterials represents a critical challenge in modern noise control applications. Global sensitivity analysis (GSA) has emerged as a powerful tool for quantifying the correlation between metamaterials' design variables and corresponding sound quality metrics (loudness, sharpness, and speech intelligibility index). However, GSA's high computational intensity limits the practical application of design optimization, especially for sound quality metrics which vary with sound stimulus. This study introduces a computational framework using radial basis function (RBF) interpolation to accelerate the GSA of acoustic metamaterials, with a periodic arrangement of micro-perforated panels (MPPs) serving as an exemplary case study. The methodology employs RBF interpolation to create a surrogate model of metamaterial acoustic response, demonstrating the potential for computational cost reduction while maintaining accuracy. It is validated across three acoustic scenarios that represent common challenges in noise control: stationary signals (broad/narrowband noise), environmental sounds (mechanical noise, traffic), and dynamic signals (speech, music). The study systematically compares sensitivity indices derived from direct numerical simulation and interpolated predictions, assessing the RBF's impact on GSA accuracy. [The European Commission is gratefully acknowledged for supporting the Marie Sklodowska Curie program through the Horizon Europe DN METAVISION project (GA 101072415)]

This study evaluates the effectiveness of Radial Basis Function (RBF) approaches, specifically Gaussian and Multiquadric RBFs, compared to Cubic and Adaptive Splines for data imputation in time-series datasets. Three datasets—multi-seasonal, non-stationary, and spikes and drops—were used, with varying data corruption levels to simulate real-world scenarios. Imputation performance was assessed using Root Mean Square Error (RMSE). The results indicate that spline-based methods, particularly Cubic and Adaptive Splines, consistently deliver lower RMSE values, showcasing superior robustness to data variability and high corruption levels compared to RBF methods. This comparative analysis offers insights into selecting appropriate techniques for effective data imputation in time-series analysis.

The work presented in this paper used rigorous 3D flow-field analysis combined with multi-objective constrained shape design optimization for the design of bladelet (winglet) configurations for a three-blade propeller type wind turbine. The fluid flow analysis in this work was performed using 3D, steady, incompressible, turbulent flow Reynolds-averaged Navier-Stokes equations in the rotating frame of reference for each combination of a given wind turbine blade and a varying bladelet geometry. The free stream uniform wind speed in all cases was assumed to be 9 m s−1 and rotational speed was 12 rpm. These were off-design conditions for this rotor. The three simultaneous design optimization objectives were: a) maximize the coefficient of power, b) minimize the coefficient of thrust, and c) minimize twisting moment around the blade axis. The bladelet geometry was fully defined by using a small number of parameters. The optimization was carried out by creating a multi-dimensional response surface for each of the simultaneous objectives. The response surfaces were based on radial basis functions, where the support points were designs analyzed using the high fidelity CFD analysis of the full blade + bladelet geometry. The response surfaces were then coupled to a multi-objective optimization algorithm. The predicted values of the objective functions for the optimum designs were then again validated using the high fidelity computational fluid dynamics analysis code. Results for a Pareto optimized bladelet on a given blade indicate that more than 4% increase in the coefficient of power at minimal thrust force penalty is possible compared to the same wind turbine rotor blade without a bladelet.

Numerical solution of differential equations using multiquadric radial basis function networks

With the development of excellent software in optimization, there is more and more wishes to use these techniques and make epoch developments in their products. However, most of us know that it costs a lot of time and costs to carry out optimization, because it needs to repeat simulations for the number of times. Therefore, most engineers would like to make mathematical models of design problems just enough to express their needs priori to optimization. Let us call these models as light models, in this paper. However, it is very difficult to make a proper light model in the first trial. Recent optimization will find the global solution, so that it will find some modeling miss and lead light model to some strange results. Such phenomenon is not a fault of optimization, but of the light model. We have to revise the light model but it is way difficult to make it rationally with the information we have had from the light model. We have to think of appropriate values of existed constraints, we have to think of new functions for constraints, and so on. It is almost hopeless to revise the light model at one time. The best thing that we can hope for is to give up the light model and make models complex to express the situation of the products. If we do not know proper values for constraints, treat them as objective functions and just go through trade-off analysis to have their relationships information, so that we can give appropriate values. In this sense, it is very important to make models in multi-objective optimization. In multi-objective optimization, it is quite important to carry out trade-off analysis and which might be a key to find epoch design. We had developed approximate multi-objective optimization method, which is composed by satisficing method, convolute radial basis function network and recommendation of new design variables. In satisficing method, we can handle trade-off analysis much easier than weighting methods. However, scalar function composed by satisficing method is not smooth when it gets close to Pareto solution that is indicated by designer’s requirement. Therefore, it is difficult even for convolute RBF to compose better approximate function although we have better approximation for each objective function. In weighting method, it is quite difficult to give proper weights for designer’s requirement. In approximate optimization, fortunately we have a number of data sets. Thus, it is possible to optimize weights which meet for aspiration level in satisficing method. Then, we can have smooth and better approximation for scalar function. In this paper, we are going to show effectiveness of the proposed method through demonstration of numerical examples.

In this study, we focus on the modelling of infiltration process in porous media. We use the meshless techniques for efficiently solving the Richards equation which describes unsaturated water flow through soils. The design of approximate numerical methods for the Richards equation remains computationally challenging and requires the development of efficient numerical techniques. This difficulty is mainly due to the nonlinearity of the unsaturated hydraulic conductivity and the capillary pressure function. In this study, we develop a new method based on the localized radial basis function (RBF) and the Kirchhoff transformation technique in order to solve Richards equation in one and two-dimensional homogeneous medium. Our approach using the multiquadric radial basis function allows us to reduce the computational time and provide accurate numerical solutions. The proposed method does not require mesh generation. Picard's iterations are used to linearize the resulting nonlinear problem obtained using the Kirchhoff transformation technique. The numerical simulations show the capability of the proposed numerical techniques in predicting the dynamics of water through unsaturated soils.

Simulation of laminar backward facing step flow under magnetic field with explicit local radial basis function collocation method