A Hybrid Method Using Response Surface and Pattern Search for Design Optimization

A hybrid method for robust and efficient optimization process is developed by integrating a new response surface method and pattern search algorithm. The method is based on: (1) multipoint approximations of the objective and constraint functions, (2) a multiquadric radial basis function (RBF) for the zeroth-order function approximation and a new RBF plus polynomial-based moving least-squares approximation for the first-order enhanced function approximation, and (3) a pattern search algorithm to impose a descent condition and applied adaptive subregion management strategy. Several numerical examples are presented to illustrate accuracy and computational efficiency of the proposed method for both function approximation and design optimization. To demonstrate the effectiveness of the proposed hybrid method, it is applied to obtain optimum designs of a microelectronic packaging system. A two-stage optimization approach is proposed for the design optimization. The material properties of microelectronic packaging system and the shape parameters of solder ball are selected as design variables. Through design optimization, significant improvements of durability performances are obtained using the proposed hybrid optimization method.

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, a hybrid metamodel using the orthogonal constraints of radial basis function and sparse polynomial chaos expansions is proposed for the global sensitivity analysis of time-consuming models. Firstly, the orthogonal conditions of radial basis functions (RBF) and polynomial chaos expansions (PCE) were derived to construct the hybrid metamodel. Then, the variance of the metamodel was decoupled into the variances of the RBF and PCE independently by using the orthogonal condition. Furthermore, the analytical formulations of Sobol indices for the hybrid metamodel were derived according to the orthogonal decomposition. Thus, the interaction items of radial basis function and polynomial chaos expansions were eliminated, which significantly simplifies the Sobol indices. Two analytical cases were employed to investigate the influence of the number of the polynomial chaos expansions items, and several analytical and engineering cases were tested to demonstrate the accuracy and efficiency of the proposed method. In the engineering cases, the proposed method yielded significant improvements in terms of both accuracy and efficiency comparing with the existing global sensitivity analysis approaches, which indicates that the proposed method is more appropriate to the global sensitivity analysis of time-consuming engineering problems.

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.

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.

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.

The mutant matrilineal (mtl) gene encoding patatin-like phospholipase activity is involved in in-vivo maternal haploid induction in maize. Doubling of chromosomes in haploids by colchicine treatment leads to complete fixation of inbreds in just one generation compared to 6–7 generations of selfing. Thus, knowledge of patatin-like proteins in other crops assumes great significance for in-vivo haploid induction. So far, no online tool is available that can classify unknown proteins into patatin-like proteins. Here, we aimed to optimize a machine learning-based algorithm to predict the patatin-like phospholipase activity of unknown proteins. Four different kernels [radial basis function (RBF), sigmoid, polynomial, and linear] were used for building support vector machine (SVM) classifiers using six different sequence-based compositional features (AAC, DPC, GDPC, CTDC, CTDT, and GAAC). A total of 1170 protein sequences including both patatin-like (585 sequences) from various monocots, dicots, and microbes; and non-patatin-like proteins (585 sequences) from different subspecies of Zea mays were analyzed. RBF and polynomial kernels were quite promising in the prediction of patatin-like proteins. Among six sequence-based compositional features, di-peptide composition attained > 90% prediction accuracies using RBF and polynomial kernels. Using mutual information, most explaining dipeptides that contributed the highest to the prediction process were identified. The knowledge generated in this study can be utilized in other crops prior to the initiation of any experiment. The developed SVM model opened a new paradigm for scientists working in in-vivo haploid induction in commercial crops. This is the first report of machine learning of the identification of proteins with patatin-like activity.

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 response surface method (RSM) is a powerful approach for carrying out reliability analysis for complicate engineering with implicit limit state functions. The quality of a response surface mainly depends on the choice of response surface function and the selection of sample points. To investigate the influence of the types of response surface functions and to reduce the computation efforts, two new sampling methods and a hybrid RSM are proposed. In the hybrid RSM, four types of response surface functions and three sampling methods are involved, and each Response surface function can be connected to each sampling method. The four response surface functions are quadratic polynomial without cross terms (PN1), quadratic polynomial with cross terms, radial basic function network (RBFN) and support vector machine (SVM). The three RSMs using the traditional sampling method, the new iterative sampling method and the new experiment design method are RSM1, RSM2 and RSM3, respectively. The accuracy and efficiency of different RSMs are illustrated through three examples. When an iterative method is used for locating sample points, the PN1-based RSM2 is proposed for its accuracy and efficiency. And when an experiment design method is used for locating sample points, the RBFN- or SVM-based RSM3 is suggested because the RBFN or SVM is suitable for global fitting.

In this article, we describe a hybrid optimizer based on a highly accurate response surface method, which uses several radial basis functions and polynomials as interpolants. The response surface is capable to interpolate linear as well as highly non-linear functions in multi-dimensional spaces having up to 500 dimensions. The accuracy, robustness, efficiency, transparency and conceptual simplicity are discussed. Based on the extensive testing performed on 296 test functions, the radial basis functions (RBFs) approach seems computationally easy to implement and results are superior, requiring small computing time. The performance of the RBF approximation is compared with wavelets neural networks for several selected test cases and the optimizer is compared with other hybrid optimizers, as well as with the IOSO commercial code.

Improved response surface method and its application to stiffened plate reliability analysis

Approximation methods such as response surface method and artificial neural network (ANN) method are widely used to alleviate the computation costs in structural reliability analysis. However most of the ANN methods proposed in the literature suffer various drawbacks such as poor choice of parameter setting, poor generalization and local minimum. In this study, a support vector machine-based radial basis function (RBF) network method is proposed, in which the improved RBF model is used to approximate the limit state function and then is connected to a reliability method to estimate failure probability. Since the learning algorithm of RBF network is replaced by the support vector algorithm, the advantage of the latter, such as good generalization ability and global optimization are propagated to the former, thus the inherent drawback of RBF network can be defeated. Numerical examples are given to demonstrate the applicability of the improved RBF network method in structural reliability analysis, as well as to illustrate the validity and effectiveness of the proposed method.

Local normalized linear summation kernel for fast and robust recognition