响应面分析法优化纳米Ti0_2的制备工艺

利用响应面分析法对TiO_2制备条件进行优化。在单因素实验基础上选取实验因素与水平,根据中心组合(Box-Benhnken)实验设计原理采用3因素3水平的响应面分析法,依据回归分析确定各工艺条件的影响因子,以亚甲基蓝降解率为响应值作响应面和等高线。结果表明,制备纳米TiO_2的最佳合成条件:TiCl_4浓度0.50mol/L,水浴温度75℃,焙烧温度500℃,亚甲基蓝的降解率可达92.31%。     

An investigation of structural optimization in crashworthiness design using a stochastic approach

In this paper the response surface methodology (RSM) and stochastic optimization (SO) are compared with regard to their efficiency and applicability in crashworthiness design. Optimization of simple analytic expressions and optimization of a front rail structure are the applications used to assess the respective qualities of both methods. A low detail vehicle structure is optimized to demonstrate the applicability of the methods in engineering practice. The investigations reveal that RSM is better compared to SO for fewer than 10–15 design variables. The convergence behaviour of SO improves compared to RSM when the number of design variables is increased. A novel zooming method is proposed which improves the convergence behaviour. A combination of both the RSM and the SO is efficient, stochastic optimization could be used in order to determine appropriate starting points for an RSM optimization, which continues the optimization. Two examples are investigated using this combined method.     

Optimization of the new Saab 9-3 exposed to impact load using a space mapping technique

The aim of this work is to illustrate how a space mapping technique using surrogate models together with response surfaces can be used for structural optimization of crashworthiness problems. To determine the response surfaces, several functional evaluations must be performed and each evaluation can be computationally demanding. The space mapping technique uses surrogate models, i.e. less costly models, to determine these surfaces and their associated gradients. The full model is used to correct the gradients from the surrogate model for the next iteration. Thus, the space mapping technique makes it possible to reduce the total computing time needed to find the optimal solution. First, two analytical functions and one analytical structural optimization problem are presented to exemplify the idea of space mapping and to compare the efficiency of space mapping to traditional response surface optimization. Secondly, a sub-model of a complete vehicle finite element (FE) model is used to study different objective functions in vehicle crashworthiness optimization. Finally, the space mapping technique is applied to a structural optimization problem of a large industrial FE vehicle model, consisting of 350.000 shell elements and a computing time of 100 h. In this problem the intrusion in the passenger compartment area was reduced by 32% without compromising other crashworthiness parameters.     

响应面法优化连翘总酚的超声辅助提取工艺

采用超声辅助提取连翘中总酚。以连翘总酚提取率为指标,研究了液料比、乙醇体积分数、超声功率和提取时间对总酚提取率的影响,在此基础上,利用响应面法对超声辅助提取连翘总酚工艺条件进行了优化,建立了二次回归模型,确定了最佳提取条件。结果表明,连翘中总酚的超声波辅助提取最佳工艺条件为:以体积分数55%的乙醇溶液为提取剂,液料比20 mL/g,提取时间34 min,超声功率240 W。在此条件下,连翘中总酚的平均提取率为3.72%,与二次回归模型理论预测值3.75%的相对误差为0.8%。实验结果与模型理论预测值相符。     

Polynomial genetic programming for response surface modeling Part 1: a methodology

The second-order polynomial is commonly used for fitting a response surface but the low-order polynomial is not sufficient if the response surface is highly nonlinear. Based on genetic programming (GP), this paper presents a method with which high-order smooth polynomials, which can model nonlinear response surfaces, can be built. Since in many cases small samples are used to fit the response surface, it is inevitable that the high-order polynomial shows serious overfitting behaviors. Moreover, the high-order polynomial shows infamous wiggling, unwanted oscillations, and large peaks. To suppress such problematic behaviors, this paper introduces a novel method, called directional derivative-based smoothing (DDBS) that is very effective for smoothing a high-order polynomial.The role of GP is to find appropriate terms of a polynomial through the application of genetic operators to GP trees that represent polynomials. The GP tree is transformed into the standard form of a polynomial using the translation algorithm. To estimate the coefficients of the polynomial quickly the ordinary least-square (OLS) method that incorporates the DDBS and extended data-set method is devised.Also, by using the classical Lagrange multiplier method, the modified OLS method enabling interpolation is presented.Four illustrative numerical examples are given to demonstrate the performance of GP with DDBS.     

Regularizing multiple kernel learning using response surface methodology

In recent years, several methods have been proposed to combine multiple kernels using a weighted linear sum of kernels. These different kernels may be using information coming from multiple sources or may correspond to using different notions of similarity on the same source. We note that such methods, in addition to the usual ones of the canonical support vector machine formulation, introduce new regularization parameters that affect the solution quality and, in this work, we propose to optimize them using response surface methodology on cross-validation data. On several bioinformatics and digit recognition benchmark data sets, we compare multiple kernel learning and our proposed regularized variant in terms of accuracy, support vector count, and the number of kernels selected. We see that our proposed variant achieves statistically similar or higher accuracy results by using fewer kernel functions and/or support vectors through suitable regularization; it also allows better knowledge extraction because unnecessary kernels are pruned and the favored kernels reflect the properties of the problem at hand.     

Parameter optimization in multiquadric response surface approximations

Multiquadric (MQ) response surface approximation uses the function C_i|X-X_i|²+h^1/2 to interpolate a given set of data. The performance of MQ approximation depends on the shift parameter h. Efficient methods of computing the optimal shift parameter based on the leave-one-out cross validation technique are presented in this paper. We also proved that the condition number of the MQ coefficient matrix is an increasing function of the shift parameter h. Two numerical examples are included to illustrate the proposed formulation.     

Analysis and optimization of surface roughness in the ball burnishing process using response surface methodology and desirabilty function

In the present study, an optimization strategy based on desirability function approach (DFA) together with response surface methodology (RSM) has been used to optimize ball burnishing process of 7178 aluminium alloy. A quadratic regression model was developed to predict surface roughness using RSM with rotatable central composite design (CCD). In the development of predictive models, burnishing force, number of passes, feed rate and burnishing speed were considered as model variables. The results indicated that burnishing force and number of passes were the significant factors on the surface roughness. The predicted surface roughness values and the subsequent verification experiments under the optimal conditions were confirmed the validity of the predicted model. The absolute average error between the experimental and predicted values at the optimal combination of parameter settings for surface roughness was calculated as 2.82%.     

响应面法优化PVA/PVDF复合膜纯化IPA的操作条件

常压下异丙醇与水形成共沸物,应用精馏法生产工艺能耗较大,本文采用自制的PVA/PVDF复合膜蒸汽渗透分离异丙醇(IPA)-水混合溶液,并应用响应面法确定最优的操作条件。首先,制备了理论交联度为8%的马来酸酐交联的PVA/PVDF复合膜:然后根据Box-Behnken中心组合实验设计原理,以膜的渗透通量和选择性为响应值,操作温度、IPA入膜浓度、膜渗透侧压力3个操作条件为影响因素进行实验:最后对实验结果进行响应面分析,得到了优化的操作条件并通过实验进行验证。综合考虑膜的渗透通量和选择性这2个响应值,求得实验所采用的PVA/PVDF复合膜在蒸汽渗透分离IPA水体系中的最优操作条件是:操作温度为95℃,IPA入膜浓度为82.5 wt.%,膜渗透侧压力为1 kPa,此时的渗透通量和选择性分别达到了1.57 kg/(m~2·h)和38.90。在此最佳操作条件下对模型预测值进行实验验证,膜的渗透通量和选择性的预测值与实验值的相对误差分别为5.7%和9.2%。     

响应面分析法优化山茱萸多糖提取工艺参数

选取山茱萸多糖提取时间、提取温度和料液比3个因素进行二次回归正交组合设计试验,利用响应面法对其提取工艺参数进行优化研究。利用MATLAB 7.0软件对山茱萸多糖提取产量的二次多项数学模型解逆矩阵分析表明:在提取时间为142 min、提取温度为84℃、水体积为276 mL的条件下,山茱萸多糖提取产量最高,最大提取产量预测值为89.7868μg·mL~(-1),实际提取值89.34μg·mL~(-1),两者基本相符。利用优化工艺参数提取山茱萸多糖时,具有最大的提取产量。     

Design of a radial basis function neural network with a radius-modification algorithm using response surface methodology

A radial basis function (RBF) neural network was designed for time series forecasting using both an adaptive learning algorithm and response surface methodology (RSM). To improve the traditional RBF networks forecasting capability, the generalized delta rule learning method was employed to modify the radius of the kernel function. Then RSM was utilized to explore the mean square error response surface so that the appropriate combination of network parameters, such as the number of hidden nodes and the initial learning rates, could be found. Extensive studies were performed on the effect of the initial values of connection weights on the accuracy of the backpropagation learning method that was employed in the training of the RBF artificial neural network. The effectiveness of the neural network with the proposed radius-modification technique and the RSM method was demonstrated with an example of forecasting intensity pulsations of a laser. It was found that, by utilizing the proposed techniques, the neural network provided a more accurate prediction of the response.     

A response surface based optimisation algorithm for the calculation of fuzzy envelope FRFs of models with uncertain properties

This paper describes a response surface based optimisation technique for the calculation of envelope frequency response functions (FRFs) of imprecisely defined structures using the interval and fuzzy finite element method.The authors developed a hybrid - global optimisation and interval arithmetic - procedure for interval and fuzzy envelope FRF calculation. This hybrid approach reduces the computational cost of the analysis compared to a full global optimisation approach and reduces the conservatism on the envelope FRF compared to a full interval arithmetic approach.Still, the optimisation step is the computationally most expensive part of the algorithm. To handle industrially sized applications, a very efficient optimisation procedure is imperative. The response surface based procedure described in this paper decreases the computational cost of a fuzzy envelope FRF calculation with a factor 50 or more compared to the commonly used two-level full factorial design of experiments, while giving a comparable accuracy.     

Nonlinear diffusion process modeling using response surface methodology and variable transformation

The nonlinear process modeling is investigated using statistical design method and response surface methodology. Three input factors are examined with respect to the response factor. In order to minimize the joint confidence region of fabrication process with varying conditions, D-optimal experimental design technique is performed and diffusion rate is characterized by response model. Then, the statistical results are used to verify the fitness of the nonlinear process model. Based on the results, this modeling methodology can be optimized process condition for semiconductor manufacturing.     

Polynomial genetic programming for response surface modeling Part 2: adaptive approximate models with probabilistic optimization problems

This is the second in a series of papers. The first deals with polynomial genetic programming (PGP) adopting the directional derivative-based smoothing (DDBS) method, while in this paper, an adaptive approximate model (AAM) based on PGP is presented with the partial interpolation strategy (PIS). The AAM is sequentially modified in such a way that the quality of fitting in the region of interest where an optimum point may exist can be gradually enhanced, and accordingly the size of the learning set is gradually enlarged. If the AAM uses a smooth high-order polynomial with an interpolative capability, it becomes more and more difficult for PGP to obtain smooth polynomials, whose size should be larger than or equal to the number of the samples, because the order of the polynomial becomes unnecessarily high according to the increase in its size. The PIS can avoid this problem by selecting samples belonging to the region of interest and interpolating only those samples. Other samples are treated as elements of the extended data set (EDS). Also, the PGP system adopts a multiple-population approach in order to simultaneously handle several constraints. The PGP system with the variable-fidelity response surface method is applied to reliability-based optimization (RBO) problems in order to significantly cut the high computational cost of RBO. The AAMs based on PGP are responsible for fitting probabilistic constraints and the cost function while the variable-fidelity response surface method is responsible for fitting limit state equations. Three numerical examples are presented to show the performance of the AAM based on PGP.     

Using surrogate models and response surfaces in structural optimization – with application to crashworthiness design and sheet metal forming

The aim of this paper is to determine if the Space Mapping technique using surrogate models together with response surfaces is useful in the optimization of crashworthiness and sheet metal forming. In addition, the efficiency of optimization using Space Mapping will be compared to traditional structural optimization using the Response Surface Methodology (RSM). Five examples are used to study the algorithm: one optimization of an analytic function and four structural optimization problems. All examples are constrained optimization problems. In all examples, the algorithm converged to an improved design with all constraints fulfilled, even when a conventional RSM optimization failed to converge. For the crashworthiness design problems, the total computing time for convergence was reduced by 53% using Space Mapping compared to conventional RSM. For the sheet metal forming problems the total computing time was reduced by 63%. The conclusions are that optimization using Space Mapping and surrogate models can be used for optimization in crashworthiness design and sheet metal forming applications with a significant reduction in computing time.     

Bayesian metrics for comparing response surface models of data with uncertainty

For many optimization applications a complicated computational simulation is replaced with a simpler response surface model. These models are built by fitting a limited number of evaluations of the full simulation with a simple function that captures the trends in the evaluated data. In many cases the values of the data at the evaluation points have some uncertainty. This paper uses Bayesian model selection to derive two objective metrics that can be used to determine which response surface model provides the most appropriate representation of the evaluated data given the associated uncertainty. These metrics are shown to be consistent with modelling intuition based on Occam’s principle. The uncertainty may be due to numerical error, approximations, uncertain input conditions, or to higher order effects in the simulation that do not need to be fit by the response surface. Two metrics, Q and G, are derived in this paper. The metric Q assumes that a good estimate of the simulation uncertainty is available. The metric G assumes the uncertainty, although present, is unknown. Application of these metrics in one and two dimensions are demonstrated. Received June 28, 2000     

Two-level optimization of airframe structures using response surface approximation

In this paper, a two-level optimization approach is developed for the preliminary and conceptual design of airframe structures. The preliminary design, involving a single objective multidisciplinary optimization, constitutes the lower level where ASTROS (Automated STRuctural Optimization System) is employed for multidisciplinary optimization. The conceptual design, which is carried out at the upper level, aims mainly at configuration design. The multiple objectives are incorporated as a single objective function by using the K-S function formulation. The objective function and constraints at the upper level are modelled through response surface approximation. During the upper level optimization process, the branch and bound method is applied for solving the problem with discrete design variables. The proposed strategy is demonstrated by the optimization of an Intermediate Complexity Wing (ICW) model. Received June 23, 1999     

响应面法优化草珊瑚总黄酮提取工艺的研究

目的:优化草珊瑚总黄酮提取工艺。方法:应用响应面实验方法,利用Box-Behnken试验设计,考察乙醇回流法提取草珊瑚中的总黄酮的工艺因素。结果:草珊瑚总黄酮的乙醇回流法最佳提取条件为:浓度为60%的乙醇,温度为80℃的水浴回流,比例为1:22的固液比,该条件下草珊瑚中总黄酮的提取率约为6.532%。结论:响应面试验方法能够有效提高草珊瑚总黄酮的提取率。