Response Surface Optimization with Discrete Variables

Population-based (gradient-free) methods have become attractive for solving optimization problems since they have been known to locate "optimal" (best) positions within a search space. In this study, we investigate the use of three different gradient-free optimization methods namely: Iterative Latin Hypercube Sampling (ILHS), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA) for well placement and well controls optimization problem of the CO2 underground storage in a 2D saline aquifer with bound constraints on the design variables. We also consider sequential optimization of well locations and well controls using the gradient-free methods. To evaluate the performance of these algorithms, we test the methods on well-known test (or toy) problems such as the multi-dimensional (40 design variables) Rosenbrock function, as well as the Traveling Salesman Problem (TSP) with 20 cities (20 design variables). We further extend the study to a more complex problem which involves finding the optimal well types, well locations, or well controls for a CO2 injection problem in a synthetic saline aquifer compositional model built in a commercial simulator. The objective function formulated for the CO2 injection problem is the net present environmental value (NPEV) involving discrete design variables (well types and well locations) and real design variables (well controls). Results obtained from the performance comparison of these gradient-free methods on the toy problems demonstrate the superior performance of the ILHS over PSO and GA for the Rosenbrock test function, while GA outperforms ILHS and PSO in solving the TSP. We also observe that for the simple well locations optimization problems involving one or two wells, all three algorithms obtained comparable results, given the same number of samples and number of iterations. However, for more complex cases such as the sequential or simultaneous optimization problems, when multiple wells which may include injectors and producers are present, we observe significant differences in the selection of the optimal well types and well locations among the three optimizers. In the simultaneous optimization of well types, well locations, and well controls, when both injectors and producers are BHP controlled, the results obtained showed that it is more optimal for at least one producer to be present, and for the injectors to be operated at the upper bound of the injector BHP, while the producers operated at the lower bound of the producer BHP to maximize the NPEV.

<div class="htmlview paragraph">To overcome the trade-off between NOx and particulate emissions for future diesel vehicles and engines it is necessary to seek methods to lower pollutant emissions. The desired simultaneous improvement in fuel efficiency for future DI (Direct Injection) diesels is also a difficult challenge due to the combustion modifications that will be required to meet the exhaust emission mandates.</div> <div class="htmlview paragraph">This study demonstrates the emission reduction capability of split injections, EGR (Exhaust Gas Recirculation), and other parameters on a High Speed Direct Injection (HSDI) diesel engine equipped with a common rail injection system using an RSM (Response Surface Method) optimization method. The optimizations were conducted at 1757 rev/min, 45% load. Six factors were considered for the optimization, namely the EGR rate, SOI (Start of Injection), intake boost pressure, and injection pressure, the percentage of fuel in the first injection, and the dwell between injections. In addition, the effects of swirl were investigated.</div> <div class="htmlview paragraph">The benefit of the combined effects of EGR with supercharging was well realized by the RSM optimization, which resulted in simultaneous reductions in both NOx and PM emissions, while even improving BSFC (Brake Specific Fuel Consumption). Through the present optimization process, it was shown that RSM optimization is an effective and powerful tool for realizing the full advantages of the combined effects of combustion control techniques. In addition, observation of the optimization process provides a more thorough understanding of HSDI diesel combustion since the observed trends can be explained by monitoring the effects of the operating parameters.</div>

The design optimization of two-stage cylindrical helical gear reducer is discussed in this paper. The volume and the center gear distance are adopted as objective functions separately, and the particle swarm optimization(PSO) method is used to improve the design quality. The design variables contain discrete variables, which are converted to discrete variables by the nearby principle in order to improve the running efficiency. Inertia weight coefficient is used in the PSO algorithm to adjust the coverage speed. The optimization result shows that our method is better than other optimization method, and it proves that PSO method is meaningful to the reducer’s design.

A novel optimization method based on a differential evolution (DE) strategy for designing low-weight truss structures is presented for both continuous and discrete variables. Applications of this technique on the optimization of a benchmark-type truss structure with continuous and discrete variables are given to evaluate its effectiveness. Results are compared with various classical and evolutionary optimization methods which show that the proposed procedure based on DE outperforms other methods and can effectively be applied to the optimization problems of truss structures with both continuous and discrete design variables.

Static and dynamic optimization of a pose adjusting mechanism considering parameter changes during construction

A new method is presented for an application of the Gaussian mixture model (GMM) to a multi-objective robust design optimization (RDO) of planar steel frame structures under aleatory (stochastic) uncertainty in material properties, external loads, and discrete design variables. Uncertainty in the discrete design variables is modeled in the wide range between the smallest and largest values in the catalog of the cross-sectional areas. A weighted sum of Gaussians is statistically trained based on the sampled training data to capture an underlying joint probability distribution function (PDF) of random input variables and the corresponding structural response. A simple regression function for predicting the structural response can be found by extracting the information from a conditional PDF, which is directly derived from the captured joint PDF. A multi-objective RDO problem is formulated with three objective functions, namely, the total mass of the structure, and the mean and variance values of the maximum inter-story drift under some constraints on design strength and serviceability requirements. The optimization problem is solved using a multi-objective genetic algorithm utilizing the trained GMM for calculating the statistical values of objective and constraint functions to obtain Pareto-optimal solutions. Since the three objective functions are highly conflicting, the best trade-off solution is desired and found from the obtained Pareto-optimal solutions by performing fuzzy-based compromise programming. The robustness and feasibility of the proposed method for finding the RDO of planar steel frame structures with discrete variables are demonstrated through two design examples.

Discrete sizing optimization method based on dividing rectangles algorithm and local response surface for steel frame structures

This paper proposes a Sequential Approximate Optimization (SAO) for the discrete design variable problems using Radial Basis Function (RBF) Network. We consider that there are two important keys for SAO: One is the parameter adjustment for good approximation, and the other is to explore the sparse region for global approximation. The authors have proposed the simple estimate of the width in the Gaussian Kernel for good approximation. In addition, in order to explore the sparse region, we have developed the density function. The density function with the simple estimate of the width works well in the case of the continuous design variables. However, a simple application of the density function to the discrete design variable problems will cause some difficulties. In order to overcome these difficulties and find the sparse region of the discrete design variables with the density function, the new variables for the discrete design variables are introduced. By using the new variables, it is possible to find the sparse region of the discrete design variables. A simple sampling algorithm is shown, in which the Discrete Differential Evolution (DDE) for the discrete design variables is employed. Through typical numerical examples, the validity of proposed approach is examined.

A large amount of radioactive waste resin will be produced in the production and operation of the nuclear industry, which needs to be treated and disposed safely. The microwave-assisted Fenton reaction is used in this study for the degradation of radioactive waste resin, and the response surface method is adopted to optimize and establish the high-efficiency degradation process. The four-factor (pH value, the dosage of FeSO4 and H2O2 reagents, and the microwave power) and three-level response surface method was used to analyze the influence degree of various factors on the experimental results, screening the key factors affecting the degradation effect, and exploring the best process conditions for the experiment. The research results show that based on the response surface optimization scheme, the optimal process conditions for degrading cationic resin are: pH = 1.637, 41.3 mL of 0.02 mol/L FeSO4, 116.1 mL of (30%) H2O2, 438.31 W of microwave power. Under the optimal condition, the cation exchange resin is rapidly degraded within 1 hour, and the COD of degradation solution is as low as 1.04 g/L. The findings indicate that the microwave-assisted Fenton reaction is an effective method to treat the cation exchange resin, and the response surface optimization method is reliable to the research of cationic resin degradation, which is conducive to design the treatment process of radioactive waste resin.

To avoid drawbacks of conventional structural op- timization approaches,a neural network(NN)response surface optimization method is proposed for the design of composite stiffened structures.Such NN-based structural analysis re- sponse surfaces can reflect the global mapping relationship between design inputs and structural response outputs.By using the orthotropic experiment method to select the appropriate structural finite element analysis samples,neural network re- sponse surfaces can be trained with reasonable accuracies.The constructed response surfaces can be either used as objective function or constraints or both.Together with other conven- tional constraints,an revised optimization design model can be formed which can be solved by using genetic algorithm(GA). Taking a hat-stiffened composite panel of blended wing-body aircraft as example,the structural weight response surface is developed as objective function,and strength and buckling factor response surfaces as constraints.All these neural net- works are trained by finite element samples computed through PATRAN/NASTRAN software.The optimization results illus- trate that it can significantly reduce the cycles of finite element model analysis and achieve highly accurate response approxi- marion results.Eventually,the approach can greatly save the com- putation time and raise the efficiency of optimization process.

Engineering design problems often involve global optimization of functions that are supplied as ’black box‘ functions. These functions may be nonconvex, nondifferentiable and even discontinuous. In addition, the decision variables may be a combination of discrete and continuous variables. The functions are usually computationally expensive, and may involve finite element methods. An engineering example of this type of problem is to minimize the weight of a structure, while limiting strain to be below a certain threshold. This type of global optimization problem is very difficult to solve, yet design engineers must find some solution to their problem – even if it is a suboptimal one. Sometimes the most difficult part of the problem is finding any feasible solution. Stochastic methods, including sequential random search and simulated annealing, are finding many applications to this type of practical global optimization problem. Improving Hit-and-Run (IHR) is a sequential random search method that has been successfully used in several engineering design applications, such as the optimal design of composite structures. A motivation to IHR is discussed as well as several enhancements. The enhancements include allowing both continuous and discrete variables in the problem formulation. This has many practical advantages, because design variables often involve a mixture of continuous and discrete values. IHR and several variations have been applied to the composites design problem. Some of this practical experience is discussed.

Research on optimization design of the heating/cooling channels for rapid heat cycle molding based on response surface methodology and constrained particle swarm optimization

Surface roughness or surface quality is considered to be one of the most crucial requirement of a machined part since it directly influences the mechanical properties of the part. However, the traditional method of choosing cutting parameters’ values to obtain a good surface finish has its own disadvantages. Therefore, an experimental study has been conducted to develop a suitable mathematical model and pair it with an optimization technique that able to produce low surface roughness of carbon steel AISI 1045. Response surface methodology (RSM) is used to develop the mathematical model whereas three types of heuristic optimization methods namely Genetics Algorithm (GA), Particle Swarm Optimization (PSO) and Simulated Annealing (SA) employed to optimize the model and find the optimal cutting parameters’ values. A brief comparison of the three optimization methods has been made to study their performance to the developed model. Experimental results indicate that the proposed modeling technique and PSO are quite efficient in determining optimal cutting parameters for CNC turning of carbon steel AISI 1045.

This paper presents design optimization technique of Axial Flux Permanent Magnet (AFPM)Brushless DC (BLDC)micromotor. The objective of the optimization process is to minimize the motor volume and improve joules efficiency with the constraints of minimum required torque and maximum back EMF using response surface modeling and Bat Algorithm (BA). Finite element computations have been used for numerical experiments on geometrical design variables in order to evaluate the coefficients of a second-order empirical model for the response surface representation. BA is used as an optimization technique to minimize volume and improve joules efficiency in separate optimization problem in terms of design variables. The optimization results were compared with other metaheuristic algorithms, including Genetic Algorithms (GA), and Particle Swarm Optimization (PSO). The bat algorithm shows a potential competitive against GA and PSO.

At present, response surface methodology (RSM) is the most preferred method for fermentation media optimization. However, in the last two decades, artificial intelligence algorithm has become one of the most efficient methods for empirical modelling and optimization. One of the popular developed approaches is Particle Swarm Optimization (PSO), which is used in optimizing a problem. This paper focuses on comparative studies between RSM and PSO in fermentation media optimization for the production of flexirubin production using Chryseobacterium artocarpi CECT 8497T. Two methodologies were compared for in terms of their modeling, sensitivity analysis, and optimization abilities. All experiments were performed accordingly to box-behnken design (BBD), and the generated data was analyzed using RSM and PSO. The sensitivity analysis performed by both methods has given comparative results. Based on the correlation coefficient, the model developed with PSO was found to be superior to the model developed with RSM. The result shows that PSO gives a better pigmentation yield with optimal fermentation concentration.