Response Surface Approximation Models Research Articles

Optimization design and experimental research on cooling performance of U-shaped latticework cooling structure with perforations used for gas turbine blade

Latticework cooling structure provides great structural strength and well heat transfer enhancement commonly used for internal cooling of turbine blades. However, traditional rectangle latticework cooling structure raises friction loss badly and reduces cooling efficiency of turbine blades due to the increase of wetted surface area. A new latticework cooling structure with U-shaped sub-channel and perforations on raffles was proposed, aiming to promote heat transfer and reduce friction loss for latticework cooling structure. Numerical simulation, optimization method and experimental test are conducted to investigate heat transfer and fluid flow characteristics of this new latticework cooling structure. Optimization design based on response surface approximation model and non-dominated sorting genetic algorithms-Ⅱ was conducted and objectives were to maximize heat transfer and minimize friction loss. The optimal U-shaped latticework cooling structure with perforations was obtained and experimental research was conducted. Characteristics of heat transfer and friction loss between the optimal U-shaped latticework cooling structure with perforations and the rectangle latticework cooling structure were compared and analyzed. The experiment results showed that the overall heat transfer and thermal–hydraulic performances of the optimal U-shaped latticework cooling structure with perforations are 21.6–37.0% and 27.9–46.7% respectively higher than those of the rectangle latticework cooling structure while friction loss performance of the former is 16.3–29.2% lower than that of the latter as Reynolds number increases from 5000 to 50000. Moreover, empirical formulas for the optimal U-shaped latticework cooling structure with perforations and the rectangle latticework cooling structure are obtained. The present study indicates that the U-shaped latticework cooling structure with perforations not only maintains the superior heat transfer performance, but also reduces friction loss, which is mainly due to the perforation ribs reduce flow distance for part of coolant and also provide an impingement effect for the heated wall and enhance flow interactions and flow mixing.

Structural analysis and optimal design of a spherical thin-walled stainless steel water tank without reinforced tie ribs

A spherical thin-walled stainless steel water tank without reinforced tie ribs has been designed to address the issues of easy fracture and corrosion of the ribs, difficulty in maintenance and cleaning, and short service life exposed during the use of thin-walled stainless steel water tanks with reinforced tie ribs. Firstly, an analytical model of a flat steel thin-walled water tank without reinforced tie ribs was established and subjected to static analysis under water pressure. The deformation and stress distribution patterns of the side molded plate of the flat box were obtained. Secondly, a spherical non ribbed thin-walled stainless steel water tank structure was designed with circular cross-section under different box bulge parameters, and its mechanical response characteristics under water pressure load were analyzed. A strengthening scheme was designed for the bottom box molded plate. Once again, optimize the combination design of the box scheme and the reinforcement scheme, and analyze their static, thermodynamic, and thermal solid coupling performance. Finally, the Latin Hypercube Sampling method was used to generate experimental design samples, and a response surface approximation model of a spherical thin-walled stainless steel water tank without reinforced tie ribs was constructed. The wall thickness of the box molded plate, skeleton, and reinforcement were used as design variables, and the maximum deformation and maximum equivalent stress were used as constraints. The lightweight design was carried out with the goal of minimizing mass. The research results indicate that the design program and parameter selection method for spherical thin-walled stainless steel water tanks without reinforced tie ribs proposed in the article are efficient and feasible, and can provide technical reference and theoretical support for the layout and overall optimization design of non ribbed thin-walled stainless steel water tank structures.

Design and optimization of a hybrid cooling configuration combining PCM and liquid cooling for Li-ion battery using data-based response surface approximation model

Effective thermal management is crucial for safe operation of electric vehicles. This study proposes a novel hybrid cooling system that using phase change material (PCM) and liquid cooling to address issues such as leakage, non-uniform temperatures under high discharge rates. A comprehensive 3D model was developed to analyze the influence of geometric parameters on the cooling system. Response surface approximation modeling was used to optimize the layout parameters of the PCM. The results show that the hybrid cooling outperforms other methods, maintaining the cell temperature with 35 ℃ and the temperature difference within 4 ℃. The thickness of the PCM is a significant factor, as it increases heat storage capacity and reduces temperature; however, exceeding a certain threshold leads to increased thermal resistance. A non-uniform PCM layout can reduce the temperature difference by 26 %. An optimal PCM thickness of 2.5 mm to 3 mm ensures that the maximum temperature remains below 32.5 ℃, with a temperature difference within 2 ℃. The lowest temperature difference is achieved with a vertical layout height of 70 mm and a horizontal layout thickness of 3 mm. These findings are valuable for enhancing the thermal management of lithium-ion battery.

Optimization of Guide Vane Centrifugal Pumps Based on Response Surface Methodology and Study of Internal Flow Characteristics

The methodologies of computational fluid dynamics (CFD) and response surface method (RSM) were integrated to uncover the optimal correlational framework for intricate hydraulic geometric parameters of guide vane centrifugal pumps. Parameters such as blade number, blade wrap angle, blade outlet angle, and relative axial distance between the guide vane and impeller, as well as radial distance, are embraced as optimization design variables. Meanwhile, pump head and efficiency were chosen as responsive variables. An analysis of 46 sets of hydraulic performance data was carried out by using the Box–Behnken experimental design method. Subsequently, response surface approximation models were established between hydraulic parameters and the efficiency, as well as the head. The optimal design point was predicted and a simulation of the hydraulic characteristics for the optimal scheme was conducted; the errors were 0.846% for head and 0.256% for efficiency between the simulation results with predicted results from RSM. The optimized model demonstrates noteworthy enhancements in hydraulic performance in comparison to the original model. By analyzing the internal flow of the optimized model under transient conditions, it was found that, as the internal flow of the flow passage components is relatively disordered at small flow rates, the amplitude of pressure pulsation is affected a lot. At other flow rates, the inside pressure pulsation waveform exhibits pronounced periodicity, and the primary causes of pressure pulsation in various flow components are not the same. Wall dissipation and turbulent dissipation emerge as significant contributors to the entropy generation in this centrifugal pump. The magnitude of entropy generation is correlated with the flow rate and the structural configuration of the pump’s components. High-entropy regions concentrate around the leading and trailing edges of the blades.

Shape optimization of a corner‐recessed square tall building to reduce mean wind pressure using a multi‐objective genetic algorithm

SummaryThe current study aims to determine how the corner recession affects tall buildings with square plans. A series of numerical simulations have been conducted to find the parametric models' wind pressure. Visualization tools, such as contour plots and streamlines, present the wind flow near the buildings. Numerical simulations are conducted using RANS k‐ℇ turbulence models considering a length scale of 1:300. Subsequently, a shape optimization study has been carried out to propose a suitable percentage of corner recession, which should minimize the wind pressure on different faces of the building. As design factors, the amount of corner recession (S) and the wind incidence angle (Ø) are taken, along with the mean pressure coefficients (Cp) on the various building faces. Due to the eight axes symmetry of the building configuration, the random sampling technique is used for the Design of Experiment while accounting for the 0°–45° wind angle of attack. The Response Surface Approximation (RSA) is used to construct surrogate models of the objective functions. The RSA models are validated with wind tunnel test results presented in previously published articles. The optimization study is carried out using the multi‐objective genetic algorithm technique.

Effect of Multifactor Interaction on the Accuracy of RV Reducers

The rotary vector (RV) reducer is one of the widely used mechanical components in industrial systems, specifically in robots. The stability of the transmission performance of the RV reducer is crucial for the efficient operation of industrial equipment. The manufacturing and assembly errors of various components of the RV reducer during the production process are important factors that affect the transmission performance. However, in previous research work, the coupling effect of multiple errors on the transmission accuracy of RV reducer has not been fully considered. Furthermore, a vague relationship between system transmission errors and various errors also has not been thoroughly discussed, which presents a challenge to analyze and optimize the errors of components using the simulation technology of virtual prototype. Therefore, we propose a novel approach to use the response surface method (RSM) to investigate the transmission accuracy of RV reducer. Firstly, based on the constructed virtual prototype of RV reducer, the individual effects of different original errors on the overall transmission error are analyzed. Secondly, a response surface approximation model using RSM is constructed to analyze the effect of multiple error interactions on the transmission accuracy of the RV reducer, and the potential functional relationship between multiple error factors and the overall transmission error is also explored. Finally, the authenticity of the proposed approach is verified by setting up some comparative experiments. This study provides a reference for the efficient analysis and optimization of the transmission accuracy of RV reducers.

Optimized Lightweight Frame for Intelligent New-energy Vehicles

In this paper, a joint optimization method based on multi-objective response surface approximation model and finite element simulation program is proposed to realize the lightweight optimization of new-energy vehicle frames. Under the premise of satisfying the constraints of strength, frequency and vibration, the thickness of different important parts is optimized to achieve the goal of minimizing the quality of intelligent vehicles. In order to obtain the stress distribution of each part and the vibration frequency of the frame, various finite element analyses of the intelligent vehicle frame are analyzed. In order to achieve optimization, this paper adopts the response surface method for multi-objective optimization. Sample data was generated by the central composite design, and the response surface optimization method was used to filter out 5 design variables that had a large impact on the frame. As a result, the weight of the frame was reduced from 25.05 kg to 19.86 kg, a weight reduction of 20.7%, achieving a significant weight reduction effect. This method provides important reference value and guiding significance for the optimization of frame and its lightweight. In this way, the design of the frame can be better optimized to make it lighter, thereby improving the performance of the smart car. At the same time, this method can also be applied to optimization problems in other fields to achieve more efficient and accurate optimization goals.

Multi-objective performance optimization of target surface of bionic blue whale-skin impinged by array jet

Jet impingement cooling technology is usually applied to combustor liner with large heat load. With the continuous improvement of high capacity and low emission of heavy-duty gas turbines,the research on low resistance and high cooling efficiency cooling technology is particularly important. To meet these urgent needs, based on the blue whale skin structure, a new array impingement target surface cooling structure was proposed from the perspective of bionics. The flow and heat transfer characteristics of the array impingement bionic target surface cooling structure were numerically simulated under high Reynolds number (Re) and high Heat flux density (q) conditions. The influence of groove depth (H, 1 to 3 mm), radius ratio (R/H, 1 to 3) and spacing ratio (P/H, 10 to 20) on the cooling performance of impingement target surface was analyzed. Multi-objective performance optimization of the bionic structure was carried out by using the second-order response surface approximation model (RSM) and the second-generation non-inferior sequencing genetic algorithm (NSGA-II), and the performance of the optimized structure was tested. The results show that the above optimization method is accurate and effective, and the optimal bionic structure is H = 1.212 mm, R/H = 1.132, P/H = 12.078. Compared with the plane target surface, the average Nusselt number (Nuave), flow friction factor (f), and comprehensive thermal coefficient (F) of the optimal bionic groove structure increased by 14.5%, 1.1% and 14.1%，respectively. The structure and multi-objective optimization method proposed in this paper can provide a certain reference for the design of new efficient cooling structure of heavy-duty gas turbines combustor liner.

Multi-Variate Optimization of Polymer Electrolyte Membrane Fuel Cells in Consideration of Effects of GDL Compression and Intrusion

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.

Design Optimization of Composite Curved Rod for Wind Tunnel Virtual Flight Test Based on Multi-island Genetic Algorithm

The curved rod of carbon fibrecomposite is the key component of the large angle three degree of freedom virtual flight test device. This article takes the composite curved rod as the research object, Establish a finite element analysis model and analyze its strength and stiffness under the initial design conditions, the RSM response surface approximation model and multi-island genetic algorithm are used to optimize the carbon fibre composite angle of each layer of the composite curved rod, so that the maximum deformation displacement of the curved rod is minimized, As far as possible, the influence of the center of mass of the Virtual flight test model on the wind tunnel test data should be reduced. The optimization results show that relative to the initial design, the failure factor of the composite curved rod is reduced by 65.4%, the rigidity is increased by 35.94%, and the maximum deformation is reduced by 44.5%. In order to realize the simulation of flight motions close to 90 ° angle of attack in a horizontal wind tunnel, it provides equipment guarantee for the research of stall deviation, initial phase of spin and stable spin.

Research on Application of Electric Vehicle Collision Based on Reliability Optimization Design Method

When mentioning multidisciplinary design optimization methods, the deterministic optimum design is frequently applied to set the constraint boundary. Furthermore, only a small amount of space tolerances (or uncertainty) is available in the process of design, manufacture and operation. Therefore, deterministic optimum design lacking uncertainty cannot meet the needs of reliability-based design optimization. In this paper, reliability optimization design method, finite element (FE) analysis, optimal Latin hypercube test design and response surface approximation model are combined to optimize the side structure of electric vehicles and improve its crashworthiness. Firstly, a side impact FE model of the electric vehicle is established and verified in this paper. Then, the dimensions and the material yield strength of the force-bearing structure in the vehicle are selected as design variables, and the impact speed in the actual collision is selected as a random variable to optimize the car crashworthiness in the side impact using the 95% reliability optimization method. The results show that the 95% reliability optimization design increases the total energy absorption of the side components by 9.45%, the intrusion of the B-pillar and the vehicle door inner panel decreased by 10.42% and 14.75%, respectively. The intrusion speed of the B-pillar and the inner panel of the vehicle door decreases by 10.35% and 17.78%, respectively. By comparing the results of traditional deterministic optimization and reliability optimization methods, the latter can better satisfy the crash safety objectives, and improve the reliability of vehicle body design.

Multi-objective optimization of the shell in autonomous intelligent argo profiling float

To improve the performance of the first-generation profiling float and ensure that the float smoothly completes the profile motion, a shell optimization method based on response surface approximation model and multi-objective genetic algorithm is proposed. The second-order nonlinear response surface models of resistance, mass and envelope volume of the shell are established based on the data from Design of Experiments (DOE). Taking the minimum resistance, the minimum mass and the maximum envelope volume of the shell as optimization objectives, the Pareto Front is obtained based on the fast elitist Non-dominated Sorting Genetic Algorithm II (NSGA-II). The influence rule and degree of each design variable on the optimization objective are gained through the main effect analysis. A set of solutions are selected as the design scheme of the shell for second-generation profiling float. The hydrodynamic performance, endurance and carrying capacity of the profiling float have been greatly improved compared with the first-generation. Finally, the feasibility and effectiveness of the optimization results are verified through simulation analysis and high pressure test. The proposed optimization process can greatly improve the performance of Argo profiling float, shorten the optimization period of shell design and improve the optimization efficiency. Therefore, it has a good reference value for the design optimization of other similar submarines. • To improve the hydrodynamic performance, endurance and carrying capacity of the first-generation Argo profiling float, a shell optimization method based on response surface approximation model and (NSGA-II) is proposed. The Pareto Front is obtained based on the NSGA-II. A set of solutions are selected as the design scheme of the shell for second-generation profiling float. • The feasibility and effectiveness of the optimization results are verified through hydrodynamic analysis by FLUENT and buckling analysis by ABAQUS. In order to prove the feasibility of the final optimization scheme and the reliability of the simulation analysis, 48 M P a high pressure test is carried out by the high pressure experimental device. • Through the analysis of the main effect plots, we can get the influence law and degree of each design variable on the optimization objective. In the design of the shell, the range of design variables can be determined according to the requirements of design indicators. • The proposed optimization process can greatly improve the performance of the second-generation Argo profiling float, shorten the optimization period of shell design and improve the optimization efficiency. Therefore, it has a good reference value for the design optimization of other similar submarines.

Multi-objective optimization of a new special-shaped tube for heating deicing fluid

The ground deicing heating system is an important equipment to ensure flight safety of civil aircraft in winter. As the main element, the heating tube directly affects the working efficiency of the system. The deicing fluid is high viscosity mixture. To improve heating efficiency of deicing fluid, a new special-shaped tube is proposed, which is named double-spiral trapezoidal grooved tube. The upper base width (m), groove depth (h) and base angle (θ) are the important structural parameters of the new tube. The optimal Latin hypercube design is applied to carry out the experiment design of m, h and θ. Nusselt number (Nu), flow resistance coefficient (f) and performance evaluation criteria (PEC) are selected as the optimization objectives. Then, the response surface approximation model is established to conduct sensitivity analysis. Based on the model, NCGA and NSGA-II algorithms are, respectively, introduced to perform the multi-objective optimization. The results show that NSGA-II is better than NCGA. When \(m \in \left[ {0.17, \, 0.33} \right]\), \(h \in [1.06, \, 1.08]\) and \(\theta \in \left[ {84.06, \, 84.75} \right]\), the best range of optimization objectives is \(Nu \in \left[ {86.04, \, 87.81} \right]\), \(f \in \left[ {0.70, \, 0.73} \right]\) and \({\text{PEC}} \in \left[ {1.60, \, 1.62} \right]\). With the growth of turbulence intensity, PEC > 1, and Nu increases by 4–27%. As a result, the heating efficiency of the new tube is superior to the original tube.

Three-objective optimization of a single-channel pump for wastewater treatment

The impeller and volute of a single-channel pump used for wastewater treatment were simultaneously optimized to improve the hydraulic efficiency and reduce unsteady radial force sources due to the impeller–volute interaction. Steady and unsteady Reynolds-averaged Navier–Stokes equations were solved with the shear stress transport turbulence model as the turbulence closure model using tetrahedral grids to analyze the internal flow in the single-channel pump. Five design variables related to the internal flow cross-sectional areas of the impeller and volute were selected to simultaneously optimize the hydraulic efficiency, the sweep area of the radial force during one revolution, and the distance of the mass center of the sweep area from the origin as the three objective functions. The response surface approximation model and genetic algorithm were employed to obtain three-dimensional Pareto-optimal solutions representing the trade-off between the efficiency and the radial force sources. The results of the three-objective optimization show that the representative clustered optimum designs enhanced the efficiency and reduced the radial force sources simultaneously in most cases, compared to the reference design. In addition, it was confirmed that the trade-off between the efficiency and the radial force sources clarifies with controlling the internal flow cross-sectional areas of the impeller and volute of the single-channel pump.

Sequential approximate optimisation for statistical analysis and yield optimisation of circularly polarised antennas

The study addresses a problem of low-cost statistical analysis and yield optimisation of circularly polarised (CP) microstrip antennas. For the sake of computational efficiency, auxiliary response surface approximation models are utilised to establish a fast representation of the expensive full-wave electromagnetic (EM)-simulation antenna model. The surrogate permits feasible Monte Carlo (MC) analysis using a large number of random samples (necessary to maintain low variance of yield estimation). The sequential approximate optimisation is then utilised to carry out a robust design, where at each iteration, a new surrogate model is established in a relocated domain. The design objectives are set for the in-band axial ratio and impedance bandwidth. For illustration purposes, a planar CP antenna with two feed line stubs is considered. Accurate yield estimation is obtained at the cost of only a hundred of EM antenna simulations despite a large number of independent geometry parameters. Yield improvement from the initial value of 63.5% at the nominal design to 99% is achieved at the cost of 300 EM analyses. Comparison with conventional MC analysis is also provided demonstrating the accuracy of the presented approach.

Robust Optimization Design for Turbine Blade-Tip Radial Running Clearance using Hierarchically Response Surface Method

Considering the robust optimization computational precision and efficiency for complex mechanical assembly relationship like turbine blade-tip radial running clearance, a hierarchically response surface robust optimization algorithm is proposed. The distribute collaborative response surface method is used to generate assembly system level approximation model of overall parameters and blade-tip clearance, and then a set samples of design parameters and objective response mean and/or standard deviation is generated by using system approximation model and design of experiment method. Finally, a new response surface approximation model is constructed by using those samples, and this approximation model is used for robust optimization process. The analyses results demonstrate the proposed method can dramatic reduce the computational cost and ensure the computational precision. The presented research offers an effective way for the robust optimization design of turbine blade-tip radial running clearance.

Shape optimization of a feedback-channel fluidic oscillator

ABSTRACTIn this study, a fluidic oscillator was optimized based on the three-dimensional unsteady Reynolds-averaged Navier-Stokes analysis to enhance peak jet velocity at the outlet and simultaneously reduce pressure drop. A multi-objective genetic algorithm performed the optimization with surrogate modeling. The ratios of the inlet nozzle width and the distance between the splitters to the throat width were chosen as the design variables. And, two objective functions related to peak jet velocity at the outlet and pressure drop through the fluidic oscillator were selected for the optimization. Ten design points were selected in the design space using a Latin hypercube sampling method; the objective functions were calculated by unsteady Reynolds-averaged Navier-Stokes analysis at these design points to construct surrogate models that were used to approximate the objective functions. Two different surrogate models, namely response surface approximation and Kriging models were tested. Pareto-optimal front representing a compromise between the two objective functions was obtained from the multi-objective optimization. The optimization results indicated that a jet velocity-oriented optimum design increased the peak jet velocity ratio at the outlet and the friction factor by 11.18% and 16.82%, respectively, when compared to those of a friction factor-oriented design.

Performance Analysis and Design Optimization of Gapped Pin-Fin in a Cooling Channel

ABSTRACTPerformance analysis and optimization of a circular pin-fin with inside gaps in a rectangular cooling channel were performed at Reynolds number, 10,000, using three-dimensional Reynolds-averaged Navier–Stokes equations and a multi-objective genetic algorithm. The low-Reynolds-number version of the shear stress transport model was used as turbulence closure. A parametric study was also performed to identify the geometrical effects of the pin-fin on heat transfer and pressure drop. The straight and reference gapped pin-fins yielded better performances than those of the circular pin-fin without the gap in terms of both heat transfer and pressure drop. The objective of the optimization was to maximize the heat transfer and minimize the pressure loss, simultaneously. The area-averaged Nusselt number and pressure loss coefficient were considered as objective functions, and three design variables related to the geometry of the gapped pin-fin were chosen for the optimization. Twenty-seven design points were generated using Latin hypercube sampling in the design space, and response surface approximation models were constructed for the objective functions. The optimization results were analyzed using five representative solutions on the Pareto-optimal front. The objective functions were found to be significantly affected by variation in the design variables, especially, the width of front gap and the rear gap angle.

Multi-objective design optimization of antenna structures using sequential domain patching with automated patch size determination

ABSTRACTIn this article, a simple yet efficient and reliable technique for fully automated multi-objective design optimization of antenna structures using sequential domain patching (SDP) is discussed. The optimization procedure according to SDP is a two-step process: (i) obtaining the initial set of Pareto-optimal designs representing the best possible trade-offs between considered conflicting objectives, and (ii) Pareto set refinement for yielding the optimal designs at the high-fidelity electromagnetic (EM) simulation model level. For the sake of computational efficiency, the first step is realized at the level of a low-fidelity (coarse-discretization) EM model by sequential construction and relocation of small design space segments (patches) in order to create a path connecting the extreme Pareto front designs obtained beforehand. The second stage involves response correction techniques and local response surface approximation models constructed by reusing EM simulation data acquired in the first step. A major contribution of this work is an automated procedure for determining the patch dimensions. It allows for appropriate selection of the number of patches for each geometry variable so as to ensure reliability of the optimization process while maintaining its low cost. The importance of this procedure is demonstrated by comparing it with uniform patch dimensions.

Optimized Reduction of Unsteady Radial Forces in a Singlechannel Pump for Wastewater Treatment

A single-channel pump for wastewater treatment was optimized to reduce unsteady radial force sources caused by impeller-volute interactions. The steady and unsteady Reynolds- averaged Navier-Stokes equations using the shear-stress transport turbulence model were discretized by finite volume approximations and solved on tetrahedral grids to analyze the flow in the single-channel pump. The sweep area of radial force during one revolution and the distance of the sweep-area center of mass from the origin were selected as the objective functions; the two design variables were related to the internal flow cross-sectional area of the volute. These objective functions were integrated into one objective function by applying the weighting factor for optimization. Latin hypercube sampling was employed to generate twelve design points within the design space. A response-surface approximation model was constructed as a surrogate model for the objectives, based on the objective function values at the generated design points. The optimized results showed considerable reduction in the unsteady radial force sources in the optimum design, relative to those of the reference design.