Multi-objective Optimization Design Of Magnesium Alloy .

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DJournal of Materials Science and Engineering B 9 (1-2) (2019) 13-24doi: i-objective Optimization Design of Magnesium AlloyWheel Based on Topology OptimizationJIANG Xin1, LIU Hai2, Yoshio Fukushima1, Minoru Otake3, Naoki Kawada1, ZHANG Zhenglai4 and JUDongying11. Department of Electronic Engineering, Graduate School of Engineering, Saitama Institute of Technology, Fukaya 369-0293,Japan2. School of Mechanical Engineering, Hebei University of Technology, Tianjin 300000, China3. Advanced Science Research Laboratory, Saitama Institute of Technology, Fukaya, 369-0293, Japan4. ZheJiang HuaShuo Technology CO., LTD, Ningbo 315000, ChinaAbstract: Lightweight of automatic vehicle is a significant application trend, using topology optimization and magnesium alloymaterials is a valuable way. This article designs a new model of automobile wheel and optimizes the structure for lightweight.Through measuring and analyzing designed model under static force, clear and useful topology optimization results were obtained.Comparing wheel performance before and after optimization, the optimized wheel structure compliance with conditions such asstrength can be obtained. Considering three different materials namely magnesium alloy, aluminum alloy and steel, the stress andstrain performances of each materials can be obtained by finite element analysis. The reasonable and superior magnesium alloywheels for lightweight design were obtained. This research predicts the reliability of the optimization design, some valuablereferences are provided for the development of magnesium alloy wheel.Key words: Magnesium alloy wheel, structural design, topology optimization, lightweight, finite element.1. Introduction Environmental and resource issues have become thefocus of attention around the world. As the automotiveindustry is increasingly demanding on energy savingand environmental protection, people are taking moreattention on the lightweight design of automobiles. Inthe United States, the Environmental ProtectionAgency (EPA) and the National Highway TrafficSafety Administration (NHTSA) issued a jointregulation in August 2012 [1, 2]. This new regulationwill be implemented on passenger cars to improveautomobile consumption standard about greenhousegases and fuels from 2017 to 2025. The emission forcombined cars and trucks has to be reduced from243g/mile in 2017 to 163 g/mile in 2025 according tonew regulation. Moreover, the fuel economy must beCorresponding author: Dongying Ju, professor, researchfields: materials and engineering.improved from 36.6 mpg in 2017 to 54.5 mpg in 2025.When designing vehicle products, it needs not only toreduce energy consumption but also to remain incompetition with peers [3, 4]. According to the data,the automotive own weight is reduced by 10%, andthe fuel consumption is reduced by about 6%-8%.Magnesium alloys are considered one of the mostpromising materials in the 21st century. In the moderndesign, it is important to improve the efficiency ofdevelopment and reduce the number of tests. Theaverage use of magnesium in cars has increased from0.1% (1.8 kg) in 1995 to 0.2% (4.5 kg) in 2007 in theUnited States according to Refs. [5, 6]. Usingmagnesium material in cars will increase by 15%(about 227 kg) by 2020 based on future vision formagnesium [7]. By understanding the efficiency ofmaterials, engineers can gain benefits throughmagnesium materials when designing wheel [8-10].Wheel is one of the most important parts of a vehicle.

14Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology OptimizationTo ensure energy efficiency, the wheels must be aslightweight as possible [11-16].Optimization design is a powerful tool formachinery design, and can produce the best layout ofstructural design. Topology optimization can providethe first optimized “design concept” of structurematerial distribution and achieve greater savings anddesign improvement in size and shape optimizations.Since Bendsoe introduces the homogenization methodof topology optimization, topology optimizationmethod has been deeply developed and applied instructural optimization design [17, 18]. Zhuang [19]carried out the topology optimization of aluminumalloy wheels, the strength and stiffness of theoptimized wheels were simulated and analyzed. Hu[20] optimized the aluminum alloy wheel use thewheel rim and flange thickness as the design variables,the maximum stress of the wheel in bending fatigueand radial fatigue conditions as the constraint, andaiming at the smallest wheel quality, the aluminumalloy wheel optimized. Based on the bending fatiguetest, Xiao [21, 22] carried out topology optimizationon steel wheels, and designed the lightweight designof the wheels with flexibility and modal frequency asthe target, and carried out stress analysis andexperimental verification. Optimization design isbeneficial to the improvement of global wheelperformance and wheel lightweight.Wheel disc and rim are two main parts of wheel.Some parameters of the vent holes such as number,position, and shape which are distributed in the wheeldisc can be changed. In this research, a kind of wheelstructure is designed, using topology optimization forwheel quality lightweight. The finite element modelsof wheels are established based on the static force.The rationality and superiority of the designedmagnesium alloy wheel are obtained.2. Structure Topology OptimizationIn this paper, wheel structure topology optimizationmethod is used to optimize the wheel, which satisfiedthe lightweight, strength and NVH requirements.2.1 Optimization MethodThe most common topology optimization is thevariable density material interpolation method, whichincludes SIMP and RAMP [23-26]. The theory ofvariable density is to convert the discrete optimizationproblem into a continuous optimization problem byintroducing an intermediate density unit.The SIMP method uses discrete element density asan optimization variable and therefore tends togenerate interlaced grayscale images of topologicaldesigns. In order to make it manufacturable, threeprocessing steps are required: identify the topologydesign, smooth the structural boundary, and thenrealize the parameterization. The advantages of aslightly modified version of SIMP were discussed bySigmund in 2007, a minimum stiffness (or othermaterial parameter) that is independent of penalizationis included.An alternative interpolation scheme known as theRational Approximation of Material Properties(RAMP) was proposed by Stolpe and Svanberg.RAMP model has nonzero sensitivity at zero density.Some numerical difficulties in problems related tovery low density values in the presence of designdependent loading could be remedied by RAMPmaterial model.From Fig. 1, the FE model before optimization wasshowed by model (a) and optimal topologyconfiguration was showed by model (b). The mostcommonly used material interpolation model method,SIMP formula is expressed as:E ( xi ) Emin ( xi ) p ( E0 Emin )(1)where E0 is the initial elastic modulus; p is thepenalty factor, p 1; E ( xi ) is the density value ofthe material at i .The theory of variable density is to convert thediscrete optimization problem into a continuousoptimization problem by introducing an intermediate

Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology Optimization(a) Before optimizationFig. 115(b) After optimizationElement model.density unit. In reality, the intermediate density unit isnot exist and cannot be manufactured. Therefore, theintermediate density unit should be reduced as muchas possible, the number of which needs to bepenalized only for the intermediate density thatappears in the design variables.2.2 Topology Optimization for Wheel StructureIn topology optimization, add draft restraint, circumsymmetry beam, minimum unit size and so on. In thewheel optimization, the lightest weight is the optimaldesign goal. Wheel spokes, disc and rim are mainparts of wheel. Several vent holes are distributed inthe wheel disc. When designing wheels, someparameters of the vent holes can be changed. Theseparameters include number, position, and shape. Manyoptimization approaches for wheel designs areconcerned with size or shape optimizations. Based ontopology optimization and the feature of the wheel,this research aims to identify wheel spokes. Whendoing topology optimization of wheels, optimize thewheel of structure by spokes for lightweight design.According to the ICM (Independent ContinuousMapping) optimization method proposed by YunkangSui [27] and the topology theory, the topologyoptimization model is established. With wheel unitdensity as design variable, weight flexibility asconstraints, the minimum quality is the objectivefunction. Topology optimization objective function isthe biggest structural stiffness or the minimumcompliance for the topology optimization, constraintis to remove the volume percentage, the topologyoptimization mathematical model is in the followingequation:T Find 1, 2, 3, n n Min C u F TU U T KU p u T k u ii0 i i 1 (2)Weight i vi v0 v0 (i 1,2,3 n) i 1 (i J1 , J 2 , J n ) i 1 F KUIn the equation: i is unit density, P is penaltyfactor, is the lower material density, is thepercentage of the volume of material removal, k0 isthe initial matrix for the structure, ki is optimizedstructure matrix, F is the load of unit structure, K is theoverall stiffness matrix, U is the displacement vector ofunit structure, V0 is the initial value of volume ofmaterial, C (u ) is compliance function of structure,J 1 , J 2 , J n are the unit number of optimizedunchanged density. Previous studies have shown thatoptimization wheel structure can be obtained. Theoptimization flowchart of the wheel is shown in Fig. 2.3. Establishment of Wheel ModelIn modern design, using finite element analysis canbe established to determine the strength of the wheelin advance and reduce the test times and cost. Staticload while vehicle stops is working conditions of thewheel that should be considered seriously [28-30].Wheel model is shown in Fig. 3.In this research, the gross weight of the vehicle isabout 1,175 kg, load on each wheel is 2,937.5 N. Twoalloy materials are used for the analysis andcalculation of wheel as Table 1 lists.

16Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology OptimizationStartEstablish multi-material wheel FE modelDefine wheel FE model optimal design regionsDefine load/boundary constraintsResetparametersTopology optimization based on wheel designparametersComprehensive performance index caculation(Lightweight/Structural strength)NoReasonYesComparisonsanalysisRealize the comprehensive performance index?EndFig. 2Flow chart of the wheel optimization.Fig. 3Wheel model.(a) 3D modelTable 1(b) FE modelMechanical properties of 2 materials.Mechanical propertiesDensity (kg/m³)Coefficient of elasticity (Mpa)Poisson ratioYield strength (Mpa)Aluminium alloy2,700690.33276Magnesium alloy1,830450.35160

Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology Optimization3.1 Verification of Finite Element ModelModel verification is necessary for finite elementanalysis. The modal analysis result is to analyze thenatural frequency, mode shape and other relatedparameters of the object, these parameters are theessential properties of any object with invariance andstability. Therefore, the finite element model isverified by modal experimental analysis.By comparing the simulation frequency of the FEanalysis and modal test frequency, experimentallymeasured modal parameters and FE analysis results ofwheel basic agreement. Wheel finite element model isaccurate, and can be applied to subsequent in depthfinite element analysis.maximum rated load (N), K is coefficient according tothe industrial standards set as 2.25. Radial load isobtained by 6,609.4 N. In this research, using StearnsJ wheel and tire contact research results, the force onthe magnesium alloy wheel from the tire can bereplaced by the radial force directly on the wheel tosimplify the modeling. The calculation formulas Wr ,W and W0 are given by the following equation:W b Fr K F(3) In the equation, Fr is radial load (N), F is 0Wr rb d (4) Wr W0 cos 2 0 (5) 0 0W0 3.2 Structural Strength AnalysisThe static force is intended to detect the wheelperformance when the total load of the vehiclecompresses the wheel radially. The radial load Fr shallbe determined from the equation: b Wr rb d 04brb 0 0 is the maximum deflection angle of radial load. Inthis way, the pressure loaded in the wheel inner ring is0.45 Mpa, the pressure loaded on the rim of the wheelis 0.785 Mpa. The load of test model is shown inFig. 5.Tap and measure pointTable 2Modal test.Comparison of simulation and experimental data.ModalFE analysis frequency/HzModal test frequency/HzError1474.514661.8%(6)In the equation, W is radial load on the wheel, b iswidth of the bead seat, rb is radius of the bead seat,supportFig. 4172480.424942.75%3948.039540.62%

18Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology Optimization3.3 Results of Structural Strength AnalysisFrom above loading conditions and finite elementtheory, in order to realize the lightweight of wheel,meanwhile ensure the strength safety, lightweightmaterial replacement and static analysis are completed.The analysis results were determined and presented inFigs. 6 and 7.Fig. 6 is the analysis results of equivalent stressbetween aluminum alloy and magnesium alloy.Through the above comparison, magnesium alloywheel and aluminum alloy wheel in the same size,magnesium alloy wheel equivalent stress is 31.67 Mpa(a)Fig. 5Wheel FE model load.Fig. 6(a) Magnesium alloyEquivalent stress of wheel.while aluminum alloy equivalent stress is 30.13 Mpa,less than the material yield stress.Fig. 7 is the analysis results before and after wheeloptimization. From the above analysis, under the premiseof the strength of wheel, the structure optimization ofmagnesium alloy wheel is carried out. The magnesiumalloy wheel deformation is 0.091 mm, aluminum alloywheel deformation is 0.058 mm. Magnesium alloy wheelhas good strength properties. The optimization effectcomparisons were in Table 3. In radial load, designedwheel model meets strength and other characteristics.Designed wheel can be further optimized.(b)(b) Aluminum alloy

Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology OptimizationFig. 7(a) Magnesium alloyDeformation of wheel.4. Optimization of the Wheel and ResultsBased on above optimization theory, structuraloptimizations of wheel were designed.4.1 Optimization of the WheelBased on above optimization theory and steps,Combined with shape and practicality of structure,optimization results of the wheel can be done.Under the premise of satisfying conditions such asstrength, the most remove and optimal model wasobtained.From the FE simulation results concerning the foursteps of wheel structure optimization in Figs. 8-10,comparisons among the wheel models can be shown:(1) Comparisons among the wheel models shown inFigs. 8 and 9, step 1 and step 4.By analyzing step 1 model and step 4 model, thestress values for critical locations of wheel under staticload, we find that the stress level of the step 4 modelis significantly higher than that of the step 1 model.Actual processing can be considered on the wheelbasis of topology optimization.(2) Comparisons among the wheel models shown inFigs. 8 and 9, step 2 and step 4.By analyzing step 2 model and step 4 model, thestress values for critical locations of wheel under staticload, we find that the stress of the step 2 is 32.52 Mpaat the same time, step 4 model is 32.35 Mpa. Step 219(b) Aluminum alloystress is significantly higher than that of the step 4model. Total deformation of step 2 model is 0.022 mmwhile step 4 model is 0.021 mm. Wheel mass of step 2model is 4.179 kg, wheel mass of step 4 model is 4.05kg. Under reasonable stress and strain conditions,wheel model of step 4 is better for optimization target.(3) Comparisons among the wheel models shown inFigs. 8 and 9, step 3 and step 4.By analyzing step 3 model and step 4 model, thestress values for critical locations of wheel under staticload, we find that the stress of the step 2 were 32.52Mpa at the same time, step 4 model is 32.35 Mpa.Step 2 stress is significantly higher than that of thestep 4 model. Total deformation of step 2 model is0.022 mm while step 4 model is 0.021 mm. Wheelmass of step 2 model is 4.097 kg, wheel mass of step4 model is 4.05 kg. Under reasonable stress and strainconditions, wheel model of step 4 is better foroptimization target.According to the stress, total deformation analysisand optimization step, the most significance model isstep 4 model, That is, the spoke reduction of 40% byvolume is combined with the influence of vent holesshape on wheel performance and inner ring of wheeldisc influence of wheel structure. The wheel structureafter parameter optimization can be done.The more removal of material of optimal topology,the more complex shape of the structure, the smallersize of the spokes, Table 3 shows spokes’ structure after

20Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology Optimization(a)Fig. 8(c)The optimization process of wheel.Fig. 9(a) Von mises stress graphOptimization of wheel.(b)(d)(b) Wheel mass graph

Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology Optimization(1) Wheel model(a) Before optimization(2) Spoke(b) After optimizationFig. 10Table 3Optimization parameters of wheel spokes.Before and after parameter optimization (mm).Wheel optimization parametersBefore optimizationAfter optimizationa6.64b925c1412d445521

22Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology OptimizationOptimization, the percentage of material removal ofthe optimal topology was chosen based on thestructure and optimization theory.4.2 Results and Discussions after OptimizationIn order to realize the lightweight of wheel,meanwhile ensure the strength safety, lightweightmaterial replacement and further structural optimizationare completed. The analysis results of wheel afteroptimization are presented in Figs. 11 and 12.Fig. 11 is the analysis results of equivalent stressbetween aluminum alloy and magnesium alloy.Through the above comparison, magnesium alloywheel and aluminum alloy wheel in the same size,magnesium alloy wheel equivalent stress is 32.35 Mpawhile aluminum alloy equivalent stress is 32.34 Mpa,Fig. 11(a) Magnesium alloyComparison of equivalent stress.Fig. 12(a) Magnesium alloyComparison of deformation.less than the material yield stress. Magnesium alloywheel has good strength properties.Fig. 12 is the analysis results before and after wheeloptimization. From the above analysis, under thepremise of the strength of wheel, the structureoptimization of magnesium alloy wheel is carried out.The magnesium alloy wheel deformation is 0.021 mm,aluminum alloy wheel deformation is 0.058 mm.Magnesium alloy wheel has good strength properties.The optimization effect comparisons were shown inTable 4.The optimized magnesium alloy wheel is muchlighter than the steel wheel and aluminum wheel,compatible with wheel lightweight design. It makessense to optimize the wheel with magnesium alloymaterials.(b) Aluminum alloy(b) Aluminum alloy

Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology OptimizationTable 4Lightweight comparisons (kg).Lightweight comparisonsWeight/kgImprovement/%Aluminium alloy6.24The optimization results meet the design targetvalue. Based on topology optimization theory, thewheel optimal structure and key dimensions areobtained while satisfying the performance of thewheel.Topology optimization method was efficient andcorrect, significance for lightweight design of wheels.5. ConclusionThe finite element analysis has been carried out onthe wheel. Through the above profound analysisfollowing research results can be acquired:(1) Using topology optimization for wheel qualitylightweight is a useful way. By optimizing wheelspokes to accurate wheel lightweight design, theoptimization designed wheel meets the strengthcondition.(2) By replacing lightweight materials, compared toaluminium alloy, the weighted reduction is 35.1%.After optimization, the weight of magnesium alloy hasreduced by 4.4%. Magnesium alloy has a betterweight reduction effect, and lightweight materialshave effective lightweight means.(3) According to the analysis results, comparison ofwheel performance of different materials, after usingthe magnesium alloy material for replacement andanalyzing of the wheel, the goal of reducing theweight of the automobile wheel can be achieved whilesatisfying the wheel strength requirements.Magnesium cknowledgmentsThis research is based on the work supported by theLight Weighting Electric Vehicle Project of SaitamaInstitute of Technology University.References[1]23Morrow, W. R., Gallagher, K. S., Collantes, G. et al.2010. “Analysis of Policies to Reduce Oil Consumption[13][14]Magnesium alloy (optimization)4.054.4and Greenhouse-gas Emissions from the USTransportation Sector.” Energy Policy 38 (3): 1305-20.EPA. 2011. “Office of Transportation and Air Quality,Regulatory Announcement: EPA and NHTSA Propose toExtend the National Program to Reduce GreenhouseGases and Improve Fuel Economy for Cars and Trucks.”EPA-420-F-11-038.Das, S. 2014. “Design and Weight Optimization ofAluminum Alloy Wheel.” Int. J. Sci. Res. Publ 4 (6).Joost, W. J., and Krajewski, P. E. 2017. “TowardsMagnesium Alloys for High-Volume AutomotiveApplications.” Scripta Materialia 128: 107-12.Pfestorf, M., and Copeland, D. 2007. “Great Designs inSteel Seminar 2007.” American Iron and Steel Institute.Ward’s Communications. 2008. “Ward’s Motor VehicleFacts and Figures 2008.” Southfield, Mich.Cole, G. S. 2007. “Magnesium Vision 2020—A NorthAmerican Automotive Strategic Vision for iumAssociation.Liu, J., and Ma, Y. 2016. “A Survey of ManufacturingOriented Topology Optimization Methods.” Advances inEngineering Software 100: 161-75.Deaton, J. D., and Grandhi, R. V. 2014. “A Survey ofStructural and Multidisciplinary Continuum idisciplinary Optimization 49 (1): 1-38.Wang, C. Q., Wang, D. F., and Zhang, S. 2016. “Designand Application of Lightweight Multi-objectiveCollaborative.” Proceedings of the Institution ofMechanical Engineers, Part D: Journal of AutomobileEngineering 230 (2): 273-88.Marin, L., and Kedziora, S. 2016. Design of AutomotiveRoad Racing Rim with Aid of Topology Optimization.Faculty of Science, Technology and CommunicationUniversity of Luxembourg.Singh, D. P. K., Mallinson, G. D., and Panton, S. M. 2002.“Applications of Optimization and Inverse Modeling toAlloy Wheel Casting.” Numerical Heat Transfer: Part A:Applications 41 (6-7): 741-56.Satyanarayana, N., and Sambaiah, C. 2012. “FatigueAnalysis of Aluminum Alloy Wheel under Radial Load.”International Journal of Mechanical and IndustrialEngineering (IJMIE) ISSN (2231-6477): 1-6.Rozvany, G. I. N. 2009. “A Critical Review on.”StructuralandMultidisciplinary

24Multi-objective Optimization Design of Magnesium Alloy Wheel Based on Topology OptimizationOptimization 37 (3): 217-37.[15] Rozvany, G. I. N. 2001. “Aims, Scope, Methods, Historyand Unified Terminology of Computer-aided TopologyOptimization in Structural Mechanics.” Structural andMultidisciplinary Optimization 21 (2): 90-108.[16] Hirano, A. 2015. “Study on Wheel Stiffness ConsideringBalance between Driving Stability and Weight.” SAEInternational Journal of Commercial Vehicles 8(2015-01-1755): 205-12.[17] Gersborg-Hansen, A., Bendsøe, M. P., and Sigmund, O.2006. “Topology Optimization of Heat ConductionProblems Using the Finite Volume Method.” Structuraland Multidisciplinary Optimization 31 (4): 251-9.[18] Kumar, C. P. V. R., and Meher, R. S. 2013. ternational Journal of Modern Engineering Research3: 1548-53.[19] Miller, W. S., Zhuang, L., Bottema, J. et al. 2000.“Recent Development in Aluminium Alloys for ering: A 280 (1): 37-49.[20] Hu, J. H., Liu, X. X., Sun, H. X. et al. 2013.“Development and Application of Light-Weight Designof the Aluminum Alloy Wheel.” Applied Mechanics andMaterials. Trans Tech Publications 310: 253-7.[21] Praveen, P., and Gopichand, D. 2014. “GeometricalOptimization and Evaluation of Alloy Wheel FourWheeler.” International Journal of Research andInnovation 1 (3).[22] Xiao, D., Zhang, H., Liu, X. et al. 2014. “Novel SteelWheel Design Based on Multi-objective TopologyOptimization.” Journal of Mechanical Science andTechnology 28 (3): 1007-16.[23] Chang, K. H., and Tang, P. S. 2001. “Integration ofDesign and Manufacturing for Structural ShapeOptimization.” Advances in Engineering Software 32 (7):555-67.[24] Chen, J., Shapiro, V., Suresh, K. et al. 2007. “ShapeOptimization with Topological Changes and ParametricControl.” International Journal for Numerical Methods inEngineering 71 (3): 313-46.[25] Sui, Y. K., and Ye, H. L. 2013. Continuum TopologyOptimization Methods ICM. Beijing: Science Press.[26] Adigio, E. M., and Nangi, E. O. 2014. “Computer AidedDesign and Simulation of Radial Fatigue Test ofAutomobile Rim Using ANSYS.” Journal of Mechanicaland Civil Engineering (IOSR-JMCE) e-ISSN 2278-1684.[27] Van Dyk, B. J., Edwards, J. R., Dersch, M. S. et al. 2017.“Evaluation of Dynamic and Impact Wheel Load Factorsand Their Application in Design Processes.” Proceedingsof the Institution of Mechanical Engineers, Part F:Journal of Rail and Rapid Transit 231 (1): 33-43.[28] Ganesh, S., and Periyasamy, D. P. 2014. “Design andAnalysis of Spiral Wheel Rim for Four Wheeler.” TheInternational Journal of Engineering and Science (IJES)3 (4): 29-37.[29] Wang, L., Chen, Y., Wang, C. et al. 2009. “Simulationand Test on Aluminum Alloy Wheel Rotary FatigueLife.” Journal of Nanjing University of Science andTechnology (Natural Science) 5: 5.[30] Papadrakakis, M., Lagaros, N., and Plevris, V. 2002.“Multi-objective Optimization of Skeletal Structuresunder Static and Seismic Loading Conditions.”Engineering Optimization 34 (6): 645-69.

doing topology optimization of wheels, optimize the wheel of structure by spokes for lightweight design. According to the ICM (Independent Continuous Mapping) optimization method proposed by Yunkang Sui [27] and the topology theory, the topology optimization model is established. With whe

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