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变压器硅钢片零件的冷冲压模设计-落料模含SW三维及8张CAD图
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Topology optimization of die weight reduction for high-strength sheetmetal stampingDongkai Xua,1, Jun Chena,n, Yucheng Tanga,1, Jian Caoa,b,2aDepartment of Plasticity Technology, Shanghai Jiao Tong University, 1954 Huashan Road, Shanghai 200030, PR China.bDepartment of Mechanical Engineering, Northwestern University, Evanston, IL 60208-3111, USAa r t i c l e i n f oArticle history:Received 4 January 2011Received in revised form6 January 2012Accepted 19 March 2012Keywords:Die weight reductionNumerical simulationTopology optimizationExperimental verificationa b s t r a c tHigh-strength steels have been increasingly used for vehicle body structures to improve fuel efficiencyand vehicle safety. In order to maintain the stiffness and forming conditions under higher formingloads, stamping dies have to be designed with larger dimensions and thicker structures which mayresult in heavier die weight. Targeting to save the die weight/cost and keep the required stiffness, atopology optimization method is proposed based on Solid Isotropic Microstructure with Penalty (SIMP)to reduce the weight of key die components. During optimization, multiple loading conditions atdifferent forming positions are considered to assure the maximum deflection at the above mentionedpositions within the limit values. Besides, the interaction behaviors between die components are alsotaken into account to reflect the real contact evolution. A step-bottomed cup is designed to testify theproposed method. Through topology optimization, the weight of blank holder is reduced by 28.1%.Based on the optimization result, the blank holder is redesigned and machined, stamping test resultsindicate that defect-free stamping parts are formed with same blank holder forces, and the thicknessdifference between the original and newly stamped parts along a cross section is less than 0.06 mm, i.e.4.29% of the initial blank thickness. This verifies that the proposed approach can effectively reduce dieweight and maintain the derived forming performance of stamped part.& 2012 Elsevier Ltd. All rights reserved.1. IntroductionLightweight products have been viewed as one promisingmeans for increasing energy efficiency since early 1990s 1.Recently, advanced high strength steels (AHSS) with initial yieldstress above 500 MPa 2 have received increasing utilization forthe purpose of achieving automotive lightweight considering theoverall performance of affordability, safety and environmentalfriendliness. Compared with mild steels, however, high strengthsteels usually exert higher loads on stamping dies. In particular,AHSS with a hardness value 45 times higher than mild steels 2significantly impacts the manufacturing tools and die design. Tocope with extra force/pressure required for forming AHSS, engi-neers usually tend to enlarge the dimensions of supportingelements in stamping dies and select better materials to guaran-tee the normal operation. Unfortunately, this directly increasesthe overall weight of dies and indirectly increases manufacturingcost and operating cost. Therefore, one of the critical require-ments is to establish a new systematic die design methodologyinstead of empirical ones to maximize the efficiency of diematerials and therefore minimize die weight.Recently researchers have qualitatively investigated the influ-ence of die structural behaviors on stamping process. Blankholder force is one of the most important parameters to affectformability and the deflection of blank holder will lead to unevendistribution of blank holder force and significantly affect blankformability, even though the deflection amount is extremelysmall 36. In addition, springback is one of the critical issuesaffecting the quality of stamped components, Zhang and Lin 7developed an analytical solution to minimize springback ofstamping part by rigid punch and elastically deformable die.Furthermore, Lingbeek and Meinders 8 summarized the defor-mations of tool and press during stamping process and definedthem into two categories. The deflection of the press or pressframe and global deformation of stamping tools is regarded asmacro-deformation, while micro-deformation means the defor-mation which occurs directly on the surfaces of the forming toolsand usually is one or two orders lower in magnitude than macro-deformation.In order to investigate the die structure deflection, Becchioet al. 9 first applied the finite element method (FEM) in sheetContents lists available at SciVerse ScienceDirectjournal homepage: /locate/ijmecsciInternational Journal of Mechanical Sciences0020-7403/$-see front matter & 2012 Elsevier Ltd. All rights reserved./10.1016/j.ijmecsci.2012.03.006nCorresponding author. Tel.: 86 21 62813425x8318; fax: 86 21 62826575.E-mail addresses: xudongkai (D. Xu),jun_chen (J. Chen), bluevincent (Y. Tang),jcao (J. Cao).1Tel.: 86 21 62813425x8313; fax: 86 21 62826575.2Tel.: 847 4671032; fax: 847 4913915.Please cite this article as: Xu D, et al. Topology optimization of die weight reduction for high-strength sheet metal stamping. Int. J.Mech. Sci. (2012), /10.1016/j.ijmecsci.2012.03.006International Journal of Mechanical Sciences () metal die design process, and predicted the die structure deflec-tion and stress induced by loading conditions. Subsequently, FEMwith full 3D discretization was used to verify the strength andstiffness of stamping tool 10,11, assist die-face design 12,13,determine the required press force/tonnage 14 and accuratelypredict springback 15 when the deflection of forming tools andpress was taken into account. However, two main disadvantagesshould be noticed. On one hand, if the forming simulation istotally coupled with structural analysis using elastic toolingdefinition instead of standard rigid tool definition, the meshmodeling effort and calculation will be really time-consuming.On the other hand, the die structural analysis using the loadboundary extracted from forming process simulation results withconventional rigid tooling definition may not be accurate enoughand it still takes much modeling effort to discretize the tools intofine 3D meshes in structure analysis. Therefore, three simplifiedmodels deformable rigid bodies (DRB), static condensation (SC)and tied contact (TC) were proposed to speed up the simulations16,17. Considering the calculation accuracy and time, Strucket al. 17 indicated that TC model was superior to other models.Experimental studies have been conducted in measuring diestructural deflection and force condition during stamping process.Zhang et al. 11 and Haller et al. 18 developed a dynamic strainDAQ system using strain gauges to track the deflection of formingtools and verify simulation results, respectively. Conle and Wang19 attached a radio controlled set of small data acquisition unitsto stamping die components and the strains at critical locationswere collected for comparison with simulation prediction. How-ever, these techniques do not have capability to measure thestrain on the contact surface between blank sheet and formingtools. To overcome this disadvantage, the distributed embeddedsensors were utilized to obtain real-time distribution of contactpressure at the toolworkpiece interface based on the thin platespline (TPS) interpolation method 2022.Previous works have shown that it is necessary to investigatethe die deflection behavior during stamping process for diestructure strength verification and forming quality improvement.To improve the energy efficiency of a stamping process, it is quitenecessary to minimize tooling weight while maintaining itsintegrity. Some researchers also pointed out that the results ofdie structure analysis could be used as die design guideline toimprove die structures and reduce the weight 11,18,19. Further-more, researchers have introduced the topology optimizationmethod in optimizing die structure. Considering the pressuredistribution at the final step of the drawing, Sheu and Yang 23established a simplified column model to obtain the optimum diestructure and reduce the die weight using the Taguchi methodand the pseudo-density method. Nilsson and Birath 24 per-formed topology optimization of the truck lid die considering twoloadconditionsstampingoperationandtransportationtoimprove the structural stiffness and reduce the weight. Themodified die structure not only showed overall more homoge-nous stress distribution, but also realized about 15% maximaldisplacement reduction and about 20% weight reduction. Theirworks strongly demonstrated that topology optimization methodcould effectively reduce the overall weight of industrial stampingdie. However, one limitation is that during forming simulation thetool parts were defined as rigid, which just provided an approx-imate structural response of the die. Chen et al. 25 successfullyachieved 54.5% mass reduction of the binder of hyperbolicbottomed cup die using the topology optimization method basedon a proposed global load mapping algorithm. The algorithmtransferred the reaction forces between tooling and blank duringforming simulation to the topology optimization model for diestructure based on the static equivalent principle. Although theglobal load mapping algorithm was validated by experiments11, it was quite expensive in terms of computational resources.Therefore, it is necessary to develop a new algorithm to overcomethis limitation to perform topology optimization of stamping die.The above works demonstrated the great potential of using thetopology optimization method in die weight reduction, but thefollowing aspects must be considered to extend the methodologyfor industrial applications. Firstly, the elastic deformation of toolsshould not be totally ignored during stamping simulation toobtain more accurate loading conditions, therefore, stamped partquality can be better assessed. Secondly, it is necessary to modelthe interaction between punch and binder, due to possible elasticdeformation or tilting of tooling during a forming process. Inaddition, the history of the load distribution over the entire diesurface needs to be considered to obtain the global optimizationthat reflects the dynamic loading condition during stampingprocess. Finally, more research in experimental verification isnecessary to check the performances of the optimized die struc-ture and the blank forming quality.In this paper, we will present a methodology for global diestructure topology optimization that models the interactionbetween tooling components and dynamic loading history duringstamping process. The method uniquely integrates the local loadmapping algorithm with the SIMP-based topology optimizationmethod. The methodology will be explained in detail first. A blankholder of a specifically designed step-bottomed cup drawing diewill then be used to evaluate the effectiveness and feasibility ofthe methodology through both numerical study and experimentalverification.2. MethodologiesThis section illustrates the proposed methodology for die structureoptimization. The methodology combines forming analysis withtopology optimization using an efficient mapping algorithm betweenthese two different numerical tools. Using this proposed method, diestructure is optimized without sacrificing forming performance. Fig. 1describes the flow chart of the methodology. The left-hand side of theflow chart indicates the main stages of the methodology and theflow chart in black wireframe on the right-hand side illustrates theiterationprocessusingtheSIMP-basedtopologyoptimizationmethod. A design will start with the current best practice, which isusually based on standards and/or designers experience. Then sheetmetal stamping simulation will be performed to obtain the surfaceload conditions on die structure. Subsequently, the surface load willbe transferred to die structure analysis model as the force boundaryconditions by using the developed local load mapping algorithm.According to the die structure analysis results, the die structurebehavior during the forming process is revealed and then the designspace for die structure topology optimization will be determined. TheSIMP-based topology optimization technology is used to achieve anoptimized die structure. Since the obtained structure is a conceptualdesign, the die structure redesign is required to ensure the manu-facturability. Finally, a check cycle will be applied to guarantee theperformance of the optimized structure after die structure redesign.2.1. Local load mapping algorithmIt is noteworthy that finite element meshes used to model thedie for stamping process simulation and for die-structure analysisare entirely different due to their specific requirements. In sheetmetal forming simulation, die faces with small local geometricfeatures, such as fillets and drawbeads, are complex but still needto be fine meshed for formability analysis. However, during die-structure analysis, some features on die face can be simplified dueto their limited influences on the stress distribution of the dieD. Xu et al. / International Journal of Mechanical Sciences () 2Please cite this article as: Xu D, et al. Topology optimization of die weight reduction for high-strength sheet metal stamping. Int. J.Mech. Sci. (2012), /10.1016/j.ijmecsci.2012.03.006structure. This simplification is helpful in reducing total elementamount and improving calculation efficiency. To overcome thehigh memory requirements of the global load mapping algorithm26, an implementation algorithm of local load mapping isdeveloped here to automatically and efficiently transfer thereaction forces on the tooling faces obtained from the stampingsimulation to the boundary elements for die structure analysisand optimization. The procedures to implement the algorithm aresummarized as follows:Step 1: Triangulate the boundary surfaces of the boundary 3Dsolid elements of tools in structure analysis, and then align itwith the corresponding tool mesh for stamping simulation inthe same coordinate system.Step 2: Divide the nodes on the mesh for stamping simulationinto different node sets according to Eq. (1).snPh i11where s is the number of node sets, n is the total amount of theboundary nodes on the die surfaces, P is the maximumallowable processing number of nodes determined by theallocated memory, and represents the rounding operation.Step 3: Project contact force is calculated by stamping processsimulation on each node in the node set Si(i1,2,.,s) of toolmesh onto the nearest point of a triangular element on the diestructure boundary.Step 4: Transfer the projected contact forces from that specificpoint to three nodes of the triangular element by the staticequivalent principle.FAm?3Si Nm?PSiFBP?3Si,i 1,2,:,s?12where m is the number of the nodes of the extracted toolingsurface element for die structure analysis, FAm?3is the total targetload matrix, FAm?3Siis the component of the target load matrix,FBP?3Siis the load matrix of node set Siand Nm?PSiis thecorresponding shape function transformation matrix.Fig. 1. Proposed methodology for die structure optimization.D. Xu et al. / International Journal of Mechanical Sciences () 3Please cite this article as: Xu D, et al. Topology optimization of die weight reduction for high-strength sheet metal stamping. Int. J.Mech. Sci. (2012), /10.1016/j.ijmecsci.2012.03.006Step 5: Sum up the mapping results, as shown in Eq. (3).FAm?3 Nm?nFBn?3Xsi 1FAm?3Si3where Nm?nis the total shape function transformation matrixand FBn?3is the total initial load matrix.2.2. SIMP-based topology optimization methodTopology optimization is one of the effective methods to findthe optimal layout of a continuum structure in a specified regionwith respect to a predefined design objective. There are essentialconceptual differences among three popular approaches: thehomogenization method, the evolutionary structural optimization(ESO) and the density method. Bendse and Kikuchi 27 andBendse 28 introduced the homogenization method in topologyoptimization of continuum structures, which has stimulated theresearch in this field. However, due to the intensity requirementof computational power, it has not been widely adopted. ESO is anumerical method of topology optimization which is developedby Xie and Steven 29. Although the ESO method is simple andcan be easily implemented into any general finite elementprograms,theobtainedoptimaltopologystructurehighlydepends on the removal ratio specified and the number of initialelements. Mlejnek and Schirrmacher 30 proposed an artificialmaterial model from the engineering point of view, in which itsrelative density (pseudo-density) was assumed to be proportionalto an artificial effective elastic modulus of the interested zone.In the density method, solid isotropic microstructure withpenalty (SIMP) 31 and rational approximation of materialproperties (RAMP) 32 are two common interpolation methods.We will adopt the SIMP method in our analysis assuming thedeformation of die structure is elastic, and this method will bebriefly summarized.Conventionally, topology optimization of a continuum struc-ture is formulated as a minimum compliance problem 31. Letthe internal virtual work of an elastic body a(u,v) at the equili-brium u and for an arbitrary virtual displacement v be expressedasa u,v ZOEijklx eiju eklv dO4whereOdenotes the design domain, Eijkl(x) is the design variableover the domain,eij(u) andekl(v) are elastic strains on the elasticbody attributed to the static state field u, and the virtualdisplacement field v, respectively. The work done by the externalload acting on the structure can be expressed asl u ZOfudOZGTtuds5where f is the body force and t is the boundary traction on thetraction partGTCG?Oof the boundary. The minimum com-pliance (maximum global stiffness) is expressed as follows:minuAU,rlus:t: : aEu,v lv,vAUEijklx rxpE0ijkl,p41ZOrxdOrV0orminrrxr1,xAO6The equilibrium equation here is written in its weak varia-tional form. U denotes the space of kinematically admissibledisplacement fields,r(x) is the relative density, E0ijklrepresentsthe elastic modulus of a given isotropic material, p is the penaltyfactor, V is the total volume of the design domain andrminis thelower-bound limit of the relative density to avoid possiblesingular solution. When solving the problems of form Eq. (6) bythe finite element method, topology optimization with a volumeconstraint is expressed asmin : C FTU UTKU XNe 1rePuTek0eues:t: : V fVV0XNe 1reveF KU0rrer17where C represents the compliance of the structure, K is theglobal stiffness matrix,redenotes the relative density of the ethelement, k0eis the stiffness matrix of the eth element, F and U arethe vectors of the force and nodal displacement, respectively, uedenotes the element displacement vector, N is the total number ofelements, V is the material volume after optimization, fVis thevolume fraction defined to constraint the structure volume, V0isthe initial design domain volume, and veis the volume of the ethelement.3. Case studyIn this section, a step-bottomed cup drawing die was devel-oped to illustrate the procedure shown in Fig. 1 and used to verifythe effectiveness of the proposed method by numerical simula-tion. Fig. 2 shows the geometry of the step-bottomed cup. Thepart is symmetric relative to the YZ plane and has a step-shapedfeature in the Z direction that may result in the punch tiltingthrough a translation along the Y direction.Fig. 2. Schematic diagram of step-bottomed cup: (a) XY view: L181 mm,W194 mmand(b) YZview:H42 mm,S16 mm,R116.5 mm,R213.5 mm, R3123.5 mm.D. Xu et al. / International Journal of Mechanical Sciences () 4Please cite this article as: Xu D, et al. Topology optimization of die weight reduction for high-strength sheet metal stamping. Int. J.Mech. Sci. (2012), /10.1016/j.ijmecsci.2012.03.0063.1. Forming process simulationMost sheet metal stamping simulation models ignore theelastic deformation of tools and the interaction between thosetools during forming process, but this kind of model has graduallybecome inapplicable with the continually increasing use of HSSand AHSS due to much higher forming force (sometimes can be 23 times higher compared to mild steels) and die structuredeflection. Therefore, it is necessary to establish an improvedstamping simulation model to overcome this drawback.The improved simulation model has been developed in LS-DYNA and 1/2 model is shown in Fig. 3. All the tools except for thepunch were modeled as rigid bodies. Hexahedral elements wereused to model the punch for the sake of investigating theinteraction behavior between the punch and the blank holderdue to the punch tilting or the elastic deformation of punch itself.BelytschkoTsay shell elements with five integration points alongthe thickness direction were used to model the blank in 1.4 mminitial thickness. In addition, the adaptive meshing strategy forblank with 3 refinement levels was applied for the whole formingsimulation. The Coulomb friction model was used to model theinteraction between the contact bodies. Barlats 3-parameter yieldfunction model 33 was chosen as follows:F a9K1K29ma9K1?K29mc92K29m 2smY8where a, c, K1and K2are parameters determined by R00, R45, R90andsY.sYis the initial yield stress. m6 is recommended for BCCmaterials. Tables 1 and 2 show the material properties.The interaction force history between the punch and the blankholder in the forming simulation is shown in Fig. 4, whichprimarily resulted from two sourcesthe punch tilting on theY-axis and the elastic deformation of punch itself. The evolutionof the interaction forces could be divided into four periods (IIV).There were no interaction forces on the both wear plates of blankholder at stage I where the shallow cup was formed (until stokeless than 16.5 mm). However, the drawing process proceeds, theinteraction forces appeared on the wear plate along the Ydirection of blank holder at stages II and III, and reached themaximum value at point 3. Then the punch tilting graduallychanged from the Y direction to its negative direction at stage III.Therefore, the contact between punch and blank holder takenplace on the wear plate at opposite side and the interaction forcesachieved the maximum value in the negative Y direction at point5. Different forming statuses of the sheet at points 2, 3 and 5 arealso shown by the illustrations in Fig. 4.3.2. Load mapping resultsThe load mapping strategy as described in Section 2.1 is the bridgebetween stamping simulation and die structural analysis and opti-mization. The load distributions on blank holder at three differentstatuses, i.e. points 2, 3 and 5 in Fig. 4, after load mapping are shownin Fig. 5 (a), (b) and (c) respectively. The arrow directions representload directions and the arrow lengths reflect load magnitudes.Compared to the evolution of the interaction forces betweenthe punch and blank holder shown in Section 3.1, it is found thatthe load mapping results in Fig. 5 properly described the char-acteristicofchangingdirectionoftheinteractionforces.Non-uniform load distributions on blank holder surface aremainly due to blank thickness variation on the flange area.3.3. Topology optimization of stamping die componentCurrently, the stamping die structure is designed according tostandard guidelines. The practice faces challenges when newmaterials or new forming processes are employed. Apparently,topology optimization is one of the most powerful enabling toolsfor optimum die structure design to achieve better performances,especially in terms of mass reduction. In this section, the blankholder of step-bottomed cup drawing die shown previously willbe used for the case study and verification. The detailed optimiza-tion procedure was illustrated in Fig. . Topology optimization modelThe objective of die structure topology optimization is to findminimum die mass expressed as the volume fraction. TheFig.3. Schematicofstampingsimulationmodelwithtoolingandblankconfiguration.Table 1Material properties.Material Thickness(mm)E(GPa)uss(MPa)K(MPa)nR00R45R90DP6001.42070.28 42111100.212 0.877 0.886 1.128Table 2Key process parameters.Friction coefficientBlank holding force (kN)Drawing depth (mm)0.1412042Fig. 4. Evolution of interaction forces between punch and blank holder.D. Xu et al. / International Journal of Mechanical Sciences () 5Please cite this article as: Xu D, et al. Topology optimization of die weight reduction for high-strength sheet metal stamping. Int. J.Mech. Sci. (2012), /10.1016/j.ijmecsci.2012.03.006objective is expressed by the following:min : fV V=V0s:t: : wnraw0V XNe 1reverV00rrer19where w*is the maximum nodal displacement after optimization,w0is the initial maximum nodal displacement, andais thecoefficient used to balance the relationship between mass reduc-tion and structure performance. A structural performance index,as shown in Eq. (10), is proposed foravalue selection.SPindexfreduction9snmax?smax9=smax9Cnmax?Cmax9=Cmax10where SPindexrepresents the structural performance index, freductionisthe reduced volume fraction after optimization,snmaxandsmaxarerespectively the maximal stress on the optimized structure and theinitial structure, Cnmaxand Cmaxare the maximal compliance of theoptimized structure and the initial structure, respectively. A higherSPindexvalue means that the structure has better behavior in bothmass reduction and structure performance.The blank holder was modeled by 25416 brick elements and1608 6-node prism elements. In order to maintain the manufac-turability of the optimized geometry, the entire blank holder wasdivided into a design space and a non-design space as shown inFig. 6. The non-design space was the area where the material incontact with the blank during forming process or some structurefeatures, at which guide sleeves and ejector pins were located.The rest of the space was defined as the design space for furthertopology optimization. According to engineering best practice, thewidth of the rib is defined within 1060 mm.3.3.2. Topology optimization based on multiple load statusesAs shown in Fig. 5, stamping process is a dynamic process, andthe maximum local load and load distribution during wholestamping process are quite variable. Therefore, considering multi-ple critical loading conditions during optimization is necessary. Inthis case, the critical stages were considered at points 2, 3 and 5as shown in Fig. 4. Fig. 7 shows topology optimization resultsunder each load case with sameavalue. Material distributionconsidering loading forces at points 2, 3 and 5 have their ownfeatures. Thus, it is necessary to take multiple load case con-straints into consideration in order to make the optimizedstructure satisfy all crucial load cases of the whole process.3.3.3. Selection ofavalueThe determination ofavalue is critical during die topologyoptimization. The relationship betweenavalue and proposedstructural performance index SPindexwas studied as shown inFig. 8, while keeping all the other parameters the same. Althoughthe mass reduction characterized by reduced volume fractionkept increasing withavalue increasing, the structural perfor-mance index rapidly decreased.a2.0 was the maximum valuein this case because the SPindexdifferences amongavalue of 1.8,1.9 and 2.0 were less than 0.5. It is clear that the SPindexachievedits maximum value whena1.2 was selected.3.3.4. Topology optimization resultThe structure of blank holder was mainly affected by blankholder force and the variable interaction forces. As a result, threetypical load cases at points 2, 3 and 5 were adopted as the forceboundary conditions during topology optimization. The conver-gence criterion was defined as Eq. (11):fVk 1?fVk?fVkoewnk 1?w0?w0oZ8:11where fV(k1)is the volume fraction at the (k1)th iteration, fV(k)is the volume fraction at the kth iteration, wnk1is the maximumnodal displacement at the (k1)th iteration, w0is the initialmaximum nodal displacement, andeandZare set as 0.005 and0.01 respectively. After 22 iterations, the computation was con-verged at volume fraction 68.6%. The optimal structure of blankholder is shown in Fig. 9.Fig. 6. Definition of non-design space and design space.Fig. 5. Load distributions on blank holder at different forming stages. (a) Point 2:stroke at 16.5 mm, (b) point 3: stroke at 39.5 mm, and (c) point 5: stroke at 42 mmD. Xu et al. / International Journal of Mechanical Sciences () 6Please cite this article as: Xu D, et al. Topology optimization of die weight reduction for high-strength sheet metal stamping. Int. J.Mech. Sci. (2012), /10.1016/j.ijmecsci.2012.03.0063.3.5. Die structure redesignBecause the topology optimization result in Fig. 9 only pro-vides a conceptual design with optimal material distribution,further efforts should be made to redesign the structure. Slicecutting technology was introduced to obtain geometry at differ-ent sections along the Z direction and non-design space was stillkept. The bottom structure of the redesigned blank holder isshown in Fig. 10. Compared to the original structure, the opti-mized blank holder realized weight reduction by about 28.1%.Fig. 11 indicates that after optimization the maximal deflection ofblank holder still meets the initial constraint condition in Eq. (9)Fig. 8. Evolution of structural performance index and volume fraction reductionwith differentavalues.Fig. 9. Topology optimization result of blank holder under multiple load caseconstraints.Fig. 10. Bottom structure of the optimized blank holder.Fig. 11. Structure performance of blank holder before and after optimization.Fig. 7. Topology optimization results under several key load cases at differentforming positions.D. Xu et al. / International Journal of Mechanical Sciences () 7Please cite this article as: Xu D, et al. Topology optimization of die weight reduction for high-strength sheet metal stamping. Int. J.Mech. Sci. (2012), /10.1016/j.ijmecsci.2012.03.006and the maximal stress has an acceptable increase, which will notdeteriorate the structure safety.4. Experimental verificationTwo blank holders with original and optimized structureswere manufactured in order to find out the real performancedifferences before and after topology optimization. Fig. 12 showsthe distribution of blind holes and cavities on the bottom side ofthe optimized blank holder following the CAD model shown inFig. 10.In addition, a dynamic strain DAQ system shown in Fig. 13 wasestablished. The dynamic strain indicator was used to convert thestrain on the blank holder to analog voltage signals. An industrialA/D acquisition card connected with the computer was used tocollect the voltage signals during the forming process. Further-more, a DAQ software was developed to display the collected datain real time and store the data for the next-step data processing.The strain gauge rosettes were attached at the locations where nocontacts between blank and tool or tool and tool were existed.Accordingly, the locations of three strain gauge rosettes areshown in Fig. 13. In particular, locations A and B are in thesymmetrical positions to check the accuracy and reliability of thisexperiment system.5. DiscussionsFig. 14 shows the comparison of measured load curves (blackline and blue line) using different blank holders. It is found thatthe general evolution trend of load curves using the original andoptimized blank holders shows good correlation but a little bit ofdifference at closing stage. The difference is directly caused by thematerial flow reduction which is possibly attributed to structurestiffness reduction of the optimized blank holder. Additionally,stamping process simulation also provides quite accurate predic-tion of load curve by using the initial blank holder (red line).Fig. 15 shows the experimental comparison of equivalent stress atlocations A and B. On the optimized blank holder, the stressincreased most in the middle of drawing process, but thedifference is less than 5 MPa. While at the early stage and finalstage of drawing process, the variation between the optimal oneand initial one is much less.Fig. 12. Optimized bottom structure of the blank holder.Fig. 13. Dynamic strain DAQ system and locations of strain gauge rosettes.Fig. 14. Experimental comparison of die load curves. (For interpretation of thereferences to color in this figure, the reader is referred to the web version of thisarticle.)D. Xu et al. / International Journal of Mechanical Sciences () 8Please cite this article as: Xu D, et al. Topology optimization of die weight reduction for high-strength sheet metal stamping. Int. J.Mech. Sci. (2012), /10.1016/j.ijmecsci.2012.03.006The optimized blank holder is evaluated in terms of structuralperformance and blank forming quality. The simulation results ofthe equivalent stress evolution at locations A and C on the twoblank holders are compared with experimental ones as shown inFig. 16. It shows that the predicted load curve or measured one atlocation A or C on the optimized die structure still follows the oldtrend, and demonstrates very small variation. This means that thestress distribution on the optimized die structure has less change.On the other hand, the thickness distribution at 14 points alongthe YZ section on stamping parts formed by stamping die withinitial blank holder or optimal one were measured as shown inFigs. 17 and 18. Compared to the results from using original blankholder, the variation of the average thickness is not changed butthe most severe thinning is larger. However, the maximumdifference of thickness reduction is only 0.06 mm at measuringpoint 5. In general, the optimized blank holder structure not onlydoes not greatly affect the die structure performance, but alsodoes not change the blank forming quality a lot in terms of thethickness reduction. Therefore, the proposed methodology for diestructure topology optimization is feasible.Fig. 15. Experimental comparison between locations A and B before and afteroptimization.Fig. 16. Comparison of experimental and optimized results. (a) Location A and (b)location C.Fig. 17. Thickness measurement at 14 positions on stamping part along sym-metric section.Fig. 18. Thickness comparison at 14 positions before and after optimization.D. Xu et al. / International Journal of Mechanical Sciences () 9Please cite this article as: Xu D, et al. Topology optimization of die weight reduction for high-strength sheet metal stamping. Int. J.Mech. Sci. (2012), /10.1016/j.ijmecsci.2012.03.0066. ConclusionsA SIMP-based topology optimization methodology for AHSSstamping die was proposed in this paper. Compared with theprevious works, the present method not only considered the elasticdeformation and the tilting of key die component, but also took intoaccount the interaction behavior of tools during stamping processsimulation modeling in order to obtain more accurate boundaryforce condition for die structure optimization. Furthermore, multiplekey load cases during forming process were involved in topologyoptimization of die structure to make the optimized structure meetthe different requirements during stamping process. Numericalsimulation and experiment for blank holder of a step-bottomedcup drawing die were implemented in detail to validate theeffectiveness and the feasibility of the proposed method. Topologyoptimization result has shown that 28.1% mass reduction wasachieved with a slight difference of the die structure performanceand blank forming quality. Therefore, it demonstrates that the dietopology optimization methodology is helpful in reducing overallweight of stamping die.AcknowledgmentsThe research work was funded by the National Key SpecificScience & Technology Program from the Ministry of Industry andInformation Technology of China through Grant nos. 2010ZX04014-072 and 2011ZX04016-051, and co-funded by the China NationalNatural Science Foundation through Grant no. 51075269.References1 Cole GS, Sherman AM. Lightweight materials for automotive applications.Mater Charact 1995;35(1):39.2 International iron and steel institute committee on automotive applications.Advanced high strength steel (AHSS) application guidelines; 2009.3 Shulkin L, Jansen SW, Ahmetoglu MA, Kinzel GL, Altan T. Elastic deflections ofthe blank holder in deep drawing of sheet metal. J Mater Process Technol1996;59(12):3440.4 Doege E, Elend LE. Design and application of pliable blank holder systems forthe optimization of process conditions in sheet metal forming. J MaterProcess Technol 2001;111(13):1827.5 Atzema EH, ten Horn CHLJ, Vegter H. Influence of tooling layout on sheetforming process analysis. In: Proceedings of the ECCOMAS; 2004.6 Lingbeek RA, Meinders T, Rietman A. Tool and blank interaction in the cross-die forming process. Int J Mater Form 2008;1(1):1614.7 Zhang LC, Lin Z. An analytical solution to springback of sheet metals stampedby a rigid punch and an elastic die. J Mater Process Technol 1997;63(13):4954.8 Lingbeek RA, Meinders T. Towards efficient modeling of macro and micro tooldeformations in sheet metal forming. In: Proceedings of the internationalconference on numerical methods in industrial forming processes; 2007.9 Becchio E, Fileccia R, Mastrocola M. Use of FEM in the drawing die structuredesign. In: Proceedings of the IBEC98; 1998. p. 3349.10 Aitharaju V, Liu M, Dong J. Integrated forming simulations and die structuralanalysis for optimal die designs. In: Proceedings of the
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