[机械模具数控自动化专业毕业设计外文文献及翻译]【期刊】可靠性建模及压铸现有的认知不确定性的优化-外文文献（定）

Int J Interact Des ManufDOI 10.1007/s12008-014-0239-yORIGINAL PAPERReliability modeling and optimization of die-casting existingepistemic uncertaintyTao yourui Duan shuyong Yang xujingReceived: 11 May 2014 / Accepted: 24 June 2014 Springer-Verlag France 2014Abstract Die-casting is a typical multidisciplinary sys-tem, in which there are many disciplines involved such ashydrodynamics, heat transfer and elastic-plastic mechanicsand their coupled relations are intricate. Moreover, thereare inherently epistemic uncertainties in die-casting. Thispaper aims at reliability modeling and optimization of die-casting. Firstly, finite element analysis is applied to simulatethe process of die-casting, and the quadratic response sur-face is used to represent the multidisciplinary optimizationmodel of die-casting, in which the evidence theory is used torepresent the epistemic uncertainty. Then, a reliability-basedmultidisciplinary optimization (RBMDO) procedure for die-casting is proposed. The results show the RBMDO procedurepossesses high computational efficiency in optimizing mul-tidisciplinary system with epistemic uncertainty.Keywords Die-casting Reliability modeling Multidisciplinary optimization Epistemic uncertainty1 IntroductionDie-casting is an economical and efficient technology forproducing metal parts. The quality of casting is determinedby the casting process parameters such as injection speed,T. youruiCollege of Mechanical Engineering, Hunan Instituteof Engineering, Xiangtan 411104, Peoples Republic of ChinaT. yourui D. shuyong Y. xujing (B)State Key Laboratory of Advanced Design and Manufacturingfor Vehicle Body, College of Mechanical and Vehicle EngineeringHunan University, Changsha 410082, Peoples Republic of Chinae-mail: injection pressure, die temperature, injection temperatureand so on. Nowadays, computer aided engineering plays animportant role in the casting industry because the simulationof casting based on numerical analysis can help engineersunderstand the process of casting 14. To increase produc-tivity and prevent defects of casting, engineers strive to opti-mize the casting process parameters with various optimiza-tion methods such as Taguchi method, neural network andlinear programming 510. Holmstrm proposes an opti-mization strategy that is based on the level-of-detail conceptto develop cost-effective components 11. Inverse analysishas also been applied in the optimization of casting indus-try by some researchers 12. The die-casting is a multi-disciplinary problem in nature. There are many disciplinesinvolved in die-casting such as hydrodynamics, heat transferand elastic-plastic mechanics, and the process parameters areinterdependent and conflict in a complicated way. The opti-mization of the combination of processes is time-consuming.Eschenauer thinks the finding of optimal layouts of cast com-ponents is a multidisciplinary optimization (MDO) task 13.Just Krimpenis claims that a unified method that can opti-mize all process parameters simultaneously regarding onecriterion or a combination of criteria is still at its infancy,although die-casting parameters have been studied by vari-ous researchers 7.In fact, there are many uncertainties that exist in die-casting such as process parameters and material property.The quality of casting is sensitive to many uncertain vari-ables. For example, the variation of injection speed or tem-perature will cause defects such as porosity, filling defects,and formation of hot cracks. Therefore, reliability model-ing, reliability analysis and reliability-based optimization isessential to die-casting. However, few researchers investi-gate reliability modeling and reliability-based optimizationof die-casting. Choi et al. 14 investigate quality engineer-123Int J Interact Des Manufing optimization of robot casting considering design robust-ness. In their research, the tolerance of material and variationof material property are considered in quality based designoptimization, and two quality engineering methods are doneabout same robot casting and compare with each other. Actu-ally, for die-casting, it is always impossible to obtain theprecise and complete information to define the probabilitydensity function for all uncertain variables. Therefore, impre-cise probability or non-probability theories may be applied inreliability modeling, reliability analysis and reliability-baseddesign such as evidence theory (ET, also named as DempsterShafer theory), interval model, and fuzzy set theory 1518.It has been pointed out by Oberkampf and Helton 19 andBae et al. 20 that ET has an intrinsic capability to handleboth aleatory and epistemic uncertainties. Recently, someresearchers have investigated reliability design based on ETand its engineering applications 2124.Just discussed above, the die-casting is a multidisciplinaryproblem, in which many uncertain variables are involved.Moreover, it is difficult to obtain the precise probability dis-tribution of all uncertain variables. Therefore, the reliabil-ity modeling and optimization of die-casting with epistemicuncertainty is a meaningful and challenging work. A reli-ability model of die-casting is established and a RBMDOprocedure is presented in this paper. We discuss the couplingrelation of die-casting and establish the RBMDO model ofdie-casting in Sect. 2. Then, a RBMDO procedure is devel-oped in Sect. 3 and this procedure is applied in optimization ofdie-casting in Sect. 4. The conclusion is presented in Sect. 5.2 Die-casting and modeling2.1 Finite element model of die-castingThe casting part is cover-shape and its finite element model isshown in Fig. 1. The finite element model includes sprue run-ner, cross gate, mould and four piece of casting. The numberof elements in this model is 80372. The material of cast-ing and mould are aluminum alloy A356AL and steel H13respectively. The finite element analysis software Procastis applied to simulate the process of casting and to calcu-late temperature field, flow field and stress field. The ini-tial process parameters are set as casting pressure 1bar, fill-ing velocity 0.25 m/s, temperature of liquid aluminum alloy700C and preheating temperature of mould 250C. TheFig. 2 shows the simulation results, which include tempera-ture of casting, filling time and solidification time. Figure 2aindicates the temperature of casting is between 243.2C and273.6C. It also shows the mold-filling is perfect. The fillingtime is 1.96 s, shown in Fig. 2b. Solidification time shown inFig. 2c is from 3.38 to 28.99s. This simulation results providebasis for MDO and RBMDO of die-casting.Fig. 1 Finite element model of die-casting2.2 Multidisciplinary model of die-castingIn die-casting, many disciplines are involved such as hydro-dynamics, heat transfer and elastic-plastic mechanic disci-pline. Moreover, the coupling relationship between the dis-ciplines is complex, which can be represented by the designstructure matrix. Although the simulation software may beapplied to analysis fluid field, temperature field etc, the cou-pling relation between the disciplines is implicit. In thispaper, we use a design structure matrix to illustrate the cou-pling relationship for die-casting, which is shown in Fig. 3.The arrows in Fig. 3 indicate the date flow, which shows dataexchange among the disciplines. There are three disciplinesinvolved in the system, including fluid dynamic discipline,heat transfer discipline and performance discipline (produc-tivity efficiency) Ti, T0, pr and v are input parameters, repre-senting pouring temperature, preheat temperature of mould,filling pressure and velocity respectively. v, tnand t repre-sent coupling variables, representing average fluid velocity,solidified time and filling time respectively. P is productivityefficiency and is selected as objective function.The input parameters in fluid dynamic discipline areTi, T0, pr and v, and the out parameters in this disciplineinclude v, Reand t. The input parameters in heat transferare Ti, T0and pr, and the discipline receives Reand v cal-culated in hydro-dynamics discipline. The performance dis-cipline receives tncalculated in heat transfer discipline andt solved in fluid dynamic discipline. The output parameterof performance discipline is P. Therefore, the die-castingis a typical multidisciplinary problem, in which the coupledrelationship among disciplines is intricate.To address the coupling relation between the disciplines,the quadratic response surface(RS) is applied in this mul-tidisciplinary system. Latin hypercube sampling method is123Int J Interact Des ManufFig. 2 Simulation resultsFig. 3 Design structure matrix of die-castingapplied in sampling. Based on the simulation results shownin Sect. 2.1 and engineering experiments, the interval of sam-pling is selected as (680C, 720C) for casting temperature,(220C, 280C) for model temperature, (0.2 m/s, 0.3 m/s)for pouring velocity and (0.8 bar, 1.2 bar) for casting pres-sure. 23 samples are selected according to Latin hypercubesampling method for constructing RS and are listed in Table1. Finite element analysis results of are also shown in Table 1.The quadratic response surface is constructed based on theleast square method and are shown in Eqs. 1, 2, 3. The multi-ple correlation coefficients R2and the adjusted multiple cor-relation coefficients R2adjare shown in Table 2.BothR2andR2adjare near to 1, which indicates that the fitting accuracy ishigh with the quadratic response surface functions. The dif-ference between R2and R2adjfor t, v and tnare 0.0083, 0.027and 0.0045 respectively, which shows there is no redundantterm in the quadratic response surface functions. Therefore,the quadratic response surface functions may be used foroptimization.Table 1 Sampling points and their responding valueProcess parameter Output parameterTi(C) T0(C) v (kg/s) pr (bar) tn(s) v (m/s) t (s)1 683.3 246.99 0.26817 1.1374 1.771 6.83 28.842 684.63 249.11 0.29778 1.0729 1.614 6.41 29.73 705.91 242.89 0.27728 0.84038 1.741 5.82 28.074 686.71 242.83 0.28507 1.0839 1.67 6.55 28.295 711.71 235.66 0.21011 1.0971 2.261 6.48 28.256 691.88 225.85 0.25746 1.1908 1.876 6.85 27.267 695.75 274.37 0.27085 0.81668 1.75 5.58 31.428 691.61 258.23 0.20109 1.1853 2.366 6.87 29.989 681.3 238.46 0.2639 1.1379 1.803 6.68 28.9110 698.12 227.64 0.28977 1.0536 1.638 7.07 27.4711 703.89 233.93 0.29214 0.93916 1.645 6.09 27.9912 708.25 240.68 0.29385 0.84126 1.653 5.81 28.0913 704.94 228.03 0.28647 0.90577 1.668 5.86 27.3314 716.58 228.97 0.24231 1.0588 1.991 6.54 27.3815 697.14 259.09 0.28743 0.8278 1.651 5.72 29.7316 716.94 233.32 0.23892 1.0533 1.987 6.55 27.4717 713.9 250.17 0.22379 0.90146 2.117 6.01 28.8718 719.52 258.09 0.205 0.97106 2.384 6.38 29.919 705.58 242.75 0.2848 0.8824 1.842 5.94 28.0120 681.53 248.29 0.28377 1.0885 1.814 6.67 29.6221 691.63 267.95 0.20756 1.0538 2.499 6.40 30.8722 699.84 251.64 0.20596 1.1093 2.371 6.60 29.3823 705.49 232.07 0.28032 0.93814 1.692 5.96 27.48Table 2 R2and R2adjof RSRS functions R2R2adjt(Ti, T0,v0, pr) 0.9733149 0.96496836v(Ti, T0,v0, pr) 0.9223122 0.8962664tn(Ti, T0, v, pr) 0.985159 0.98069123Int J Interact Des Manuft = 1.9808 0.0356(Ti 700)/20+ 0.0248(T0 250)/30 0.4126(v0 0.25)/0.05 0.0344(pr 1)/0.2 0.3583(Ti 700)/20(T0 250)/30 0.1959(Ti 700)/20(v0 0.25)/0.05 0.4128(Ti 700)/20(pr 1)/0.2 0.3452(T0 250)/30(v0 0.25)/0.05 0.4839(T0 250)/30(pr 1)/0.2 0.3524(v0 0.25)/0.05(pr 1)/0.2 0.1079(Ti 700)/202 0.2369(T0 250)/302 0.0332(v0 0.25)/0.052 0.3545(pr 1)/0.22(1)v = 6.4320 + 0.3702(Ti 700)/20+ 0.3820(T0 250)/30+ 0.2777(v0 0.25)/0.05+ 1.0104(pr 1)/0.2 0.1275(Ti 700)/20(T0 250)/30 0.2597(Ti 700)/20(v0 0.25)/0.05+ 0.0115(Ti 700)/20(pr 1)/0.2 0.6787(T0 250)/30(v0 0.25)/0.05 0.2353(T0 250)/30(pr 1)/0.2 0.1030(v0 0.25)/0.05(pr 1)/0.2 0.1842(Ti 700)/202 0.2845(T0 250)/302 0.2955(v0 0.25)/0.052+ 0.0141(pr 1)/0.22(2)tn= 29.0322 + 0.2341(Ti 700)/20+3.999(T0 250)/30+0.8184(v 6)/1+ 0.6408(pr 1)/0.2+ 1.8535(Ti 700)/20(t0 250)/30 1.5503(Ti 700)/20(v 6)/1+ 2.4753(Ti 700)/20(pr 1)/0.2 4.3747(T0 250)/30(v 6)/1+ 3.9005(T0 250)/30(pr 1)/0.2v+ 2.0332(v 6)/1(pr 1)/0.2+ 1.2675(Ti 700)/202+ 1.3475(T0 250)/302 4.5343(v 6)/12+ 0.7778(pr 1)/0.22(3)The productivity efficiency is decided by solidificationtime, therefore, the authors select solidified time tnas objec-tive function. The filling time t is selected as constrain and isless than 2. The MDO model is shown in Eq. 4.InthisMDOmodel, Ti, T0, pr and v are input parameters.Min tns.t. t 2 0()2.3 Reliability-based optimization model of die-castingThere are many uncertainties involved in die-casting sys-tem, such as process and material parameters. For exam-ple, although the injection pressure and velocity are set asconstant, they fluctuate in the process of casting and maybe considered as uncertain variables. At the same time, theengineer always can not obtain the precise and completeinformation to define uncertain variables probability densityfunction in engineering practice, especial in the early stageof design. In other words, the epistemic uncertain variablesalways exist in die-casting system. Moreover, the propaga-tion of uncertainty through coupled subsystem and the strongnonlinearity of the multidisciplinary system make the relia-bility analysis and RBMDO difficult and computational costexpensive.In this paper, ET is used to represent the epistemic uncer-tainties in the die-casting. ET measures uncertainty with twomeasuresbelief (BEL) and plausibility (PL). These mea-sures can be thought of as lower and upper bounds on prob-ability, defining a probability interval. ET defines a frame ofdiscernment to determine the values of BEL and PL. Moredetails of ET can be found in reference 14.In the MDO model, the inputs parameters include the fill-ing temperature Ti, the temperature of mould T0, the fillingvelocity v and the filling pressure pr. In the process of cast-ing, the temperature of mould and molten metal can be easilycontrolled by heating equipment. Therefore, the two inputparameters are selected as optimization variables. However,the filling velocity and pressure always fluctuate within somerange, which make the die-casting system uncertain. Whilethe probability distribution function of uncertain variables isdifficulty to obtain due to experimental cost, ET is an alter-native way for uncertain quantification. In this paper, theuncertain variables v and pr are presented by ET, and theirBPA is shown in Table 3 based on engineering experiments.The constraint in MDO model is transformed to reliabilityconstraint, and the RBMDO model of die-casting is shownin Eq. 5, in which R is set as R = Phi1() = 0.90.Min tns.t. Pr(t 2 0) R (5)123Int J Interact Des ManufTable 3 BPA of v and pv (m/s) pr (bar)Focal BPA Focal BPA0.20 0.23 0.1 0.8 0.9 0.10.23 0.28 0.8 0.9 1.0 0.40.28 0.30 0.1 1.0 1.1 0.41.1 1.2 0.1InitializationDeterministicMDOReliability analysisby the modified J-CmethodEndConvergenceYNFig. 4 Flow chart of RBMDO3 Procedure of RBMDOIn reference 24, the authors presented a novel reliabilityanalysis procedure for multidisciplinary system with epis-temic and aleatory uncertain variables. First, the epistemicuncertain variables are represented by ET and their prob-ability density function is assumed piecewise uniform dis-tribution based on Bayes method, and approximate mostprobability point is solved by equivalent normalizationmethod(named as the modified J-C method). In this paper,the modified J-C method is used in the RBMDO procedure.The flow chart of the RBMDO procedure is shown in Fig. 4.The detail of this RBMDO procedure is shown as follow:(1) Initialize design variables, coupled variables yij(in mul-tidisciplinary system, yijis a output variable form i-thsubsystem, it is simultaneously an input variable to j-thsubsystem) 25 and reliability index ,(2) Deterministic MDO,(3) Converge or not, if converge, go to step (4), else go tostep (5),(4) Obtain the optimal value of design variables and objec-tive function, end,(5) Solve the most probability points (MPPs) by the modifiedJ-C method, and go to step (2).In this procedure, deterministic MDO and reliability analy-sis are the main two steps and conducted with sequentialcycles. This strategy is called as sequence optimization andreliability analysis (SORA). Compared with the conventionalRBMDO strategy, in which the number of reliability analy-ses is equal to the number of function evaluations consumedby the overall optimization, the strategy of SORA is a way toreduce computational 26. The model of deterministic MDOis formulated based on the reliability analysis results from theprevious cycle such that the violated reliability constraintscan be improved. In SORA, the aim of reliability analysisis to calculate MPP. The modified J-C method presented inRef. 24 may be used to solve MPP for uncertainties. In thisprocedure, conventional MDO method such as multidiscipli-nary feasible method and bi-level integrated system can beused in this RBMDO method. Variation rate or differenceof objective function value may be selected as convergencecriterion in step(3).4 Results and discussionThe strategy presented in this paper is used to optimize thecasting parameters. To compare with the nondeterministicoptimization result, the deterministic optimization result iscalculated too. Both the nondeterministic and the determin-istic optimization results are shown in Table 4. The optimiza-tion results shows that the values of Tiand T0are 680Cand 220C respectively in deterministic optimization, andtheir values increase to 718C and 269C in RBMDO. Inother words, the temperature of molten metal and mould inRBMDO is higher than the one in deterministic