Application of Six Sigma Robust Optimization in Sheet Metal Forming

Transcription

1 Application o Six Sigma Robust Optimization in Sheet Metal Forming Y. Q. Li, Z. S. Cui, X. Y. Ruan, D. J. Zhang National Die and Mold CAD Engineering Research Center, Shanghai Jiao Tong University, Shanghai, 23, China Abstract. Numerical simulation technology and optimization method have been applied in sheet metal orming process to improve design quality and shorten design cycle. While the existence o luctuation in design variables or operation condition has great inluence on the quality. In addition to that, iterative solution in numerical simulation and optimization usually take huge computational time or endure expensive experiment cost. In order to eliminate eect o perturbations in design and improve design eiciency, a CAE-based six sigma robust design method is developed in this paper. In the six sigma procedure or sheet metal orming, statistical technology and dual response surace approximate model as well as algorithm o Design or Six Sigma (DFSS) are integrated together to perorm reliability optimization and robust improvement. A deep drawing process o a rectangular cup is taken as an example to illustrate the method. The optimization solutions show that the proposed optimization procedure not only improves signiicantly the reliability and robustness o the orming quality, but also increases optimization eiciency with approximate model. INTRODUCTION Sheet metal orming is one o the most widely used manuacturing processes in industries. However, in the cases o complicated sheet metal deormation, improper design o process parameter will lead to deects such as racture, wrinkling and springback, etc. In order to predict deects and avoid time consuming in trial and error tryout procedure, numerical simulation and optimization method have been separately applied or decades. Many researches have been recently reported on integrating numerical simulation and optimization technology [1-4] together. While with the increase o the design variables and complexity o the problem, the optimization will take too long computational time or even impossible to obtain optimized solutions. In addition to that, perturbations existed in material properties, geometry and processing parameters may have signiicant eects on the orming quality [5]. Traditional deterministic optimization methods can not take into account o perturbations, thereore may lead to unreliable or non-robust solutions. Based on numerical simulation technique and approximation approach o response surace model, this paper develops a sixsigma robust design method or sheet metal stamping optimization. The philosophy o DFSS [6, 7] in quality engineering is applied in this optimization procedure to improve process quality and design eiciency. A deep drawing process o a rectangular cup is taken as an example to illustrate the proposed method. DFSS ROBUST OPTIMIZATION In conventional optimization problems, the obective is to minimize/maximize a linear or nonlinear unction o many variables subect to a set o constraints. In contrast, robust optimization aims to optimize not only the unction value but also its sensitivity to the variation o the design variables. Robust optimization can be ormulated as a multiobective optimization problem shown as ollowing: Minimize Subect to [ μ, σ ] (1) hk ( x ) = k = 1,2,..., K (2) 819

2 g g ( x) + p Δx = 1,2,..., J (3) n i i= 1 xi x +Δx x x Δ x i = 1, 2,... N (4) L U i i i i Where μ andσ are the mean and standard deviation o the obective unction ( x ), respectively. The mean and the variance o design variables are identiied as x and Δ x. hk and g are constraint unction. p is penalty actor. Six sigma methodology was proposed at Motorola and developed into DFSS at General Electric (GE). The main purpose o DFSS is to prevent deects at design stage instead o ixing them at later stages, and also to improve parts quality up to 6σ level. The perormance level o 6σ is equivalent to 3.4 deect parts per million (PPM), while at 3σ level (the average sigma level or most companies) the deect ratio is about 668PPM. Figure 1 shows a graphical illustration o probability distribution o part quality at 3σ level and 6σ level. It can be seen that at 6σ level the part quality is more steady than at 3σ level. model and robust optimization strategy, the six-sigma robust design ormulation can be established as ollows: Minimize Subect to ( ) F μ, σ = λμ ( x) + (1 λ) σ ( x) (5) hk ( x ) = k = 1,2,..., K (6) ( ) g μ ( X), σ ( X) + nσ = 1, 2,..., J (7) g X + nσ μ X nσ i = 1, 2,... N (8) L U i x x i x Where λ is a weight deined by the designer to decrease the sensitivity o quality to the deviation o design variables (controllable variables) or the noise actors (uncontrollable variables). n denotes the desired sigma level. The six-sigma design is achieved when setting n = 6. The dual response surace models or the mean and standard derivation are established ollowing equation (9) and (1), k k k 2 i i ii i i i i= 1 i= 1 i< (9) ˆ μ = β + β x + β x + β x x + R μ k k k 2 i i ii i i i i= 1 i= 1 i< (1) ˆ σ = γ + γ x + γ x + γ x x + R σ FIGURE 1. 3σ design and 6σ design Taguchi method is widely used to achieve certain obectives in a mean response while simultaneously minimizing the variance. However, Taguchi method relies heavily on the measure o signal-to-noise ratio and encounters diiculties when trying to optimize the obect unction while satisying the limitation to the variance. For this reason, a methodology o dual response surace model was proposed by Vining and Myers to realize Taguchi philosophy, in which the response surace or the mean and variance was established simultaneously. Upon the dual response surace, one can directly sees which actors primarily inluence the mean and which actors primarily inluence the variance. It leads a way to minimize the obective unction while keeping the variance to a given limit. By integrating the dual response surace Where ˆμ is the response o the mean μ and ˆ σ is the response o standard derivation σ. β i and γ i are the unknown coeicients determined by least square method. R μ and R σ are random errors. DFSS PROCEDURE FOR SHEET METAL FORMING The main steps to achieve DFSS can be summarized as DMADOV which stands or Deine, Measure, Analyze, Design, Optimize and Veriy. As shown in igure 2, the steps o DFSS in sheet metal orming can be explained below: Deine: This step is to setup optimization problem, including deinition o obective unction, constraint conditions, controllable actors and noise actors. In 82

3 addition, the statistic distribution o noise actors should also be speciied. Measure: The measure step takes a role to separate the signiicant actors rom all would-be actors that aect the orming quality, and reduce the number o input variables. The common technique or measure step is screening design, which includes design o experiment (DOE), analysis o variance (ANOVA) and signiicant test. For this sheet metal orming problem, the inite element simulation, instead o physical test, was applied to implement the DOE process. Analyze: Signiicant actors include both design variables and noise actors are sampled in this step. The design o experiment upon these signiicant actors is conducted and the perormance in each case study is simulated by inite element. Upon the response o obective unction and standard derivation, the approximate dual response surace model is established or the interests including obect and constraints. Design and Optimize: Based on the dual response surace model, the deterministic optimization or six sigma robust optimization is perormed by applying optimization method such as sequential quadratic programming (SQP) or genetic algorithm (GA). The solutions in each iteration can be converted to sigma level n to evaluate the robustness and reliability. In act, by deining dierent value or n, the optimization can be evaluated at dierent sigma level. Veriy: The optimum result should be veriied by experiment or numerical simulation; currently we veriy the results by numerical simulation. Since the optimum result is obtained rom the response surace model, sometimes it is dierent rom the result given by simulation in the case o same design variables. In this situation, the design work goes back to the measure stage. The simulation result at this pseudo optimum point is then added to the response model to make it more approximate to the real perormance. FIGURE 2. DFSS or sheet metal orming. APPLICATION IN DEEP DRAWING A deep drawing process o a square cup is taken as an example to illustrate the six sigma robust design method. The main deect in this process is wrinkle and racture. The thickening o the blank will leads to wrinkle while the thinning leads to racture. To decrease the possibility o wrinkle, the thickening o blank ater orming, described using a natural logarithm unction, is taken as the obective to be minimized, i.e. Minimize: ln( tmax / t ) (11) 821

4 To avoid racture, the thinning o the blank ater drawing should be less than a critical value that obtained rom experiment; thereore the constraint or this optimization problem is ormulated H t tmin T L = ln( / ) < (12) where t max and tmin are maximum and minimum thicknesses o blank ater orming, t is initial thickness, and T L is the upper limit o thinning. In this work, the inite element simulation instead o physical test was conducted. The simulation work was perormed by using commercial sotware LS-DYNA. The material o the blank is medium steel. The Young s modulus is E=26MPa and initial yielding stress is 2MPa. The FEM model or this drawing process is shown in Fig 3. The ANOVA and signiicant test shows that the blank holding orce (BHF) and the radius o die (D) are the maor design variables, and the riction coeicient m and initial thickness t are the maor noise actors. Assume that the design variables are die diameter (D) and blank holding orce (BHF), and the design limit or D is 4.7~5.3mm while or BHF is 2~4kN. The noise actors are riction coeicient m and blank thickness t, where m has a mean value o.14 and varies in.12~.16, while t has a mean value o.78mm and varies in.74~.78mm. FIGURE 3. FEM model o rectangular cup orming In the initial design, the variables are D = 4.7 mm and BHF=35kN. Fig 4 demonstrates the thickness changes ater drawing. It can be seen that the thinnest thickness is.61235mm and the corresponding H =.125, which show a tendency o racture. FIGURE 4. Thickness distribution or initial design FIGURE 5. Thickness distribution or 6 sigma optimum design In the optimization, the response o thickening as well as the racture characteristics H is simulated at each design point in the approach o design o experiment. Upon these results, the dual response surace model or thickening and racture characteristic are established through equation (9) and (1), respectively. In this way the optimization can be perormed by calculating the obect and constraint value rom dual response surace model, instead o running the simulation sotware in each iteration o design. This is very important to computational time saving or the situation that each simulation will take a long time. Based on the DRSM, deterministic optimization and robust optimization with dierent sigma level are perormed. The comparison o the design results are shown in table 1. In the results o deterministic optimization, the mean value o H which denotes the thinning characteristics is very close to, or the allowable upper limit o constraint. When the design variables and noise actors have luctuation within their normal distributed probability, the probability o exceeding the upper limit o constraint is about 44.13%, which implies a high possibility o 822

5 ailure. In addition to that, the value o standard deviation in this case is very big, which shows that H is sensitive to the luctuation o design variables. With the increasing o design sigma level, the possibility o ailure decreases rapidly. For example, at 3σ level, the ailure probability o the design decreases to 2.28%. When the quality level increases up to 6σ level, nearly zero ailure probability can be achieved, and at the same time, the sensitivity o H to the perturbations o design variables and noise actors is reduced. Fig 5 shows the thickness distribution ater 6 sigma robust optimization. Fig 6 demonstrates the probability distribution H in dierent optimization results. It can be seen that in the deterministic optimization, the mean o H is close to the ailure boundary and the distribution covers a wide range, while with increasing sigma level, the mean value goes away gradually rom the allowable upper limit, and the probability distribution shrinks much thinner. By this way, the optimization increases the robustness o the design. TABLE 1. Solutions o dierent optimum methods Optimization method Optimum points (D, BHF) μ H σ H P[H>] (%) Deterministic optimization (29.6, 5.8 ) σlevel (25.572, 5.2) σlevel (24.132, 5.26) σlevel (22.914, 5.29) σlevel (21.516, 5.18) (a) deterministic optimization (b) six sigma level design FIGURE 6. Histogram o deterministic optimization and six sigma robust design CONCLUSION A CAE-based six sigma robust design method was proposed in this paper. This method combines six sigma quality philosophy and computer simulation to perorm robust design. The optimum solutions o square cup orming illustrated that the proposed method is superior to the traditional optimization method in improving design eiciency and quality. However, as the increasing o sigma level, the design cost and product cost may increase too. The sigma level should be careully determined by designer according to the importance and cost o products. REFERENCES 1 Browne, M.T., Hillery, M.T., J. Mater. Proc. Technol, 136, 64-71(23). 2 Nakamura, Y., Ohata, T., Katayama, T. and Nakamachi, E., Optimum die design or sheet metal orming process by using inite element and discretized optimization methods in Simulation o materials processing: Theory, Methods and Applications, edited by J. Huetink, Rotterdam: Balkema, 1998, Huh, H., Kim, S.H., J. o Engng. Mater. and Technol, 123, (21). 4 Gasper, G., Pepelnak, T. and Kuzman, K., J. Mater. Proc. Technol, , 54-59(22). 823

6 5 Kleiber, M., Roek, J., Stocki, R., Comput. Methods Appl. Mech. Engrg., 191, (22). 6 Breyogle, F.W., Implementing Six Sigma: Smarter Solutions Using Statistical Methods, John Wiley & Sons, Koch, P.N., Yang, R.J., Gu, L., Struct Multidisc Optim., 26, (24). 8 Vining, G.G., Myers, R.H., J. Quality Technol., 22, (199). 824