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A simplified column model for the automatic design of the stamping die structure
Journal of Materials Processing Technology 177 (2006) 109–113
A simplified column model for the automatic design of the stamping die structure Jinn-Jong Sheu ∗ , Ching-Hsun Yang Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences, No. 415, Chienkung Road, Kaohsiung 807, Taiwan, ROC
Abstract An automatic design method of a stamping die structure was proposed in this paper. The design parameters of the stamping die structure were studied first via the Taguchi method. The effects of the parameters were analyzed in order to establish the design rules. A die structure analysis method using a simplified column model was proposed to design the optimum die topology automatically. The solid die block under the die surface was divided into small rectangular columns. The surface pressure of the part after drawing was converted into the boundary conditions of the die structure design. A pseudo density of the column model was defined as a function of pressure and die surface slope. The topology optimization of the die structure was obtained by using the required pseudo density and the constraint of the buckle condition. The design results were compared with the ANSYS topology optimization to verify the proposed model. The similar topology results demonstrated that the proposed method is feasible. © 2006 Elsevier B.V. All rights reserved. Keywords: Simplified column model; Die structure design automation; Taguchi method; CAE
1. Introduction In the traditional stamping die structure layout, uniformly distributed ribs were designed for the supporting of the die face. The die structure designed in according to the experienced rules ignored the pressure distribution of the die face. In case of over design, the material cost of die and the energy consumption of stamping were increased. On the other hand, design with fewer structural ribs can save the cost of die material but may have the problem of die failure. Kusiak [1] proposed a nongradient optimization technique for the axi-symmetric closed forging die design. The uniform distribution of austenite grain is chosen as the criterion of optimization. Chen and Chiang [2] adopted a commercial package to analyze the bathtub stamping process and proposed a better die design to avoid fracture of blank. Mamalis et al. [3] used DYNA 3D to simulate the deep-drawing of cylindrical cups and compared the analysis data with the experimental results. The simulation results of load, strain and stress were in good agreement with the experimental data. The effects of the scaling parameters of density and velocity were studied. Shin and Gandhi [4] proposed an integration algorithm of the interval method and the FEM analysis. The ∗
Corresponding author. Tel.: +886 7 3494880; fax: +886 7 3835015. E-mail address: [email protected] (J.-J. Sheu).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.04.087
bound of the optimization search was found and the goal function was built with a polynomial. The global structural optimization was obtained by the proposed interval method. Guo and Cheng [5] proposed an first-order extrapolation method to obtain the optimal solution when the singular condition occurs. Yang and Soh [6] proposed a genetic programming method to optimize the structure of truss. The allowance stress and buckling condition were considered in the fitness function. Swaminathan et al. [7] divided the product geometry into topological elements and coded them into a two-dimensional array of gene. The optimal design was obtained by minimizing the volume, the maximum stress and the strain. In this paper, a simplified column model was proposed to calculate the die stress and build the pseudo density of the die structure design. The least important columns were given a lower density and will be removed during the topological structure design process. 2. Theoretical analysis The proposed method uses two steps to establish the design rules and the evaluation criterion. The first step is to determine the most important parameters of die structure design from all of the considered factors. The next step is to establish the evaluation function which is composed of the design parameters. The pressure distribution of the blank at the final step of the drawing was
110
J.-J. Sheu, C.-H. Yang / Journal of Materials Processing Technology 177 (2006) 109–113
Fig. 1. The geometrical factors for the design parameter study. Fig. 2. The simplified column model created from the entire die block.
read into the die structure design program. The simplified column models were created using the die face geometry. The stress, strain and displacement of these column models were calculated from the given pressure boundary condition. The optimization process of die structure design was carried out in the domain of these column models.
where yi are the measured data and n is the testing number. The normalized quality indices (Qi ) were multiplied by a weight (Wi ) and summed up to obtain the cost function CF, i.e., the measured data yi : CF = Q1 W1 + Q2 W2 + Q3 W3
2.1. Step 1: determination of the most important parameters of die structure design The design parameters were studied with a design of experiment (DOE) method to determine the importance of parameters at the first step. The Taguchi method was adopted to study the main effect of the design factors and build the design rules. The design factors represented by a cylindrical cup drawing die were shown in Fig. 1. The width of rib (W), the thickness of die face (T) and the span distance of the structure in the x- and y-direction (Dx and Dy ) were taken into consideration. These factors were labeled as A, B, C and D for the analysis of variance (ANOVA) and given in Table 1. The L9 (34 ) orthogonal array was chosen to set-up the parameter combinations of the simulation. The die stress analysis was carried to obtain the stress, strain and displacement information. The quality indices of each design are composed of the ratio of the maximum von Mises stress to yield stress, the maximum effective strain and the maximum displacement. The optimum criterion of the DOE is smaller-the-better. The M.S.D. (mean square deviation) and the S/N (signal to noise) ratio of the smaller-the-better were calculated by MSD =
1 2 (y + y22 + · · · + yn2 ) n 1
(1)
S/N = −10 log M.S.D.
The stress and strain are more important than the displacement in the die design. To emphasize the importance of these two factors, W1 (weight of stress) and W2 (weight of strain) were both set to 1. W3 (weight of displacement) was set to 0.5 to tune down the contribution of displacement. The FEM code LS-DYNA was adopted to analyze the drawing process with the combination of Table 1. The pressure distribution on the blank at the final stage of the forming was considered as the boundary conditions of the die structure design. 2.2. Step 2: creation of the column model and evaluation function of the design The proposed column model was shown in Fig. 2. The entire die structure was composed of many simplified columns with inclined top face. The static analysis was performed to obtain the maximum stress of column. The pressures on the top face of each column were accumulated and an equivalent force was obtained. In order to evaluated the necessity of keeping a column or not, the pseudo density of each column was calculated by using Eq. (4): ρi = Pi Wp + Si Ws + K
(2)
Table 1 Design factors and levels for the parameter study of die structure design Label
Factors
Level 1 (mm)
Level 2 (mm)
Level 3 (mm)
A B C D
W T Dx Dy
3 4 5 5
4 5 10 10
5 6 20 20
Note: ‘W’ is the rib width, ‘T’ the thickness of die face, ‘Dx ’ and ‘Dy ’ are the span distance in x- and y-direction.
(3)
Fig. 3. The column models with different top inclined angles.
(4)
J.-J. Sheu, C.-H. Yang / Journal of Materials Processing Technology 177 (2006) 109–113
111
Table 2 The levels of section area, pressure and slope adopted for the weighting value study of the pseudo density
Table 5 The maximum von Mises stress of the column model for different combination of the section area (E), pressure (F) and slope (G)
Label
Factors
Level 1
Level 2
Level 3
Run
E
F
G
von Mises stress (MPa)
S/N ratio
E F G
Area (mm2 ) Pressure (MPa) Slope (◦ )
30 × 30 10 0
40 × 40 20 22.5
50 × 50 30 45
1 2 3 4 5 6 7 8 9
1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3
1 2 3 2 3 1 3 1 2
15.02 391.30 1310.00 166.13 724.15 50.29 309.96 36.49 442.04
23.536 51.850 62.345 44.409 57.197 34.029 49.826 31.244 52.909
Table 3 The orthogonal array and the simulation results for structure layout study Run
Factors A
1 2 3 4 5 6 7 8 9
1 1 1 2 2 2 3 3 3
Results of analysis B 1 2 3 1 2 3 1 2 3
C 1 2 3 2 3 1 3 1 2
D 1 2 3 3 1 2 2 3 1
σ v /σ y 0.7202 0.7711 1.1986 0.9658 0.7333 0.7431 0.9479 0.7208 0.6526
ε
Displacement (mm)
9.00E−04 9.63E−04 1.50E−03 1.21E−03 9.17E−04 9.29E−04 1.18E−03 9.01E−04 8.16E−04
2.12E−02 2.52E−02 3.36E−02 3.20E−02 2.26E−02 2.03E−02 2.43E−02 2.10E−02 1.94E−02
Table 6 ANOVA of the stress analysis of the simplified column model Label Factors
E F G
Sum of squares
Area (mm2 ) 2.3576 Pressure (MPa) 175.671 Slope (◦ ) 1171.20
Error variance
Pure sum of square
Significance of maximum stress
1.179 87.836 585.604
– 172.909 1168.446
(Pooled) 13% 87%
3. Results and discussion where Wp and Ws are the weighting values of the normalized pressure and slope of column, respectively. K is the penalty consideration of buckling. The default value of K is 0 means no buckle occurred. If the load of simplified column model is larger than the half of the limit of buckle condition (Euler’s theory) of a column fixed at two ends, K is set to 1 to ensure this column will be kept. The weighting values in Eq. (4) will be solved by using the Taguchi method. A column in 200 mm height was built with different section areas and slopes of top face. The boundary conditions of pressure and slopes of column were shown in Fig. 3. The equivalent pressure was accumulated from the pressure of meshes of blank within the area of each column. The factors and levels adopted for solving the weighting values of pseudo density were given in Table 2. The factors considered were section area of column, pressure and slope of top face. The labeled E, F and G are given for analysis of variance. An ANSYS command file was created automatically to perform the topology optimization of die. The die designs of the proposed method were compared with the ANSYS results.
3.1. The importance analysis of the design parameters The orthogonal array (OA) and the simulation results of the importance analysis of parameters were given in Table 3. The mechanical properties of the blank in 1 mm thickness were given as follows: • • • •
Young’s module, E = 2.1E5 MPa; density = 7.8E−9 tonnes/mm3 ; Poison ratio, ν = 0.3; flow stress, σ = 514.73ε0.277 MPa; yield stress = 250 MPa.
Table 4 ANOVA of the structure layout factors for different consideration Label
Factors
Significance of stress ratio (%)
Significance of strain (%)
Significance of displacement (%)
A B C D
W T Dx Dy
8.92 13.69 34.37 43.02
9.06 13.64 34.05 43.26
19.79 5.81 30.16 44.23
Note: ‘W’ is the rib width, ‘T’ the thickness of die face, ‘Dx ’ and ‘Dy ’ are the span distance in x- and y-direction.
Fig. 4. The optimal die structure of a cylindrical cup drawing designed by the proposed system.
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Fig. 5. The topology optimization result of the ANSYS analysis.
The results in Table 3 shown that case 3 may have the die failure problem. The further ANOVA analysis was carried out to calculate the significance of each factor. The significances of the design parameters were shown in Table 4. The significances of the x- and y-direction span were larger than the other factors dramatically. The ANOVA analysis shows the x- and y-direction span are the most important design parameters.
Fig. 6. The smoother circular die structure design refined from the automatic design.
nificances of the section area of column and the error were too small and pooled. The weights Wp and Ws of pseudo density in Eq. (4) are assigned to 0.13 and 0.87 according to the significance of pressure and slope in Table 6. The evaluation function (pseudo density) was determined.
3.2. The weighting values of the evaluation function 3.3. The verification of the proposed method The maximum von Mises stresses and S/N ratios of the simplified column model analysis were shown in Table 5. The ANOVA analysis was given in Table 6. The significances of the pressure and slope are 13% and 87%, respectively. The sig-
The automatic die structure design of the cylindrical cup drawing was shown in Fig. 4. The columns were kept if the pseudo density is larger than 61%. The less the pseudo density
Fig. 7. The maximum von Mises stress distribution of the circular die structure design.
J.-J. Sheu, C.-H. Yang / Journal of Materials Processing Technology 177 (2006) 109–113
113
means the less importance of column. The ANSYS topology optimization result was shown in Fig. 5. The kept volume rate of the ANSYS topology optimization was 20%. Figs. 4 and 5 demonstrated the locations and shapes of the die structure designed by the proposed method were in good agreement with the ANSYS result.
the established simplified column model is capable of designing a reasonable die structure automatically. The comparison of the results of the proposed method and the ANSYS topology optimization demonstrated a good agreement.
3.4. The refined smooth die structure design and distress analysis
This work was supported by National Science Council of Taiwan and the JuiLi Company. The project grant number is NSC 92-622-E-151-005-CC3.
The design shown in Fig. 4 is in the zigzag profile of die geometry. The smoothing process was carried out to obtain a more reasonable die design and shown in Fig. 6. The smoother die design is in a circular shape. The die stress analysis was performed to verify the feasibility of design. The von Mises stress distribution of die was shown in Fig. 7. The maximum value 659 MPa is much smaller than the yield stress of the cast iron or steel. This is a safer and lighter design of die. 4. Conclusions The proposed die design method considered the importance of design parameters. The pseudo density of column model was proposed to evaluate the necessity of the supporting design. The pseudo density is also an evaluation function of design. A lower pseudo density can reduce the weight of the die but the buckling limit should also be considered. The design examples show
Acknowledgements
References [1] J. Kusiak, A technique of tool-shape optimization in large scale problems of metal forming, J. Mater. Proc. Technol. 57 (1996) 79–84. [2] F.K. Chen, B.H. Chiang, Analysis of die design for the stamping of a bathtub, J. Mater. Proc. Technol. 72 (1997) 421–428. [3] A.G. Mamalis, D.E. Manolakos, A.K. Baldoukas, Simulation of sheet metal forming using explicit finite-element techniques: effect of material and forming characteristics. Part 1. Deep-drawing of cylindrical cups, J. Mater. Proc. Technol. 72 (1997) 48–60. [4] Y.S. Shin, R.V. Grandhi, A global structural optimization technique using an interval method, Struct. Multidisc. Optim. 22 (2001) 351–363. [5] X. Guo, G.D. Cheng, An extrapolation approach for the solution of singular optima, Struct. Multidisc. Optim. 19 (2000) 255–262. [6] Y. Yang, C.K. Soh, Automated optimum design of structures using genetic programming, Comput. Struct. 80 (2002) 1537–1546. [7] N. Swaminathan, et al., Optimal topology design of products using genetic algorithm, Mater. Proc. Des. (2004) 2138–2142.
A simplified column model for the automatic design of the stamping die structure Jinn-Jong Sheu ∗ , Ching-Hsun Yang Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences, No. 415, Chienkung Road, Kaohsiung 807, Taiwan, ROC
Abstract An automatic design method of a stamping die structure was proposed in this paper. The design parameters of the stamping die structure were studied first via the Taguchi method. The effects of the parameters were analyzed in order to establish the design rules. A die structure analysis method using a simplified column model was proposed to design the optimum die topology automatically. The solid die block under the die surface was divided into small rectangular columns. The surface pressure of the part after drawing was converted into the boundary conditions of the die structure design. A pseudo density of the column model was defined as a function of pressure and die surface slope. The topology optimization of the die structure was obtained by using the required pseudo density and the constraint of the buckle condition. The design results were compared with the ANSYS topology optimization to verify the proposed model. The similar topology results demonstrated that the proposed method is feasible. © 2006 Elsevier B.V. All rights reserved. Keywords: Simplified column model; Die structure design automation; Taguchi method; CAE
1. Introduction In the traditional stamping die structure layout, uniformly distributed ribs were designed for the supporting of the die face. The die structure designed in according to the experienced rules ignored the pressure distribution of the die face. In case of over design, the material cost of die and the energy consumption of stamping were increased. On the other hand, design with fewer structural ribs can save the cost of die material but may have the problem of die failure. Kusiak [1] proposed a nongradient optimization technique for the axi-symmetric closed forging die design. The uniform distribution of austenite grain is chosen as the criterion of optimization. Chen and Chiang [2] adopted a commercial package to analyze the bathtub stamping process and proposed a better die design to avoid fracture of blank. Mamalis et al. [3] used DYNA 3D to simulate the deep-drawing of cylindrical cups and compared the analysis data with the experimental results. The simulation results of load, strain and stress were in good agreement with the experimental data. The effects of the scaling parameters of density and velocity were studied. Shin and Gandhi [4] proposed an integration algorithm of the interval method and the FEM analysis. The ∗
Corresponding author. Tel.: +886 7 3494880; fax: +886 7 3835015. E-mail address: [email protected] (J.-J. Sheu).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.04.087
bound of the optimization search was found and the goal function was built with a polynomial. The global structural optimization was obtained by the proposed interval method. Guo and Cheng [5] proposed an first-order extrapolation method to obtain the optimal solution when the singular condition occurs. Yang and Soh [6] proposed a genetic programming method to optimize the structure of truss. The allowance stress and buckling condition were considered in the fitness function. Swaminathan et al. [7] divided the product geometry into topological elements and coded them into a two-dimensional array of gene. The optimal design was obtained by minimizing the volume, the maximum stress and the strain. In this paper, a simplified column model was proposed to calculate the die stress and build the pseudo density of the die structure design. The least important columns were given a lower density and will be removed during the topological structure design process. 2. Theoretical analysis The proposed method uses two steps to establish the design rules and the evaluation criterion. The first step is to determine the most important parameters of die structure design from all of the considered factors. The next step is to establish the evaluation function which is composed of the design parameters. The pressure distribution of the blank at the final step of the drawing was
110
J.-J. Sheu, C.-H. Yang / Journal of Materials Processing Technology 177 (2006) 109–113
Fig. 1. The geometrical factors for the design parameter study. Fig. 2. The simplified column model created from the entire die block.
read into the die structure design program. The simplified column models were created using the die face geometry. The stress, strain and displacement of these column models were calculated from the given pressure boundary condition. The optimization process of die structure design was carried out in the domain of these column models.
where yi are the measured data and n is the testing number. The normalized quality indices (Qi ) were multiplied by a weight (Wi ) and summed up to obtain the cost function CF, i.e., the measured data yi : CF = Q1 W1 + Q2 W2 + Q3 W3
2.1. Step 1: determination of the most important parameters of die structure design The design parameters were studied with a design of experiment (DOE) method to determine the importance of parameters at the first step. The Taguchi method was adopted to study the main effect of the design factors and build the design rules. The design factors represented by a cylindrical cup drawing die were shown in Fig. 1. The width of rib (W), the thickness of die face (T) and the span distance of the structure in the x- and y-direction (Dx and Dy ) were taken into consideration. These factors were labeled as A, B, C and D for the analysis of variance (ANOVA) and given in Table 1. The L9 (34 ) orthogonal array was chosen to set-up the parameter combinations of the simulation. The die stress analysis was carried to obtain the stress, strain and displacement information. The quality indices of each design are composed of the ratio of the maximum von Mises stress to yield stress, the maximum effective strain and the maximum displacement. The optimum criterion of the DOE is smaller-the-better. The M.S.D. (mean square deviation) and the S/N (signal to noise) ratio of the smaller-the-better were calculated by MSD =
1 2 (y + y22 + · · · + yn2 ) n 1
(1)
S/N = −10 log M.S.D.
The stress and strain are more important than the displacement in the die design. To emphasize the importance of these two factors, W1 (weight of stress) and W2 (weight of strain) were both set to 1. W3 (weight of displacement) was set to 0.5 to tune down the contribution of displacement. The FEM code LS-DYNA was adopted to analyze the drawing process with the combination of Table 1. The pressure distribution on the blank at the final stage of the forming was considered as the boundary conditions of the die structure design. 2.2. Step 2: creation of the column model and evaluation function of the design The proposed column model was shown in Fig. 2. The entire die structure was composed of many simplified columns with inclined top face. The static analysis was performed to obtain the maximum stress of column. The pressures on the top face of each column were accumulated and an equivalent force was obtained. In order to evaluated the necessity of keeping a column or not, the pseudo density of each column was calculated by using Eq. (4): ρi = Pi Wp + Si Ws + K
(2)
Table 1 Design factors and levels for the parameter study of die structure design Label
Factors
Level 1 (mm)
Level 2 (mm)
Level 3 (mm)
A B C D
W T Dx Dy
3 4 5 5
4 5 10 10
5 6 20 20
Note: ‘W’ is the rib width, ‘T’ the thickness of die face, ‘Dx ’ and ‘Dy ’ are the span distance in x- and y-direction.
(3)
Fig. 3. The column models with different top inclined angles.
(4)
J.-J. Sheu, C.-H. Yang / Journal of Materials Processing Technology 177 (2006) 109–113
111
Table 2 The levels of section area, pressure and slope adopted for the weighting value study of the pseudo density
Table 5 The maximum von Mises stress of the column model for different combination of the section area (E), pressure (F) and slope (G)
Label
Factors
Level 1
Level 2
Level 3
Run
E
F
G
von Mises stress (MPa)
S/N ratio
E F G
Area (mm2 ) Pressure (MPa) Slope (◦ )
30 × 30 10 0
40 × 40 20 22.5
50 × 50 30 45
1 2 3 4 5 6 7 8 9
1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3
1 2 3 2 3 1 3 1 2
15.02 391.30 1310.00 166.13 724.15 50.29 309.96 36.49 442.04
23.536 51.850 62.345 44.409 57.197 34.029 49.826 31.244 52.909
Table 3 The orthogonal array and the simulation results for structure layout study Run
Factors A
1 2 3 4 5 6 7 8 9
1 1 1 2 2 2 3 3 3
Results of analysis B 1 2 3 1 2 3 1 2 3
C 1 2 3 2 3 1 3 1 2
D 1 2 3 3 1 2 2 3 1
σ v /σ y 0.7202 0.7711 1.1986 0.9658 0.7333 0.7431 0.9479 0.7208 0.6526
ε
Displacement (mm)
9.00E−04 9.63E−04 1.50E−03 1.21E−03 9.17E−04 9.29E−04 1.18E−03 9.01E−04 8.16E−04
2.12E−02 2.52E−02 3.36E−02 3.20E−02 2.26E−02 2.03E−02 2.43E−02 2.10E−02 1.94E−02
Table 6 ANOVA of the stress analysis of the simplified column model Label Factors
E F G
Sum of squares
Area (mm2 ) 2.3576 Pressure (MPa) 175.671 Slope (◦ ) 1171.20
Error variance
Pure sum of square
Significance of maximum stress
1.179 87.836 585.604
– 172.909 1168.446
(Pooled) 13% 87%
3. Results and discussion where Wp and Ws are the weighting values of the normalized pressure and slope of column, respectively. K is the penalty consideration of buckling. The default value of K is 0 means no buckle occurred. If the load of simplified column model is larger than the half of the limit of buckle condition (Euler’s theory) of a column fixed at two ends, K is set to 1 to ensure this column will be kept. The weighting values in Eq. (4) will be solved by using the Taguchi method. A column in 200 mm height was built with different section areas and slopes of top face. The boundary conditions of pressure and slopes of column were shown in Fig. 3. The equivalent pressure was accumulated from the pressure of meshes of blank within the area of each column. The factors and levels adopted for solving the weighting values of pseudo density were given in Table 2. The factors considered were section area of column, pressure and slope of top face. The labeled E, F and G are given for analysis of variance. An ANSYS command file was created automatically to perform the topology optimization of die. The die designs of the proposed method were compared with the ANSYS results.
3.1. The importance analysis of the design parameters The orthogonal array (OA) and the simulation results of the importance analysis of parameters were given in Table 3. The mechanical properties of the blank in 1 mm thickness were given as follows: • • • •
Young’s module, E = 2.1E5 MPa; density = 7.8E−9 tonnes/mm3 ; Poison ratio, ν = 0.3; flow stress, σ = 514.73ε0.277 MPa; yield stress = 250 MPa.
Table 4 ANOVA of the structure layout factors for different consideration Label
Factors
Significance of stress ratio (%)
Significance of strain (%)
Significance of displacement (%)
A B C D
W T Dx Dy
8.92 13.69 34.37 43.02
9.06 13.64 34.05 43.26
19.79 5.81 30.16 44.23
Note: ‘W’ is the rib width, ‘T’ the thickness of die face, ‘Dx ’ and ‘Dy ’ are the span distance in x- and y-direction.
Fig. 4. The optimal die structure of a cylindrical cup drawing designed by the proposed system.
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J.-J. Sheu, C.-H. Yang / Journal of Materials Processing Technology 177 (2006) 109–113
Fig. 5. The topology optimization result of the ANSYS analysis.
The results in Table 3 shown that case 3 may have the die failure problem. The further ANOVA analysis was carried out to calculate the significance of each factor. The significances of the design parameters were shown in Table 4. The significances of the x- and y-direction span were larger than the other factors dramatically. The ANOVA analysis shows the x- and y-direction span are the most important design parameters.
Fig. 6. The smoother circular die structure design refined from the automatic design.
nificances of the section area of column and the error were too small and pooled. The weights Wp and Ws of pseudo density in Eq. (4) are assigned to 0.13 and 0.87 according to the significance of pressure and slope in Table 6. The evaluation function (pseudo density) was determined.
3.2. The weighting values of the evaluation function 3.3. The verification of the proposed method The maximum von Mises stresses and S/N ratios of the simplified column model analysis were shown in Table 5. The ANOVA analysis was given in Table 6. The significances of the pressure and slope are 13% and 87%, respectively. The sig-
The automatic die structure design of the cylindrical cup drawing was shown in Fig. 4. The columns were kept if the pseudo density is larger than 61%. The less the pseudo density
Fig. 7. The maximum von Mises stress distribution of the circular die structure design.
J.-J. Sheu, C.-H. Yang / Journal of Materials Processing Technology 177 (2006) 109–113
113
means the less importance of column. The ANSYS topology optimization result was shown in Fig. 5. The kept volume rate of the ANSYS topology optimization was 20%. Figs. 4 and 5 demonstrated the locations and shapes of the die structure designed by the proposed method were in good agreement with the ANSYS result.
the established simplified column model is capable of designing a reasonable die structure automatically. The comparison of the results of the proposed method and the ANSYS topology optimization demonstrated a good agreement.
3.4. The refined smooth die structure design and distress analysis
This work was supported by National Science Council of Taiwan and the JuiLi Company. The project grant number is NSC 92-622-E-151-005-CC3.
The design shown in Fig. 4 is in the zigzag profile of die geometry. The smoothing process was carried out to obtain a more reasonable die design and shown in Fig. 6. The smoother die design is in a circular shape. The die stress analysis was performed to verify the feasibility of design. The von Mises stress distribution of die was shown in Fig. 7. The maximum value 659 MPa is much smaller than the yield stress of the cast iron or steel. This is a safer and lighter design of die. 4. Conclusions The proposed die design method considered the importance of design parameters. The pseudo density of column model was proposed to evaluate the necessity of the supporting design. The pseudo density is also an evaluation function of design. A lower pseudo density can reduce the weight of the die but the buckling limit should also be considered. The design examples show
Acknowledgements
References [1] J. Kusiak, A technique of tool-shape optimization in large scale problems of metal forming, J. Mater. Proc. Technol. 57 (1996) 79–84. [2] F.K. Chen, B.H. Chiang, Analysis of die design for the stamping of a bathtub, J. Mater. Proc. Technol. 72 (1997) 421–428. [3] A.G. Mamalis, D.E. Manolakos, A.K. Baldoukas, Simulation of sheet metal forming using explicit finite-element techniques: effect of material and forming characteristics. Part 1. Deep-drawing of cylindrical cups, J. Mater. Proc. Technol. 72 (1997) 48–60. [4] Y.S. Shin, R.V. Grandhi, A global structural optimization technique using an interval method, Struct. Multidisc. Optim. 22 (2001) 351–363. [5] X. Guo, G.D. Cheng, An extrapolation approach for the solution of singular optima, Struct. Multidisc. Optim. 19 (2000) 255–262. [6] Y. Yang, C.K. Soh, Automated optimum design of structures using genetic programming, Comput. Struct. 80 (2002) 1537–1546. [7] N. Swaminathan, et al., Optimal topology design of products using genetic algorithm, Mater. Proc. Des. (2004) 2138–2142.