Optimal design of longitudinal conformal cooling channels in hot stamping tools

Applied Thermal Engineering 106 (2016) 1176–1189

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Optimal design of longitudinal conformal cooling channels in hot stamping tools Bin He, Liang Ying, Xianda Li, Ping Hu ⇑ School of Automotive Engineering, Dalian University of Technology, Dalian 116024, China

h i g h l i g h t s
A new longitudinal conformal cooling channel design method is proposed.
An integrated process is used to optimize the layout of cooling channels.
Longitudinal conformal design is meaningful only for the tool with bending shape.
Cooling performance after optimization is improved.
Optimal longitudinal conformal design does not depend on the initial design.

a r t i c l e

i n f o

Article history: Received 4 March 2016 Accepted 17 June 2016 Available online 18 June 2016 Keywords: Hot stamping tool Longitudinal conformal cooling channel Experimental design Response surface model Multi-objective optimization

a b s t r a c t A new longitudinal CCC (conformal cooling channels) design in a B-pillar tool is proposed in this study. Three kinds of design parameters, the radius (Rad) of cooling channels, the distance (H) from channel center to tool work surface and the ratio (rat) of each channel center, are defined in a parameterized CAD model with the new design. For optimizing the layout of the longitudinal CCC which is determined by the above design parameters, an integrated process established by multiple software platforms based on the response surface methodology and multi-objective optimization is put forward. A design matrix with 17 factors and 50 levels is generated through OLH (optimal Latin hypercube) method. The average temperature and temperature deviation of work surface are used to evaluate the cooling performance. Quadratic response surface models are established to describe the relationship between design parameters and evaluation objectives. The accuracy of fitting models is checked by error analysis. The layout of the longitudinal CCC is optimized to find the Pareto-optimal frontier which consists of some optimal combinations of design parameters. Besides, the influence of initial design is discussed and a novel manufacture method of hot stamping tools with longitudinal CCC is briefly introduced. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction To meet the expectations of safety improvement and vehicle lightweight for current automobiles, the hot stamping technology of high strength boron steel offers the possibility to produce structural components of the car body with complex shape, high strength and less springback [1,2]. In the hot stamping process, the boron steel blank is heated up to the austenitic temperature (about 900 °C) and then cooled rapidly at a cooling rate above 27 °C/s until the martensitic transformation occurs [3,4]. During ⇑ Corresponding author at: School of Automotive Engineering, Dalian University of Technology, Linggong Road 2#, Ganjingzi District, Dalian 116024, Liaoning Province, China. E-mail address: [email protected] (P. Hu). http://dx.doi.org/10.1016/j.applthermaleng.2016.06.113 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.

the serial production, it is necessary to maintain the hot stamping tool’s temperature below 200 °C to ensure the blank cooling and prolong the tool life [5,6]. It is reported that the water cooling is a method of high efficiency yet energy saving [7]. Water cooling can be classified into direct cooling (water spraying) [8], indirect cooling (internal water channels) [9] and hybrid cooling (water and die quenching) [10]. Generally, the indirect cooling arranged channels inside tool is the preferred method in industry. It is meaningful to explore the optimal cooling layout in the hot stamping tool to achieve superior cooling performance. Hoffmann and Steinbeiss [5] studied the straight hole design in the hot stamping tool and the evolutionary algorithm was adopted to optimize the cooling system based on thermal analysis and thermal mechanical analysis. Lei et al. [11] investigated the effects of pressure holding time and water velocity on cooling performance

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by CFX software, and the simulation results were properly verified by actual experiments. However, channels with straight design have to be placed as near to the tool work surface as possible for efficient cooling at the expense of strength reduced due to its shape restriction. Liu et al. [12] put forward a novel method to design and manufacture the channels which were pre-embedded in a casting mold with pre-bending conformal stainless pipes, and uniform temperature distributions of both tool and component were obtained. Lim et al. [13] determined the channel number and location on each 2D cross-section by energy balance principle and triangular method and then connected the channels’ profiles on all 2D cross-sections to realize uniform cooling. Hu et al. [14] simulated the quenching process of hot stamping tools with five different cooling configurations by CFD method, and the longitudinal CCC (conformal cooling channel) was proved as the optimal cooling design when the mass flow rate of inlet was high. In consideration of the high cost and complicated manufacture of the tool with conformal cooling channels, the conformal design was not widely studied and applied in hot stamping tools. But it has been reported in many literatures of injection mold. Sachs et al. [15] manufactured an injection mold with conformal cooling channels through three-dimensional printing process. Xu et al. [16] proposed a systematic and modular approach for conformal design which was determined by five criteria, including transient heat transfer, maximum distance from the mold surface to the cooling channel, pressure drop, temperature drop and mold strength. The conformal design reduced the production time by 15% and the component distortion by 9%. Park and Dang [17] presented a cooling configuration with an array of baffles to achieve the conformal cooling performance. Agazzi et al. [18] developed a robust methodology based on morphological surfaces to design a high efficiency cooling system. Li et al. [19–23] continuously put forward different mathematical methods, to name just a few, feature-based approach, graph traversal algorithm, part segmentation by superquadric fitting, configuration space method and C-space method to automatically search for the optimal cooling layout of injection mold. Wang et al. [24] also developed a program for the automating conformal cooling design in the basis of centroidal Voronoi diagram and geometric modeling algorithm. Au et al. [25] proposed a methodology called visibility-based cooling channel generation for automatic preliminary cooling channel design. Schieck et al. [8] pointed out two manufacturing methods including laser cladding and rapid tooling to substantiate the conformal cooling design. Ahn et al. [26] provided a state of the art introduction to the applications of laser assisted metal rapid tooling process in the manufacture of forming tools. Furthermore, the conformal cooling channel design especially serpentine design was also broadly applied in the cooling design of battery cells [27,28]. Jarrett et al. [27,29,30] integrated GAMBIT and FLUENT with a MATLAB optimization program package to optimize the cooling layout in battery cells. Wang et al. [31] created a quadratic response surface fitting model with sample data from Box-Behnken design and then optimized the regression model by constrained particle swarm optimization method. Lin et al. [32] established a thermal-fluid-mechanical coupled model to analyze the cooling performance and the deformation of tools by MPCCI (mesh based parallel code coupling interface). Lam et al. [33] explored an approach to optimize the cooling design and production process at the same time through an evolutionary algorithm. Although there have been some researches related to conformal cooling design in the field of injection molds and battery cells, few achievements are implemented in the cooling design of hot stamping tools. There is still room and potentiality for us to study the optimal design of conformal cooling channels in hot stamping tools. In this paper, a new longitudinal CCC design of a B-pillar tool insert is put forward. The radius (Rad) of the cooling channel, the

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distance (H) from the channel center to tool work surface and the ratio (rat) of each channel center are considered as the design parameters to build the parameterized CAD model. The parameterized CAD model and heat transfer FE (finite element) model are integrated by ISIGHT to execute the OLH (optimal Latin hypercube) experimental design and obtain the response surface models. The accuracy of fitting models is verified by the error statistical analysis. Taking the average temperature and temperature deviation of tool work surface as the optimization objectives, NSGA-II (Nondominated Sorting Genetic Algorithm), a multi-objective optimization method, is utilized to find the Pareto-optimal frontier which reveals the optimal parameters combination. Furthermore, the sensitivity of the initial design is discussed and an advanced manufacture combined ceramic core technology and casting technique of a B-pillar tool insert with longitudinal CCC design is briefly introduced.

2. Longitudinal CCC design A typical set of hot stamping B-pillar tool is illustrated in Fig. 1. The punch and die are the key service parts to form the component shape. The two fixed plates are used to fasten the punch and die on the press machine. The blank holder force is provided by several gas springs which are assembled on the binder. The stripper plate is used to assist the components after stamping to leave die cavity. The guideposts are installed to restrain the close direction and guarantee the precision. Compared with cold stamping tools, the biggest difference in hot stamping tools is the water manifold. The cooling water flowing into each punch and die insert is assigned by water manifold. From Fig. 1b, the inlet and outlet sinks are designed for each insert to ensure the water circulation. There are almost no bending characteristics on the work surface of the punch insert 1 and 3, but the surface of the punch insert 2 is curved obviously. Hence, the cooling design will become complex in the punch insert 2. Fig. 2 presents the traditional straight hole design and the longitudinal CCC design of the hot stamping B-pillar tool. The cooling water passes from the inlet on the manifold into the inlet sink on the tool insert, and then goes out from the outlet on the manifold. It is necessary to ensure that the water unobstructed since the insert 2 shares the same inlet and outlet with the other two inserts. The traditional method is to drill holes from face to face direction to connect the whole channel. By this method, the distance between the channel wall and tool work surface is non-isometric along the longitudinal direction. However, it can achieve the isometric design by the longitudinal CCC design which has identical direction with tool work surface. Take the punch insert 2 as an example, the longitudinal CCC [14] design process is described in Fig. 3. For simplicity, the structures such as inlet/outlet sink and other assembly holes which have negligible impacts on the design and analysis are not considered. First, splitting the tool work surface along the shape characteristic by several planes to obtain the intersecting lines of the tool work surface and the discrete planes. It is noted that the intersecting lines on the two edge surfaces are just the feature lines on them. Second, offsetting all intersecting lines on the corresponding discrete planes, and the offset value is set as a design parameter which stands for the distance from the channel center to tool work surface. According to the expected number of cooling channels, the same number of ratio points are seeded on each offset lines to act as the channel center. The location of each point on the offset lines is controlled by ratio value. These ratio values are defined as parameters which indirectly describe the distance of adjacent channels. Third, connecting the ratio points on the offset lines in sequence by tangent continuous splines to create the so-called

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(a) Assembly structure

(b) Punch structure

Fig. 1. A typical set of hot stamping B-pillar tool.

3. Method 3.1. Definition of design parameters

(a) Straight hole design

(b) Longitudinal CCC design Fig. 2. Straight (non-isometric) and longitudinal CCC (isometric) design of the hot stamping B-pillar tool.

conformal splines. Finally, using circular profiles on the edge surface to sweep and cut the tool insert along the conformal splines, and then the conformal channels are generated. The radius of the circle is also designated as design parameters. In order to adapt to the integrated optimization, the longitudinal CCC design process is converted to a parameterized model generation routine. The design parameters controlling the geometry shape and location of cooling channels in the routine are easy to be modified by the optimization program.

The definition of design parameters is depicted in Fig. 4. It is noted that Rad symbol denotes the radius of the cooling channel, H symbol refers to the distance from the channel center to tool work surface and rat symbol stands for the ratio of each channel center. Depending on the energy conservation theorem, eight conformal cooling channels are arranged inside punch insert 2. As a result, a total of 17 parameters including 8 radius values (Rad1Rad8), 8 ratio values (rat1rat8) and 1 distance value (H) are defined. It is worth mentioning that the longitudinal CCC design process including surface discretization, ratio points seeded, conformal splines created and conformal channels generated is written into a CAD routine file which is convenient to be invoked by the optimization program. Considering the design and assembly constraints, the design parameters including the radius, distance and ratio must be restricted to a range to avoid the intersection and interference. Whether in the experimental design or in the optimization process, design parameters always change within the range listed in Table 1. The type of H and Rad1Rad8 is integer, but rat1rat8 is real. 3.2. FE model building In the hot stamping process, the initial stamping temperature of hot blank is 800–850 °C and a stamping cycle is 10–14 s. The

Fig. 3. Longitudinal CCC design process.

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Fig. 4. The schematic diagram of design parameters.

Table 1 Range of design parameters. Design parameters

Lower value

Upper value

Unit

H

13

17

mm

Rad1Rad8

4

6

mm

rat1 rat2 rat3 rat4 rat5 rat6 rat7 rat8

0.05 0.203 0.313 0.423 0.533 0.643 0.753 0.863

0.127 0.237 0.347 0.457 0.567 0.677 0.787 0.95

– – – – – – – –

temperature of blank sheet is 100–200 °C when it is captured out of the stamping tool. Assuming that the heat is evenly absorbed by the punch and die, then after a time-average estimation, a constant heat flux of 190,000 Wm2 is prescribed on the tool work surface. So when only concerning the cooling performance of the hot stamping tool, the model can be simplified as transient heat transfer model with Neumann boundary condition [14]. 3.2.1. Governing equation According to the Fourier’s law and energy conservation theorem, the governing equation of transient heat transfer is given as,




 @ @T @ @T @ @T @T kx þ ky þ kz þ Q_ ¼ qc @x @x @y @y @z @z @t

ð1Þ

where q is density (kgm3), c is specific heat (Jkg1K1), kx , ky , kz refer to the thermal conductivity (Wm1K1) along the x, y, z direction of Cartesian coordinate system, Q_ is strength of heat source (Wm3), T is temperature (K), t is time (s). The Neumann condition on the work surface is described as,

@T @T @T  on the tool work surface nx þ ky ny þ kz nz ¼ q kx @x @y @z

stable range after several stamping cycles under the influence of the cooling channels. The better the cooling design, the lower the tool temperature and the less stamping cycles are needed to achieve stability. Therefore, the stable temperature distribution should be used to measure the stand or fall of cooling designs. As the hot stamping process mentioned above, assuming that the total time of a stamping cycle is 14 s and among that the stamping and holding time, feeding time and reclaiming time are 10 s, 2 s, 2 s, respectively. Generally, the temperature of the tool work surface will become stable after six stamping cycles. Therefore, six stamping cycles, i.e. the total time of 84 s, are taken into account in transient heat transfer simulation. The heat boundary condition in the entire hot stamping process is assumed as a constant heat flux of 190,000 Wm2 calculated by time-average estimation. Based on the time assumption, the heat flux of 190,000 Wm2 is imposed in the stamping and holding stage of 10 s. But during the feeding and reclaiming stages which are in total 4 s, the heat flux is regarded as 0 Wm2. As is seen in Fig. 5, a uniform heat flux load which complies with the cyclic heat flux boundary shown in Fig. 5b is prescribed on the tool work surface marked as red (Fig. 5a). Moreover, in order to avoid the abrupt change of heat flux load at different stages, the smooth step definition is used to ramp up or down smoothly from one amplitude value to another. As shown in Fig. 5c, the heat flux expressed as q between two consecutive data points ðti ; qi Þ and ðtiþ1 ; qiþ1 Þ can be calculated from the Eqs. (3) and (4) as follows. It is noted that the heat flux curve related to time is tangent continuous smooth after using this method. Although the smooth characteristic of the heat flux load is not evident in the entire process shown in Fig. 5b, the influence on the stability of convergence should not be underestimated.

q ¼ qi þ ðqiþ1  qi Þn3 ð10  15n þ 6n2 Þ

ð3Þ

n ¼ ðt  t i Þ=ðt iþ1  ti Þ

ð4Þ

3.2.4. Forced convection boundary The other important boundary condition is the heat transfer coefficient on the cooling channel wall which is caused by the forced convection of cooling water. The Reynolds number has significant influence on the CHTC (convective heat transfer coefficient), and it is dependent on the hydraulic diameter which is directly determined by the geometry shape and size of cooling channels. It means that CHTC will change when the channel diameter and length change. Depending on the theory of similitude, the CHTC is related to the Nusselt number (Eq. (5)) and it can be calculated from Gnielinski equation which has not only considered the effects of large temperature difference between the channel wall and the cooling water, but also the limitation of l=d ratio,

h¼

ð2Þ

where nx , ny , nz stand for the direction cosine of outward normal on boundary.

kf Nuf d

ð5Þ

Gnielinski equation; Nuf ¼

"
2=3 # ðf =8ÞðRe  1000ÞPrf d pffiffiffiffiffiffiffiffi 2=3 1þ ct l 1 þ 12:7 f =8ðPrf  1Þ ð6Þ

3.2.2. Physical property In this research, the tool material is H13 steel and its material property, initial temperature condition and mesh property are listed in Table 2. It is noted that the thermal conductivity is dependent on tool temperature. 3.2.3. Heat flux boundary (Neumann condition) The tool temperature rises up with the increment of stamping cycles in the practical production and it will fluctuate within a


ct ¼

Pr f Pr w

0:01 ;

Prf ¼ 0:05  20 Prw

Filonenko equation; f ¼ ð1:82 lg Re  1:64Þ

ð7Þ 2

ð8Þ

where kf is thermal conductivity of cooling water, l is channel length, d is channel diameter, f is Darcy resistance coefficient of turbulent flow, Prf is Prandtl number calculated at the average

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Table 2 The material property, initial temperature condition and mesh property of FE model. Material parameter

Symbol

Value

Unit

Density Thermal conductivity

k

q

7900

kgm3 Wm1K1

Special heat Initial tool temperature

c T0

k ¼ 3  105 T 2 þ 5:7  103 T þ 42:624 450 20

Mesh type

DC3D4: a 4-node linear heat transfer tetrahedron.

Jkg1K1 °C

Fig. 5. The heat flux boundary on the tool work surface.

temperature of cooling water, Pr w is Prandtl number achieved at the average temperature of channel wall. In this paper, Pr f ¼ 7:0, Prw ¼ 1:95. In the basis of engineering experience, the radius of cooling channel is controlled in a range from 4 mm to 6 mm. Moreover, it is measured that the length of cooling channels in the punch insert 2 is between 180 mm to 260 mm. Hence, in order to simplify the boundary condition, an average CHTC constant of 5417 Wm2K1 calculated by Eqs. (5)–(8) is set to replace the CHTC value changing with the radius and length of cooling channels.

3.3. Experimental design by using Optimal Latin Hypercube method 3.3.1. Model verification Generally, the accuracy of FE analysis increases with the increase of element number. However, computing expense should also be considered, especially for optimization which requires hundreds of optimal computations. Therefore, a mesh independence study is performed to ascertain a mesh size that would achieve convincible results at an acceptable computing time.

Before the model verification, two evaluation indicators are defined. They are: average temperature Tave and temperature deviation Tdev of the tool work surface. In order to quantify these indicators, they are expressed by

Pn T t At Tav e ¼ Pt¼1 n t¼1 At rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xn Tdev ¼ ðT t  Tav eÞ2 t¼1 n1

ð9Þ

ð10Þ

where T t and At refer to node temperature and element area on the tool work surface respectively, and n denotes element number. Taking the punch insert 2 which is more sensitive to the design structure as an example, the same material property and boundary conditions are set and different models with a mesh range of 2–25 mm are simulated. The relative error of tool work surface’s temperature and the computing time for those FE models with different mesh base size are shown in Fig. 6. The Re-Tave and Re-Tdev in Fig. 6 stand for the relative error of average temperature and temperature deviation on the tool work surface. Comparing the model with 2 mm mesh base size, it is

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Fig. 6. Analysis of the relative error and computing time for models with different mesh base size.

demonstrated that Re-Tdev increases with the increase of mesh base size, but the Re-Tave presents random fluctuation. It is not surprising that the computing time reduces with the increase of mesh base size and the computing efficiency is improved sharply for the mesh size of 2–4 mm. Based on the mesh independence analysis, it is easy to find that the optimal mesh base size is 8 mm for the present problem. The Re-Tave, Re-Tdev and computing time at this mesh base size are only 2.3%, 2.5% and 382 s respectively.

transfer the data information among the CATIA, ABAQUS and data process model. The workflow is shown in Fig. 7. The parameterized CAD model is created by CATIA, and then the CAD routine is mapped to ABAQUS. Next the temperature data simulated by ABAQUS are extracted to perform the data process. The response, i.e. evaluation indicators are solved in the Calculator. According to the design matrix including 17 design parameters and 50 sample points, the workflow will auto run 50 times in total and the response values corresponding to the 50 sample points are listed in Table 4. These sample points will be used to determine the response surface model which can describe the characterization of the mathematical relationship between the design parameters and responses. It is noted that Tave and Tdev are different for different designs. In order to show more images to design differences, two different designs are illustrated in Fig. 8. Obviously, in compliance with the principles of the longitudinal CCC design, the channels’ shape size and location of Des.3 and Des.47 are different. 3.4. Response surface model establishment Response surface methodology (RSM) is a statistical and mathematical technique for relationship evaluation between design parameters and responses based on the Taylor formula’s expansion. The quadratic polynomial model can be expressed as,

y ¼ b0 þ

X

bi xi þ

3.3.2. Design matrix Design of experiment (DOE) is an approach widely applied in industry that can reveal the relationship of design parameters and responses using a small amount of sample points. The OLH (optimal Latin hypercube) is a method of DOE in which the number of levels of each factor is equal to the number of design points evenly spread within an n-dimensional space defined by the n factors through combination optimization. A design matrix with 17 factors i.e. design parameters and 50 levels displayed in Table 3 is generated by the OLH method. Due to the space limitations, only some design points are listed in Table 3. The range of the design parameters is placed in Table 1. 3.3.3. Design results The sample points in the design matrix are executed automatically by an integration flow established by ISIGHT which can

Table 3 Design matrix generated by the OLH method. Design parameters

Des.1

Des.2

Des.3Des.48

Des.49

Des.50

H

14

14

...

15

14

Rad1 Rad2 Rad3 Rad4 Rad5 Rad6 Rad7 Rad8

5 4 5 6 5 5 5 5

4 5 5 6 4 5 5 4

... ... ... ... ... ... ... ...

5 5 5 5 6 4 4 5

6 5 6 5 5 6 6 5

rat1 rat2 rat3 rat4 rat5 rat6 rat7 rat8

0.0516 0.208 0.324 0.429 0.563 0.661 0.785 0.8648

0.1066 0.209 0.343 0.424 0.543 0.661 0.774 0.9269

... ... ... ... ... ... ... ...

0.1019 0.211 0.332 0.450 0.534 0.674 0.756 0.9447

0.1003 0.207 0.340 0.440 0.564 0.665 0.754 0.934

X

bii x2i þ

i–j X bij xi xj þ e

ð11Þ

Fig. 7. Integration workflow of the OLH process.

Table 4 Response values corresponding to the 50 sample points. Sample points

Tave

Tdev

Sample points

Tave

Tdev

Des.1 Des.2 Des.3 Des.4 Des.5 Des.6 Des.7 Des.8 Des.9 Des.10 Des.11 Des.12 Des.13 Des.14 Des.15 Des.16 Des.17 Des.18 Des.19 Des.20 Des.21 Des.22 Des.23 Des.24 Des.25

71.93 70.95 85.95 83.38 78.80 79.10 71.57 70.26 73.60 72.98 72.48 76.13 84.08 82.31 71.00 77.89 79.48 71.12 83.65 85.44 71.12 85.23 67.80 82.14 80.80

11.97 12.18 17.63 16.89 15.23 15.85 11.15 12.02 11.95 14.48 13.92 13.55 16.16 15.66 13.12 12.65 14.32 12.69 15.80 17.29 13.05 15.03 10.89 16.18 16.00

Des.26 Des.27 Des.28 Des.29 Des.30 Des.31 Des.32 Des.33 Des.34 Des.35 Des.36 Des.37 Des.38 Des.39 Des.40 Des.41 Des.42 Des.43 Des.44 Des.45 Des.46 Des.47 Des.48 Des.49 Des.50

75.94 78.09 86.38 75.98 66.74 78.56 65.91 66.74 63.15 75.41 84.59 85.82 70.63 74.66 81.59 68.84 78.89 68.64 77.12 73.15 70.35 65.70 70.21 75.63 70.43

14.78 14.90 17.32 14.59 10.61 14.56 10.48 11.50 10.08 14.78 15.26 17.53 13.38 13.95 14.81 10.60 15.62 10.39 14.05 12.57 13.00 9.87 13.51 15.12 11.09

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Des. 3

Des. 47 Fig. 8. CAD models of Des.3 and Des.47.

where b0 is constant term, bi is linear coefficient, bii is quadratic coefficient, bij is interaction coefficient, i, j are the number of design variables, and e is statistical error. In this study, xi refers to the design parameters of H, Rad1Rad8 and rat1rat8, and y refers to the response values. The key problem of establishing the response surface model is to solve a series of coefficients including b0 , bi , bii and bij . Meanwhile, the error analysis must be performed to verify the accuracy of fitting models. 3.5. Multi-objective optimization

Fig. 9. Optimization process.

Tave and Tdev are two conflicting responses. When the average temperature attains the minimum value, the temperature uniformity is more obvious, and vice versa. Therefore, the multiobjective optimization method which will compromise with the two objectives is utilized to explore the optimal layout of longitudinal CCC. The mathematical description of the optimization problem is given as,

Fig. 10. Flow chart of integration optimization.

B. He et al. / Applied Thermal Engineering 106 (2016) 1176–1189

Minimize Tav e Eq: ð9Þ

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4. Results and discussion

Minimize Tdev Eq: ð10Þ

:

Parameters range fH;Rad1  Rad8;rat1  rat8g abide by Table 1

4.1. Comparative analysis of traditional and longitudinal CCC design

NSGA-II (Non-dominated Sorting Genetic Algorithm), which is well-suited for highly non-linear and discontinuous design spaces, is used in the present study. By this method, each objective is treated separately. Standard genetic operation of mutation and crossover is performed on the designs. Selection process is based on two main mechanisms, ‘‘non-dominated sorting” and ‘‘crowding distance sorting”. The population size which controls the number of solutions generated at each iteration is set to 400 and the number of generations which controls the number of iterations that the algorithm will execute before termination is 40. The total number of function evaluations exhausted by the algorithm equals the product of the population size and the number of generations. The process model created in ISIGHT is shown in Fig. 9. The design parameters of Des.1 from the OLH design matrix are used to initialize the optimization model. To sum up, the whole optimization process of longitudinal CCC based on the integration of multiple software platforms can be shown in Fig. 10 as follows.

According to the description in Section 2, comparing with the straight hole design, longitudinal CCC design is an isometric design which can ensure the distance between the cooling channel center and tool work surface always equal along the longitudinal direction to realize the uniform cooling performance. To reach this conclusion, two simulations using two B-pillar punch models with straight hole design and longitudinal CCC design are carried out respectively. The material property, boundary condition and mesh property have been introduced in Section 3.2. The temperature distributions of two models including the tool work surface and cooling channel wall are illustrated in Fig. 11. Obviously, the heat always forms ‘‘hot spot” at round corners and edge corners for any type of design due to the fact that the heat dissipated area of convex shape is not sufficient. From Fig. 11a, it is noted that the heat is significantly concentrated on the round corner at convex position of insert 2. Moreover, a low temperature area caused by non-isometric cooling design presents clearly on the insert 2 and it will reduce the temperature

(a) Temperature distribution of straight hole design

(b) Temperature distribution of longitudinal CCC design Fig. 11. Cooling performance of a B-pillar punch with straight and longitudinal CCC design.

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uniformity. Similarly, the heat is also gathered at the round corner when using the longitudinal CCC design (Fig. 11b), but the temperature uniformity of insert 2 has been improved. From Fig. 11, it is easy to find that when the longitudinal CCC design is applied, the Tave decreases from 77.5 °C to 74.3 °C, and Tdev from 20 °C to 15.5 °C respectively. Besides, the maximum temperature of the channel wall also decreases from 75.9 °C to 71.9 °C and it is helpful to improve the cooling efficiency of unit power. Therefore, the improvement of cooling performance is confident with the longitudinal CCC design. For further study, the independent analysis of single insert’s cooling performance is performed. As is shown in Fig. 12, despite the different cooling designs are also applied in the insert 1 and 3, but the temperature distributions on the work surface of them are almost no difference. It means that the longitudinal CCC design is meaningless for the tool model with no bending shape. However, from Fig. 13a, it is clearly noted that the longitudinal CCC design has a great improvement on the cooling performance of the insert 2. Not only the maximum temperature (Tmax) of the tool work surface after six hot stamping cycles drops to 108.8 °C from 149.2 °C, but also the temperature deviation (Tdev) reduces to 13.9 °C from 24.3 °C. It is actually more intuitive that the hot spot area at convex round corner of the tool work surface takes place when using the straight hole design, but disappears when applying the longitudinal CCC design. The temperature-time curves of a specific node located at the hot spot area in two designs are monitored in Fig. 13b. As it turns out, the node temperature of two designs reaches fluctuation stability when six hot stamping cycles are finished and the speed towards stability of the longitudinal CCC design is faster than the other design after the first cycle. During the six stamping cycles, the maximum temperature of the straight hole design will reach up to about 180 °C, but that of the longitudinal CCC design is less than 140 °C. It means a 20% reduction of the maximum temperature once the longitudinal CCC design is applied in the tool. Therefore, it is certain that the longitudinal CCC design makes more contribution to the cooling performance of hot stamping tool. The present study will focus on how to obtain the optimal layout of longitudinal CCC design.

4.2. Response surface model fitting and error analysis Depending on the design parameters in Table 3 and the responses in Table 4, the regression response surface models of Tave and Tdev computing by ISIGHT are expressed as Eqs. (12) and (13). Both of the two mathematical models described Tave and Tdev contain 25 terms. The Tave model is composed by 1 constant term, 9 linear terms, 6 quadratic terms and 9 interactive terms. The Tdev model is composed by 1 constant term, 8 linear terms, 9 quadratic terms and 7 interactive terms. The positive coefficient is on behalf of the positive effect, whereas the negative coefficient stands for the negative effect.

Tav e ¼ 873:766  86:03H þ 9:04Rad1  0:37Rad2  1:08Rad5 þ 10:59Rad8 þ 137:11rat1 þ 6099:08rat4  5885:66rat5 þ 41:51rat7 þ 0:16H2  0:68Rad82  1526:11rat22  6939:08rat42 þ 4336:8rat52  662:83rat62  0:62H  Rad1  0:11H  Rad3  0:03H  Rad4  0:06H  Rad7  0:32H  Rad8  10:18H  rat1 þ 44:25H  rat2 þ 77:99H  rat5 þ 60:29H  rat6

ð12Þ

Tdev ¼ 919:38 þ 0:25Rad1  3:47Rad5  2:23Raad7 þ 1563:37rat2 þ 2451:49rat4  895:89rat5  2381:71rat6 þ 3201:53rat7  0:24Rad32 þ 0:21Rad72  245:01rat12  3609:53rat22  2775:08rat42 þ 675:75rat52 þ 1825:76rat62  1978:71rat72 þ 4:91rat82 þ 0:11H  Rad3 þ 0:22H  Rad5  0:04H  Rad6 þ 2:67H  rat1 þ 0:61H  rat3 þ 10:16H  rat5  7:76H  rat7

ð13Þ

The cross validation method is employed for error analysis. For cross validation error analysis, a number of data points will be removed from the sampling data set, one at a time. For each of

Fig. 12. Cooling performance of punch insert 1 and 3 with the straight and longitudinal CCC design.

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the removed points, the approximation coefficients will be recalculated, and the exact and approximate output values will be compared. The removed point is then put back into the data set and the next point is removed. In the whole process, the points are selected randomly. Here 25 points specified from the total 50 sampling points are used for the cross validation error analysis. Fig. 14 is plotted by the predicted value from response surface model on the horizontal axis and the actual value simulated by FE model on the vertical. Those red points are the 25 points used for error analysis. The 45 degree diagonal line named no error line means the predicted value and the actual value are exactly the same and no error for any point landed on this line. The average line refers to the average value of those 25 actual data. It is noted that both for Tave and Tdev, the validation points are scattered around the no error line. This indicates the response surface models are matched good and reliable, and the adjusted R square coefficient of Tave is 0.941 and Tdev is 0.954. The two models present a good characterization of the relationship between design parameters and response values in the limited range of design parameters shown in Table 1. 4.3. Pareto graph analysis

(a) Temperature distribution of the insert 2

The Pareto graph drawn with the percentage of influence on the horizontal axis and design parameters on the vertical reflects the contribution degree of all items in the fitting model for each response. Fig. 15 shows the Pareto graphs of Tave and Tdev. The red bar stands for a negative effect, and the blue bar represents a positive effect. It is noted that whether for Tave or Tdev, the design parameters of rat have the most significant effect. The effect of H is smaller than rat but bigger than all Rad variables. For Tave, rat4 and rat5 show a significant impact and the effect degrees reach up to 35% and 39% respectively. However, effect degrees of other parameters are all below 11%. For Tdev, effect degrees of rat7, rat4 and rat2 are 28%, 24% and 23% respectively. Similarly, effect degrees of other parameters are all under 11%. 4.4. Pareto-optimal frontier and feasibility analysis

(b) Node temperature-time curve Fig. 13. Cooling performances of punch insert 2 with the straight and longitudinal CCC design.

(a) Error analysis of Tave

A total of 16,000 evaluations are operated for objective functions. Due to the hot stamping simulation has been regressed as mathematical models by OLH and response surface method before optimization, the optimization efficiency is improved greatly and only 5 min is spent. A total of 329 Pareto-optimal solutions are obtained and the Pareto frontier results from

(b) Error analysis of Tdev

Fig. 14. Error analysis of Tave and Tdev.

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(a) Pareto graph of Tave

(b) Pareto graph of Tdev

Fig. 15. Pareto graph analysis of Tave and Tdev.

Pareto-optimal solutions which present a convex shape is displayed in Fig. 16. It is noted that after multi-objective optimization Tave will be controlled within the range of 54–58 °C, and Tdev is kept from 3.5 °C to 6.0 °C. The Pareto solutions are uniformly continuous and the trend line of these solutions is a quadratic polynomial. The point 1 marked as black in Fig. 16 is the optimal solution revealed by ISIGHT. The H, Rad1Rad8 and rat1rat8 of point 1 are, in order, 13, 4, 6, 6, 6, 6, 6, 6, 6, 0.127, 0.237, 0.313, 0.423, 0.562, 0.677, 0.753, 0.8631. According to the Pareto solutions, it can be found that design parameters of H and Rad1Rad8 tend to unique peak values, but rat1rat8 have more options. It is a further validation that design parameters of rat1rat8 have more effect on response values. In order to verify the accuracy of the optimization results, the design parameters of three points including point 1, 2 and 3 are taken back to CAD routine and FE model to carry out the hot stamping simulation separately. Tave and Tdev values getting from simulation and mathematical model are shown in Table 5. The results demonstrate that the optimization using regress models are not accurate enough comparing with direct simulation. But relatively speaking, Tave obtained by the model optimization, whose relative error is always less than 15%, is more accurate than Tdev. The relative error of Tdev is below 45%, which means that though the R square of Tdev model reaches up to 0.95, the design space is still not entirely covered by the regress model. By increasing sample points or adjusting the experimental design process, the prediction precision and optimization accuracy will be improved. Although the prediction error is inevitable when using the response surface model to perform optimization, this method has a high cost-effective compared to the direct simulation optimization which represents a high computational cost. The temperature distributions before and after optimization are illustrated in Fig. 17. The Des.1 listed in Table 3 is used as the initial

Fig. 16. Pareto frontier graph.

Table 5 Feasibility analysis of the optimization results. Point 1

Simulation Mathematical model Relative error

Point 2

Point 3

Tave

Tdev

Tave

Tdev

Tave

Tdev

63.5 54.7 14%

8.6 4.8 44%

62.1 54.5 12%

8.8 5.2 41%

61.0 54.4 11%

8.1 5.6 31%

design before optimization and the Point 1 is the optimal solution recommended by Pareto solutions in ISIGHT. Obviously, the temperature distribution including average temperature (from 71.93 °C to 63.55 °C) and temperature uniformity (from 11.97 °C to 8.6 °C) has been improved after optimization. Tave and Tdev are separately reduced by 11.7% and 28.2%. Moreover, the design parameters of cooling channels after optimized become more uniform and helpful to manufacture the channels. The whole points on Pareto frontier are the optimal combinations of design parameters, however, the design combinations which contribute to the convenience of processing and the low cost should be prioritized.

4.5. Effect of initial design In terms of different design problems, the optimization results may depend on the choice of the initial design. In the research of Jarrett, etc. [27], the Latin hypercube sample of eight initial designs was used for the optimizations of all objective functions. They found that although the optimization of the eight initial designs produce almost equivalent objective function values, their converged designs are in some cases quite different from each other. That means in the searching space there are many local minima which show almost identical performance. Therefore, it is meaningful to study the effect of initial design for the optimization of longitudinal CCC design. Eight designs chosen from the OLH design matrix are used as initial designs for the multi-objective optimization. These eight designs are listed in Table 6 and the Des.1 has been used in the previous research. After changing the initial design of the original optimization model and performing all the optimization processes, it is revealed that no matter what kind of initial design is chosen, the optimal designs recommended by Pareto solutions in ISIGHT are always the same with the optimal results which using Des.1 as the initial design. The optimal parameters combination of H, Rad1Rad8 and rat1rat8 are 13, 4, 6, 6, 6, 6, 6, 6, 6, 0.127, 0.237, 0.313, 0.423, 0.562, 0.677, 0.753, 0.8631. The optimal values of Tave and Tdev are 54.7 °C and 4.8 °C. It can be concluded that the optimal longitudinal CCC design obtained by the OLH experimental design,

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(a) Temperature distribution of initial design.

(b) Temperature distribution of the optimal design. Fig. 17. Temperature distribution before and after optimization.

Table 6 Eight initial designs for optimization. Design parameters

Des.1

Des.7

Des.15

Des.23

Des.30

Des.37

Des.44

Des.50

H

14

15

14

13

13

17

15

14

Rad1 Rad2 Rad3 Rad4 Rad5 Rad6 Rad7 Rad8

5 4 5 6 5 5 5 5

4 6 6 6 6 6 5 5

6 6 6 6 4 5 5 6

5 5 5 5 4 6 6 5

5 5 5 4 6 6 5 6

4 6 5 5 4 4 5 5

5 5 5 6 4 5 4 5

6 5 6 5 5 6 6 5

rat1 rat2 rat3 rat4 rat5 rat6 rat7 rat8

0.0516 0.208 0.324 0.429 0.563 0.661 0.785 0.8648

0.0877 0.215 0.339 0.426 0.553 0.655 0.757 0.8879

0.1097 0.218 0.322 0.434 0.538 0.669 0.773 0.8861

0.061 0.229 0.336 0.425 0.541 0.644 0.771 0.9216

0.1207 0.215 0.328 0.438 0.549 0.658 0.784 0.95

0.0704 0.222 0.339 0.444 0.540 0.648 0.775 0.9127

0.0956 0.234 0.321 0.448 0.559 0.644 0.760 0.9411

0.1003 0.207 0.340 0.440 0.564 0.665 0.754 0.934

response surface methodology and NSGA-II does not depend on the initial design. 4.6. Advanced manufacture method When adopting the longitudinal CCC design in hot stamping tools, how to realize the low-cost manufacture process is a tough issue. An advanced manufacture technology is proposed in this paper and its manufacturing process is displayed in Fig. 18. The key step is to use the ceramic core model by extrusion to realize the channels’ structure with conformal design. This advanced manufacture method is seen as a combination of ceramic core making and casting process. As is seen in Fig. 18, first, the sand mold cavity and ceramic core with longitudinal conformal shape are prepared.

Second, the cavity, core, runner and riser are assembled into a gating system. Third, the casting process and the heat treatment after casted are performed. The survival state of ceramic core should be paid close attention in the casting process to prevent fracture. Fourth, after the cast mold cooling down, the ceramic core wrapped inside mold is cleaned by alkaline corrosion with a high temperature caldron. Fifth, ultrasonic inspection is used to find the severe defects, and if there are no serious defects, the casting mold will be annealed and polished subsequently, otherwise further evaluation of that mold will be carried out. Sixth, to operate the surface finishing and to quench hardened the casting mold. Finally, after a dimensional accuracy test, a finished hot stamping tool insert with longitudinal CCC design is produced successfully. By this method, the tool insert with any conformal structure is

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Cavity manufacture

Ceramic core extrusion

Assembly Casting Ceramic core corrosion and clean up Ultrasonic inspection

Annealing and polishing Finish machining Heat treatment and precision test

Final product

Fig. 18. Manufacture process of hot stamping tools with longitudinal CCC design.

not hard to be manufactured. Further verification tests of temperature distribution will be carried out in our future studies. 5. Conclusions Comparing with the traditional straight hole design method, a new longitudinal CCC (conformal cooling channels) design in a Bpillar tool insert is proposed and the design process has been described in detail. The longitudinal CCC design can realize the distance between the channel center and the tool work surface is equidistant which provides a more uniform cooling performance. Three kinds of design parameters, the radius (Rad) of the cooling channel, the distance H from the channel center to tool work surface and the ratio (rat) of each channel center, are defined in a parameterized CAD model of the B-pillar tool insert with the new longitudinal CCC design. In order to optimize the layout of the longitudinal CCC which is determined by the H, Rad1Rad8 and rat1rat8, the OLH (optimal Latin hypercube) experimental design method, response surface methodology and multiobjective optimization technique are used in combination on the basis of an integrated optimization process by multiple software platforms. Due to the longitudinal CCC design process including surface discretization, ratio points seeded, conformal splines created and conformal channels generated is written into a CAD routine file, the design parameters can be invoked conveniently by the optimization program. A design matrix with 17 design parameters and 50 levels is generated according to the OLH (optimal Latin hypercube) experimental design method. The average temperature (Tave) and temperature deviation (Tdev) of tool work surface are defined to evaluate the cooling performance. According to the OLH design results, quadratic response surface models which describe the relationship between the design parameters and the response values are established. The accuracy of response surface models is verified by error analysis using cross validation method. And not only that, Pareto graphs of Tave and Tdev are also described to discuss the influence of design parameters. Then the layout of longitudinal CCC is optimized by a multi-objective optimization to find the Pareto-optimal frontier. Moreover, the effect of initial design is discussed and an advanced manufacture method of the hot stamping tool insert with longitudinal CCC design is briefly introduced.

For proving the superiority of the longitudinal CCC design, two FE simulations using two B-pillar punch models with straight hole design and longitudinal CCC design are performed. It is proved that the heat always forms ‘‘hot spot” at round corners and edge corners for any type of design due to the fact that the heat dissipated area of convex shape is not sufficient. However, the Tave decreases from 77.5 °C to 74.3 °C, and Tdev from 20 °C to 15.5 °C when adopting the longitudinal CCC design. Besides, the maximum temperature of the channel wall also decreases from 75.9 °C to 71.9 °C. Furthermore, the longitudinal CCC design is meaningful only for the tool model with bending shape like the punch insert 2. With regard to the insert 2, the maximum temperature (Tmax) of the tool work surface after six hot stamping cycles decreases from 149.2 °C to 108.8 °C and the temperature deviation (Tdev) reduces to 13.9 °C from 24.3 °C. It is also found that the speed towards stability of the longitudinal CCC design is faster than the straight hole design after the first cycle. During the six stamping cycles, the Tmax of the straight hole design will reach up to 180 °C, but that of the longitudinal CCC design is less than 140 °C. Both of the two response surface models of Tave and Tdev established by the OLH results contain 25 terms. According to the error analysis, the adjusted R square coefficient of Tave is 0.941 and Tdev is 0.954 which means the models are matched good and reliable. In the case of Pareto graphs of Tave and Tdev, for Tave, the effect degrees of rat4 and rat5 reach up to 35% and 39% respectively, and for Tdev, the effect degrees of rat7, rat4 and rat2 are 28%, 24% and 23% respectively. However, the effect degrees of other parameters are all below 11%. After the multi-objective optimization, a total of 329 Pareto-optimal solutions are obtained and the Pareto frontier results from Pareto-optimal solutions whose trend line is a quadratic polynomial. The H, Rad1Rad8 and rat1rat8 of the optimal design recommended by ISIGHT are, in order, 13, 4, 6, 6, 6, 6, 6, 6, 6, 0.127, 0.237, 0.313, 0.423, 0.562, 0.677, 0.753, 0.8631. For confirming the feasibility of the fitting models, the predicted values computed by fitting models and the actual values simulated by FE models are compared at three specified points. Tave obtained by the model optimization, whose relative error is always less than 15%, is more accurate than Tdev. But the relative error of Tdev is below 45%, which means that though the R square of Tdev model reaches up to 0.95, the design space is still not entirely covered by the regress model. However, this whole solution method has a high cost-effective compared to the direct simulation optimization which represents a high computational cost. Comparing the temperature distribution of the initial design and the optimal design, Tave and Tdev are separately reduced by 11.7% and 28.2%. The optimal longitudinal CCC design obtained by the whole design and optimization process does not depend on the initial design through the comparative analysis of eight initial designs. Finally, an advanced manufacture method combined the ceramic core making and casting process is described for the manufacture of cooling channels with any bending shape. For realizing the homogeneous cooling and low manufacture cost of hot stamping tools, the optimization and manufacture process of tools with the longitudinal CCC design is well worth studying. Though we have investigated this problem by the multiobjective optimization of the fitting models established according to the experimental design, there are some shortages of the fitting models used in this study. How to get more accurate mathematical models and plan the temperature proof test of the tools with longitudinal CCC design will be discussed in future researches.

Acknowledgements This research is financially supported by the Key Project of the National Natural Science Foundation of China (No. 11272075),

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