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Chinese Journal ofAeronautics 23(2010) 260-267

Chinese Journal of Aeronautics

www.elsevier.com/locate/cja

Optimization of Preform Shapes by RSM and FEM to Improve Deformation Homogeneity in Aerospace Forgings

Yang Yanhui, Liu Dong*, He Ziyan, Luo Zijian

School of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an 710072, China Received 27 March 2009; accepted 29 July 2009

Abstract

This article, in order to improve the deformation homogeneity in aerospace forgings, proposes an approach that combines the finite element method (FEM) and the response surface method (RSM) to optimize the preform shapes. New expressions that take into account the influences of equivalent effective strain distribution are developed to evaluate the homogeneity of deformation distribution in aerospace forgings. In order to reduce the number of design variables, the domain-division method is put forward to determine the optimal design variables. On the basis of FEM results, the RSM is used to establish an approximate model to depict the relationship between the responses (deformation homogeneity and die underfilling) and the design variables represented by geometric parameters of the preform shape. With a typical aeroengine disk as an example, the proposed method is verified by achieving an optimal combination of design variables. By comparing the preform shape obtained with the proposed method to that with the existing one, it is evidenced that the former could achieve more homogeneous deformation in forging.

Keywords: response surface method; optimization; preform design; finite element method

1. Introduction

Homogeneous distribution of deformation is one of the key requirements for the aerospace forgings. This is especially true of the forgings made of difficult-to-deform materials, such as superalloys and titanium alloys, due to the rugged environment in which they work for the most time. Generally, ununiform distribution of deformation results from diversified forms of workpieces and dies and the constraints imposed by the friction between them. All these factors might cause inhomogeneous distribution of the thermome-chanical parameters (equivalent strain J, equivalent

strain rate e and temperature T ), and, in its wake, of the microstructures and mechanical properties in for-gings. Given some satisfied lubrication, improving the preform design might be one of the main resorts to homogenize deformation distribution apart from its contribution to lowering forming load, eliminating die underfilling and reducing die wear.

S. Kobayashi and his colleagues[1] firstly developed the backward tracing method in the 1980s for the preform design. Some works were done on it and several

*Corresponding author. Tel.: +86-29-88460545. E-mail address: liudong@nwpu.edu.cn

1000-9361/$ - see front matter © 2010 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(09)60214-4

techniques such as optimization[2] and inverse die contact tracking method[3-5] were introduced and used successfully in some simple forging processes. However, the effectiveness of the computed results significantly relied on the accuracy of the criterion of boundary nodes released from dies in the backward simulation. Accordingly, the preform optimization based on sensitivity analysis[6-8] was developed by incorporating the forward simulation with the optimization method. For a forging process, the optimization objective of this method was expressed as a function of design variables describing the preform shape through rigorous mathematical derivation. Several optimization objective functions had been proposed by researchers[6-9] to solve the problems such as die underfilling and control of forging quality. In order to overcome the difficulty to obtain the sensitivity information and simplify the iterative calculation, non-gradient optimization technique was introduced to the preform optimiza-tion[10-12]. With this method, finite element method (FEM) simulation was separated from the optimization program and only acted as a solver to calculate the objective function for each trial vector of design variables. Though the efficiency is improved to a certain extent, the procedure is still time-consuming due to frequent calculation of the objective function using FEM, especially in the cases with many design variables.

Different from the above-mentioned methods, response surface method (RSM)[13] combines mathematical and statistical techniques which employs an approximate model on the basis of function fitting concept to replace the accurate objective function during optimization. Therefore, the FEM simulation need not be performed during the optimization process and merely be used to calculate the response of the approximate model, by which the laborious computation can be greatly alleviated. Thanks to its practicality, RSM characteristic of high efficiency and easy operation has found wide application in a variety of industrial practices, such as chemical, semiconductor and electronic manufacturing, machining, and metal cut-ting[14-15]. In the field of metal sheet forming, many researchers applied RSM to solving the problems like cracks, wrinklings and springbacks[16-18]. Recently, Y. C. Tang, et al.[19] tried to use RSM to optimize the preform to solve the problem of die underfilling.

The present study is aimed to develop an approach to optimize the preform shapes with the help of the combination of RSM and FEM to improve the deformation homogeneity in aerospace forgings. Furthermore, the details of optimizing preform shapes are presented with a typical aeroengine disk forging as an example.

2. Brief Characterization of RSM

In general, the procedure of RSM consists of the following steps:

Step 1 Design of experiment (DOE) and implementation.

Step 2 Development of approximate model.

Step 3 Optimization of the approximate model involving design variables to obtain the maximum or minimum values of the response.

Step 4 Representation of the direct and interactive effects of the process parameters through two and three dimensional plots and determination of the optimized results.

In Step 2, the polynomial function is usually assumed to be the approximate model for the response surface in order to simplify the procedure. When the number of experiments is n, a second-order model can

be expressed by the following matrix equation:

Y = Xfi+£ (1)

y1 1 x11 x1m A S1

y2 , X = 1 x21 ' x2m , P = A ,£ = S2

_ yn _ 1 xn1 ' x nm Pn Sn

where Y is the response vector, X the design variable matrix, p the regression coefficient vector, s the random error term, and m the number of design variables.

In this method, the response concerned is taken as the objective with design variables as its influencing

factors. The goal of optimization is to find out the best combination of design variables for determining the optimal response characterized by a simple relationship between the response and design variables. It allows the users to judge the importance of each design variable to the response by checking the significance of desired terms using analysis of variance (ANOVA). Especially, a complex optimization problem with a lot of design variables can be dealt with RSM with great efficiency.

The method of RSM is well described in Ref.[13], to which readers may refer for detail.

3. Objective Function

3.1. Evaluation of deformation homogeneity in aerospace forgings

As a rule, the deformation homogeneity in forgings are evaluated[20] by

f1 =X (s> "£avg)2

/— F — F

2 max min

where is the equivalent strain of element i, eavg the

average of equivalent strain for all elements, N the total number of the elements, emax and are the

y iiidx. mill

maximum and minimum of the equivalent strain, respectively.

However, it is inadequate to evaluate the deformation homogeneity in forgings with f1 or f2 alone since the deformation homogeneity is often diverse even in the cases with different distribution modes of the equivalent strain but the same value of f1 or f2.

Thus, taking into account the distribution of the equivalent strain, the following equations are tenable.

'= - Y^i

Aë. =-

ei -F:

( j * i)

j=1,j*i

V( xi - xj)

2 + (y - y,)2

( j * i)

where d, is the distance between centroids of the two elements i and j; x„ yt and x,, yj are the coordinates of the centroid of the element i and j.

According to Eq.(5), the deformation homogeneity ^ is equal to the arithmetical average of Kf of all of the elements. Thus, Àëj is viewed as the deformation homogeneity of the element i.

In order to validate the contribution of the distribution of equivalent strain, the deformation homogeneity in a simple domain (a rectangular region of 3x3 unit-distance with mesh division of 3x3, see Fig.1) is analyzed. It is supposed that only one of all of the elements has the equivalent strain equal to 1; the others are equal to 0. Because of symmetry, the three cases

with different modes of equivalent strain distribution can be obtained and labeled as sn = 1 , s12 = 1

and ¿T22 = 1, respectively (see Fig.1).

0 Ü 0

{*)£u=\

{b)£l2=\

Fig.1 Three cases with different modes of distribution of equivalent strain in a simple domain.

Table 1 lists the evaluated results of the deformation homogeneity with Eqs.(3)-(7) for above-cited three cases. The results show that the deformation homogeneity calculated with Eq.(3) equals 0.89 for all cases, but 1.00 with Eq.(4). In contrast, the calculated deformation homogeneity is 0.13, 0.16 and 0.19 for three different cases by using the proposed Eqs.(5)-(7). This indicates that the larger the calculated deformation homogeneity becomes, the closer the element with an equivalent strain 1.00 to the center of the domain is. It can be also seen that the distribution of the equivalent strains does exert effects upon the deformation homogeneity in the domain which can be quantitated by the proposed Eqs.(5)-(7).

Table 1 Calculated results for three cases in Fig.1 with different equations

Case Eq.(3) Eq.(4) Eqs.(5)-(7)

1 0.89 1.00 0.13

2 0.89 1.00 0.16

3 0.89 1.00 0.19

3.2. Formulation of objective function

The deformation homogeneity calculated with Eqs.(5)-(7) in forgings is assumed to be the first response ^i. Furthermore, the filling ratio of die cavities, another important factor to evaluate the processing feasibility of forgings and their quality, is regarded as the second response ^ denoted by

a _ i _ contact

fc -1 a

where A is the total surface area of die cavities and Acontact the contact area between the workpiece and the die cavities.

Then, the objective function is formulated as follows:

¥ = pA + P202

where px and fr are weighting coefficients.

After trials and references to the earlier studies, different optimal shapes of a preform can be acquired through optimization in pursuit of the best deformation

homogeneity or the best filling ratio of die cavities alone. They are frequently inconsistent and the latter one is easy to achieve. As above-mentioned, p1 and p2 are determined as 0.7 and 0.3 since the main target of this work is to obtain forgings of improved deformation homogeneity.

4. Example of Preform Shape Optimization

Fig.2 shows the form of a typical aeroengine disk forging. It consists of three main segments—hub, rim and web—with significant differences in height. As it is very easy for inhomogeneous deformation to happen in it from the viewpoint of metal flow during forging, the disk is chosen as an example to show the preform shape optimization to improve the deformation homogeneity in forgings by using the proposed method.

Fig.2 Form of a disk forging.

4.1. Design variables

To describe a preform shape, the B-spline curve is often employed with the control points as design variables. However, there are two main problems confronted in operation.

(1) A large number of design variables lead to substantial increase in time for the numerical simulation.

(2) Highly complex shape possessed by the optimal preform described by the B-spline curve is usually quite hard to machine and often needs to be further simplified.

As a result, the domain-division method is proposed instead to describe the preform shape, in which the whole cross-section of the preform is divided into several parts, each represented by a simplified form such as a rectangle, a trapezoid and otherwise.

For example, the cross-section of the preform under study can be divided into three parts represented by rectangles and trapezoids (see Fig.3). The volume of each part is calculated by

V = no? h1 (10)

V2 = 2ka2 d2 = na2 (h1 + h3)(a1 + a2 / 2) (11)

V3 = 2nA3d3 = 2na3h3(a1 + a2 + a3 /2) (12)

v = v + V2 + v3 (13)

where V1, V2 and V3 are volumes of part 1, part 2 and part 3, respectively; A2 and A3 cross-section areas of part 2 and part 3; d2 and d3 distances from the centroids

of part 2 and part 3 to the symmetric axis; a1, a2, a3, h1 and h3 geometric parameters of the three parts.

Fig.3 Cross-sections of a preform.

In this way, the four design variables a, b, c and d are determined. The relationships between the design variables and the geometric parameters are given by

a = VJ V (14)

b = a1 (15)

c = h3 (16)

d = V3 /(V - V1) (17)

Then, the constrained optimization of the preform shape is defined as

Variables a, b, c, d Min y/(a, b, c, d )-

s.t. a > 0.1

0.7^ + 0.3^2

where the constraint means that the volume of part 1 is at least able to meet the requirement for locating the preform in the die cavity during forging.

4.2. Experiment and FEM simulation schedule

The experiments of FEM simulation were carried out on DEFORM 2D. The parameters for the FEM simulation are listed in Table 2.

A central composite design (CCD)[9] matrix with four factors and five levels was adopted in the experiment. The samples include one central point, 2k axial points and 2k vertices, where k is the number of factors

and equals 4 in the present work. Table 3 presents the levels of factors and Table 4 presents the 31 sets of coded conditions and raw experimental data including six repeatable ones. It shows that the results are identical without random error (s = 0) thanks to the absence of repeatability errors in numerical simulations.

Eq.(19)[21] defines the relationship between the natural variable and the coded symbol. It can be used to calculate the coded values of any intermediate levels.

(^max ^ ^min )

(^max ^min ) / 2

where ^ is the required coded value of a variable with value £ <fmin and <fmax are the lower and upper levels of the variable.

Table 2 Parameters for FEM simulation

Parameter

Workpiece

Material Temperature/K Density /(kg-m~ )

Heat conductivity/(W-m"1-K"1)

Specific heat /(J-kg"1-K"1) Ambient temperature/K Friction factor

Velocity of upper die/(mm-s-1)

Constitutive relationship of IN718 developed by A. J. Brand, et al.[22]

9.304+0.014 19T (T: temperature/K)

293 0.25 1

5CrNiMo 673 7 860

34.6 448

exp I 1 - —

(C is the work harderning exponent)

Table 3 CCD factor levels

Natural Coded Level(coded)

variable Symbol -2 -1 0 1 2

a 0.100 0.325 0.550 0.775 1.000

b/mm 13.0 15.5 18.0 20.5 23.0

c/mm £ 13.00 24.75 36.50 48.25 60.00

d £ 0.20 0.35 0.50 0.65 0.80

Table 4 Design matrix and experimental results

No. 6 (a) 6 (b) £ (c) £ (d) w

1 -2 0 0 0 0.349 9 0.217 8 0.310 3

2 0 2 0 0 0.461 3 0 0.322 9

3 -1 -1 -1 -1 0.494 3 0 0.346 0

4 1 1 -1 1 0.480 0 0 0.336 0

5 0 0 2 0 0.461 4 0 0.323 0

6 0 0 0 -2 0.564 9 0 0.395 4

7 0 0 0 0 0.561 9 0.234 3 0.463 6

8 -1 1 1 1 0.188 9 0.605 8 0.314 0

9 0 0 0 0 0.561 9 0.234 3 0.463 6

10 0 0 0 0 0.561 9 0.234 3 0.463 6

11 -1 1 -1 -1 0.176 8 0.717 4 0.339 0

12 0 -2 0 0 0.290 0 0.881 3 0.467 4

Continued

No. 6 (a) 6 (b) 6 (c) 6 (d) <h <h W

13 0 0 0 0 0.561 9 0.234 3 0.463 6

14 0 0 0 0 0.561 9 0.234 3 0.463 6

15 -1 1 1 -1 0.267 2 0.516 4 0.342 0

16 1 1 1 -1 0.538 1 0 0.376 7

17 1 -1 -1 -1 0.290 3 0.865 5 0.462 9

18 -1 -1 -1 -1 0.516 3 0.149 7 0.406 3

19 0 0 -2 0 0.505 3 0.104 2 0.385 0

20 -1 -1 -1 1 0.522 8 0.237 7 0.437 3

21 1 -1 -1 1 0.307 6 0.874 8 0.477 8

22 1 1 -1 -1 0.520 9 0.562 2 0.533 3

23 0 0 0 0 0.561 9 0.234 3 0.463 6

24 -1 1 -1 1 0.172 1 0.699 3 0.330 3

25 -1 -1 1 1 0.539 4 0 0.377 6

26 2 0 0 0 0.675 3 0 0.472 7

27 1 -1 1 -1 0.595 4 0 0.416 8

28 0 0 0 2 0.530 2 0.326 8 0.469 2

29 1 1 1 1 0.514 8 0 0.360 4

30 1 -1 1 1 0.462 5 0.551 0 0.489 1

31 0 0 0 0 0.561 9 0.234 3 0.463 6

4.3. Approximate model

Given complexity of the problem, the article decides to adopt a second-order analysis. In the case with four factors, the quadratic polynomial can be expressed as follows:

f = p0+№ + ++ +M2 +

^ + ^7#32 + A#42 + +

+ + + PlA^A (20)

In order to examine the fitness of the model and identify the significance of the terms, the ANOVA is conducted. Table 5 demonstrates the significance level of various terms. Table 6 evinces the ANOVA table of the response model. Some statistic data are included in Table 5 and Table 6 such as standard errors (S.E.) of different coefficients, T-value (the value of T-test), F-value (the value of F-test), the degrees of freedom of each source (DF), adjusted sum of squares (SSadj), adjusted squares of mean (MSadj), confidence probabilities P for two tests and PRESS (prediction sum of squares).

In general, the more appropriate regression model is, the higher the values ofRa2dj (R is correlation coefficient) and the smaller the values of S (standard errors of samples) are. Therefore, the fitting model can provide an adequate approximation as shown in Table 5. However, the value P of interaction terms ¿^x^

and #2x^3 is relatively high when the desired confidence is a = 0.05. It shows that these terms (labeled as "x" in Table 5) are insignificant for the response, which alludes to the possibility of neglecting them in the following analysis. With the least square method, the regression equation can be built up as follows:

Table 5 Significance levels of various factors

Term Coefficient S.E. T-value P

Constant 0.463 620 0.011 560 40.110 0.000

6 0.036 878 0.006 243 5.907 0.000

6 -0.032 138 0.006 243 -5.148 0.000

6 -0.017 674 0.006 243 -2.831 0.012

6 0.001 950 0.006 243 0.312 0.759

-0.017 479 0.005 720 -3.056 0.008

£ -0.016 564 0.005 720 -2.896 0.011

£ -0.026 854 0.005 720 -4.695 0.000

£ -0.007 275 0.005 720 -1.272 0.222

6*6 0.000 104 0.007 646 0.014 0.989

6*6 -0.002 096 0.007 646 -0.274 0.787

6*6 -0.009 519 0.007 646 -1.245 0.231

6* 6 0.000 590 0.007 646 0.077 0.939

6* 64 -0.025 005 0.007 646 -3.270 0.005

6*6 0.013 735 0.007 646 1.796 0.091

Table 6 ANOVA for response model (quadratic)

Source DF SSadj MSadj F-value P

Regression 14 0.110 896 0.007 921 8.47 0.00

Linear 4 0.065 016 0.016 254 17.38 0.00

Square 4 0.031 332 0.007 833 8.37 0.001

Interaction 6 0.014 548 0.002 425 2.59 0.060

Residual error 16 0.014 967 0.000 935

Lack-of-fit 10 0.014 996 7 0.001 497

Total 30 0.125 863

R2 88. 11% Radj 77.70

PRESS 0.086 211 2 S 0.030 585 2

f = 0.463 62 + 0.039 816^1 - 0.029 2^2 -0.014 752£ + 0.000 976^4 - 0.016 744£2 -0.015 829^2 - 0.026122£2 - 0.006 54^42 -0.013 908££4 - 0.029 394^4 + 0.009 337££4 (21)

Table 7 shows the ANOVA table for the reduced quadratic model. Compared to the corresponding statistical data in Table 6, the value of Rdj in Table 7 is

higher while the value of S is lower, which indicates that the reduced quadratic model is more appropriate. The value of P for the interaction term is 0.004 (a = 0.05), which indicates the noticeable significance of the remaining interaction term. The statistical results show that the approximate model Eq.(21) is adequate to predict the response to the design variables with an accuracy of 88.05%.

Table 7 ANOVA for response model (reduced quadratic)

F-value P

12.73 0.000

20.54 0.000

9.89 0.000

6.09 0.004

Source DF SSadj MSadj

Regression 11 0.110 789 0.010 072

Linear 4 0.065 016 0.016 254

Square 4 0.031 302 0.007 825

Interaction 3 0.014 471 0.004 824

Residual error 19 0.015 038 0.000 791

Lack-of-fit 13 0.015 038 0.001 157

Total 30 0.125 827

R2 88.05% Radj

PRESS 0.048 718 0 5

81.13% 0.028 132 7

4.4. Results and discussion

Fig.4 illustrates the distribution of the equivalent strain and the deformation homogeneity A^ calculated with Eq.(6) in the forging in No.27 experiment and Fig.5 those in No.16. It can be discovered from Fig.4(a) that, in No.27 experiment, the deformation occurring in the web seems larger than in the hub with an extreme difference of the equivalent strain over 2. Fig.4(b) evinces the rim possesses the highest deformation homogeneity A^ with a value of A^ ranging from 0.20 to 0.45. The worst deformation homogeneity takes place in the hub with the value of A^ up to 0.95. No.16 experiment shows a trend similar to No.27 but having smaller values of equivalent strain and extending the region in which A^ =0.20-0.45 to the hub. Moreover, the maximum A^ is 0.825 in the hub in No.16 experiment. Thus, it is reasoned that No.16 experiment shows a better deformation homogeneity than No.27. The same conclusion can be drawn from Table 4, where the deformation homogeneity (fa calculated with Eqs.(5)-(7) is 0.595 4 for No.27 experiment and 0.538 1 for No.16 which appears somewhat lower.

Fig.4 Distribution of s and As in No.27 experiment.

Fig.5 Distribution of s and As in No.16 experiment.

From Eq.(21), it is observed that the objective response decreases with the increase in variable <f2 when and <f4 remain unchanged. The results from No.27 and No.16 experiments (see Table 4) displays that the response ^decreases from 0.416 8 to 0.376 7 with the variable <f2 increasing from -1 to 1 when 1, 1 and & = 1.

Figs.6-8 illustrate the 3D surface graphs and the 2D contours of the objective response y/. It is clear from Fig.6 that the objective response ^decreases with the decrease in variable ^ when <f2 and <f3 are at the middle level. When -1, the objective response declines to about 0.3.

Fig.6 3D surface graph and contour for the objective response as ^ and ¿4 varies (¿2=0, ¿3=0).

Fig.7 3D surface graph and contour for objective response as ¿2 and £4 vary (£=0, &=0).

Fig.7 shows the 3D surface graphs and contours of the objective response against <f2 and <f4 when ^1= 0, <f3 = 0 and Fig.8 those against <f3 and <f4 when <^= 0, <f2 = 0. As shown in Fig.7, the value of the objective response firstly increases followed by decreasing as values of <f2 and <f4 increase. It is found from Fig.8 that when <f3 > 0.4, the value of the objective response increases as <f4 increase. When £3> 0.75 and <f4<-1.25, the objective response decreases to about 0.3. Thus, the optimal range of objective response values can be set to be 0.2-0.4.

Fig.9 shows the optimizing chart obtained with the least square method, in which the objective response varies in line with the corresponding variables and the dash line represents the value of the desired objective response 0.3. With the help of the optimizing chart, a reasonable combination of design variables can be attained by taking into account the practical conditions such as the equipment capacity, manufacturing cost and so on.

Fig.9 Optimizing chart.

For example, assuming 0.3 as the desired value of the response objective, Fig.10 offers the optimal shape of the preform by resorting to the optimizing chart. The corresponding coded design variables are equal to -1.07, 0.50, 1.32 and 0.40 which have been transformed into geometric parameters (see Fig.(10) with the unit in mm) using Eq.(19) and Eqs.(10)-(18). Fig.11 shows the distribution of the equivalent strain and A^ in the forging with the optimal preform. The predicted response y/ is 0.322 7 from Eq.(21) and the homogeneity of deformation is 0.312 2 from FEM simulation.

Fig.8 3D surface graph and contour for objective response as ¿¡3 and £4 vary (£=0, &=0).

Fig.10 An optimal shape of preform.

(a) £ (b) A £

Fig. 11 Distribution of s and As with an optimal preform.

5. Conclusions

(1) The proposed Eqs.(5)-(7) are suitable for describing deformation homogeneity in forgings taking into account the equivalent strain distribution.

(2) The number of design variables can be reduced greatly with the domain-division method, in which a shape of preform is represented with a set of simple ones. In addition, the optimally-shaped preform can be easily machined without profile modification.

(3) The approximate model (Eq.(21)) developed with RSM is suitable for evaluating the deformation homogeneity with an accuracy of 88.05% if there is no die underfilling.

(4) The optimal range of design variables a, b, c and d and their optimal combination can be obtained by resorting to the optimizing chart.

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Biography:

Yang Yanhui Born in 1979, she received M.S. degree from

Northwestern Polytechnical University in 2006, and now is a

Ph.D. candidate at the same school. Her main research interest is material processing technology.

E-mail: yangyanhui@gmail.com