prediction of the localized necking phenomena based on theâ€¦

International Journal of Forming Processes. Volume 10 – No. 3/2007, pages 277 to 301

Prediction of the localized necking phenomena based on the MMFC and optimization of blankholder forces L. Ben Ayed* — A. Delamézière* — J.L. Batoz* — C. Knopf-Lenoir**

*Institut Supérieur d’Ingénierie de la Conception, Equipe de Recherche en Mécanique et Plasturgie 27 rue d’Hellieule, 88100 Saint-Dié-des-Vosges, France [email protected] [email protected] [email protected] **Université de Technologie de Compiègne, Laboratoire Roberval, FRE 2833 UTC-CNRS, BP 2052 60205 Compiègne Cedex, France [email protected]

ABSTRACT. Numerical simulation of deep drawing and optimization of some process parameters have been the topic of numerous studies by the authors during the last five years. In this work, an optimization procedure based on the response surface method (RSM) and on an adaptation strategy of the research space is proposed and used for the optimization of the blankholder forces design of a square box and a front door panel. In order to solve these forming problems and to predict undesirable defects, the software ABAQUS code was used. The optimization procedure was completely coupled with the finite element analysis to find the optimum forming parameters. The goal is to control the flow of the metal, by adjusting locally the blankholder force profiles during the forming process in order to improve the reliability of the process and the product quality (without wrinkling and fracture of the workpiece). To avoid the failure, a criterion of localised necking is used and tested on some applications. This criterion is based on work of Hora (Hora et al., 1996) and of Brunet (Brunet et al., 2001). KEY WORDS: Deep drawing process, optimization, blankholder force, necking, wrinkle.

DOI:10.3166/IJFP.10.277-301 © 2007 Lavoisier, Paris

278 International Journal of Forming Processes. Volume 10 – No. 3/2007

1. Introduction

Sheet metal stamping is widely used and is one of the most important manufacturing processes in the automotive industries. Great efforts have been made towards the development of new materials and of new technologies. According to a study of Kim (Kim et al., 2002), the manufacturing time in automotive stamping was shortened almost by half recently thanks to numerical simulations. Indeed, the lead time of a side panel outer die was reduced from 16.5 months to 8.5 months within two years and the quality was improved by 30%.

In the last years the international competition of the industries is extremely severe, all companies try to reduce manufacturing costs on one hand and to increase productivity, robustness of the forming process and quality on the other. High-strength steels have been developed recently, allowing to reduce considerably the sheet thickness and then the weight of car body.

Numerical simulations and mathematical methods of optimization ((Labergère., 2003), (Ohata et al., 2003), (Naceur et al., 2003), (Shi et al., 2004), (Jansson et al., 2005), (Forsberg et al., 2006)) are increasingly used to evaluate the forming difficulties in sheet metal forming and to achieve these goals. In sheet metal forming various and sometimes contradictory criteria must be satisfied, therefore different constraint and objective functions are necessary in order to obtain proper quality and to reduce the product cost. Good product quality is related with proper part characteristics: without wrinkling in the flange or on the wall of the part, without fracture of the workpiece and without geometrical defects due to springback, etc.

The following section is devoted to the prediction of necking phenomena. A mathematical criterion was studied and validated on two applications. The last part of this paper concerns the optimization of the blankholder forces in time and/or in space with two applications. An optimization methodology is proposed and coupled with a numerical simulation.

2. Localized necking criterion

When the deformations are sufficiently large, the material flow can become unstable. Then unacceptable and irrevocable defects can appear on the final part. The formability of the sheet is very difficult to quantify since it depends both on the material used and on the stress and strain states imposed upon the workpiece during the forming process. The formability can be defined as the ability of the material to be formed into a useful shape without defects.

Various approaches have been developed in the literature to predict localized necking. Most of them are used to calculate a Forming Limit Curve (FLC). For example, the perturbation method which was first introduced by Molinari to study the necking phenomenon (Molinari, 1985). This phenomenon was considered as an

Necking prediction and optimization of the BHF 279

instability of the local mechanical equilibrium. The linearised perturbations technique is used to detect this instability. It consists in the introduction of a small perturbation in the local equilibrium state at each integration point and to look for the development of an instability along a certain plane. The perturbation method is presented in (Boudeau et al., 1998). A necking criterion was proposed and used to predict necking during stamping process. This approach has been then used by (Lejeune et al., 2003) in hydroforming process. But they noted that this criterion is too severe and a positive constant value must be used as a tolerance parameter. Furthermore, the necking criterion value depends of the material parameters (Keltz, 2004) and on the orientation angles which define the plane band at the equilibrium.

Marciniak and Kuczynski (Marciniak and Kuczynski, 1967) have proposed another method, that we refer as MK method, by assuming the presence of an initial defect. This defect is characterised by a band in which the thickness of the sheet is smaller. The Forming limit condition is reached when the ratio of the strain outside the band to the strain inside is very low. The MK method was used and improved by several authors. Some of them have introduced and studied the effect of the damage on the forming limit curve.

In the following section, a necking criterion is proposed, implemented in the ABAQUS finite element code and tested on some sheet metal applications.

2.1. The modified maximum force criterion

The first theoretical analysis of the localized necking phenomenon was made by Considère (Considère, 1885). Based on the load/displacement curve of the homogeneous tensile specimen, the unstable condition is characterized by a stationary point where the load is maximal. Considère’s assumption of unstable plastic flow was then generalized by Swift (Swift, 1952) in the state of biaxial tensile stresses using the Von Mises yield criterion:

II

I σdεdσ

≤ [1]

Hora and al. (Hora et al., 1996) modified this unstable condition by considering that the state of deformation is plane at the moment of the localized necking appearance and that the maximum principal stress Iσ depends on the maximum principal strain Iε and of the rate of deformation:

I

IIdεdεβ = [2]

The unstable condition proposed by Hora is:

II

I

I

I σdεdβ

βσ

εσ

≤∂∂

+∂∂ [3]


This condition known under the name Modified Maximum Force Criterion, noted MMFC, is applied in relation with the Hill48 quadratic anisotropic plasticity criterion, which is used in the ABAQUS code. To take into account the orientation, noted φ , between the principal stress directions and the anisotropy axes, the yield function and the equivalent stress are expressed as :

22yeq σσf −= [4]

If 0<f then the solution is elastic.

If 0=f then the plastification has occurred.

where yσ is the current flow stress and eqσ is the equivalent stress:

232

21

2 2 IIIIIIeq σAσσAσAσ ++= [5]

where:

( ) ( ) ( )( ) ( ) ( )[ ] ( )( )

( ) ⎥⎦

⎤⎢⎣

⎡+

−+

+++

++

+=0

0

090

900452

090

900441 1

21

211

1r

rrr

rrrφ SinφCosrr

rrφSinφCosA [6]

( ) ( ) ( )( )

( )( )( ) ( ) ( )[ ]φSinφCos

rr

rrrrr

rrrrφ SinφCos A 44

0

0

090

90045

090

900222 11

211

11 ++

−⎥⎦

⎤⎢⎣

⎡+

++−

++

+= [7]

( ) ( ) ( )( ) ( ) ( )[ ] ( )( )

( ) ⎥⎦

⎤⎢⎣

⎡+

−+

+++

++

+=0

0

090

900452

090

900443 1

21

211

1r

rrr

rrrφ SinφCosrr

rrφCosφSin A [8]

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=2211

122atan21

σσσφ [9]

Using the normality law, equation [2] leads to:

αAAAαA

β21

23++

= [10]

where:

III σσα = [11]

The first and the second part of the equation [3] are written as follow:

( )

( ) ( ) ( )[ ]( ) I

I

I

I

I

I

εσ

AβAβ-AAAAβAAαAAβΨ

dεdβ

βσ

HΨαβεσ

232

12232132

1

−

+−+−=

∂∂

+=∂∂

[12]

where:


2321 2

1αA αAA

Ψ++

= [13]

H is the hardening function which has the following form:

eq

eq

εσ

H∂

∂= [14]

The hardening function depends of the yield function sign. It is equal to the elasticity modulus, E , when the yield function is negative.

According to the previous equations, the unstable condition [3] may be written:

( ) ( ) ( ) ( )[ ]( ) I

I

Icrn σ

εσ

AβA β-AAAAβAA αAA

βΨ HΨαβσ ≤−

+−+−+= 2

32

122321321 [15]

That expression is the basis of our MMFC.

2.2. Results and validation of the localized necking criterion

This section is devoted to the presentation of necking prediction obtained with the post-processor and the FLC based on the MMFC. The numerical simulations are carried out with the dynamic/explicit and/or static/implicit formulations of ABAQUS software (Hibbit et al., 2004). The equation [15] is used to define a necking indicator which is calculated at each step time (or increment) and for each element:

crnI

eln σσR = [16]

When elnR is higher than one, the behavior of the material is unstable and there is

a risk of failure.

The MMFC was also used to calculate Forming Limit Curves (FLC) (in a general case) without supposing that the strain paths, characterized by β , are linear or non-linear and that the principal stress directions coincides with the anisotropy directions.

In order to compare the two approaches, the principal logarithmic strains ( Iε , IIε ) corresponding to the critical area of the MMFC post-processing results, on which the el

nR is higher than one, are reported on the FLD.

2.2.1 Cylindrical cup

The geometry of the application is given in figure1. It concerns a steel with 221=E GPa and 3.0=ν . The coefficients of the Swift hardening law are:


729.09k = MPa, 1657.0=n and 01345.00 =ε . The Lankford coefficients are: 831.00 =r , 949.045 =r , 07.190 =r .

The failure was observed experimentally for a punch displacement of 22.5 mm (Arrieux, 1990).

Figure 1. Geometry parameters of the cylindrical cup

The distributions of the MMFC indicator on the external, average and internal, surfaces are shown respectively in figure2-a, figure3-a, and figure4-a. On the one hand, the maximum values of MMFC indicator are largely higher than one and thus there is a risk of failure. The critical stress which is equivalent to the diffuse necking is reached for a punch displacement of 17.2 mm. The largest maximal value is on the external surface and the weakest value is on the average surface. This result is completely coherent with the distribution of the maximum principal stress in a shell. On the other hand, the necking indicator distributions are not uniform since the material used is anisotropic. This result can be easily combined, for a complex geometry, with an optimization procedure, through a constraint or objective function, in order to find the best position of the blank or the optimum anisotropic coefficients allowing to improve the formability of the material and therefore the quality of the final part.

The critical logarithmic strains on the external, average and internal surfaces, where the MMFC indicator is higher than one, are compared to the FLC in figure2-b, figure3-b, and figure4-b. The Forming Limit Curves (FLC) was built by supposing that the strain paths are linear and that the principal stress directions coincide with the anisotropy directions. A good correlation is obtained between the critical logarithmic strains on the average surface and the FLC. But, some critical points on the external and internal surfaces are under the FLC although the necking indicator is higher than one.

mm 75R0 = ; mm 37.5R1 = ; mm 3.42R 2 = ; mm 7.8R3 = ; mm 3.6R 4 =


Figure 2. MMFC indicator distribution (a) and FLD (b) on the external surface after a punch stroke of 22.5 mm

Figure 3. MMFC indicator distribution (a) and FLD (b) on the average surface after a punch stroke of 22.5 mm

Figure 4. MMFC indicator distribution (a) and FLD (b) on the internal surface after a punch stroke of 22.5 mm

(a) (b)

Iε

IIε

(a) (b)

Iε

IIε

(a) (b)

Iε

IIε


2.2.2 Nakazima test

The Nakazima test is a typical test used to build the Forming Limit Curve (FLC). It consists in stamping blanks of different widths, in order to obtain strain paths for biaxial stretch, plane strain, deep drawing (in-plane pure shear) and intermediate strain paths, with a hemispherical punch in a cylindrical die (Knockaert, 2001). The widths considered in the present work are 110mm and 240mm which correspond to positive strain paths.

The material parameters of the aluminium sheet are: 70=E GPa, 3.0=ν , 1=oh mm (thickness), hardening law ( =eqσ 506.93(0.0245+ P

eqε )0.367 MPa), Lankford coefficient 45.00 =r , 85.045 =r , 35.090 =r . For this test drawbeads are present under the blankhoder to limit the flow of the material towards the die. These drawbeads are not modeled, they are replaced by zero in-plane displacement conditions. The simulations are carried out with the dynamic\explicit and static\implicit formulations of ABAQUS code.

Figure5-a and figure5-b, show the distributions of the necking indicator on the external surface obtained after a punch stroke of 40 mm for a width of 110 mm and which were carried out respectively using dynamic\explicit and static\implicit formulation. Satisfactory results are obtained showing good agreement with the experimental result (figure7-a).

Although the maximum values of the necking criterion at the end of the deep-drawing operation are slightly different, the critical stress was reached almost at the same punch stroke, 37.5 mm.

A similar analysis was carried out for a blank width of 240 mm and for a punch stroke of 45 mm where the failure was observed experimentally. The numerical results (figure6-a and figure6-b) obtained respectively using a dynamic\explicit and static\implicit formulations are in agreement with the experimental result (figure7-b) and prove that the MMFC criterion allows to predict the failure zone with precision.

Figure 5. Necking indicator distributions (a) and (b) after a punch stroke of 40 mm

(b) Static/Implicit

(a) Dynamic/Explicit


Figure 6. Necking indicator distributions (a) and (b) after a punch stroke of 45 mm

Figure 7. Experimental results for blank widths of 110 mm (a) and of 240 mm (b)

The results presented in this section show a good correlation between the numerical necking predictions and the experimental observations. The post-processing of the localized necking criterion enables us on one hand, to hold account of the real conditions of the forming operation (without supposing that the strain paths are linear or non-linear and that the principal stress directions coincide with the anisotropy directions) and on the other hand to follow the propagation of the necking phenomenon. The presented criterion could then be used to control a remeshing strategy during the simulation and then allows to delete element and to see failure bands appearing naturally.

(b)

(a)

(b) Static/Implicit(a) Dynamic/Explicit


3. Optimization of the blankholder forces

During the past 20 years, numerical simulation techniques of sheet metal stamping made continuous progress due to significant increase in computer hardware capacities and in improvements on the finite elements procedures.

Numerical simulations of the process can efficiently help to predict the behavior of the sheet by detecting defects like failure and wrinkling. The difficulties to achieve defect free parts in sheet metal stamping are highlighted by new materials such as dual phase steel or aluminium.

To obtain parts without defects several design variables can be optimized: sheet material properties and dimensions, blank holding force profiles and location, drawbead restraining forces, friction law coefficients, etc.

The achievement of a part could be obtained by controlling the material flow during the forming phase : in some areas the sheet should be almost fixed, in others it should be let free. That control is achieved through the blankholder pressure or the restraining forces of the drawbeads. Siegert, Häussermann and Haller ((Siegert, 2000), (Häussermann, 2000), (Haller, 2000)) proposed the design of a deformable flexible blankholder. A uniform pressure is applied on a classical blankholder, the force is imposed on all contact surface of the blank. For a well-designed deformable blankholder, different forces can be applied on different zones of the contact surface.

The aim of our work is to control the flow of the blank under the blankholder by a better regulation of the blankholder pressures. Too strong blankholder forces prevent the sheet from draw-in and may cause necking, but insufficient forces may lead to wrinkling.

3.1. Optimization problem

In sheet metal forming various and sometimes contradictory criteria must be satisfied, so several constraints and objective functions are necessary in order to obtain proper quality product. The optimization problem can be stated as follows:

( )

( )

( )( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≤≤≤≤=≤≤≥

: 1 01 0

:

vui

i

withsihrig

tosubjectJMin

P

xxxxx

x

[17]

where r and s are respectively the number of inequality, ( )xig , and equality, ( )xih , constraints, ( )xJ is the objective function and x represents the design

variables vector, which is defined as follows:


( ) ( ) ( ) ( ) ( ) T1 1 11 nsmcmcjins d, . . ., Fd, . . ., Fd, . . ., Fd, . . ., FdF=x [18]

where ( )ji dF is the value of the ith blankholder force at the jth punch stroke jd . mc and ns are respectively the number of the blankholder forces to optimize and the number of the punch displacement steps (or equivalently, the time), ux and vx are the lower and upper bounds vectors of the design variable x .

The objective function to minimize depends of the external work, w, and is expressed as:

minmax

minww

wwJ−

−= [19]

where minw and maxw are respectively the minimal and maximal external work.

In order to avoid necking and wrinkling three inequality constraints functions are formulated. The first function is built to avoid the formation of strong wrinkles under the blankholder zones. It is defined according to the angle of inclination θel between an element of the blank under the blankholder and the surface die (figure8):

( ) ( )( )

( ) ( )( )⎪⎪⎩

⎪⎪⎨

⎧ ≤∈∀=

∑

∑

∈

∈

p

n

Delel

Delelnel

else θ-θ

θ θD el if θ-θntg

sinsin

sinsin1

max

maxmax1 [20]

Dn and Dp are respectively the set of the total elements under the blankholder, and the set of the elements under the blankholder which have an angle of inclination higher than θmax at the end of the stamping operation. nt is the total elements number under the blankholder.

Figure 8. Geometrical criterion to detect the development of wrinkles under the blankholder surfaces

In the useful part of the workpiece a wrinkle criterion, proposed by Brunet, Batoz and Bouabdallah (Brunet et al., 1997) based on bifurcation plastic theory was used. For an element the criterion is defined by:

crwI

elwR σσ= [21]

Indeed, a second constraint function is then formulated according to elwR :


( )( )

⎪⎪⎩

⎪⎪⎨

⎧ ≤∈∀=

∑

∑

∈ wDel

elww

w

elw

elww

elww

u

else -RRn

R RD el if -RRn

glim

limlim

2 1

1

[22]

where wD is the set composed of the un elements in the useful part of the workpiece, wn is the number of elements which have a wrinkle criterion greater than lim

wR .

To avoid necking, the major stress of the blank is limited to a value which is determined using the modified maximum force criterion. A constraint function is then formulated according to el

nR :

( )( )

⎪⎪⎩

⎪⎪⎨

⎧

∑

∑ ≤∈∀=

∈Del

elnn

n

eln

eln

elnn

t

else -RRn

RD R el if -RRng

lim

limlim

3 1

1

[23]

D is the set composed of the total number of elements, nt, and nn is the number of elements which have a necking criterion greater than the imposed limit lim

nR .

3.2. Optimization procedure

The quality functions are implicit compared to the optimization parameters and their evaluations need a complete numerical analysis of the deep-drawing operation, which requires a large amount of computing time. In order to find the optimum parameters with a good precision and in a minimum CPU time, the Response Surface Method (RSM) is used and coupled with a strategy of actualization of the research space. The RSM allows to replace the initial optimization problem which is implicit by an explicit problem.

3.2.1. Approximation of the response surface

Various methods can be used to approach the nonlinear optimization problem associated to Eq [17]. Here the Moving Least Squares (MLS) method ((Villon, 1991), (Naceur et al., 2003), (Liew et al., 2004), (Zhou et al., 2006)) is used to obtain an explicit expression of the objective and constraints functions:

( ) ( ) ( ) ( ) ( )xaxpxxx T .apJm

iii == ∑

=1.~ [24]

where ( )xJ~ is the approximate value of ( )xJ , ( ) ( ) ( ) Tmp,,p ][ 1 xxxp K= is a

finite set of m polynomial basis functions, and ( ) ( ) ( ) Tma,,a ][a 1 xxx K= is the

unknown coefficients.


The MLS technique consists to determinate the coefficients allowing to reduce locally (in the vicinity of x) the gap, ( )xR , between the exact and the approximated values of the function:

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )[ ] ( )[ ]JxaPWJxaP

xxaxpxx

xxxxx

T

−−=

−−=

−−=

∑

∑

=

=

....

Jw

JJwR

T

N

1I

2III

N

1I

2III

.

~

[25]

The matrices J , P and W are defined as:

( ) ( ) ( )N21T xxxJ ,.....,J,JJ= [26]

( )( )

( ) m×⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

NNT

2T

1T

xp

xpxp

PM

[27]

( )

( )( )

( ) NNN

2

1

00

0000

×⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

−−

=

xx

xxxx

xW

w

ww

K

MMMM

L

L

[28]

where ( ) N,...,2,1II =x , are the N grid points in the research space, ( )Ixx −w is the weight function used in the MLS fitting which can take different forms as shown in ((Liew et al., 2004), (Zhou et al., 2006), (Ben Ayed, 2005)). The following weight function (Häussler-Combe, et al., 1998) is used:

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≤−⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−=−

−−⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

0

1 I

11.

I

22

2I

else

difeeewd xxxx

xx

ααα [29]

This function ( )Ixx −w is positive and its value decreases with the distance, Ixx − , between the sampling point x and the node Ix . It always takes unit value at the sampling point ( )Ixx = and zeros value outside the influence domain, which is defined by the value, d.

The parameter α allows to control the weight function curve in the domain of influence (figure9). When the value of α decreases, the influence domain in the MLS technique will be more concentrated around the region near the fitting point, x . The parameter α is fixed to 1/3.


Figure 9. Influence of the value of α on the MLS weight function.

The Euclidean norm is used to compute the weight function. Difficulties may occur for high dimensional problems, but in the present applications the dimension is lower than seven and moreover, the domain is localised (see section 3.2.2). No problem occurred in those cases, and the approximate function remains precise.

The minimisation of R(x) in the equation [25] compared to ( )xia leads to the following linear relation between ( )xa and J :

⎪⎩

⎪⎨⎧

=

=∂∂

JxBxaxAaR

).()().(

0 [30]

where the matrices )(xA and )(xB are defined by:

PxWPxA )()( T= [31]

)()( T xWPxB = [32]

The unknown coefficients )(xa can be achieved by solving equation [30], which results in:

JxBxAxa )( )()( -1= [33]

Substituting the unknown coefficient from equation [33] into equation [24] leads to the MLS approximation of the estimated value )(~ xJ as:

JxBxAxpx )(J - )()()(~ 1T= [34]

dIxx −

( )Ixx −w


3.2.2. Optimization strategy

An efficient optimization strategy using response surfaces has to manage several opposite objectives (figure11):

– to build precise approximations in order to reach a true optimum (defined by the exact functions) and if possible the global optimum,

– due to the high computational cost of simulation, use a low number of evaluation points.

The first step is a global one, using a Design of Experiments technique (DoE, (Goupy, 1999)) defined on the whole domain. The objective and constraints functions will be computed into a limited evaluation points number. Successive approximate problems are then solved, using an optimization procedure based on the RSM and an SQP algorithm, providing the optimum. After that, a new research domain, DJ, is defined around the optimum and the optimization procedure is repeated.

Since the MLS approximation is accurate locally, it’s possible that the SQP algorithm find a local optimum. To avoid this difficulty, a global approximation is made initially. The weight functions are equal to one. After that, successive local approximations are built, in the vicinity of the optima.

During the progression of the procedure, the region of interest moves and zooms on each optimum, k

optx , (figure10). The influence region is more and more concentrated around the optimum, k

optx , by decreasing the diameter, d, in the following way:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−⎟⎠⎞

⎜⎝⎛

+

≤−⎟⎠⎞

⎜⎝⎛

+=+

else 11

11

if

2

min

2

min

1

kkd

dk

kdd

d

k

k

k [35]

where mind is a minimal diameter to be respected.

The optimization procedure is stopped (figure 11, "condition n°1") when the number of approximations exceeds ten or when the optima of the last two approximations are very close [36]:

4kopt

1kopt 10−+ ≤− xx [36]

The number of the discrete points in the interest region is determined by the size of the influence domain, d, and it must be always higher than the unknown coefficients number. Since the MLS approximation is precise only in the vicinity of, x, the lower and upper bounds of each design variable, ix , are limited (figure 10-b) at each approximation as follows:


⎪⎩

⎪⎨⎧ >−−=

else if

min

min

i

iiiui x

xdxdxx [37]

⎪⎩

⎪⎨⎧ <++=

else if

max

max

i

iiivi x

xdxdxx [38]

where the scalars uix and v

ix are the actualized lower and upper bounds of the ith design variable, ix . min

ix and maxix are the lower and upper bounds of the DoE.

Figure 10. Progression of the optimization procedure for 2D case: Global (a) and successive local (b) approximations

The actualization of the research domain is stopped (figure 11,"condition n°2") when the size of the domain reaches a smallest step [39] or when the optima,

1JDopt

+x

and JD

optx , of the last two DoE are very close [40]:

nixxx ii , ... ,1 minminmax =∀∆≤− [39]

lxmin

Dopt

Dopt

J1J ∆≤−

+xx [40]

where minx∆ is the minimal step between the lower and upper bounds of the DoE and l is the minimal number of the design variable levels. It should be noted that the central composite design of experiment uses three levels for each variable.

The goal of this strategy is to improve the precision of the RSM in the vicinity of the optimum and to lead the optimization procedure to the global optimum with

(b) (a) Validity domain of the initial global approximation Validity domain of the first local approximation Validity domain of the second local approximation Validity domain of the kth local approximation


lower cost. A Python Script is used to create the ABAQUS finite element model, to run the computations and to post-process the results (extraction of the objective and constraints functions). They are exported in an ASCII file. This file is read by a Fortran program, in which the response surface approximation and SQP method are implemented. The methodology is shown in figure11-b.

Figure 11. The flowchart of the optimization strategy

3.2. Applications and results

The optimisation strategy described previously is applied on two deep drawing applications: a square box of Numisheet’93 (Nakamachi, 1993) and a front door panel benchmark of Numisheet’99. The material and process parameters used in these applications are given in Table 1.

The numerical simulations are carried out using the ABAQUS explicit code. These numerical parameters (mesh, punch speed) have been selected after several runs in order to evaluate their influences on the total CPU time and the precision. Good results have been obtained by comparison with both experimental and numerical results (Ben Ayed, 2005).

Evaluation of the objective function and constraints at the DoE points xI

Exploration of the research space

SQP Method

Upd

ate

of th

e re

sear

ch sp

ace

Global approximation

Condition n°1?

Condition n°2 ?No

Optimal solution

Yes

Yes

Initialization

No (a)

(b)

Loca

l app

roxi

mat

ion

(MLS

)


Table 1. Material and process parameters Square box Front door panel Young’s modulus (GPa) 71 220 Poisson coefficient 0.33 0.3 Thickness (mm) 0.81 1 Hardening law (MPa) ( ) 3593.001658.079.576 pε+ ( ) 214.000626.016.521 pε+

ro 0.71 1.73 r45 0.58 1.23 Lankford coefficients r90 0.7 2.02

Blankholder force (kN) 19.6 300

Three blankholder forces with two intervals of control are considered for the square box application (figure12-a). For the front door panel seven constant blankholder forces are defined (figure12-b). Thus, six and seven design variables have to be optimized, respectively, for the square box and front door panel applications.

Figure 12. Blankholder surfaces of the square box (a) and of the front door panel (b)

The same objective function [19] (external work) was used. Two constraints functions [20], [23] are considered for the square box in order to avoid the risk of failure and the wrinkles in the zone under the blankholder. For the front door panel benchmark, a third constraint function [22] was used in order to avoid wrinkles in the useful part of the workpiece.

A central composite design of experiment was adopted to define the design variables at each evaluation point xI. Indeed, for n independent variables, the central composite design requires 122 ++ nn function evaluations: n2 factorial designs augmented by n2 axial points and one centre point. The total number of numerical simulations to carry out at each design of experiments is thus equal to 77 and 120 respectively for the square box and the front door panel.

For the square box application the optimum solution is reached after two actualizations of the research space. In figure13, the optimum BHF profiles obtained

(a) (b)


for each blankholder are presented. Weak blankholding forces are applied at the beginning to allow more draw-in of the material. In the second stage high blankholding forces are applied in order to avoid wrinkling under the blankholder surfaces. After optimization, the maximum inclination angle is limited to 8° (15° before optimization). The indicator of necking el

nR is limited to 0.975. The total external work required by the forming operation was reduced by 110 J (-2.5%). The Forming limit diagrams before and after optimization are presented in figure14. After optimization all the points are below the FLC with a security margin S=0.075.

Figure 13. Optimum BHF profiles

The regulation of the blankholder forces during the forming operation allows to control and to improve the flow of the material between the die and the blankholder surfaces. This leads to an increase of the material formability, of the part quality and of the reliability of the forming process. It should be noticed that the material is the Aluminium proposed for Numisheet’93. A failure was experimentally observed for a punch displacement of 20 mm. Here the punch displacement is set to 25 mm and, thanks to the modification, no failure is detected.

Figure 14. Forming limit diagram

IIε

Iε

BH

F (k

N)

Punch displacement (mm)


For the industrial application (front door panel) the optimization results were obtained after three actualizations of the research domain. The optimum forces obtained for each blankholder are presented in figure 15.

The major and minor strain distributions before and after optimization are given in figure 16. All the points are below the FLC with a security margin (S=0.015). After optimization the maximum inclination angle is limited to one degree (five degrees before optimization). The indicator of wrinkling in the zone under the punch el

wR is limited to 4.5 and the indicator of necking el

nR is limited to 0.975. In figure17-a and figure17-b, the thickness distributions are plotted respectively for a total unique blankholder force equal to 300 kN and for the seven optimum blankholder forces obtained through the optimization procedure.

Figure 15. Optimum blankholder forces distribution (N)

Figure 16. Forming limit diagram before and after optimization

It should be noted that the sum of the optimum blankholder forces (351 kN) is slightly higher than the initial blankholder force (300 kN). However the total

IIε

Iε


external work required by the forming process was reduced by 1.6% (250 kJ). Furthermore, by adjusting the blankholder forces in space, the formation of strong wrinkles under the blankholder is limited. The minimum and maximum thickness is now equal to 0.788 mm (-21.2%) and 1.1 mm (+10.4%). The reduction of the total external work could be found small. But on the other hand it should be noted that the increase of the total blankholder forces did contribute to the disappearance of wrinkles leading also to a possible reduction of the wear under the blankholder surface.

Figure 17. Thickness distributions before (a) and after (b) optimization

4. Conclusions

The aim of this work is to make a contribution to increase the reliability and the product quality of the deep-drawing process by the optimization of the blankholder force in time and/or in space. A mathematical criterion allowing to predict localized necking was validated (MMFC) and implemented into ABAQUS to identify by numerical simulation the most critical areas.

(a)

(b)


Two optimization problems with constraints have been carried out using the response surface methodology and a strategy of actualization of the research space. An automatic optimization procedure is presented.

According to the optimization results, there are many advantages in using several blankholder forces compared to the conventional stamping process:

1. It is possible to form more complicated deep drawn parts;

2. With the capability to make more complicated shapes it is possible to reduce the number of stamping operations;

3. The ability of the deep drawing process is increased by reducing the forming cost;

4. The formability, the quality of the shapes and the productivity are improved.

Acknowledgements

We duly acknowledge the support of the Ministry of Research, France, (2002-2005 OPTIMAT RNTL project decision number 02 V0584).


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