Journal of Materials Processing Technology 80–81 (1998) 76–82

Three dimensional multi-step inverse analysis for the optimumblank design in sheet metal forming processes

C.H. Lee *, H. HuhDepartment of Mechanical Engineering, Korea Ad6anced Institute of Science and Technology, 373-1 Kusongdong, Yusonggu,

Taejon 305-701, South Korea

Abstract

An inverse finite element approach is employed for more capability to design the optimum blank shape from the desired finalshape. Since one-step analysis is not accurate enough to calculate the initial blank shape for complicated parts, multi-step analysisis introduced to obtain more accurate blank shapes and strain distribution. Multi-step analysis deals with the sliding surface forintermediate shapes and the coordinate transformation for appropriate Newton–Raphson iterations. The present algorithm hasbeen applied to square cup drawing and oil-pan drawing. The numerical results confirm the validity and versatility of thealgorithm with the initial blank shapes calculated and comparison to experimental results. © 1998 Elsevier Science S.A. All rightsreserved.

Keywords: Sheet metal forming process; Optimum blank; Multi-step inverse analysis

1. Introduction

Design of sheet metal forming processes needs toevaluate many process parameters such as the die ge-ometry, the blank shape, the sheet thickness, the blankholding force, the bead force, friction, lubrication, andso on. These process parameters have to be determinedfor the optimum forming condition before the processdesign. The determination still relies on the experience,intuition or the time-consuming computer analysis suchas incremental finite element methods for small modifi-cation after the process design. The determination ofblank shapes has been attempted in various methodssuch as the slip-line method [1,2], the characteristic ofplane stress [3,4], the geometric mapping [5], and in-verse finite element analysis [6–8]. The inverse finiteelement analysis is considered preferable in the estima-tion of process parameters.

The one-step inverse analysis, however, is not accu-rate enough to calculate the optimum blank shape sinceit causes some amount of error. The amount of errorgenerally increases as the deformation path becomesmore complicated. In this paper, multi-step inverseanalysis is introduced to obtain more accurate blank

shapes and strain distribution for complicated parts.The analysis has more capability to design the fullythree dimensional optimum blank shape from the de-sired final shape.

The present algorithm has been applied to square cupand oil-pan drawing in order to confirm its validity ofoptimizing the initial blank shape. The calculation di-rectly estimates a blank shape and strain distributionfrom the desired final shape. The developed algorithmis also capable of calculating the curved blank shapes asthe initial state of binder wrap condition for large scaleparts such as a fender or a rear quarter part.

2. One step inverse scheme

The inverse finite element method predicts an initialblank shape from a final deformed shape in one stepcalculation with aid of the deformation theory which inreality should be restricted to infinitesimal deformation.The problem reduces to minimize the potential energywhich relates the initial state to the final state. In theproblem, the given variables are the geometry of thefinal state and the thickness of the initial state, whilethe unknowns are the coordinates of the initial stateand the thickness of the final state. These unknowns are* Corresponding author.

0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.

PII S0924-0136(98)00178-2

C.H. Lee, H. Huh / Journal of Materials Processing Technology 80–81 (1998) 76–82 77

determined from minimization of the functional re-lated to the principle of virtual work as a function ofthe coordinates of the initial state [6–8].

2.1. Plastic deformation energy

The geometry of the initial and final state is dis-cretized by linear triangular membrane elements. Thelogarithmic strain components oi(X) are expressed asEq. (1). l1, l2 are defined by the principal values ofthe Green deformation tensor and the u designatesthe direction of the l1 with respect to the X coordi-nates.

[oi(X)]=ÃÆ

È

oxx

oyy

oxy

ÃÇ

É=ÃÆ

È

ln l1 cos2 u+ ln l2 sin2 u

ln l1 sin2 u+ ln l2 cos2 u

ln(l1l2−1) sin u cos u

ÃÇ

É(1)

The stress–strain rate relation, which is derivedfrom the associated flow rule and the Hill’s an-isotropic yield criterion for plane stress, is integratedfrom the initial state to the final state assuming thatdeformation follows the minimum work path [7].Then, the integrated relation is expressed as Eq. (2)which is equivalent to the Hencky’s deformation the-ory.

ÃÆ

È

sxx

syy

sxy

ÃÇ

É=

23

s

o

2+r1+2r

ÃÆ

È

1+r10

11+r

0

001ÃÇ

ÉÃÆ

È

oxx

oyy

oxy

ÃÇ

É(2)

where r in Eq. (2) is the Lankford value.When the domain of the final state is discretized by

finite elements, the sum of the plastic deformationenergy for each element can be calculated by Eq. (3).

Wp(X)= %E

e=1

&Ve

�& of

0

s : do�

dV

= %E

e=1

&Ve

�& of

0

23

s

oDaboa dob

�tdA (3)

where the stress and strain components are defined byEqs. (1) and (2), respectively. Eq. (3) shows the plas-tic work, Wp(X), is also the function of the nodalcoordinates X at the initial state.

2.2. Boundary conditions

While the boundary condition applied to the sheetsurface alters continuously during the deformation,the present analysis avoids a rigorous modeling of theboundary conditions. Instead, a simplified method

that considers the friction force, the blank holdingforce, and the draw-bead force is proposed in thefollowing form.

Friction force induced by the normal force is con-sidered for the punch and die interface surface. Thenormal force Fn at a node can be obtained from theequilibrium of sheet metal. The frictional work isthen written as follow:

Wp(X)= %E

e=1

&Ge

�& uf

0

tf · du�

dG

= − %node

�mFn tanh

�a

U U0

� U U

n· U (4)

U(X)= (x−X)− (ni · (x−X))ni (5)

where tf, m, Fn, U are the friction force, the frictioncoefficient, the normal force and the tangential dis-placement. a is a control number greater than 1 andU0 is the reference displacement averaged within eachelement. ni is the unit normal vector at the nodalpoint of the initial state.

The work done by the blank holding force couldbe evaluated at the nodes in the outer edge of ablank sheet.

Wb(X)= %Eb

e=1

&Ge

�& uf

0

tb · du�

dG

= − %node

�mFb tanh

�a

U U0

� U U

n· U (6)

where Fb is the blank holding force converted intothe nodal values at the outer edge.

The work done by the draw-bead force can beevaluated with similar concept as the blank holdingforce.

Wd(X)= %Ed

e=1

&Ge

�& uf

0

td · du�

dG

= − %node

�mFd tanh

�a

Ud U0

� Ud

Ud n

· Ud (7)

Ud(X)= [nd · (x−X)]nd (8)

where the draw-bead force Fd could be obtained fromeither the experiment or the numerical simulation.The unit vector nd toward the drawing direction isperpendicular to the bead line and tangent to diesurface.

2.3. Minimization of the functional

The principle of virtual work can be expressed asthe minimum of the functional as follows:

min. C(X)=Wp(X)−Wf(X)−Wb(X)−Wd(X) (9)

UR(X) (C(X)(X

=0 (10)

C.H. Lee, H. Huh / Journal of Materials Processing Technology 80–81 (1998) 76–8278

R(X) = %E

e=1

&Ve

s :(o

(XdV− %

E

e=1

&Ge

tf ·(u(X

dG

− %Eb

e=1

&Ge

tb ·(u(X

dG− %Ed

e=1

&Ge

td ·(u(X

dG

(11)

The minimization procedure adopts the Newton–Raph-son method as follows:�(R(X)(X

n(n)

{DX} [K(X)](n){DX}= −{R(X)}(n) (12)

X(n+1)=X(n)+b · DX (13)

3. Multi-step inverse scheme

Multi-step inverse scheme is employed to obtain moreaccurate blank shapes and strain distribution of fullythree dimensional problems. A similar numerical proce-dure for multi-step inverse analysis of axisymmetricproblems has been introduced by Majlessi and Lee [9].

In order to extend one step to multi-step inversescheme and to calculate the blank shape and intermedi-ate shapes of arbitrary curved surfaces, they need thelocal coordinates (X( i, Y( i) to define the constraint of thesliding surface for each nodal point and the materialcoordinates (S, T) to calculate the integration of theplastic deformation energy within each element asshown in Fig. 1.

Since the local coordinates are defined at the nodalpoints and the material coordinates are defined at theelements, the stiffness matrix K e and the residual forcevector F e of the plastic deformation energy calculated

with respect to the material coordinates should betransformed into the local coordinates for the Newton–Raphson iteration as shown in Eq. (14).

K( eTTK eT, F( e=TTF e (14)

T=ÃÆ

È

T1

00

0T2

0

00

T3

ÃÇ

É6×6

(15)

Ti=�cos(X( i, S)

cos(X( i, T)cos(Y( i, S)cos(Y( i, T)

n(16)

where K( e and F( e are the stiffness matrix and residualforce vector with the local coordinates. The subscript iof the Ti means the node number in an element.

The boundary stiffness matrix Kn and the boundaryresidual force vector Fn from the boundary conditionscan be constructed at the global coordinates as shown inEqs. (17a) and (17b). After evaluation in the globalcoordinates, Kn and Fn should be transformed into thelocal coordinates using the transformation matrix Q asshown in Eq. (18).

Fn= − %node

�F tanh

�a

U U0

� U U

n·(U(X

(17a)

Kn= − %node

(

(X�

F tanh�

a U U0

� U U

n·(U(X

(17b)

K( n=QTKnQ, F( n=QTFn (18)

The resultant matrix equation can be stated in the localcoordinates as Eq. (19).

[K( e+K( n]{DX( }= −{F( e+F( n} (19)

Then, the displacement in Eq. (19) is restricted withinthe sliding surface during the Newton–Raphsoniteration.

The procedure of the Newton–Raphson iteration ateach step in the multi-step inverse scheme has somedifferent features compared to that of the one stepinverse scheme as shown in Fig. 2. The most differentfeatures are the sliding surface, three kinds of coordi-nate, two kinds of transformation. After convergence ofnon-linear iteration for each step, the global procedureis carried out in the same manner with Majlessi andLee’s procedure.

4. Numerical results

The present algorithm described above has been im-plemented in a finite element code, CASHE-Inverse, andapplied to several examples of sheet metal forming. Thispaper concerns square cup drawing and oil-pan drawingin order to confirm its validity in optimizing the initialblank shape.

Fig. 1. Schematic description of the material and local coordinates ofan element used in the multi-step inverse scheme.

C.H. Lee, H. Huh / Journal of Materials Processing Technology 80–81 (1998) 76–82 79

Fig. 2. Procedure of non-linear iteration for each step in the multi-step inverse scheme.

Fig. 3. Desired mesh geometry and assumed sliding surfaces of asquare cup for multi-step inverse analysis: (a) desired final deformedshape, cup height=40 mm; (b) sliding surface, cup height=30 mm;(c) sliding surface, cup height=20 mm; (d) flat sliding surface for theinitial blank.

step inverse analysis is compared with that obtainedfrom one-step inverse analysis in Fig. 4. The blank sizeof three step inverse analysis is smaller than that of onestep inverse analysis in the direction of 45° and largerthan in the direction of 0 and 90° of square cupdrawing. After rigid-plastic incremental finite elementsimulation using the blank shapes obtained from thetwo inverse analyses, the flange contours are comparedin Fig. 5. The flange contour using the blank shapefrom three-step inverse analysis is in better agreementwith the desired contour than that of one-step analysis.Fig. 5 shows that three-step inverse analysis calculates

4.1. Square cup drawing

The tool geometry of a square cup drawing for abench mark test by Huh and Han [10] is adopted foranalysis. The material properties and the process vari-ables for the analysis of square cup drawing are asfollows:

Stress–strain curve s=54.5(0.004+o)0.263 kgfmm−2

Lankford value r=1.87Initial sheet thickness t0=0.8 mm

m=0.15Friction coefficientFb=40 kNBlank holding force

The geometry of square cup discretized by 1528 ele-ments and 813 nodes is shown in Fig. 3(a). Slidingsurfaces of Fig. 3(b)–(d) are assumed for three stepinverse analysis using punch and die geometry. Onestep inverse analysis, however, requires only flat slidingsurface of Fig. 3(d).

A calculated initial blank shape obtained from threeFig. 4. Comparison of initial blank shapes between the one stepinverse method and the multi-step inverse method.

C.H. Lee, H. Huh / Journal of Materials Processing Technology 80–81 (1998) 76–8280

Fig. 5. Comparison of flange contours among the desired CAD modeland the results from rigid-plastic FEM simulations using the blankobtained from the inverse method.

Fig. 7. Comparison of the thickness strain distributions among theresults from rigid-plastic FEM using the blank obtained from theinverse method and the results obtained directly from the inversemethod in the diagonal direction of square cups.

Figs. 6 and 7 show the thickness strain distributionobtained from inverse analyses and incremental analy-ses in the rolling direction and the diagonal direction,respectively. Figs. 6 and 7 demonstrate that the resultsfrom the three-step inverse analysis are in better agree-ment with those from incremental analysis than thosefrom one-step inverse analysis. However, the thicknessstrain distributions from three-step inverse analysis inthe wall and flange region still disagree with that ofincremental analysis, as shown in Fig. 7. The incrementof the number of step is required to consider moreaccurate deformation path and the boundary condi-tions, and improve the accuracy of the results frominverse analysis.

4.2. Oil pan drawing

In order to calculate the initial blank shapes for oilpan drawing, one- and two-step inverse analysis werecarried out and the results were compared with thatobtained from experimental trial and error method. Theoptimum blank shape for an oil pan in Fig. 8 has beenfound experimentally by Hahn [11].

The material properties and the process variables forthe analysis of the oil pan drawing are as follows:

s=794o0.292 kgfStress–strain curvemm−2

r=1.132Lankford valueInitial sheet thickness t0=0.975 mm

m=0.1Friction coefficientFb=19 kNBlank holding force

The finite element geometry of the oil pan is con-structed using CAD model of Fig. 8. The part domainis discretized by 3966 elements and 2063 nodes as

the optimum blank shape better than one-step inverseanalysis. The CPU time with HP715/75 workstationtook 163 s to calculate the blank shape by one-stepinverse analysis and 1120 s by three-step inverse analy-sis. The computing time is compared to the conven-tional finite element analysis. The rigid-plasticincremental analysis took 15788 s to calculate the flangecontour in Fig. 5. Although the computing time withmulti-step inverse analysis increases with increment ofthe number of step, the above results show that multi-step inverse analysis is more appropriate to calculationof accurate blank shapes than one step inverse analysisor incremental analysis.

Fig. 6. Comparison of the thickness strain distributions among theresults from rigid-plastic FEM using the blank obtained from theinverse method and the results obtained directly from the inversemethod in the rolling direction of square cups.

C.H. Lee, H. Huh / Journal of Materials Processing Technology 80–81 (1998) 76–82 81

Fig. 8. Geometric description of an oil pan.

Fig. 10. Comparison of initial blank shapes among the experimentand the two types of inverse analyses.

shown in Fig. 9(a). Intermediate sliding surfaces arealso assumed using CAD model of Fig. 8 and used fortwo-step inverse analysis.

Fig. 9(b) and (c) show the intermediate shape and theinitial blank shape after calculation of two-step inverseanalysis. The calculated initial blank shapes obtainedfrom one- and two-step inverse analyses are comparedwith the experimental results. The right hand side ofblank shape from two step inverse analysis is nearly the

same with the experimental result as shown in Fig. 10.The left hand side of the blank shapes from one andtwo inverse analysis have some difference from theexperimental one. Although there is small differencebetween the numerical result and the experimental one,Fig. 10 demonstrates that inverse analysis is valuablefor process design problems in sheet metal formingprocesses. The multi-step inverse analysis has moreversatility in calculation of the optimum blank shape.

5. Conclusions

A multi-step inverse finite element method has beendeveloped to improve the accuracy of the initial blankshape calculated and the strain distribution in sheetmetal forming processes. This method also calculatesintermediate shapes as well as the initial blank shapeusing assumed intermediate sliding surfaces. As thenumber of step is increased, this approach can be moreaccurate following more accurate deformation path andboundary conditions. The numerical results of squarecup and oil pan drawing demonstrate that the presentalgorithm improves the accuracy of the initial blankshape and the strain distribution with rapid computingtime.

References

[1] T. Jimma, Deep drawing convex polygon shell researches on thedeep drawing of sheet metal by the slip line theory 1st report,Jpn. Soc. Tech. Plast. 11 (1970) 653–670.

Fig. 9. Calculated intermediate shape and initial blank shape from thedesired mesh geometry of an oil pan: (a) desired final deformedshape, height=40 mm; (b) intermediate shape, height=20 mm; (c)initial blank shape.

C.H. Lee, H. Huh / Journal of Materials Processing Technology 80–81 (1998) 76–8282

[2] V.V. Hazek, K. Lange, Use of slip line field method in deepdrawing of large irregular shaped components, Proceedings of theSeventh NAMRC, 1979, pp. 65–71.

[3] J.H. Vogel, D. Lee, An analysis method for deep drawing processdesign, Int. J. Mech. Sci. 32 (1990) 891–907.

[4] X. Chen, R. Sowerby, The development of ideal blank shapes bythe method of plane stress characteristics, Int. J. Mech. Sci. 34(1992) 159–166.

[5] R. Sowerby, J.L. Duncan, E. Chu, The modeling of sheet metalstamping, Int. J. Mech. Sci. 28 (1986) 415–430.

[6] Y.Q. Guo, J.L. Batoz, J.M. Detraux, P. Duroux, Finite elementprocedures for strain estimations of sheet metal forming parts, Int.J. Numer. Methods Eng. 30 (1990) 1385–1401.

[7] K. Chung, O. Richmond, Ideal forming—I. Homogeneous defor-

mation with minimum plastic work, Int. J. Mech. Sci. 34 (1992)575–591.

[8] C.H. Lee, H. Huh, Blank design and strain prediction of automo-bile stamping parts by an inverse finite element approach. J. Mater.Process. Technol. 65 (1997) 645–650.

[9] S.A. Majlessi, D. Lee, Development of multistage sheet metalforming analysis method, J. Mater. Shaping Technol. 6 (1988)41–54.

[10] H. Huh, S.S. Han, D.Y. Yang, Modified membrane formulationconsidering bending effects in sheet metal forming analysis, Int.J. Mech. Sci. 36 (1994) 659–671.

[11] Y.H. Hahn, B.C. Jeong, J.H. Choi, An experimental study on thedeep drawing of a rectangular container with stepped bottom,Proceedings of the Korea Society of Mechanical Engineer, 1994,pp. 761–765.

.