# Finite-element simulation of springback in sheet metal forming using local interpolation for tool surfaces

**ABSTRACT** This paper describes new techniques for the sheet metal forming simulation using a local interpolation for tool surfaces proposed by Nagata [Simple local interpolation of surfaces using normal vectors. Computer Aided Geometric Design 2005;22:327–47] and the effect of tool modeling accuracy on springback simulation of a high strength steel sheet. The Nagata patch enables the creation of tool models that are much more accurate, in terms of not only shape but also normal vectors, than those of conventional polyhedral representations. Besides allowing an improved description of the contact between the sheet nodes and the tool surfaces, the proposed techniques have the advantage of relatively straightforward numerical implementation. Springback simulations of a two-dimensional draw bending process of a high strength steel sheet are then carried out using the polyhedral and Nagata patch models. It is found that the simulation results are largely influenced by the tool mesh when using polyhedral representations, while they are rather independent when using the Nagata patch representations. This demonstrates the efficiency and reliability of the numerical solution using the Nagata patch model.

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**ABSTRACT:**This study deals with the new strategy currently implemented in DD3IMP in-house code to describe the forming tools using Nagata patches. The strategy is based on the use of the Nagata patch interpolation to generate smooth contact surfaces over coarse faceted finite element meshes. The description of the adopted algorithm is briefly presented, highlighting the contact search algorithm employed. The reverse deep drawing of cylindrical cups, proposed as benchmark at the Numisheet’99 conference, is selected to examine the accuracy and robustness of the proposed approach. The effect of the gap between the blank-holder and the die is studied, adopting two distinct strategies: fixed gap and variable gap. The numerical results are compared with the experimental ones, previously presented and discussed in [1]. It is shown that the agreement is very good both in terms of punch force evolution and thickness distribution.Key Engineering Materials 07/2013; 554-557:2277-2284. - SourceAvailable from: D.M. Neto
##### Article: Applying Nagata patches to smooth discretized surfaces used in 3D frictional contact problems

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**ABSTRACT:**The accurate solution of large deformation frictional contact problems using the finite element method is still a challenging task due to the strong nonlinearities involved. This paper presents a smoothing method applicable to 3D contact surfaces discretized with an arbitrary mesh topology. The quadratic Nagata patch interpolation is adopted to define the smooth surface. The resulting contact surface passes through all nodes of the mesh while providing a smooth description, with at least G1 continuity at the nodes and quasi-G1 continuity between the patches. Thus, the proposed method avoids the non-physical oscillations in the contact force, which are induced by the traditionally used faceted contact surfaces description, when slave nodes slide over several master segments. Moreover, it allows the accurate evaluation of kinematic variables, leading to important improvements in terms of convergence rate within the Newton–Raphson iteration loop. The developed global and local contact search algorithms, designed for contact surfaces described by Nagata patches, are described in detail. Three numerical examples were selected to illustrate the advantages of the proposed smoothing method, including a complex industrial example of sheet metal forming process. The results show the significant improvements attained with the proposed approach, in terms of efficiency, robustness and accuracy, when compared with the traditional faceted contact surfaces description.Computer Methods in Applied Mechanics and Engineering 01/2013; · 2.62 Impact Factor - SourceAvailable from: D.M. Neto[Show abstract] [Hide abstract]

**ABSTRACT:**This study deals with different tool surface description methods used in the finite element analysis of sheet metal forming processes. The description of arbitrarily-shaped tool surfaces using the traditional linear finite elements is compared with two distinct smooth surface description approaches: (i) Bézier patches obtained from the Computer-Aided Design model and (ii) smoothing the finite element mesh using Nagata patches. The contact search algorithm is presented for each approach, exploiting its special features in order to ensure an accurate and efficient contact detection. The influence of the tool modelling accuracy on the numerical results is analysed using two sheet forming examples, the unconstrained cylindrical bending and the reverse deep drawing of a cylindrical cup. Smoothing the contact surfaces with Nagata patches allows creating more accurate tool models, both in terms of shape and normal vectors, when compared with the conventional linear finite element mesh. The computational efficiency is evaluated in this study through the total number of increments and the required CPU time. The mesh refinement in the faceted description approach is not effective in terms of computational efficiency due to large discontinuities in the normal vector field across facets, even when adopting fine meshes.International Journal of Material Forming 01/2014; · 1.42 Impact Factor

Page 1

International Journal of Mechanical Sciences 50 (2008) 175–192

Finite-element simulation of springback in sheet metal forming

using local interpolation for tool surfaces

Takayuki Hamaa,b,?, Takashi Nagatab, Cristian Teodosiub,c,

Akitake Makinouchib, Hirohiko Takudaa

aGraduate School of Energy Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan

bVolume-CAD System Research Program, The Institute of Physical and Chemical Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

cLPMTM-CNRS, University Paris 13, 93430 Villetaneuse, France

Received 11 April 2007; received in revised form 6 July 2007; accepted 8 July 2007

Available online 13 July 2007

Abstract

This paper describes new techniques for the sheet metal forming simulation using a local interpolation for tool surfaces proposed by

Nagata [Simple local interpolation of surfaces using normal vectors. Computer Aided Geometric Design 2005;22:327–47] and the effect

of tool modeling accuracy on springback simulation of a high strength steel sheet. The Nagata patch enables the creation of tool models

that are much more accurate, in terms of not only shape but also normal vectors, than those of conventional polyhedral represen-

tations. Besides allowing an improved description of the contact between the sheet nodes and the tool surfaces, the proposed techniques

have the advantage of relatively straightforward numerical implementation. Springback simulations of a two-dimensional draw bending

process of a high strength steel sheet are then carried out using the polyhedral and Nagata patch models. It is found that the simu-

lation results are largely influenced by the tool mesh when using polyhedral representations, while they are rather independent when

using the Nagata patch representations. This demonstrates the efficiency and reliability of the numerical solution using the Nagata

patch model.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Sheet metal forming; Springback; Finite element method; Contact; Tool modeling; Quadratic polynomial patch

1. Introduction

In the automotive and die manufacturing industries,

prior predictions of the formability of sheet metals using

finite-element methods (FEM) have enabled lowering cost

and shortening delivery periods, as well as the achievement

of a higher quality [1–4]. Their contribution is recently

increasing and became indispensable in many industrial

applications. Prediction accuracy for shape defects includ-

ing springback and surface deflection is, however, still

inadequate, and empirical trial-and-error methods are

presently used in the field operations. Moreover, high

tensile strength steels are nowadays commonly used, which

makes the amount of springback remarkably large, and

renders the prediction of springback much more important.

Therefore, the establishment of simulation technologies for

improving the prediction accuracy is an urgent and

challenging task.

For the purpose of dealing with the above problems,

many attempts have been made to improve the accuracy of

sheet metal forming simulation by FEM over the last

decade, including the development of new finite elements

[5–13], the improvement of material constitutive models

[14–21], and the coupled analysis of the elastoplastic sheet

deformation with the elastic deformation of tools [22,23].

Although the treatment of the contact between a sheet and

tools is one of the most important issues in order to achieve

accurate simulations, attempts to improve the tool model-

ing accuracy are scarce [24,25], thus the effect of tool

modeling accuracy on simulation is not clear.

ARTICLE IN PRESS

www.elsevier.com/locate/ijmecsci

0020-7403/$-see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijmecsci.2007.07.005

?Corresponding author. Graduate School of Energy Science, Kyoto

University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan.

Tel.: +810757535418; fax: +810757535428.

E-mail address: hama@energy.kyoto-u.ac.jp (T. Hama).

Page 2

For sheet metal forming simulations, the tools are in

general assumed rigid and only their surfaces are modeled

by either a set of polyhedral patches (i.e., finite-element

meshes), points, or parametric representations [24–26].

Although the mesh approach is employed in most cases due

to its capability of wide application and simplicity in use,

this approximation sometimes leads to large errors in the

geometry [27,28]. Since the contact formulation generally

considers the tools approximated as polyhedral surfaces

[29–34], such geometrical errors may affect in turn the

simulation accuracy [35]. For instance, considering simula-

tions of surface deflection, it is generally estimated that the

elastic recovery, as main origin of this defect, should be

predicted with the accuracy of the order of 10mm, since the

defect itself is the order of dozens to several hundred mm

[4,36–38]. Clearly, tool models with higher level of

accuracy should be used for this purpose.

In order to improve the contact surface, various

techniques to smooth the discretized surfaces, such as

NURBS, Be ´ zier patch, and Gregory patch, have been

proposed by many researchers [35,39,40]. The conventional

smoothing involves free parameters or assumptions on the

derivatives, which should be given a priori. This is,

however, hardly possible since the analytic properties of

the original surface are generally unknown [27]. Therefore,

the geometrical accuracy of the conventional smooth

surface interpolators is not assured, and has not even been

studied [27]. Furthermore, since the smoothed surfaces are

generally described by cubic or higher polynomials or

rational functions, the contact search requires significant

effort, which is sometimes the central factor for increasing

computational time and instability [30].

Recently, Nagata [27] has proposed a simple algorithm

for interpolating discretized surfaces to recover the original

geometry with good accuracy (subsequently called the

Nagata patch). The Nagata patch is a quadratic parametric

interpolator determined only by the position and normal

vectors given at the vertices of polyhedral meshes. The

algorithm involves no free parameters, and the interpolated

surface rapidly converges to the original one when

decreasing the mesh size. The Nagata patch yields an

accurate and simple geometrical description that can

hardly be obtained by the standard polyhedral approxima-

tions without drastically increasing the number of tool

elements.

For the purpose of improving the tool modeling

accuracy and clarifying its effect on the quality of

numerical solution, this paper proposes new techniques

of the sheet metal forming simulation using the Nagata

patch for describing the tool surfaces. A contact search

algorithm and a consistent contact tangent stiffness matrix

considering tool models represented by the Nagata patch

are formulated. Simulations of a two-dimensional draw

bending process are then carried out using both tool

models represented by the Nagata and polyhedral patches,

and the effect of the tool modeling accuracy on the

springback simulation is examined in detail.

2. Finite-element formulation

2.1. Virtual work and material constitutive equations

New techniques of the sheet metal forming simulation

using the Nagata patch for the tool surfaces are introduced

in the static explicit FEM code STAMP3D [5,41]. The

formulation applied in STAMP3D may be briefly described

as follows. An updated Lagrangian rate formulation is

used to describe the finite deformation. The rate form of

the equilibrium equations and boundary conditions at time

t are equivalently expressed by the principle of virtual

velocity in the form:

Z

where V and S denote, respectively, the domain occupied

by the body and its boundary at time t. S1is the part of the

boundary S on which the nominal surface traction f is

prescribed, while S3is that on which contact conditions

between the sheet and the tools are prescribed. dv is the

virtual velocity field satisfying the conditions dv ¼ 0 on the

velocity boundary S2 and dvN

boundary S3, where the subscript N denotes the normal

component. P is the first Piola–Kirchhoff stress tensor.

The superscript T denotes the transpose. Assuming that all

the rate-form relations are preserved from time t to t+Dt,

where Dt is a small time increment, the incremental form of

Eq. (1) can be derived as

Z

¼

S1

S3

Z

or

Z

¼ Dt

S1

S3

Z

where the following relations:

V

_PT: dv ? rx

ðÞdV ¼

Z

S1

_f ? dvdS þ

Z

S3

_f ? dvdS;

(1)

¼ 0 on the contact

V

D sJ

Z

þ

?? 2r ? DD

??: dD þ DL ? r

Df ? dvdS þ

Z

ðÞ : dL

??dV

f ? dvdS

Df ? dvdS þ

ZZ

S1

S3

f ? dvdS ?

V

r : dLdV,

ð2Þ

Dt

V

sJ

Z

?? 2r ? D

??: dD þ L ? r

_f ? dvdS þ Dt

ðÞ : dL

??dV

Z

r : dLdV,

_f ? dvdS þ

Z

S1

f ? dvdS

þ

S3

f ? dvdS ?

Z

V

ð2Þ0

Df ¼_f Dt

and

DP ¼_PDt ¼ DsJ

are introduced. r is the Cauchy stress tensor sJ

Jaumann rate of the Kirchhoff stress tensor, L is the

velocity gradient tensor, and D is the strain rate tensor,

which is the symmetric part of L. The third, fourth, and

fifth terms of the right-hand side in Eq. (2) remain in order

to cancel the non-equilibrated forces arising from the

(3)

?? DD ? r ? r ? DD þ r ? DLT

(4)

?is the

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T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

176

Page 3

explicit time-marching algorithm, as explained in the next

section.

Small-strain linear elasticity and large deformation, rate-

independent work-hardening plasticity are assumed for the

constitutive equations. In order to take into account the

anisotropy of sheet, Hill’s quadratic yield function [42] and

the associated flow rule with isotropic hardening are used.

The elastoplastic constitutive equations are given in the

form:

sJ

?¼ Cep: D ¼ Cep: L,

where Cepis the tensor of the tangent elastoplastic moduli.

(5)

2.2. Discretization

A four-node isoparametric shell element [5,43] is

employed in this study. Assembling the discretized virtual

work over the entire domain, the tangent stiffness equation

is obtained as

?

where [K] is the tangent elastoplastic stiffness matrix, {Du}

is the displacement increment vector, {Df} is the equivalent

nodal force increment vector due to the surface traction

and {Dfc} is that due to the contact, which is formulated in

the next chapter. {Fneq} is the non-equilibrated nodal force

vector, i.e., the difference between the external and internal

forces. The tangent stiffness matrix [K] is calculated at time

t and considered as constant over the increment Dt. To

preserve the approximate linearity during the increment,

the generalized rmin-strategy [44] is employed to appro-

priately limit the size of the increment.

Using the tangent solution described above, the nodal

equilibrium between the external and internal forces at the

end of increment is not guaranteed and the non-equili-

brated force vector {Fneq} remains. To overcome the

difficulty, a special algorithm called ‘‘ALGONEQ’’ [45–47]

is employed in STAMP3D, in order to systematically

cancel the remaining non-equilibrated nodal forces when-

ever their norm exceeds a prescribed admissible tolerance.

K

½ ? Du

fg ¼ Df

?þ Dfc

??þ Fneq

fg,(6)

3. Contact treatment

3.1. Virtual work

The master-slave algorithm is employed, in which sheet

nodes and tool elements are considered to be slave nodes

and master segments, respectively. The tools are considered

rigid and their surfaces are discretized by a set of triangular

patches. The virtual work due to the contact, which is the

second term in the right-hand side of Eq. (2)0, can be

written as

Z

S3

_f ? dvdS

??

Dt ¼

Z

S3

_^fN^ e3þ^fN_^ e3þ_^f

i

T^ eiþ^f

i

T_^ ei

??

? dvdS

??

Dt,

(7)

where ^ ej, j ¼ 1, 2, 3, denote the base vectors of a local

Cartesian frame defined at each contact point, where the

unit vector normal to the tool surface at the contact point

is taken as ^ e3, while ^ e1and ^ e2lie within the tangent plane.

The superscripts4denotes the components with respect to

this local frame, the subscripts N and T denote the normal

and tangent components of the contact force, respectively,

and i ranges over 1 and 2. For simplicity, friction is not

taken into consideration in this study so that the last two

terms of the integrand in the right-hand side of Eq. (7) are

omitted in the following. Therefore, the contact force f

should have only a normal component and thus Eq. (7)

yields

Z

Since a penalty method is employed to satisfy the

impenetrability condition in this study, the condition

d^ vN¼ 0 no longer holds and Eq. (8) is discretized in a

rather standard way (see e.g. Ref. [48]) as

S3

_f ? dvdS

??

Dt ¼

Z

S3

_^fN^ e3þ^fN_^ e3

??

? dvdS

??

Dt.(8)

Z

S3

_f ? dvdS

??

Dt ¼

X

X

l

l_f

c? dlvDt

¼

l

al^ vrel

Ndl^ vNþl^f

c

N

l_^ e3? dlv

??

Dt,

ð9Þ

where l denotes the number of the contact nodes,l_f

concentrated contact force vector at node l,

normal component of the relative velocity of sheet node l

with respect to the tool, and a is the penalty number. The

second term in the right-hand side of Eq. (9) arises from

rotations of the contact force vectors when the contact

nodes are sliding on curved parts of the tool surface.

cis the

Nis the

l^ vrel

3.2. Contact treatment using the Nagata patch for tool

surfaces

The Nagata patch is employed for the tool surfaces

discretized by a set of triangular patches. In this section,

the formulation of the Nagata patch is briefly described in

Section 3.2.1, followed by Sections 3.2.2 and 3.2.3

formulating contact search algorithms between sheet nodes

and tool elements (the Nagata patch). The second term in

the right-hand side of Eq. (9) is then discretized and the

consistent tangent stiffness matrix for the contact taking

into account the Nagata patch is derived in Section 3.2.4.

3.2.1. Nagata patch formulation

The Nagata patch is a quadratic parametric interpolator

for polyhedral meshes [27]. The algorithm recovers the

curvature of surfaces using the position and tool normal

vectors at the element nodes in the following procedure. It

is noted that the normal vectors that are consistent with

original CAD surfaces are supposed to be given at the

element nodes.

ARTICLE IN PRESS

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

177

Page 4

(a) Interpolation of edges of a tool element

An edge of a tool element is replaced by a curve

given in the form:

x x ð Þ ¼ x1þ x2? x1? c

where x is a parameter satisfying the condition

ðÞx þ cx2,(10)

0pxp1.(11)

x1and x2are the position vectors of the ends of the

edge (i.e., the element nodes), and c is the vector

adding curvature to the edge. Assuming that the curve

given by Eq. (10) is orthogonal to the normal vectors

n1and n2given at nodes x1and x2, respectively, and

minimizing the norm of c, the vector c can be

determined as

c ¼

n1;n2

1 ? c2

n1;?n1

2

½?

1

?c

1

?c

?

??

n1? x2? x1

?n2? x2? x1

Þ

Þ

ðÞ

ð

)

Þ

()

ca ? 1

ðÞ

½

n1? x2? x1

?n1? x2? x1

ð

ð

(

¼ 0

c ¼ ?1

ðÞ

8

<

>

>

>

>

>

>

>

>

>

>

:

,

(12)

where c ¼ n1?n2. All the edges of the tool element are

replaced by the curves given by Eq. (10).

(b) Interpolation of the surface of a tool element

Curvatures of a tool element can be recovered by

interpolating the curves given by Eq. (10). In the case

of a triangular patch, the interpolated surface is given

by the quadratic polynomial:

x Z;z

ðÞ ¼ C1þ C2Z þ C3z þ C4Zz þ C5Z2þ C6z2,

(13)

where Z and B are the parameters defined on the surface

element region, and satisfy the next condition:

0pzpZp1.(14)

The coefficient vectors are given by

C1¼ x1;

C3¼ x3? x2þ c1? c3,

C4¼ c3? c1? c2;

where c1, c2, and c3 are the vectors defined by

Eq. (12) for the edges (x1, x2), (x2, x3), and (x3, x1),

respectively.

By replacing all the triangular patches by the Nagata

patch (Eq. (13)), the discretized tool surface can be

more precise without increasing the number of

elements. The reader is referred to the original paper

[27] for details concerning the Nagata patch and the

assessment of its accuracy. Techniques for detecting

C2¼ x2? x1? c1,

C5¼ c1;

C6¼ c2,

ð15Þ

contact conditions between the Nagata patches and the

sheet nodes, i.e., the contact search algorithm, and for

the consistent contact tangent stiffness matrix taking

into account the Nagata patch are newly formulated,

and are described in Sections 3.2.2, 3.2.3, and 3.2.4.

3.2.2. Contact search algorithm for free nodes

In STAMP3D, the contact search for free sheet nodes is

done by tentatively calculating the intersection point xcof a

relative displacement increment vector Durelof a free sheet

node x0with a tool element [32]. It should be noted that,

when using shell elements for the sheet, the assumed

surface half-thickness away from the mid-surface of the

shell element x0

account the thickness of the sheet in the contact search [49],

where

his used instead of x0in order to take into

xh

0¼ x0?h

where V0is the fiber vector of the shell element at node x0

and h is the length of V0. The sign is determined by which

side of the surface is considered. The contact search for free

nodes is schematically shown in Fig. 1, where x0

denote the position vectors of the assumed sheet node xh

the beginning and end of the increments, respectively.

First, the local Cartesian frame f~ e1; ~ e2; ~ e3g is defined as

Durel

Durel

~ e3? C6

where ~ e1can arbitrarily be chosen in the plane perpendi-

cular to ~ e3if ~ e3? C6¼ 0. The sheet node never intersects

the Nagata patch if Durel¼ 0, and hence is being

considered to remain free without conducting the following

calculation. Writing Eq. (13) in a component form with

respect to f~ e1; ~ e2; ~ e3g yields

C1

C2

C3

2V0,(16)

tand x0

t+Dt

0at

~ e3¼

????;

~ e1¼

~ e3? C6

kk;

~ e2¼ ~ e3? ~ e1,(17)

1? x1

1? x2

1? x3

where the superscripts of the coefficients denote their

components, and xc1, xc2, and xc3are the components of xc.

The condition C6

definition of ~ e1. Since the relative displacement increment

cþ C1

cþ C2

cþ C3

2Z þ C1

2Z þ C2

2Z þ C3

3z þ C1

3z þ C2

3z þ C3

4Zz þ C1

4Zz þ C2

4Zz þ C3

5Z2¼ 0,

5Z2þ C2

5Z2þ C3

6z2¼ 0,

6z2¼ 0,

ð18Þ

1¼ 0 is always attained due to the

ARTICLE IN PRESS

e1

~

e2

~

e3

~

x0t

x1

x2

x3

Nagata patch

Triangular patch

xc

Δurel

x0t+Δt

Fig. 1. Schematic of the contact search between a free node and the

Nagata patch.

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

178

Page 5

vector Durelhas only the third component in the local

frame, xc1and xc2correspond, respectively, to the first and

second components of x0

Hence Eq. (18) can be solved for the unknown quantities Z,

z, and xc3. Moreover, as explained below, the problem is at

most a quartic, and hence its closed-form solution exists

[50,51], thus enabling a stable and robust calculation.

The parameters Z and z are obtained through solving the

first and second Eq. (18). Depending on the coefficients in

the first equation, next two cases are considered.

t, which are known quantities.

(a) The case C3

Solving the first Eq. (18) for z yields

16¼0 or C4

16¼0

z ¼ ?C1

1? x1

cþ C1

C1

2Z þ C1

4Z

5Z2

3þ C1

.(19)

Substituting Eq. (19) into the second Eq. (18) gives

C2

6C1

? C1

? C2

Hence the next equation is obtained

1? x1

1? x1

1? x2

cþ C1

cþ C1

cþ C2

2Z þ C1

2Z þ C1

2Z þ C2

5Z2

??2? C1

?¼ 0.

3þ C1

3þ C1

4Z

??

C2

3þ C2

4Z

??

5Z2

5Z2

??þ C1

4Z

??2

?

ð20Þ

MZ4þ NZ3þ OZ2þ PZ þ Q ¼ 0,

where

(21)

M ¼ C1

N ¼ 2C1

4

3C1

?

? C1

?

? C1

þ 2 C1

?

þ C1

?

?2C2

5? C1

4C2

4C1

3C1

?

5C2

5C2

?2C2

4þ C1

4? C1

2þ 2C1

3C2

5

2C1

??2C2

6,

5? C1

5C2

?2C2

?

?2C2

4C2

4

4C1

3þ C1

5? C1

5C2

42C1

5C2

?C1

6,

O ¼ C1

32C1

4? C1

4C2

1? x1

3C1

?2C2

?þ 2 C1

c

?

?

4C2

?

4

3C1

3? C1

1? x1

1? x1

2þ 2C1

1? x1

2C1

?C1

3þ 2C1

5C2

3C2

?

4C2

2þ C2

6,

1? x1

?

c

1? x2

c

?

C1

4

??2

c

6þ C1

4? C1

3C1

?

2

P ¼ ? C1

c

?C1

?C1

2C1

3C2

3? C1

c

??C1

c

4C2

?C1

3

3

?

4C2

1? x2

?2C2

c

1? x1

?þ C1

2C2

6,

?2C2

ð22Þ

Q ¼ ? C1

c

3C2

3þ C1

31? x2

?

1? x1

c

?

6.

Eq. (21) is generally a quartic in Z and is solved

analytically [50,51] in this study. Eq. (21) can

degenerate to be cubic, quadratic or linear depending

on the coefficients and hence the analytical roots can

be obtained in any case. Substituting the root of Z into

Eq. (19) gives z, and then substituting the roots of Z

and z into the third Eq. (18) gives xc3, thus at most four

sets of Z, z, and xc3are attained.

(b) The case C3

The first Eq. (18) becomes a quadratic in Z which

directly gives two roots of Z. Two roots of z are then

derived for each Z by substituting the root of Z into the

1¼ 0 and C4

1¼ 0

second Eq. (18). Consequently, still at most four sets of

Z, z, and xc3are derived.

The roots of Z should fulfill the next condition due to

Eq. (19) when C3

1+C4

1¼ 0 is satisfied

C1

1? x1

cþ C1

2Z þ C1

5Z2¼ 0. (23)

Note that Eq. (21) is equivalent to Eq. (20) and

hence degenerates to Eq. (23) in the case C6

C3

only for C6

the Nagata patch and the physical requirements, the

set of Z, z, and xc3for the intersection point xcshould

satisfy not only Eq. (14) but also the conditions:

26¼0 and

1+C4

1Z ¼ 0. Therefore Eq. (23) needs to be checked

2¼ 0. In addition, due to the definition of

Im Z ð Þ ¼ 0

Therefore, sets of Z, z, and xc3which do not conform

to these conditions are checked off from the following

calculations. Although in most cases only a set of Z, z,

and xc3becomes valid, more sets can remain after testing

Eqs. (14), (23), and (24) when two intersection points lie

within the Nagata patch, as shown in Fig. 2. In such

case, the set for which xc3has the smaller absolute value,

i.e., the intersection point is closer to the assumed sheet

node at the beginning of the increment x0

For instance, in the case of Fig. 2, xc1is adopted as the

tentative intersection point xc. If all the sets of Z, z and

xc3do not satisfy the above conditions, it is considered

that the intersection point xcdoes not lie on the Nagata

patch in question and hence the above calculations are

repeated with other Nagata patches until the tentative

intersection point is attained.

Finally, whether the assumed sheet node x0

the tool or not can be decided from the condition

andIm z ð Þ ¼ 0. (24)

t, is selected.

hpenetrates

n ? xc? xtþDt

0

?

where n is the outward tool normal vector at the

intersection point xc, and can be calculated as

?X0,(25)

n ¼

t1? t2

t1? t2

k k¼

r

r k k,(26)

ARTICLE IN PRESS

x1

x2

x3

x0t

xc1

xc2

x0t+Δt

Δurel

Fig. 2. A case where two intersection points lie on the Nagata patch.

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

179

Page 6

where ? k k denotes the Euclid norm,

t1¼qx

t2¼qx

qZ¼ C2þ C4z þ 2C5Z,

qz¼ C3þ C4Z þ 2C6z,

r ? t1? t2.

The sheet node is considered to penetrate and hence to

contact the tool in question if Eq. (25) is satisfied, while

otherwise this sheet node remains free.

(27)

The above-mentioned contact search can be used for the

calculation of rminfor contact, say rc, in the generalized

rmin-strategy, which is given as

??

Hence, the penetration of the sheet nodes into the tools can

be prevented by taking into account Eq. (28) when

determining the step size, i.e., the rminvalue. In such case,

furthermore, the contact search for free nodes that is

conducted at the end of the increment can be done by

simply comparing the magnitude of rcwith the rminvalue,

which can be stored for all the candidate contact nodes.

rc¼

xc? xt

Durel

0

??.

??

??

(28)

3.2.3. Contact search for contact nodes

The contact search for a sheet node which is already in

contact with a tool and is sliding on the tool surface is done

by the normal projection of the sheet node at the end of the

increment on the tool surface [29], and hence can be done

by calculating the intersection point xcof the tool normal

vector for the sheet node at the end of the increment with a

Nagata patch. This is schematically shown in Fig. 3, where

the tool normal vector defined at the intersection point xcis

denoted by nt+Dt. On the other hand, according to Eq. (26),

nt+Dtis a non-linear function of the parameters Z and z of

the initially unknown projection. To avoid this difficulty,

the intersection point xcmay be calculated iteratively, by

taking the ith approximation xcias intersection of the

Nagata patch with the line of direction niand passing

through x0

direction to the patch at xci. The iteration is initialized by

taking n1¼ ntand stops when ni?ni+1is sufficiently close

to 1.

Clearly, at each iteration, we have to solve Eq. (18), in

which the local frame is defined as

t+Dtand then determining ni+1as a normal

~ e3¼ ni;

~ e1¼

~ e3? C6

~ e3? C6

kk;

~ e2¼ ~ e3? ~ e1. (29)

As the time increments in STAMP3D are relatively small

due to the use of the rmin-strategy, even the first

approximation xc1and n2is usually sufficiently accurate.

If all sets of Z and z do not satisfy Eqs. (14) and (24) (and

Eq. (23) when C6

that the intersection point xcdoes not lie on the Nagata

patch in question and the above calculations are repeated

with other Nagata patches until the proper intersection

point is obtained.

As described above, the contact search is formulated

based on solving Eq. (18) analytically, thus allowing a

robust and accurate contact search.

2¼ 0 and C3

1+C4

1Z ¼ 0), it is considered

3.2.4. Consistent contact tangent stiffness matrix

Since STAMP3D employs the static-explicit approach

with the use of rmin-strategy, it is essential to follow the

change of tool normal vectors during an increment

explicitly when contact nodes are sliding on curved parts

of tool surfaces. This section describes the discretization of

the second term in the right-hand side of Eq. (9), i.e., the

consistent contact tangent stiffness matrix, by taking into

account the Nagata patch.

First, taking an increment of Eq. (26) yields

D^ e3¼ Dn ¼Dr

r k kþ rD

r k k?1

??¼ I ? n ? n

ðÞ ?Dr

r k k,(30)

where I is the unit tensor. From Eq. (27), Dr is given in the

form:

Dr ¼ 2C5DZ þ C4Dz

¼ r1DZ þ r2Dz,

where

ðÞ ? t2þ t1? C4DZ þ 2C6Dz

ðÞ

ð31Þ

r1? 2C5? t2þ t1? C4;

Writing Eq. (31) in a component form with respect to

f^ e1; ^ e2; ^ e3g yields

D^ ri¼ ^ rjiDZj,

where i and j take over 1, 2, 3, and 1, 2, respectively, Z1?Z,

and Z2?z. Substituting Eq. (33) into Eq. (30) gives:

T¼D^ ri

r2? C4? t2þ 2t1? C6. (32)

(33)

D^ ni

T

r k k¼

^ rji

r k kDZj,(34)

where i takes over 1 and 2. Eq. (34) can be written in a

matrix form as

(

r k k

D^ n1

D^ n2

T

T

)

¼

1

^ r11

^ r12

^ r21

^ r22

"#

DZ1

DZ2

()

.(34)0

ARTICLE IN PRESS

x3

x2

x1

x0t+Δt

xc

Δurel

nt+Δt

x0t

Fig. 3. Schematic of the contact search for a sheet node sliding on the

Nagata patch.

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

180

Page 7

Note that D^ nN¼ D^ eN¼ 0 is fulfilled due to the relation

D n ? n

Due to the assumption of the tangent solution, the

relative displacement increment vector Durelof the contact

node is equivalent to the increment of the position vector

on the Nagata patch Dx, thus obtaining

ð Þ ¼ 2n ? Dn ¼ 0.(35)

Durel¼ Dx ¼qx

qZDZ þqx

qzDz ¼ t1DZ þ t2Dz.(36)

Writing in a component and matrix form gives

D^ urel;k

T

¼^tjkDZj, (37)

D^ urel;1

T

D^ urel;2

T

()

¼

^ t11

^t12

^t21

^t22

"#

DZ1

DZ2

()

,(37)0

where k and j take over 1 and 2. The normal component

D^ urel

Solving Eq. (37) for DZj and substituting into Eq. (34)

yields

N¼ 0 is fulfilled by the contact boundary condition.

D^ ni

T¼^ rji^tjk

?

r k k

??1

D^ urel;k

T

,(38)

where i, j, and k take 1 and 2. Writing Eq. (38) in a matrix

form gives

8

:

¼

^k

2122

D^ n1

T

D^ n2

T

<

9

=

;¼

1

r k k

2

^ r11

^ r21

^ r12

^ r22

3

"#

^ t11

^t21

^ t12

^t22

9

"#?1

D^ urel;1

T

D^ urel;2

T

8

:

<

9

;

=

^k

N

11

^k

N

12

N

^k

N

64

75

D^ urel;1

T

D^ urel;2

T

8

:

<=

;,

ð38Þ0

where

2

^k

N

11

N

21

^k

N

12

N

22

^k

^k

4

Eq. (38) is the consistent contact tangent stiffness matrix

taking into account the Nagata patch. Substituting Eq. (38)

into Eq. (9) yields the final form of the discretized virtual

work due to the contact taking into account the Nagata

patch as

Z

X

where i, j, and k takes over 1, 2, and p takes over 1, 2,

and 3. dij is the Kronecker delta. Writing in a matrix

3

5?

1

r k k

^ r11

^ r12

^ r21

^ r22

"#

^t11

^t12

^ t21

^ t22

"#?1

.(39)

S3

_f ? dvdS

??

Dt

¼

l

dl^ vp aDl^ urel

Ndp3þl^f

c

N

l^ rji

l^tjk

????

???1

lr

Dl^ urel;k

T

dpi

!

,

ð40Þ

form gives

Z

2

S3

_f ? dvdS

??

Dt ¼

X

N

12

l

dl^ v1

dl^ v2

dl^ v3

n

0

o

?

l ^f

c

N

l ^k

N

11

l ^f

c

N

l ^k

l ^f

c

N

l ^k

N

21

l ^f

c

N

l ^k

N

22

0

00

a

6664

3

7775

Dl^ urel;1

T

Dl^ urel;2

T

Dl^ urel

N

8

>

>

>

>

>

>

:

<

9

>

>

>

>

>

>

;

=

.

ð41Þ

The consistent contact tangent stiffness matrix is the

function of the relative displacement increment of sheet

node l and Dlurel, thus incorporated into the tangent

stiffness matrix in the tangent stiffness equation.

The formulation stated above cannot completely follow

the change of tool normal vector since this is only the first

approximation, while the Nagata patch is the quadratic

parametric surface. Hence, the canceling algorithm de-

scribed in Section 2.2 is simultaneously employed such that

the remaining non-equilibrated forces due to the highly

non-linear phenomena are cancelled explicitly.

Although the contact treatment taking into account the

Nagata patch stated above is formulated in the framework

of the tangent solution in which iterations for nodal

equilibrium are not employed, the key techniques are of

course applicable to other finite-element codes without loss

of generality.

3.3. Contact treatment used in conventional STAMP3D

Contact

STAMP3D are described in the original papers [32,33,46]

in detail. Only their outlines are described briefly here.

treatments employed inconventional

3.3.1. Polyhedral approximation of surface

In conventional STAMP3D, the tool surfaces approxi-

mated by a set of planar triangular patches are directly

used. A unit tool normal vector nkat tool node k is

uniquely defined by averaging the tool normal vectors

defined on all adjacent elements as

P

m

where nk

mis the unit normal vector at node k as the cross

product of two tangential vectors to the adjacent element

m. Smis the weighting factor, which is taken equal to the

area of the element m. Thereafter, a tool normal vector at

an arbitrary point is defined by interpolation with use of

shape functions3Nkfor a 3-node triangular element as

P3

nk¼

mSmnk

mSmnk

m

P

????,(42)

n ¼

k¼1

k¼13Nknk

3Nknk

P3

??????

¼

V

V

k k,(43)

where

V ?

X

3

k¼1

3Nknk.(44)

ARTICLE IN PRESS

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

181

Page 8

3Nk, k ¼ 1, 2, 3 are linear functions of the area coordinates

of the tool element. It should be noted that, using the

above definitions, the normal vectors change continuously

over the tool surface, while they are not necessarily

consistent with the shape of the tool surface.

3.3.2. Contact search algorithm

The contact search is carried out between sheet nodes and

planar triangular patches using the similar procedures

explained in Sections 3.2.2 and 3.2.3 both for the free and

contact nodes, thus coming to calculating the intersection

point of a projection vector from a sheet node with a planar

triangular patch. Clearly, the intersection point can be

calculated analytically as well as the Nagata patch model.

3.3.3. Consistent contact tangent stiffness matrix

Discretizing the second term in the right-hand side of Eq. (9)

using Eq. (43) in the similar way as that in Section 3.2.4, the

discretized virtual work due to the contact is derived as

Z

X

where

X

X

i, j, k takes over 1, 2, and p takes over 1, 2, 3. Writing in a

matrix form gives

Z

l

l ^f

N

6664

S3

_f ? dvdS

??

Dt

¼

l

dl^ vp aDl^ urel

Ndp3þl^f

c

N

l ^Aij

l^Bkj

??

???1

lV

??

Dl^ urel;k

T

dpi

!

, ð45Þ

^Aij¼

3

h¼1

3

q3Nh

qxj

^ nh

i,(46)

^Bkj¼

h¼1

q3Nh

qxj

^ xh

k,(47)

S3

_f ? dvdS

??

Dt ¼

X

2

dl^ v1

dl^ v2

dl^ v3

no

0

?

c

l ^k

C

11

l ^f

c

N

l ^k

C

12

l ^f

c

N

l ^k

C

21

l ^f

c

N

l ^k

C

22

0

00

a

3

7775

Dl^ urel;1

T

Dl^ urel;2

T

Dl^ urel

N

8

>

>

>

>

>

>

:

<

9

>

ð48Þ

>

>

>

>

>

;

=

,

where

^k

C

11

^k

C

12

^k

C

21

^k

C

22

2

64

3

75 ¼

1

V

k k

P

P

q3Nh

qz1

3

h¼1

3

q3Nh

qz1

^ nh

1

P

P

q3Nh

qz2

3

h¼1

3

q3Nh

qz2

^ nh

1

h¼1

q3Nh

qz1

^ nh

2

h¼1

q3Nh

qz2

^ nh

2

2

66664

P

P

3

77775

?1

?

3

h¼1

3

^ xh

1

P

P

3

h¼1

3

^ xh

1

h¼1

q3Nh

qz1

^ xh

2

h¼1

q3Nh

qz2

^ xh

2

2

66664

3

77775

.

ð49Þ

4. Simulation of a two-dimensional draw bending process of

a high strength steel

4.1. Simulation conditions

In order to validate the proposed contact treatment and

to examine the effect of tool modeling accuracy on

springback analysis, simulations of a two-dimensional

draw bending process [47] of a high strength steel sheet

are carried out. The mechanical properties of the sheet

material are shown in Table 1. In order to emphasize the

effects of contact modeling, the anisotropy of sheet is not

taken into consideration. The geometries of tools employed

in the simulation are shown in Fig. 4 [47]. The size of the

blank is 5mm wide, 200mm long, and 1.0mm thick. Due

to the symmetry of the process, only the half part is

modeled. The plane strain condition is assumed in the

width direction, to which corresponding boundary condi-

tions are given. Four-node degenerated shell elements with

assumed-strain integration [5,43] are being used and the

number of elements are 1 and 100 along the width and

longitudinal directions, respectively. Sheet discretization is

one of the important factors for springback simulations.

From our experiences, it would be rather worse to make

the element size be smaller than the shell thickness. The

sheet mesh in the longitudinal direction described above is

therefore chosen although the element size may be

considered minimal to describe the punch and die radii.

ARTICLE IN PRESS

Table 1

Material properties used in the simulationsa

E (GPa)

n

sy(MPa)

F (MPa)

n

e0

200

0.3

760

1600

0.12

0.0021

aThe true stress–true plastic strain curve is approximated by the Swift

law s ¼ F(e0+ep)n.

x

z

27.6

R3

R5

30

Blank holder

Die

Punch

Fig. 4. Schematic of tool geometries in mm.

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

182

Page 9

On the other hand, we adopt the sheet mesh with the plane

strain constraint in the width direction in order to reduce

the effect of the difference in the tool modeling accuracy in

the width direction on the simulated results as much as

possible, since the deformation in the present two-dimen-

sional draw bending process should be uniform in the

width direction except for the vicinity of the sheet edge.

The tool modeling accuracy in the present tool models will

be described in the next section in detail. Fifteen Gauss

integration points are introduced through thickness.

Friction is not taken into consideration, while, as will be

described below in detail, stretching forces equivalent to

the friction force due to the blank holding force are given

to the sheet prior to the drawing process.

Calculation procedures adopted in the present draw-

bending simulation consist of the following three steps.

(i) Stretching forces are applied on the sheet edge until the

tensile stress s reaches a prescribed value, which is

varied from around 1% to 40% of the initial yield

stress sy.

(ii) With keeping the initial stretching forces, the sheet is

drawn up to the punch stroke of 70mm.

(iii) Finally, nodal forces opposite to the internal forces are

applied until the latter are cancelled. (This process

corresponds to springback calculation.)

Comparisons of the simulation results with the experi-

mental ones are not conducted in this paper from the

following reasons: (a) simple simulation conditions of

friction and constitutive models are chosen in this study in

order to clarify the effect of tool model accuracy on the

results and (b) the scattering in experiments may cover

the difference observed in the simulation results. Instead,

the simulation results are compared with the analytical

ones based on the elastoplastic incremental strain theory

proposed by Kuwabara et al. [52,53]. This analysis models

a metal sheet subjected to bending–unbending deformation

under tension, which corresponds to the deformation at the

die shoulder in the present process. It has been clarified

that the analytical results agree rather well with the

experimental ones. It is considered that, if the simulations

are properly carried out, the simulation results will agree

with the analytical ones unless contact between the sheet

and the sidewall of the punch takes place.

This forming process is rather simple, but involves the

bending–unbending deformation, a large influence of

stretching force at the flange, and sometimes the so-called

over-run phenomenon. Hence it is difficult to predict a

deformed profile after springback accurately.

4.2. Tool modeling accuracy

Since the sheet slides continuously on the die shoulder

during the process, the modeling accuracy at the die

shoulder may largely affect the simulation results. There-

fore, two different discretized die models, dies A and B

(Fig. 5), are used. The die shoulder is finely discretized in

die A, and somewhat roughly in die B. Three types of tool

modeling are employed in this study. The first approach is

a tool modeling used in STAMP3D, which is subsequently

called the conventional polyhedral model (see Section 3.3).

The second approach employs the Nagata patch for the

tool surfaces. In the third approach, the polyhedral

surfaces are directly employed for the shape description,

while the tool normal vectors that are consistent with the

original CAD surfaces are given at the tool nodes instead

of the averaged normal vectors defined in Eq. (42). A tool

normal vector at an arbitrary point is defined by the

interpolation (Eq. (43)) as in the conventional polyhedral

model. This tool modeling is subsequently called the new

polyhedral model. These three approaches are employed

both for dies A and B, thus attaining six die models. As for

the punch and the blank-holder, finely discretized models

as they are usually employed in the conventional poly-

hedral model are used (Fig. 6). In the calculations, all the

tools, i.e., the punch, the die, and the blank holder, are

represented by a same tool modeling.

ARTICLE IN PRESS

10

x, ex

y, ey

D

C

z, ez

G

G

H

H

10

135

100

D

C

G

G

H

H

6.25

100

100

135

Fig. 5. Die models employed in the simulations in mm. The dotted lines

GG and HH correspond to the boundaries of the shoulder and straight

parts in the exact die profile. The points C and D are on the lines GG and

HH, respectively. (a) Die A and (b) die B.

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

183

Page 10

The shape and normal vector error distributions at the

die shoulder are compared between dies A and B, and

between the Nagata patch, conventional polyhedral, and

new polyhedral models. The shape and normal vector

errors dsand dnat point p are respectively defined as

ds¼ rds? x ? o

?

where x is the position vector of point p, rds¼ 5 [mm] is the

die shoulder radius, and o is the position vector of the

center of the die shoulder radius. n ¼ x ? o

the die shoulder, while n ¼ ?ez at the upper part and

n ¼ ?exat the sidewall, V is the unit tool normal vector at

point p calculated on the die model, and t is the unit vector

tangent to the exact tool surface and satisfies both the

relations n?t ¼ eyand n?t ¼ 0, where ex, ey, and ezare the

base vectors shown in Fig. 5.

Fig. 7 shows the shape error distributions along line CD

shown in Fig. 5. Note that the shape description for the

new polyhedral and conventional polyhedral models are

the same. Line CD is chosen since the sheet nodes slide on

this line. In the polyhedral model, the negative shape errors

distribute periodically depending on the tool mesh both in

dies A and B. The shape error is smaller around the

element boundary, and larger around the center of the ele-

ment. Although somewhat periodical errors exist in the

Nagata patch model as well, they are too small to be visible

in Fig. 7. Clearly, the Nagata patch model yields a much

more accurate shape than the polyhedral model both in

dies A and B. Comparing die A with die B, die A yields a

more precise shape than die B in the polyhedral model,

while die B is as accurate as die A in the Nagata patch

model except for the ends of the die shoulder.

Fig. 8 shows the normal vector error distributions along

line CD. The level of accuracy for the four models, the

ð Þ ? n

ðÞ=rds? 100

?

½%?,(50)

dn¼ sgn t ? V ? n

ðÞ

V ? n

k k ? 100

½%?,(51)

ðÞ= x ? o

kk at

Nagata patch models for dies A and B and the new

polyhedral models for dies A and B, are almost the same

except at the ends of the die shoulder. The new polyhedral

models yield rather better distributions at the die shoulders

as shown in the emphasized figures. On the other hand, the

conventional polyhedral model yields much larger errors

than the others both in dies A and B. The conventional

polyhedral model for die A yields a negative normal vector

error distribution in the whole region, while the normal

vector errors for die B change the signs and their

magnitude is larger than that of die A. These results show

that the interpolation of normal vectors (Eq. (43))

practically gives an accurate distribution as that of the

Nagata patch when accurate normal vectors are defined at

the tool nodes, and that the averaging of normal vectors

(Eq. (42)) gives a worse result compared to the others in the

present models.

The large errors are shown at the ends of the die

shoulder both in the shape and in the normal vector in all

the models. These errors are due to the fact that the

original CAD data is represented by one surface so that the

element boundary is not on the boundary of the straight

and curved parts both in dies A and B. Therefore, to reduce

such large errors, CAD data with explicit specification of

ARTICLE IN PRESS

Punch

Blank holder

Die (die A)

Fig. 6. Discretized punch and blank holder models employed in the

simulations.

-0.6

-0.4

-0.2

0

-0.500.51 1.5

-0.6

-0.4

-0.2

0

-0.50 0.5 1.5

Shape error ?s%

Dimensionless length from C l

Nagata patch

Polyhedral

Shape error ?s%

Dimensionless length from C l

Nagata patch

Polyhedral

Die shoulderUpper part Sidewall

Die shoulderUpper part

CD

CD

1

Sidewall

Fig. 7. Shape error distributions at die shoulders. (a) Die A and (b) die B.

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

184

Page 11

curvature discontinuities should be used not only for

the polyhedral models but also for the Nagata patch

models.

4.3. Results and discussion

4.3.1. Simulation results for die A

Fig. 9(a) shows the deformed profiles after springback

for die A for the ratio of initial tensile stress and initial

yield stress s/sy¼ 0.14. The simulation result for the new

polyhedral model agrees well with that of the Nagata patch

model, while that of the conventional polyhedral model

differs from the other two results and yields much smaller

springback at the sidewall. The simulation is apparently

affected by the tool modeling.

To examine the simulation results more in detail, the

relationship between the sidewall curvature after spring-

back and the initial tensile stress is shown in Fig. 9(b). The

sidewall curvature is calculated as follows. Fig. 10 shows an

example of the curvature distribution along the long-

itudinal direction of the sheet after springback by

approximating three nodes for every three nodes with a

circle. Since the sidewall curvature is fairly uniform as

shown in Fig. 10, an average curvature of a region

circumscribed by a dotted square is adopted in Fig. 9(b).

The deformed profiles shown in Fig. 9(a) correspond to the

results for s/sy¼ 0.14 in Fig. 9(b). The peak curvature

occurs at around s/sy¼ 0.14 for the Nagata patch and new

polyhedral models, and their results agree fairly well with

each other. On the other hand, the peak curvature occurs

at around s/sy¼ 0.36 for the conventional polyhedral

model, and the result differs from the other two results in

the low tensile stress region. Generally speaking, the

curvature decreases as the initial tensile stress decreases

in the low tensile stress region since the reverse bending due

to the over-run phenomenon increases as the initial tensile

stress decreases [37,54]. Therefore, Fig. 9(b) indicates that,

in the conventional polyhedral model, the reverse bending

starts occurring at higher initial tensile stress than in the

other two models.

To confirm this rationale, the variations of the distance

from the sidewall of the punch to the upper assumed

surface of the sheet at the punch stroke of 70mm for

s/sy¼ 0.04, 0.14, and 0.26 are examined in Fig. 11. Two

points a and b are chosen on the sheet and the longitudinal

stress distributions through the thickness at the punch

stroke of 70mm for s/sy¼ 0.04, 0.14, and 0.26 are at the

same time examined in Figs. 12–14. The analytical stress

distributions [52,53] are also shown in the figures. It is

noted that point a is that immediately after experiencing

the unbending. When using the conventional polyhedral

model, the sheet already contacts the sidewall of the punch

for s/sy¼ 0.26, inducing the apparent reverse bending for

all the conditions (Fig. 11(a)). Consequently, although the

simulated stresses are in good agreement with the

analytical ones at point a (z ¼ 8.9mm) (Figs. 12(a), 13(a),

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-0.50 0.511.5

-10

-5

0

5

10

-0.50 0.51 1.5

-0.04

-0.02

0

0.02

0.04

00.51

Normal vector error ?n%

Dimensionless length from C l

Dimensionless length from C l

Nagata patch

Conventional polyhedral

Normal vector error ?n%

New polyhedral

Nagata patch

Conventional polyhedral

New polyhedral

-0.04

-0.02

0

0.02

0.04

00.51

Normal vector error ?n%

Dimensionless arc length from C toD l

Nagata patch

New polyhedral

Normal vector error ?n%

Dimensionless arc length from C to D l

Nagata patch

New polyhedral

Die shoulderSidewallUpper part

Die shoulder Sidewall Upper part

CD

CD

CD

CD

Fig. 8. Normal vector error distributions at die shoulders. (a) Die A and (b) die B. (Figures on the right side are emphasized figures for the ranges of

?0.04odno0.04 and 0olo1.0. Only the results of the new polyhedral and Nagata patch models are shown.)

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Page 12

14(a)), they decrease at the sidewall (point b, z ¼ 48.3mm)

(Figs. 12(b), 13(b), 14(b)) for all the conditions. Since the

reverse bending increases as the initial tensile stress

decreases, the simulated stress also decreases at the sidewall

as the initial tensile stress decreases.

When using the Nagata patch and new polyhedral models,

the reverse bending is not observed for s/sy¼ 0.26, while is

observed for s/sy¼ 0.14 and 0.04 (Fig. 11(b), (c)). Conse-

quently, the simulated stresses remain unchanged even at the

sidewall (point b) for s/sy¼ 0.26 (Fig. 14(b)), while decrease

largely for s/sy¼ 0.04 (Fig. 12(b)). Since the reverse bending

shown for s/sy¼ 0.14 is minute, the simulated stresses

hardly decrease at the sidewall (Fig. 13(b)). The occurrence

timing of the reverse bending in the conventional polyhedral

model is clearly different from the others, resulting in the

different stress distributions at the sidewall, and ultimately in

the smaller curvature after springback for s/syo0.36, as

shown in Fig. 9.

4.3.2. Simulation results for die B

Fig. 15(a) shows the deformed profiles after springback

for die B for s/sy¼ 0.14. Unlike the case of die A

(Fig. 9(a)), the simulation result for the conventional

polyhedral model agrees well with that of the Nagata patch

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Nagata patch

Conventional

polyhedral

New polyhedral

Profile before springback

0

5

10

15

20

0 0.20.40.6

Ratio of initial tensile stress and yield stress ?/?y

Sidewall Curvature ?/m-1

: Nagata patch

: Conventional polyhedral

: New polyhedral

Fig. 9. Simulated results for die A. (a) Deformed profiles after springback

for s/sy¼ 0.14 in mm. (b) Relationship between the sidewall curvature

after springback and the initial tensile stress.

0

0.04

0.08

0.12

0.16

0.2

0.24

020406080100

Distance from the symmetric plane l /mm

Curvature ?/m-1

Sidewall Flange

Die shoulder

Punch shoulder

Punch bottom

×1000

Fig. 10. Distribution of curvature along the deformed profile after

springback.

Sheet

z

Punch

d

0

0.05

0.1

0.15

0.2

0.25

02030 40506070

0

0.05

0.1

0.15

0.2

0.25

010 20 3040 50 6070

0

0.05

0.1

0.15

0.25

0102030 40506070

z/mm

0.04

0.26

?/?y = 0.14

d/mm

Sidewall Punch bottomFlange

z/mm

?/?y = 0.04

0.26

0.14

d/mm

SidewallPunch bottom Flange

z/mm

?/?y = 0.04

0.14

0.26

d/mm

SidewallPunch bottom Flange

0.2

10

Fig. 11. Variation of distance from the sidewall of the punch to the sheet

at the punch stroke of 70mm for die A. (a) Conventional polyhedral

model, (b) new polyhedral model, and (c) Nagata patch model.

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

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Page 13

model, while that of the new polyhedral model differs from

the others and yields much smaller springback at the

sidewall. The relationship between the sidewall curvature

after springback and the initial tensile stress is examined in

Fig. 15(b). The result for the conventional polyhedral

model tolerably agrees with that of the Nagata patch

model, while that of the new polyhedral model largely

differs from the other two results and the peak curvature

occurs at higher initial tensile stress.

As is the case with the conventional polyhedral model for

die A, this difference is due to the fact that the reverse bending

starts occurring at higher initial tensile stress for the new

polyhedral model. This is supported by Figs. 16–19, which

show the deformed profiles at the sidewall and the longitudinal

stress distributions through the thickness at points a and b at

the punch stroke of 70mm for s/sy¼ 0.04, 0.14, and 0.26,

respectively. For the new polyhedral model, the reverse

bending is not induced for s/sy¼ 0.26 but for s/sy¼ 0.04

and 0.14 (Fig. 16(b)). Consequently, the simulated stress at the

sidewall (point b, z ¼ 48.3mm) remains unchanged from point

a (z ¼ 8.9mm) for s/sy¼ 0.26 (Fig. 19), while decreases for

s/sy¼ 0.04 and 0.14 (Figs. 17 and 18). For the conventional

polyhedral and Nagata patch models, the reverse bending

and the decrease in the simulated stress are clearly induced

only for s/sy¼ 0.04, and these results fairly agree with each

other.

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Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

a

b

0

Fig. 12. Longitudinal stress distributions at points a and b for s/sy¼ 0.04 for die A. The solid line is the analytical result [52,53]. J, m, and ~ are the

results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.

-0.5

0

0.5

-20002000

-0.5

0

0.5

-200002000

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

a

b

0

Fig. 13. Longitudinal stress distributions at points a and b for s/sy¼ 0.14 for die A. The solid line is the analytical result [52,53]. J, m, and ~ are the

results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.

-0.5

0

0.5

-20002000

-0.5

0

0.5

-20000 2000

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

a

b

0

Fig. 14. Longitudinal stress distributions at points a and b for s/sy¼ 0.26 for die A. The solid line is the analytical result [52,53]. J, m, and ~ are the

results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

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Page 14

4.3.3. Discussion

We recall that the die shoulder is finely discretized in die

A, and somewhat roughly in die B, as shown in Fig. 5. The

results described in Sections 4.3.1 and 4.3.2 clearly show

that the simulation results including the deformed profile

after springback are strongly affected by the tool modeling.

It is interesting that both the new and conventional

polyhedral models show opposite tendencies between dies

A and B. The occurrence of the reverse bending is in

general governed by forming conditions, such as die radius

and blank holding force which corresponds to the initial

stretching force in the present simulation. However, the

differences shown in the present results can be rather

explained by the tool modeling errors shown in Figs. 7

and 8. Fig. 20 shows the variations of the distance from the

exact die shoulder to the sheet nodes at the punch stroke of

70mm for dies A and B for s/sy¼ 0.14. Note that negative

distance indicates that the sheet node penetrates the exact

die shoulder. As for die A, the shape errors for the

conventional polyhedral model are same as those of the

new polyhedral model, while the normal vector errors are

not only larger than those of the new polyhedral model

but also negative in the whole region (Fig. 8(a)). Due to

these large negative errors, when using the conventional

polyhedral model, the contact nodes slide in the direction

releasing from the die shoulder, and hence the sheet does

not fit the die shoulder firmly, as shown in Fig. 20(a).

Therefore it becomes easier to induce the reverse bending.

On the other hand, when using the new polyhedral model,

the contact nodes slide almost exactly tangent to the die

shoulder in each increment since the normal vectors are

much more accurate, thus fitting the die shoulder well.

Therefore the reverse bending is more difficult to be

induced.

As for die B, in the new polyhedral model, the normal

vectors are as accurate as that of die A (Fig. 8), while the

shape errors are larger than those of die A (Fig. 7). Due to

these large shape errors, the sheet slightly penetrates the

exact die shoulder as shown in Fig. 20(b), thus being

geometrically easier to release from the die shoulder and

ultimately to induce the reverse bending. In the conven-

tional polyhedral model, although both the shape and

normal vector errors are much larger than those of die A,

the effect of the normal vector errors is dominant to the

deformation due to their magnitude. Since the normal

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New polyhedral

Conventional

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Nagata patch

Profile before springback

0

5

10

15

20

00.20.40.6

Ratio of initial tensile stress and yield stress ?/?y

Sidewall Curvature ?/m-1

: Nagata patch

: Conventional polyhedral

: New polyhedral

Fig. 15. Simulated results for die B. (a) Deformed profiles after spring-

back for s/sy¼ 0.14 in mm and (b) Relationship between the sidewall

curvature after springback and the initial tensile stress.

Sheet

z

0

0.05

0.1

0.15

0.2

0.25

0 203040506070

0

0.05

0.1

0.15

0.2

0.25

010 2030 40506070

0

0.05

0.1

0.15

0.25

0 1020 304050 6070

z/mm

d/mm

SidewallPunch bottomFlange

z/mm

?/?y = 0.04

d/mm

SidewallPunch bottomFlange

z/mm

d/mm

SidewallPunch bottomFlange

0.2

10

?/?y = 0.04

0.26

0.14

0.14

0.26

0.26

0.14

?/?y = 0.04

Punch

d

Fig. 16. Variation of distance from the sidewall of the punch to the sheet

at the punch stroke of 70mm for die B. (a) Conventional polyhedral

model, (b) new polyhedral model, and (c) Nagata patch model.

T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192

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Page 15

vector errors take positive and negative values almost

equally, the effect of the normal vector error may be

cancelled by chance, thus fitting the die shoulder well

(Fig. 20(b)). Therefore the reverse bending is more difficult

to be induced.

The above mechanisms show that not only the shape but

also the normal vectors apparently play quite important

roles in the simulation accuracy of springback, and hence it

is clear that the use of polyhedral model is not always

acceptable since this cannot escape from the possibility of

affecting the simulation result in any manner.

The simulation results for the Nagata patch model yield

quite different tendencies from those of the polyhedral

models. As shown in Figs. 9, 11(c), 15, and 16(c), the

results for dies A and B show similar tendencies. This

correspondence can also be seen in Fig. 20, which shows

that the sheet fits the die shoulder well both for dies A and

B. Fig. 21 shows the comparisons of the relationship

between the sidewall curvature after springback and the

initial tensile stress between dies A and B for each tool

modeling. The large differences are observed both for the

conventional and new polyhedral models as explained

above. On the other hand, when using the Nagata patch

model, the two results fairly agree with each other,

although a slight difference is observed at the low initial

tensile stress region. These results show the level of

accuracy of the shape and the normal vectors are adequate

in the Nagata patch model both for dies A and B for the

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Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

a

b

0

Fig. 18. Longitudinal stress distributions at points a and b for s/sy¼ 0.14 for die B. The solid line is the analytical result [52,53]. J, m, and ~ are the

results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.

-0.5

0

0.5

-20002000

-0.5

0

0.5

-20000 2000

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

a

b

0

Fig. 19. Longitudinal stress distributions at points a and b for s/sy¼ 0.26 for die B. The solid line is the analytical result [52,53]. J, m, and ~ are the

results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.

-0.5

0

0.5

-20002000

-0.5

0

0.5

-200002000

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

Longitudinal stress ?l/MPa

Distance from the

mid-surface h /mm

a

b

0

Fig. 17. Longitudinal stress distributions at points a and b for s/sy¼ 0.04 for die B. The solid line is the analytical result [52,53]. J, m, and ~ are the

results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.

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189