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Materials Science and Engineering A 479 (2008) 197–212

Bulk metal forming process simulation based onrigid-plastic/viscoplastic element free Galerkin method

Ping Lu, Guoqun Zhao ∗, Yanjin Guan, Xin Wu

Mold & Die Engineering Technology Research Center, Shandong University, Jinan 250061, China

Received 1 February 2007; received in revised form 14 June 2007; accepted 19 June 2007

Abstract

A rigid-plastic/viscoplastic element free Galerkin method is established based on the flow formulation for rigid-plastic/viscoplastic materials to

simulate bulk metal forming processes. According to incomplete generalized variational principle, stiffness equations and solution formulas arederived. The transformation method is adopted to impose the essential boundary condition in the local coordinate system. The arctangent frictional

model is used to implement the frictional boundary conditions, and the transform matrix from the global coordinate system to local coordinate

system is given. The analysis software is developed. The method for dealing with the rigid region is introduced. Pressure projection method is used

to solve volumetric locking and pressure oscillation problems. The contact and detachment criterion for the workpiece and die is given to judge the

contact state. Numerical examples of bulk metal forming process are analyzed based on rigid-plastic/viscoplastic element free Galerkin method.

The material flow, field variable distributions, etc. are analyzed. The numerical analysis results obtained by the method proposed in the paper are

in good agreement with those obtained by the finite element method and experiment. The correctness of the presented method is demonstrated.

© 2007 Elsevier B.V. All rights reserved.

Keywords: Element free Galerkin method; Meshless; Rigid-plastic/viscoplastic; Bulk metal forming; Numerical simulation analysis

1. Introduction

Nowadays, numerical simulation plays a more and more

important role in metal forming process with the development

of numerical analysis and computer technology. Among these

numerical simulation methods for metal forming, finite ele-

ment method (FEM) is used widely. However, because of high

mesh-dependence, FEM encounters some difficulties that the

precision and efficiency of the simulation degrade when the

meshes become severely distorted, especially in handling large

deformations. In this case, remeshingthat is regardedas aneffec-

tive way is necessary to complete the whole process simulation.

Although remeshing techniques have made some development,

remeshing usually leads to not only the loss of the accuracy but

also the increase of time-consuming. In addition, the remeshing

techniques in the three-dimensional metal forming simulation

still need further investigation.

In order to avoid the above problems, a new numerical

computational method—mesh free (meshless) [1] method was

∗ Corresponding author. Tel.: +86 531 88393238; fax: +86 531 88395811.

E-mail address: [email protected] (G. Zhao).

proposed. The main characteristic of the method is that thedomain of the problem is discretized by a set of nodes (par-

ticles). The approximation field function is obtained through

information of nodes not meshes.

In recent years, the meshless method develops promptly, and

many international computational mechanics researchers have

devoted themselves to the development of meshless method,

and made lots of innovative contributions not only to theoreti-

cal researches [2–14] but also engineering applications [15–26].

Many progresses of meshless applications are made in the field

of metal forming processes [27–48].

Some researchers have applied meshless method to solve

metal forming problems and made certain achievements. Chen

et al. [27–29] applied reproducing kernel particle meshless

method (RKPM) in metal forming problems such as two-

dimensional ring compression test and upsetting, in which

material was considered as perfectly plastic material. Com-

parison with experiments demonstrated the performance and

effectiveness of themethod in metal forming. Bonet andKulase-

qaram [30] addressed corrected smooth particle hydrodynamics

(CSPH) method and applied it to metal-forming simulations.

The effectiveness of themethod insimulating metal forming was

demonstrated by the numericalexamples:planestrainupsetting,

0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved.

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198 P. Lu et al. / Materials Science and Engineering A 479 (2008) 197–212

plane strain/axisymmetric forging. Xiong et al. [31–35] used

reproducing kernel particle meshless method and element free

Galerkin (EFG) method for analyzing slightly compressible

rigid-plasticmaterialsplanestrain rolling.Thepenalty technique

wasutilizedfor enforcingtheessentialboundary conditions. The

main variables to the rolling process were calculated, and the

results were in agreement with experimental data. Park [36,37]

usedthe lagrangianreproducing kernel particle method (RKPM)

to simulate material processing. A comparative study between

elasto-plastic and rigid-plastic RKPM methods was made, and

consistent results from elasto-plastic and rigid-plastic simula-

tions for the metal forming application were obtained. Guo et

al. [38,39] applied the rigid-plastic point collocation method

to the analysis of plane strain forging and backward extrusion

processes, and verified the obtained results by comparing with

a rigid-plastic finite element solution. Alfaro et al. [40] pre-

sented an application of the meshless natural element method in

simulating three-dimensional aluminum extrusion, and used ␣-

shape-based approach to extract the geometry of the domain at

each time step, and illustrated the potential of the method usingsome examples. Kwon et al. [41,42] introduced the least-squares

meshfree method for the analysis of elasto-plastic and rigid-

plastic metal forming. The residuals were represented in a form

of first-order differential system using displacement/velocity

and stress components as nodal unknowns. The main benefit

of the method was that it did not employ structure of extrin-

sic cells for any purpose. Li et al. [43] utilized a lower order

integration scheme and a particle to segment contact algorithm

in 3D bulk forming numerical simulation analysis by using

reproducing kernel particle method. Liu et al. [44] used the

rigid-plastic reproducing kernel particle method for the anal-

ysis of three-dimensional bulk metal forming. The results werein good agreement with conventional finite element predictions.

Wang et al. [45] presented a parallel meshless method based

on the theory of the reproducing kernel particle method for

3D bulk forming process. The parallel contact search algorithm

was also presented, and the simulation results demonstrated the

efficiency of the parallel reproducing kernel particle method.

Li and co-workers [46] employed reproducing kernel particle

method for simulating upsetting and rolling process based on

the formulation of slightly compressible rigid plastic materials.

The meshless method is a node-based numerical method and

does not need mesh. From the current researches mentioned

above, the meshless method shows obvious merits, such as

simple preprocessing and post-processing as well as high com-putational precision in the numerical simulation of bulk metal

forming processesespecially large deformation bulkmetal form-

ing processes in which severe distortion of meshes often occurs

duringconventional finiteelement method analysis.Researchers

achieved remarkable progresses on the application of mesh-

less method to metal forming simulations. In the study done by

researchers, themain material modelsareperfectlyplastic mate-

rial, slightly compressible rigid-plastic material, elasto-plastic

and rigid-plastic material. And the main meshless method is

RKPM.

For bulk metal forming processes, such as forging, rolling,

extrusion,drawing,etc.,the portion of theworkpieceundergoing

plastic deformation is much larger than the portion undergoing

elastic deformation. Thus, the elastic deformation can be gen-

erally neglected. The rigid-plastic/viscoplastic material model

that neglects elastic deformation and simplifies the solution

of equations is widely used in the simulations of bulk metal

forming processes [49]. Element free Galerkin (EFG) has the

character of high stability and high accuracy. The paper com-

bines the flow theory and element free Galerkin (EFG) method,

and establishes rigid-plastic/viscoplastic element free Galerkin

(EFG) method. The related mathematic formulas, as well as key

algorithms and technologies are given according to flow for-

mulation for rigid-plastic/viscoplastic materials. Equal channel

angular pressing process and cubic billet upsetting process are

analyzed by rigid-plastic/viscoplastic EFG method numerically.

The related mechanics parameters are calculated and detailed

comparisons with the results obtained by finite element method

and experiment are made to validate the effectiveness of the

method presented in the paper.

2. Rigid-plastic/viscoplastic element free Galerkin

method

2.1. Approximation of velocity field

Using moving least-squares (MLS) scheme, the velocity

approximation uh(x) at any point x in the sub-domainx can be

defined by

uh(x) =

N I =1

I (x)dI = T (x)d (1)

where d I is “generalized” velocity vector, I (x) is the shapefunction of node x I and N is the total number of nodes, I is the

I th node.

(x) = [1(x)2(x) . . . N (x)]T (2)

d = [d1 d2 . . . dN ]T (3)

with

I (x) =

⎡

⎢⎣ΦI (x) 0 0

0 ΦI (x) 0

0 0 ΦI (x)

⎤

⎥⎦(4)

dI = [ d xI d yI d zI ]T

(5)

Due to the lack of Kronecker delta properties in the ele-

ment free Galerkin methodshape functions, namelyΦ I (x J ) = δ IJ

and d I = uh(x I ), the essential boundary conditions cannot be

imposed directly. In order to exert the boundary conditions

directly, the transformation method presented by Chen et al.

[47] is adopted to modify shape function. Then the relationship

between the nodal value u I and the “generalized” velocity d I is

established and expressed as

u =

T

d (6)



P. Lu et al. / Materials Science and Engineering A 479 (2008) 197–212 199

where

u = [uhx(x1) uh

y(x1) uhz (x1) . . . uh

x(xN ) uhy(xN )

uhz (xN )] (7)

=

⎡⎢⎢⎢⎢⎣Φ1(x1)I Φ1(x2)I . . . Φ1(xN )I

Φ2(x1)I Φ2(x2)I . . . Φ2(xN )I

......

. . ....

Φn(x1)I Φn(x2)I . . . Φn(xN )I

⎤⎥⎥⎥⎥⎦ (8)

where I is a unit vector. From Eq. (6), the following equation

can be derived.

d = −T u (9)

Substituting Eq. (9) into Eq. (1), we obtain

uh(x) =

N

I =1

I (x)uI = T (x)u (10)

where

(x) = (x)−T (11)

2.2. Meshless formulation for rigid-plastic/viscoplastic

models

2.2.1. Variational formulation for rigid-plastic/viscoplastic

materials

When the plastic deformation for the rigid-plastic/visco-

plastic material occurs, basic equations [49]: equilibrium equa-

tions, compatibility conditions, constitutive equations, yieldcriterion, incompressibility condition and boundary conditions

should be satisfied.

By employing penalty function method to constrain

incompressibility condition, incomplete generalized variational

principle considers that among the admissible velocities which

satisfy velocity boundary conditions and compatibility condi-

tions, the actual solution make the total energy rate functional

have an minimum value. It is expressed as

Π =

Ω

σ̄ ̇̄ε dΩ +α

2

Ω

(ε̇V )2 dΩ −

S F

F̄ iui dS

for rigid-plastic materials (12a)

Π =

Ω

U (ε̇ij )dΩ +α

2

Ω

(ε̇V )2 dΩ −

S F

F̄ iui dS

for rigid-viscoplastic materials (12b)

where σ̄ denotes the effective stress. ˙̄ε is the effective strain-rate.

α is a very large positive constant called penalty constant, a rec-

ommended value for α is 105–106. ε̇V represents the volumetric

strain-rate, F̄ i is surface tractions.

Eqs. (12a) and (12b) can be written as,

Π = Π D + Π P + Π f (13)

where

Π P =α

2

Ω

(ε̇V )2 dΩ (14)

Π f = −

S F

F̄ iui dS (15)

Π D =

Ω

σ̄ ̇̄ε dΩ for rigid-plastic materials (16a)

Π D =

Ω

U (ε̇ij )dΩ U (ε̇ij ) =

̇̄ε

0σ̄ d˙̄ε


Introducing a similar method as in finite element method, the

stiffness equations and other equations for meshless solution of

metal forming process can be established.

Considering the relationship of velocity and strain-rate and

using Eq. (10), strain-rate ε̇ can be expressed as

ε̇ = [ B1(x) B2(x) . . . BN (x) ]u = B(x)u (17)

where B(x) is strain-rate matrix and expressed as

B(x) = [ B1(x) B2(x) . . . BN (x) ] (18)

With

BI (x) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ψ I,x(x) 0 0

0 Ψ I,y(x) 0

0 0 Ψ I,z(x)

Ψ I,y(x) Ψ I,x(x) 0

0 Ψ I,z(x) Ψ I,y(x)

Ψ I,z(x) 0 Ψ I,x(x)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

Ψ I , x (x), Ψ I , y(x) and Ψ I , z(x) are the derivatives of transformed

shape function of node I with respect to x , y and z, respectively.

The matrix of the effective strain-rate ˙̄ε is defined as

˙̄ε = (ε̇T Dε̇)1/2

(20)

Introducing Eqs. (17)–(20), the following equation is

obtained

˙̄ε = (uT BT DBu)1/2

= (uT Pu)1/2

(21)

where D is a matrix relating stresses with strain-rates, namely

D =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

2/3 0 0 0 0 00 2/3 0 0 0 0

0 0 2/3 0 0 0

0 0 0 1/3 0 0

0 0 0 0 1/3 0

0 0 0 0 0 1/3

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(22)

P(x) = BT (x)DB(x) ≡

⎡⎢⎢⎢⎢⎣

P11 P12 . . . P1N

P21 P22 . . . P2N

......

. . ....

PN 1 PN 2 . . . PNN

⎤⎥⎥⎥⎥⎦

(23)




PIJ = BT I (x)DBJ (x) ≡

⎡⎢⎣

P IJxx P IJxy P IJxz

P IJyx P IJyy P IJyz

P IJzx P IJzy P IJzz

⎤⎥⎦ (24)

Volumetric strain-rate ε̇V can be expressed as follows

ε̇V = ε̇x + ε̇y + ε̇z (25)

Substituting the corresponding strain-rate components in Eq.

(17) into Eq. (25) results in

ε̇v = [Ψ 1,x(x) Ψ 1,y(x) Ψ 1,z(x) . . .

Ψ N,x(x) Ψ N,y(x) Ψ N,z(x)] u (26)

The above equation can be rewritten as

ε̇V = [ CT 1 CT

2 . . . CT N ]u = CT u (27)

where

C = [ CT 1 CT

2 . . . CT N ]

T (28)

CI = [ Ψ I,x(x) Ψ I,y(x) Ψ I,z(x) ]T

(29)

2.2.2. The variation of D

The first-order partial derivative of D is defined as follows

∂Π D

∂uI

=

∂Π D

∂uxI

∂Π D

∂uyI

∂Π D

∂uzI

T

(30)

By introducing Eqs. (16)–(24) and Eq. (30), the following

equations can be obtained.

∂Π D

∂uI

= Ω

σ̄ ∂ ˙̄ε

∂uI

dΩ for rigid-plasticmaterials (31a)

∂Π D

∂uI

=∂

∂uI

Ω

U (ε̇ij )dΩ

=

Ω

σ̄ ∂ ˙̄ε

∂uI

dΩ forrigid-viscoplasticmaterials (31b)

where

∂ ˙̄ε

∂uI

=1

˙̄ε[ PI 1 PI 2 . . . PIN ]u (32)

Substituting Eq. (32) into Eq. (31) results in

∂Π D∂uI

=

Ω

σ̄ ˙̄ε

[ PI 1 PI 2 . . . PIN ]u dΩ (33)

Therefore

δΠ D = δuT I

Ω

σ̄

˙̄ε[ PI 1 PI 2 . . . PIN ]u dΩ (34)

The second-order partial derivative of D can be evaluated

as

∂2Π D

∂uI ∂uJ

=

Ω

σ̄

˙̄εPIJ dΩ +

Ω

∂

∂ ˙̄ε

σ̄

˙̄ε

1

˙̄ε[ PI 1 . . . PIN ]uuT

⎡⎢⎢⎣

P1J

...

PNJ

⎤⎥⎥⎦

dΩ (35)

where

∂

∂ ˙̄ε

σ̄

˙̄ε

= −

σ̄

˙̄ε2

for rigid-plasticmaterials (36a)

∂

∂ ˙̄ε

σ̄

˙̄ε

=

1

˙̄ε

∂σ̄

∂ ˙̄ε−

σ̄

˙̄ε2


According to Eq. (35), the second-order variation of D isexpressed as follows

δ2Π D = δuT I

⎧⎪⎪⎨⎪⎪⎩

Ω

σ̄

˙̄εPIJ dΩ +

Ω

∂

∂ ˙̄ε

σ̄

˙̄ε

×1

˙̄ε[ PI 1 . . . PIN ]uuT

⎡⎢⎢⎣

P1J

...

PNJ

⎤⎥⎥⎦ dΩ

⎫⎪⎪⎬⎪⎪⎭

δuJ (37)

2.2.3. The variation of P

The substitution of Eq. (27) in Eq. (14) yields

Π P =α

2

Ω

(CT u)2

dΩ (38)

By employing a similar treating procedure as that of D, the

first-order partial derivative and variation can be evaluated as

∂Π P

∂uI

= α

Ω

CI CT u dΩ (39)

δΠ P = δuT I α

Ω

CI CT u dΩ (40)

and the second-order partial derivative and variation of P can

be evaluated as

∂2Π P

∂uI ∂uJ

= α

Ω

CI CT J dΩ = α

Ω

C̄IJ dΩ (41)

δ2Π P = δuT I α

Ω

CI CT J dΩδuJ = δuT

I α

Ω

C̄IJ dΩδuJ (42)

where

C̄IJ = CI CT J

=⎡⎢⎣

Ψ I,x(x)Ψ J,x(x) Ψ I,x(x)Ψ J,y(x) Ψ I,x(x)Ψ J,z(x)

Ψ I,y(x)Ψ J,x(x) Ψ I,y(x)Ψ J,y(x) Ψ I,y(x)Ψ J,z(x)Ψ I,z(x)Ψ J,x(x) Ψ I,z(x)Ψ J,y(x) Ψ I,z(x)Ψ J,z(x)

⎤⎥⎦(43)




2.2.4. Frictional contact conditions

In metal forming processes, the frictional conditions along

the workpiece–die interface are complex. It is difficult to han-

dle the frictional conditions by a standardized mathematical

formula. Commonly, the frictional force can be calculated by

some simple formulas. Theusually used friction models include

Coulomb law f = μ p, and the constant friction factor law f = mk ,

where, μ denotes frictional coefficient, p denotes the pressure

on the workpiece-die interface, m is called frictional factor, and

k represents shear yield stress.

The direction of the frictional stress is opposite to the direc-

tion of the relative velocity. There is a point (or a region) along

the die–workpiece interface where the velocity of the deform-

ing material relative to the die becomes zero, and the location of

this point (or region) depends on themagnitudes of the frictional

stress itself. The frictional stress at the neutral point (or region)

changes suddenly. The sudden change of the frictional stress

near the neutral point can cause difficulties in numerical calcu-

lation. The arctangent friction law [49] defined as follows can

solve the difficulties effectively. So it is employed in the paperto treat the friction conditions on the workpiece–die interface.

f = −mk

2

πtan−1

ws

u0

(44)

where ws is the sliding velocity of material relative to the die

velocity, u0 is an very small positive constant and recommended

as 10−3 to 10−4.

In the three-dimensional problems, establish the tangent-

normal localcoordinatesystem s, t , n,whichmakeupoftheright

hand coordinate system for contact node. And s denotes the first

tangential direction, t represents the second tangential direction,

and n denotes the normal direction and its positive direction isfrom workpiece to die, respectively. The first and second tan-

gential direction components of relative slipping velocity on the

contact interface for the contact node are defined as us(x) and

ut (x). So ws(x) can be defined as

ws(x) =

u

s(x)2 + ut (x)2 (45)

And us(x) and u

t (x) can be approximated as follows using

moving least-squares (MLS) scheme, namely

us(x) =

N

I =1

Ψ I (x)usI (46)

ut (x) =

N I =1

Ψ I (x)utI (47)

where usI and u

tI indicate the first and second tangential direc-

tion components of relative slipping velocity for node I which

is on the frictional contact interface, and can be evaluated as

usI = u

xI − sT ud (48)

utI = u

yI − tT ud (49)

where uxI and u

yI are the first and second tangential direc-

tion components of slipping velocity for node I , ud represents

the velocity of die, s is the first tangential direction unit vec-

tor on the contact interface expressed as s = ( sx sy sz )T

,

t is the second tangential direction unit vector expressed as

t = ( t x t y t z )T

.

The relationship for velocity field between global coordinate

system and local coordinate system are given by

u = Tu (50)

where

u = [u1 u

2 . . . uN ]

T (51)

with

uI = [ u

xI uyI u

zI ]T

(52)

and

T =

⎡⎢⎢⎢⎢⎣

T1 0 . . . 0

0 T2 . . . 0

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

0 0 0 TN

⎤⎥⎥⎥⎥⎦ (53)

with

TI =

⎡⎢⎣

t I 11 t I 12 t I 13

t I 21 t I 22 t I 23

t I 31 t I 32 t I 33

⎤⎥⎦ (54)

In the above matrix, t Ijm is the cosine value of angle from

direction j of local coordinate system to direction m of global

coordinate system. Because T is an orthogonal matrix, it can be

obtained

u = T−1u = TT u (55)

Substituting Eq. (44) into Eq. (15), the following Eq. (56) is

obtained

Π f =

S c

ws

0mk

2

πtan−1

ws

u0

dws

dS (56)

The first-order partial derivative of f in local coordinate

system can be evaluated as

∂Π f

∂uxI

= S c

mkΨ I

2

πtan−1

ws

u0 u

s

ws

dS (57)

∂Π f

∂uyI

=

S c

mkΨ I

2

πtan−1

ws

u0

u

t

ws

dS (58)

and the second-order partial derivative of f can be obtained as

follows

∂2Π f

∂uxI ∂u

xJ

=

S c

mk2

π

u0

u20 + w2

s

Ψ I Ψ J

usu

s

w2s

+ tan−1

×

ws

u0

Ψ I Ψ J

ws

− tan−1

ws

u0

Ψ I Ψ J

usu

s

w3s

dS

(59)




∂2Π f

∂uxI ∂u

yJ

=

S c

mk2

π

u0

u20 + w2

s

Ψ I Ψ J u

sut

w2s

− tan−1

ws

u0

Ψ I Ψ J

usu

t

w3s

dS (60)

∂2Π f

∂uyI ∂u

xJ

=

S c

mk2

π

u0

u20 + w2

s

Ψ I Ψ J

ut u

s

w2s

− tan−1

ws

u0

Ψ I Ψ J

ut u

s

w3s

dS (61)

∂2Π f

∂uyI ∂u

yJ

=

S c

mk2

π

u0

u20 + w2

s

Ψ I Ψ J

ut u

t

w2s

+ tan−1

ws

u0

×Ψ I Ψ J

ws

− tan−1ws

u0Ψ I Ψ J

ut u

t

w3

s dS (62)

In the two-dimensional problems, s denotes the tangential

direction, n denotes the normal direction and its positive direc-

tion is also from workpiece to die. Thus, the following Eqs. can

be obtained for two-dimensional problems.

ws(x) = us(x) (63)

∂Π f

∂uxI

=

S c

mkΨ I

2

πtan−1

ws

u0

dS (64)

∂2Π f

∂u

xI

∂u

xJ

= S c

mk2

π

u0

u2

0+ w2

s

Ψ I Ψ J dS (65)

TI =

t I 11 t I 12

t I 21 t I 22

(66)

2.2.5. Stiffness equation

Applying incomplete generalized variation principle, make

thefirst-order variation of the total energyrate functional vanish,

namely

δΠ = δΠ D + δΠ P + δΠ f = 0 (67)

Introducing Eqs. (34) and (40) into above equation and con-

sidering arbitrariness of δu, results in Ω

σ̄

˙̄ε[ PI 1 . . . PIN ]u dΩ + α

Ω

CI CT u dΩ +

∂Π f

∂uI

= 0

(68)

The above equation is called the meshless stiffness equation

of rigid-plastic/viscoplastic metal forming problems. The New-

Raphson iterative procedure is implemented to solve the above

non-linear equation until converged results are obtained. The

Eq. (68) can be rewritten as

∂Π

∂uI u=u

(n−1)

+ ∂2Π

∂uI ∂uJ u=u

(n−1)

u(n)J = 0 (69)

Introducing Eqs. (35) and (41) to Eq. (69), Eq. (69) can be

rewritten and re-arranged as⎧⎪⎪⎨

⎪⎪⎩

Ω

σ̄

˙̄εPIJ dΩ+

Ω

∂

∂ ˙̄ε

σ̄

˙̄ε

1

˙̄ε[ PI 1 . . . PIN ]u(n−1)u(n−1)T

×

⎡⎢⎢⎣

P1J

...

PNJ

⎤⎥⎥⎦ dΩ+α

Ω

C̄IJ dΩ+∂2Π f

∂uI ∂uJ

⎫⎪⎪⎬⎪⎪⎭u

(n)J

= −

Ω

σ̄

˙̄ε[ PI 1 . . . PIN ]u(n−1) dΩ

− α

Ω

CI CT u(n−1) dΩ−

∂Π f

∂uI

(70)

namely

S +

∂2Π f

∂u∂uT

u(n)

= f −

∂Π f

∂u (71)

where

SIJ =

Ω

σ̄

˙̄εPIJ dΩ +

Ω

∂

∂ ˙̄ε

σ̄

˙̄ε

×1

˙̄ε[ PI 1 . . . PIN ]u(n−1)u(n−1)T

⎡⎢⎢⎣

P1J

...

PNJ

⎤⎥⎥⎦ dΩ

+ α Ω C̄IJ dΩ (72)

f I = −

Ω

σ̄

˙̄ε[ PI 1 . . . PIN ]u(n−1) dΩnonumber (73)

− α

Ω

CI CT u(n−1) dΩ (73)

In order to employ frictional contact conditions directly, it

is need to assemble the stiffness equation in local coordinate

system. Hence, Eq. (71) is rewritten as

TSTT +∂2Π f

∂u∂uT u(n)= Tf −

∂Π f

∂u(74)

and further simplified as

K(n) u(n)= F(n) (75)

where

K(n) = TSTT +∂2Π f

∂u∂uT (76)

F(n) = Tf −∂Π f

∂u(77)

The expressions of ∂Π f

∂u and∂2Π f

∂u∂uT are obtained from Eqs.

(57)–(65).




The integrals of all above equations can be evaluated by

means of Gaussian quadrature through the utilization of back-

ground cells.

2.3. Treatment of the rigid region

In metal forming processes, the rigid region that is charac-terized by a very small value of effective strain-rate exists, in

which the value of the first term in Eq. (12a) or (12b) cannot be

calculated when the effective strain-rate approaches zero.

This difficulty is removed by the following linearconstitutive

equation (78) when the effective strain-rate of the I th node is

smaller than ˙̄ε0

ε̇ij =3

2

˙̄ε0

˙̄σ 0σ ij (78)

where σ̄ 0 = σ̄ (ε̄, ˙̄ε0), ˙̄ε0 takes an assigned limiting value.

Thus, in the rigid region the Eq. (31a) or (31b) is replaced by

the following equation (79).

∂Π D

∂uI

=

Ω

σ̄ 0˙̄ε0

˙̄ε

∂ ˙̄ε

∂uI

dΩ (79)

Then,

δΠ D = δuT I

Ω

σ̄ 0˙̄ε0

˙̄εδ˙̄ε dΩ (80)

Substituting Eq. (32) into Eq. (79) yields

∂Π D

∂uI

=

Ω

σ̄ 0˙̄ε0

[ PI 1 PI 2 . . . PIN ]u dΩ (81)

The first-order variation of D is expressed as follows

δΠ D = δuT I

Ω

σ̄ 0˙̄ε0

[ PI 1 PI 2 . . . PIN ]u dΩ (82)

Then the second-order partial derivative of D and the

second-order variation of D are replaced as follows

∂2Π D

∂uI ∂uJ

=

Ω

σ̄ 0˙̄ε0

PIJ dΩ (83)

δ2Π D = δuT I

Ω

σ̄ 0˙̄ε0

PIJ dΩδuJ (84)

2.4. Treatment of volumetric locking

In order to obtain satisfactory accuracy, a high-order integra-

tion rule is often used in meshless method. This often leads to an

over-constrained condition and causes volumetric locking and

pressure oscillation problems. Although increasing the size of

the influence domain can release volumetric locking, it cannot

solve the problem of pressure oscillation and deviates from the

local fitting character owned by meshless method. In addition,

thecomputational efficiency is reduced as theincreaseof thesize

of influence domain. In the paper, a releasing algorithm is estab-

lished based on pressure projection method that is proposed by

Chen [48] to solve volumetric locking and pressure oscillation

problems encountered in rigid-plastic/viscoplastic element free

Galerkin method.

Thealgorithm is realized by modifying the following penalty

function items p of total energy rate functional.

Π p =α

2 Ω(ε̇V )

2 dΩ =α

2 Ω(CT u)

2dΩ (85)

Volumetric strain-rate ε̇V is mapped onto each integration

domain Ωsx which isa lower order space spanned bya set offunc-

tions Q = {Q1(x), Q2(x),. . .,Qn(x)}. ε̇sV = {ε̇s

V 1, ε̇sV 2, . . . , ε̇s

Vn}T

is obtained by minimizing R(̇sV ) as follows

R(ε̇sV ) = ||ε̇V − Qε̇s

V ||2L2(Ωsx) (86)

where || · ||2L2(Ωs

x) is least-square norm in the integration domain

Ωsx.

And

∂R(ε̇sV )

∂ε̇sV

= 0 (87)

The following equation can be obtained

Msε̇

sV = Zs (88)

where

Ms =

Ωs

x

QT Q dΩ (89)

Zs =

Ωs

x

QT ε̇V dΩ (90)

and the projected volumetric strain-rate ε̇∗sV in the integration

domain Ωsx is calculated as follows

ε̇∗sV = Qε̇s

V = QMs−1Zs (91)

When thevolumetric strain-rate is projectedonto a constant field

Q = {1, 1,. . .,1}, the following is obtained

ε̇∗sV =

Ωs

xCT u dΩ

As

(92)

where As is the volume of Ωsx.

2.5. Dynamic adjustment of the boundary nodes

Thecontact anddetachment state between workpiece bound-

ary and die surface changes continually with the increase of deformation during metal forming processes. Therefore, it is

necessary to give the contact and detachment criterion for the

workpiece and die.

After the velocity field is convergent in one incremental step,

it needs to judge whether the free boundary nodes contact die

or not in this step, and whether the contact nodes separate from

the die or not.

The time increment t I for each boundary free node to con-

tact die should be calculated. Where I represents the I th node.

The problem can be solved by treating with the intersection of

a ray and the die surface. The starting point coordinate of the

ray is the current location coordinate expressed as [ x I , y I , z I ], and




the direction of the ray can be expressed by the velocity vec-

tor as [u xI − u xd , u yI − u yd , u zI − u zd ], in which [u xI ,u yI ,u zI ] is

the velocity components of the node, and [u xd ,u yd ,u zd ] is the

velocity components of the die. Assuming the coordinate of the

intersecting point is [ x c, y x , zc], the value of t I can be calculated

as follows

t I = [(xc − xI )2 + (yc − yI )2 + (zc − zI )2]1/2

[(uxI − uxd )2 + (uyI − uyd )

2 + (uzI − uzd )2]

1/2(93)

If t I is smaller than the time increment t , the I th node is

regarded as the contact one and imposed contact restriction.

Otherwise, the node is still free.

The normal stress σ n is used as the detachment and contact

criterion for the nodes that have been contact nodes in the last

step. If σ n ≥ 0, the nodes which have contacted with the die

surface in the last step separate from the die and become free in

the current step. And the restriction should be released. Or else,

the nodes are still regarded as the contact ones, and move along

the die surface.After thedetachmentandcontact judgment isdone, thegeom-

etry of the workpiece can be obtained by updating from the

previous value by

xI (t + t ) = xI (t ) + uxI t (94)

yI (t + t ) = yI (t ) + uyI t (95)

zI (t + t ) = zI (t ) + uzI t (96)

The effective strain of the I th node can be updated and

expressed as follows

ε̄I (t + t ) = ε̄I (t ) + ˙̄εI t (97)

where ε̄I is the effective strain of the I th node, and ˙̄εI is the

effective strain-rate.

3. Numerical examples

3.1. Equal channel angular pressing (ECAP)

The schematic diagram of ECAP is illustrated in Fig. 1. Φ

is die channel angle and usually ranges from 90◦ to 150◦, Ψ is

die corner angle and ranges from 0◦ to (180◦- Φ). The billet is

Fig. 1. The schematic diagram of ECAP process.

Fig. 2. The metal flow patterns by using EFG method at different strokes of

punch. (a) Stroke is 0 mm, (b) stroke is 15 mm, (c) stroke is 45 mm and (d)

stroke is 75 mm.

a cuboid solid with the length of 80 mm and the cross-section

dimension of 10mm × 10 mm. And the equal channel angular

pressing belongs to a plane strain problem. The material is pure

aluminum (Al 99.99%), and the material model is regarded as

rigid-plasticity. The flow stress of the material is

σ̄ = Cε̄n (98)

where C = 170MPa and n = 0.24.

In the simulation, the workpiece is discretized by 11 × 81nodes, friction factor m = 0.12, Φ is 90◦, Ψ is 37◦. The speed

of the punch is 2 mm/s, the time increment is 0.25 s and the

temperature is 20 ◦C. During the analysis, the Gauss quadrature

order is 3 × 3, linear basis function is employed.

Fig. 2 gives the metal flow patterns when the strokes of the

punch are 0, 15, 45, and 75 mm, respectively. Because of no

remeshing, the discrete nodes with the deformation increasing

can clearly form flow lines that can reflect the material locus. So

the flowing states of material are more visual by using the EFG

method.

Fig. 3. The distributions of effective strain in the workpiece by using EFG

method. (a) Stroke is 15 mm, (b) stroke is 45 mm and (c) stroke is 75 mm.




Fig. 4. The distributions of effective stress in the workpiece by using EFG

method. (a) Stroke is 15mm, (b) stroke is 45 mm and (c) stroke is 75 mm.

Fig. 3 shows effective strain contours of the workpiece when

the strokes of the punch are 15, 45 and 75 mm. The effective

strain is accumulated relying on the shear force when extrusion

part goes through the die corner with the increasing of punch

stroke to refine grains. It canbe observed that the effective strain

distributes more uniformly in the horizontal direction except

stub bar and two channels intersection regions of the extrusion

part. This is because shear deformation only occurs in the two

channels intersectionzones andit hardlyoccurs in other regions.

In the vertical direction the effective strain of the regions that

are close to the upper surface and the center portion is large, the

Fig. 5. The distributions of effective strain-rate in the workpiece by using EFG

method. (a) Stroke is 15 mm, (b) stroke is 45 mm and (c) stroke is 75 mm.

effective straindecreaseswith thedistance getting smaller to thelower surface.

Fig. 4 gives thedistribution contours of effective stressfor the

extrusion part atdifferentstrokes15,45 and75 mm,respectively.

Obviously, the effective stress in the die corner is more concen-

trated, and the value and the gradient of the effective stress are

large. The effective stress in the inner corner is larger than that

in the outer corner. These indicate the workpiece will suffer the

large force when it goes through the die corner. Therefore, the

appropriate materials should be selected during the die design,

and heat treatment process should be carried out to ensure the

carrying capacity and strength requirements of the dies.

Fig. 6. Different field variable distributions in the workpiece by using Deform (2D) when the stroke of the punch is 45 mm. (a) Distribution of effective strain, (b)

distribution of effective stress and (c) distribution of effective strain-rate.




Fig. 5 shows effective strain-rate contours of the workpiece

when thestrokesof thepunchare 15,45 and75 mm,respectively.

Theeffective strain-rate reduces gradually from the inner corner

to outer corner. This explains why the effective strain decreases

from the upper surface to the lower surface displayed in Fig. 3.

The numerical model is also simulated by the finite element

method software Deform (2D) to verify the correctness of the

established method. The workpiece is discretized by 800 quad-

rangle elements with 891 nodes. Other parameters are similar

with those during EFG method simulation. During the FEM

simulation, mesh distortion occurs when the stroke is 8.5mm,

and three times remeshing are needed in order to enable the

simulation analysis continue.

The effective strain, effective stress and effective strain-rate

obtained by Deform (2D) are shown in Fig. 6 when the stroke of

the punch is 45mm. Comparing Fig. 3(b), Fig. 4(b) and Fig. 5(b)

with Fig. 6(a)–(c), it finds that thedistributionsof themechanics

variables are in good agreement.

The equal channel angular pressing experiment is carried

out. The billet geometry with the dimension 10 mm × 10mm ×80 mm is similar with the simulation model, and the other

process parameters are almost the same as in the numerical

simulation. Fig. 7 gives the experimental dies. The forming

equipment is 300 kN electronic universal testing machine in the

experiment (shown in Fig. 8).

Fig. 9(a) and (b) give the geometric shapes of the work-

piece obtained by EFG analysis and experiment. The symbol

A-B signed in Fig. 9(a) and (b) indicates the shearing direction,

and the shearing direction gained from numerical simulation is

identical with that gained from actual experiment. The label C

indicates the zone in the circle is stub bar portion of the extrusion

part, and the geometric shapes obtained from numerical simula-tion and actual experiment, respectively, are similar. Label D in

Fig. 9(a) denotes the metal flowing direction calculated by EFG

method, and label D in Fig. 9(b) denotes texture streamline. It is

clear that both of the labels have the some trend. It can be seen

that the simulation results agree with the experiment results.

Fig. 7. The experimental die of square-section ECAP.

Fig. 8. The equipment for experiments—300kN testing machine.

3.2. Cubic billet upsetting

Upsettingprocess is a typical metal forming process. A three-

dimensional cubicbillet upsettingprocesswasanalyzedbyusing

the EFG method proposed in the paper. The predicted results

were compared with those given by commercial FEM software-

Deform (3D).

The initial geometrical shape of the billet is shown in

Fig. 10. The size of the billet is 20 mm × 20mm × 20 mm and

11 × 11 × 11 divided nodes are used for the simulation. The

upper die moves downward with a velocity of −1 mm/s, while

the lower die is stationary. The arctangent friction model was

used for modeling the friction behavior on the workpiece–die

interface. The constant frictional factor m is selected as 0.3.

Due to the symmetry of geometry and deformation, only one

eighth of the billet discretized with 6 × 6 × 6 nodes is consid-

ered. In the analysis using EFG method, the Gauss quadrature

order is 4 × 4 × 4, linear basis function is used, the weight func-

tion is cubic spline weight function, and the sphere domains of

influence are employed. In order to verify the validity of the pre-

dicted solution of the EFG, the problem is simulated by FEM

software-Deform (3D) too. The FEM model is discretized by

Fig. 9. The numerical and experimental analysis of the workpiece. (a) The geo-

metric shapes of the workpiece obtained by EFG analysis. (b) The geometric

shapes of the workpiece obtained by experiment.




Fig. 10. The billet shape and the initially discretized meshless model.

999 tetrahedral elements with 233 nodes. The processes from

0 to 40% reduction in height are analyzed, and the incremen-tal step in reduction is 2% of the initial billet height. The billet

material is AISI 8620 steel and the temperature is 1100 ◦C. The

relationship between strain and stress is given as

σ̄ = C ˙̄εm

(99)

where C = 75.3 MPa and m = 0.134.

Fig. 11 describes the geometrical shapes of the deforming

material at different stages of deformation. Fig. 11(a)–(d) give

Table 1

Displacement of maximum bulge at 20% reduction in height

Displacement

(mm)

Absolute

error (mm)

Relative error

(%)

EFG 1.508070.03465 2.35167%

FEM-Deform (3D) 1.47342

Notes: Absolute error= EFGresult − FEMresult.Relativeerror = AbsoluteerrorFEMresult ×100%.

Table 2

Displacement of maximum bulge at 40% reduction in height

Displacement

(mm)

Absolute

error (mm)

Relative error

(%)

EFG 3.486730.00986 0.28359

FEM-Deform (3D) 3.47687

Notes: Absolute error= EFGresult − FEMresult.Relativeerror = AbsoluteerrorFEMresult

×

100%.

the geometrical shapes of the deforming material obtained byEFG and FEM-Deform (3D) at 20 and 40% reduction in height,

respectively. Clearly, the bulge at the stage of 20% reduction

in height is not very obvious. Meanwhile, the shape of the

formed part obtained from EFG is in good agreement with

that obtained from FEM-Deform (3D). When the deformation

becomes bigger, the bulge is obvious, and the agreement of the

shapes between EFG and FEM is still good. The displacement

values of maximum bulge analyzed by EFG and FEM-Deform

Fig. 11. Material flow patterns at different reductions in height. (a) The shape of the deforming body obtained by EFG at 20% reduction in height. (b) The shape of

the deforming body obtained by FEM-Deform (3D) at 20% reduction in height. (c) The shape of the deforming body obtained by EFG at 40% reduction in height.

(d) The shape of the deforming body obtained by FEM-Deform (3D) at 40% reduction in height.




Fig. 12. Velocity vector distributions of all nodes at 20 and 40% reduction in height. (a) Velocity vector distributions obtained by EFG at 20% reduction in height.

(b) Velocity vector distributions obtained by FEM-Deform (3D) at 20% reduction in height. (c) Velocity vector distributions obtained by EFG at 40% reduction in

height. (d) Velocity vector distributions obtained by FEM-Deform (3D) at 40% reduction in height.

(3D), and corresponding absolute errors and relative errors at

20% and 40% reduction in height are shown in Tables 1 and 2,

respectively. It can be seen that the displacement values of

maximum bulge analyzed by EFG is larger than those ana-

lyzed by FEM both at 20 and 40% reduction in height. But

the corresponding relative errors of the results calculated by

two methods are small, the relative error is 2.35167% at 20%

reduction in height and it is only 0.28359% at 40% reduction in

height.

Fig. 12 shows the velocity vector plots of all nodes at 20

and 40% reduction in height. It can be seen from these plots

that, the velocity fields of nodes evaluated by EFG match very

well with those evaluated by FEM. The directions and values

of velocity vector and the material flow trends for both EFG

and FEM are very close, too. The positions, velocity directions

and magnitudes of the characteristic nodes numbered as 1–8

in Fig. 12 are listed in Tables 3–6. The numerical comparisons

show that the results of EFG and FEM are in good agreement.

Table 3

The coordinates, velocity directions and values of the characteristic nodes (CN) at 20% reduction in height evaluated by EFG

Sequential number of CN Coordinate Velocity value (mm/s) Velocity direction

x y z α (◦) β (◦) γ (◦)

1 10.67954 0 8 1.10233 65.1172 90 155.117

2 0 0 8 1 90 90 180

3 0 10.68996 8 1.10577 90 64.7355 154.736

4 10.87233 10.87127 8 1.24936 64.8948 64.946 143.169

5 11.50735 0 0 0.83697 0 90 90

6 0 0 0 0 – – –

7 0 11.50807 0 0.83788 90 0 90

8 11.23201 11.23119 0 0.99152 44.9681 45.0319 90

Notes: α represents the angle between velocity direction of the node and x direction in global coordinate system; β represents the angle between velocity direction

of the node and y direction in global coordinate system; γ represents the angle between velocity direction of the node and z direction in global coordinate system.




Table 4

The coordinates, velocity directions and values of the characteristic nodes (CN) at 20% reduction in height evaluated by FEM


x y z α (◦) β (◦) γ (◦)

1 10.77209 0 8 1.10698 64.6035 90 154.604

2 0 0 8 1 90 90 180

3 0 10.75682 8 1.10275 90 65.0696 155.074 10.97056 10.98967 8 1.31254 63.1518 62.3356 139.63

5 11.47342 0 0 0.82969 0 90 90

6 0 0 0 0 – – –

7 0 11.46725 0 0.82473 90 0 90

8 11.24257 11.25626 0 0.97959 45.3089 44.6911 90



The comparisons of effective strain distributions obtained

by using the two methods at 20 and 40% reduction in height

are shown in Fig. 13, respectively. For two methods, the value

of effective strain in the central upper zone in contact with

the die is very small and usually called the dead zone, thematerial in this zone almost has no deformation. The material

in the middle zone and the corner area has a large effective

strain distribution and this zone or area is usually called large

deformation zone. The distributions of effective strain calcu-

lated by EFG is accordant with that calculated by FEM. The

deformation distribution is in good agreement with the practicalsituation.

Fig. 13. Comparisons of effective strain distributions obtained by EFG and FEM at 20 and 40% reduction in height. (a) Color contours of effective strain obtained

by EFG at 20% reduction in height. (b) Color contours of effective strain obtained by FEM-Deform (3D) at 20% reduction in height. (c) Color contours of effective

strain obtained by EFG at 40% reduction in height. (d) Color contours of effective strain obtained by FEM-Deform (3D) at 40% reduction in height.




Table 5

The coordinates, velocity directions and values of the characteristic nodes (CN) at 40% reduction in height evaluated by EFG


x y z α (◦) β (◦) γ (◦)

1 11.87739 0 6 1.21122 55.6505 90 145.651

2 0 0 6 1 90 90 180

3 0 11.9226 6 1.22152 90 54.9503 144.954 12.27201 12.26884 6 1.52761 57.3439 57.1316 130.065

5 13.48566 0 0 1.1349 0 90 90

6 0 0 0 0 – – –

7 0 13.48673 0 1.13331 90 0 90

8 12.83488 12.83913 0 1.24863 45.843 44.157 90



Fig. 14 gives the isoline contours of effective strain on

the cross-section of x = 3.5 and z = 2.0 of the deforming body

obtained by EFG and FEM at the reduction in height of 40%,

respectively. Through comparing the isolines, it canbe also seen

that thechange rulesof effectivestrainobtainedby thetwo meth-ods are the same. The effective strains in the central zone of the

deforming material are large, the effective strains in the central

upper zone in contact with the die are smallest, and the effective

strains in the middle zone close to the surface of the deforming

body are small.

Fig. 15 shows the load-stroke curves of upsetting process forboth EFG and FEM. The curves are in good agreement.

Fig. 14. Isoline contours of effective strain on the different cross-sections given by EFG and FEM. (a) Isoline contours of effective strain on the cross-section of

x = 3.5 given by EFG. (b) Isoline contours of effective strain on the cross-section of x = 3.5 given by FEM-Deform (3D). (c) Isoline contours of effective strain on the

cross-section of z = 2.0 given by EFG. (d) Isoline contours of effective strain on the cross-section of z = 2.0 given by FEM-Deform (3D).




Table 6

The coordinates, velocity directions and values of the characteristic nodes (CN) at 40% reduction in height evaluated by FEM


x y z α (◦) β (◦) γ (◦)

1 12.05036 0 6 1.28427 51.1374 90 141.137

2 0 0 6 1 90 90 180

3 0 12.01026 6 1.27541 90 51.6342 141.6344 12.49641 12.5749 6 1.67225 56.3474 54.6154 126.726

5 13.47687 0 0 1.18324 0 90 90

6 0 0 0 0 – – –

7 0 13.4566 0 1.17561 90 0 90

8 12.91056 12.9308 0 1.39875 44.9163 45.0837 90



Fig. 15. Load–stroke curves obtained by EFG and FEM.

4. Conclusions

Based on the flow formulation for rigid-plastic/viscoplastic

materials, element free Galerkin (EFG) method is applied to

the simulation of rigid-plastic/viscoplastic bulk metal forming

processes. The EFG mathematic model, the related equations,

key algorithms and technologies are proposed in this paper. In

order to validate the feasibility andeffectiveness of EFGin deal-

ing with rigid-plastic/viscoplastic bulkmetal forming problems,

equal channel angular pressing and cubic billet upsetting as the

examplesare simulatednumerically. Theresults suchas material

flow, forming load and different field variable distributions arecalculated, respectively, and the results are in good agreement

with those obtained by commercial finite element software and

experiment.

The meshless method exhibits some advantages for solving

bulk metal forming problems since it requires no mesh gen-

eration and remeshing in the simulation. For the future lots

jobs should be done to promote the application for element free

Galerkin (EFG) method in bulk metal forming problems, espe-

cially in complex three-dimension metal forming processes. In

addition, the computational efficiency should be improved on

the basis of ensuring the precision. Reliable and user-oriented

EFG based analysis software for bulk metal forming simulation

is also needed to provide a tool for guiding the process and die

design in industrial production.

Acknowledgement

The research work was supported by National Natural Sci-

ence Foundation for Distinguished Young scholars of China

(Approval No.: 50425517) and National Natural Science Foun-

dation of China (Approval No.: 50575125).

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