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Three-dimensional finite-element simulation of the dynamic Brazilian tests on concrete cylinders

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng 2000; 48:1089}1106
Mismatching re"nement with domain decomposition for
the analysis of steady-state metal forming process
K. Park and D. Y. Yang*R
Department of Mechanical Engineering, Korea Advanced Institute of Science and ¹echnology,
ME3214, Kusung-dong, 171-1, >ousung-gu, ¹aejon, 305-701, Korea
SUMMARY
The "nite element analysis of three-dimensional metal forming processes is generally subject to large
computational burden due to its non-linearity. For economic computation, the mismatching re"nement, an
e$cient domain decomposition method with di!erent mesh density for each subdomain, is developed in the
present study. A modi"ed velocity alternating scheme for the interface treatment is proposed in order to
obtain good convergence and accuracy in the mismatching re"nement. As a numerical example, the analysis
of the axisymmetric extrusion processes is carried out. The results are discussed for the various velocity
update schemes and for the variation of the length of overlapped region. The three-dimensional extrusion
processes for a rectangular section and an E-section are analysed in order to verify the e!ectiveness of the
proposed method. Comparing the results with those of the conventional method of full region analysis, the
accuracy and the computational e$ciency of the proposed method are then discussed. Copyright 2000
John Wiley & Sons, Ltd.
KEY WORDS:
mesh re"nement; domain decomposition; iterative calculation; rigid-plastic "nite element
method; extrusion process
1. INTRODUCTION
In order to solve large-scale "nite element problems with good accuracy and high e$ciency, there
have been various studies on adaptive mesh re"nement schemes. Much of the focus of the studies
has been put on the development of error estimators and following re"nement schemes to reduce
error [1}8]. The basic constraint of the conventional mesh re"nement methods is the continuity
of nodes between re"ned regions and unre"ned regions. For the re"nement using triangular mesh
or tetrahedral mesh [5}7], it has been known that the continuity is fully maintained. For the
re"nement of quadrilateral mesh or hexahedral mesh, however, special techniques are required to
control the mesh density.
* Correspondence to: Dong-Yol Yang, Department of Mechanical Engineering, Korea Advanced Institute of Science and
Technology, ME3214, Kusung-dong, 373, Yousung-gu, Taejon 305-701, Korea
R E-mail: dyyang@mail.kaist.ac.kr
Copyright 2000 John Wiley & Sons, Ltd.
Received 24 August 1998
Revised 23 September 1999
1090
K. PARK AND D. Y. YANG
There have been various studies on the re"nement using quadrilateral mesh; Babuska and
Rheinboldt [1}3] introduced a bisectionally re"ned mesh structure by imposing some geometric
constraints on the irregular nodes. Choi et al. [8, 9] used transition elements for the interface
region between re"ned mesh and unre"ned mesh so as to remove the arti"cial constraints. In spite
of e!ective interface treatment, this method has a disadvantage that the bandwidth is highly
increased for hexahedral transition elements which can possess 26 nodes at most [9]. It is not
adequate for the analysis of large-scale problems, such as the analysis of three-dimensional metal
forming processes.
Domain decomposition has become an important subject to obtain parallelism when solving
a partial di!erential equation (PDE). It is useful for reducing the problem size, and distributed
processing enhances the reduction of computation time. The basic idea is to subdivide the domain
of the de"nition of the PDE into a set of subdomains and then to solve the PDE problems on each
subdomain. These solutions for all the subdomains are then assembled to obtain the solution for
the whole domain. Schwarz [10] equated the alternating procedure which solves each subdomain
concurrently with data exchange between two adjacent overlapped subdomains. The convergence
of the procedure was established by Sobolev [11]. Rodrigue and Shah [12] imposed pseudoboundary conditions to accelerate the parallel Schwarz alternating procedure. Fahrat and Wilson
[13] implemented recursive substructuring method which does not require overlapped region.
Funaro et al. [14] introduced an iterative solving scheme between subdomains as imposing the
continuity of the solution and the continuity of the normal derivative. In the "nite element
analysis of metal forming processes, Doltsinis and NoK lting [15] implemented parallel substructuring method to analyse the rolling processes.
The domain decomposition method is e!ective in reducing computation time by reducing the
problem size as well as to ensure parallelism of the problem. Adaptive mesh re"nement can be
easily implemented by imposing di!erent mesh density to each subdomain. In this study,
mismatching re"nement, a mesh re"nement scheme combined with the domain decomposition
method, is developed. Iterative calculation with an error minimization scheme is carried out
between the two subdomains which have di!erent mesh densities. As numerical examples, the
extrusion processes are analysed in order to verify the e!ectiveness of the proposed method.
Many numerical parameters which a!ect the accuracy and the convergence of the solution are
discussed.
2. THE RIGID-PLASTIC FINITE ELEMENT METHOD
2.1. Basic formulation
In this work, the rigid-plastic "nite element method [16] is employed to simulate metal forming
processes. The rigid-plastic analysis requires considerably less computation time than the elastoplastic analysis. The equilibrium demands the following relations:
Copyright 2000 John Wiley & Sons, Ltd.
pGHH"0 in )
(1)
pGH nH"fG on !D
(2)
Int. J. Numer. Meth. Engng 2000; 48:1089}1106
STEADY-STATE METAL FORMING PROCESS
1091
where nH is the unit normal vector to a given surface, and fG is the force vector. The incompressibility condition for a rigid-plastic material is given as
eR 4"eR GG"0
(3)
Since the rigid-body rotation and elastic deformation of material are neglected in the rigid-plastic
"nite element method, the constitutive equation is given as follows:
2 pN
p "
eR
GH 3 eNQ GH
(4)
where
pN "( p p ,
GH GH
eNQ "( eR eR
GH GH
(5)
and p and eR are components of the deviatoric stress and strain rate, respectively. From the
GH
GH
variational principle, the functional for a rigid-plastic material can be written as follows
pN deNQ d)#K* eR 4 deR 4 d)!
!
D
f du d!"0
G G
(6)
where u denotes the velocity component, and K* is the penalty constant for the incompressibility
G
condition. Since discretization of Equation (6) is a non-linear system of equations with respect to
u , the Newton}Raphson iteration is utilized with an initial guess by assuming a linear viscous
G
material [17].
For the computation, bilinear quadrilateral elements are used for two-dimensional analysis,
and trilinear hexahedral elements are utilized for three-dimensional analysis. In order to avoid
locking due to incompressibility, selective reduced integration is employed for numerical integration. That is, the second term in Equation (6) is integrated with a lower-order rule while the other
terms are integrated with a normal Gauss quadrature rule.
2.2. Error estimation
In the iterative solving method with domain decomposition, there occurs some numerical error
for connection between subdomains. In the present study, the error estimation is considered in
view of that error caused by the iterative calculation. The result of the iterative calculation is
compared with the solution which is obtained from the fully re"ned mesh. That is, the solution of
the fully re"ned mesh is taken as the reference solution. For the comparison of accuracy of the
iterative calculation, two error measures, such as velocity error and strain energy error, are
considered. The velocity error at a speci"c point is de"ned as follows:
#u!uL # #u*!uL #
e"
:
S
#u#
#u*#
Copyright 2000 John Wiley & Sons, Ltd.
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K. PARK AND D. Y. YANG
where superscripts * and denote the results of the full domain analysis and the results of the
iterative calculation, respectively. In the rigid}plastic formulation, the energy dissipation norm is
given as follows:
p2 : eR d)
#;#"
(8)
where p and eR denote deviatoric stress and strain rate, respectively. Thus, the energy error norm is
given by the following forms:
#E#"
(p!pL )2 : (eR !eRL ) d)
:
(p*!pL )2 : (eR *!eRL ) d)
(9)
The error measure for strain energy then can be written as the ratio of the energy dissipation
norm to the energy error norm [5, 6].
#E#
e "
# # ;#
(10)
3. ITERATIVE CALCULATION WITH DOMAIN DECOMPOSITION
Let us consider the matrix equation after "nite element discretization of Equation (6)
Ku"f, in )
(11)
u"g, on !
(12)
where K and f are sti!ness matrix and load vector, respectively, and ! is the boundary of
domain ). Domain ) is subdivided into two subdomains, ) and ) , as shown in Figure 1.
For the subdomain ) , by introducing the concept of Schwarz alternating procedure, [10]
Equations (11) and (12) can be rewritten as follows:
K uI" f ,
in )
(13)
Figure 1. Schematic description of domain decomposition.
Copyright 2000 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2000; 48:1089}1106
STEADY-STATE METAL FORMING PROCESS
uI"g,
on !
uI"uI\, on ! !!
1093
(14)
(15)
where subscript 1 and 2 denote the subdomain 1 and the subdomain 2, respectively, and
superscript (k) means the current iteration step. In order to improve the convergence of the
iterative calculation, the continuity condition of the normal derivative is taken as follows:
*uI *uI\
" , on ! !!
*n
*n
(16)
where n denotes the outward normal direction. For the subdomain ) , the equations are given in
the same manner
K uI"f ,
in )
(17)
uI"g,
on !
(18)
uI "uI,
on ! !!
(19)
*uI *uI
" , on ! !!
*n
*n
(20)
As shown above, velocity components of the overlapped interface region are obtained using the
results of the neighboring subdomain. The iteration is carried out until the following criterion is
satis"ed:
#uI!uI\#
eI"
)d
# uI#
(21)
where eI is an amount of error at the current step, # z # is a Euclidean vector norm, and d is a very
small constant called as a limit of the fractional norm.
4. MISMATCHING REFINEMENT WITH DOMAIN DECOMPOSITION
The main idea of the mismatching re"nement is that the solution is obtained by iterative
calculation between several subdomains which have di!erent mesh densities. At the interfaces
between subdomains, there exist mismatching overlapped regions where the continuity of nodes
is not guaranteed due to the di!erence of mesh density or mesh con"guration. A domain for
axisymmetric extrusion is divided into two subdomains, a coarse subdomain and a "ne subdomain. In the overlapped region, a projection operator from the coarse mesh to the "ne mesh is
de"ned as
P : RL P RL
Copyright 2000 John Wiley & Sons, Ltd.
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Int. J. Numer. Meth. Engng 2000; 48:1089}1106
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K. PARK AND D. Y. YANG
where n and n mean the numbers of DOFs in the overlapped region for the coarse subdomain
and for the "ne subdomain, respectively. The reverse operation can be de"ned in the same manner
Q : RL P RL
(23)
The velocity alternating procedure (Equations (15), (16), (19), and (20)) can thus be rewritten by
introducing the projection operators
uI"QI\uI\,
on ! !!
(24)
*uI
*uI\
"QI\ ,
*n
*n
on ! !!
(25)
uI"PIuI,
on ! !!
(26)
*uI
*uI
"PI ,
*n
*n
on ! !!
(27)
where subscripts c and f denote the coarse subdomain and the "ne subdomain, respectively. The
iterative alternating procedure has been known to render no problem of convergence [11].
However, it should be modi"ed so as to consider the di!erence of mesh density for the
mismatching re"nement. For the consideration, let eI and eI be the velocity error measures for
the coarse subdomain and the "ne subdomain, respectively
#u!QI\uI\ #
#u!uI #
"
eI"
# u!uI\ #
#u!uI\ #
(28)
# u!uI #
# u!PI uI #
eI"
"
#u!uI\ #
# u!uI\ #
(29)
Since uI\ ensures better accuracy than uI\, the convergence criterion is ful"lled for the
alternating procedure from "ne subdomain to coarse subdomain (Equations (24) and (25)), that is,
eI(1. The reverse procedure (Equations (26) and (27)), however, may cause some convergence
problem since uI is not always more accurate than uI\ .
In the present work, for the characteristics of mismatching re"nement, a modi"ed velocity
update scheme is proposed considering the solution of the previous step as well as that of the
other subdomain. For both subdomains, Equations (24)}(27) can be rewritten as follows:
uI"aI QI\ uI\#(1!aI) uI\,
on ! !!
(30)
*uI
*uI\
*uI\
"aIQI\ #(1!aI) ,
*n
*n
*n
on ! !!
(31)
Copyright 2000 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2000; 48:1089}1106
STEADY-STATE METAL FORMING PROCESS
1095
uI"bI PI uI#(1!bI) uI\,
A
on ! !!
(32)
*uI
*uI\
*uI
"bIPI #(1!bI) ,
*n
*n
*n
on ! !!
(33)
where a and b are the weight factors to consider solution of the previous step. In the modi"ed
alternating scheme, eI can be revised as follows:
# u!uI #
# u!aI QI\ uI\!(1!aI) uI\ #
eI"
"
# u!uI\ #
#u!uI\ #
(34)
if aI'(1!aI), that is, 0.5(aI(1, the weighting factor for uI\ becomes higher than the
weighting factor for uI\. For such a condition, eI can be rewritten as follows, which satis"es
the convergence criterion
eI:
# u!aI QI\ uI\ #
(1
#u!uI\ #
(35)
On the other hand, eI can be expressed in the same manner as follows:
# u!uI #
#u!bI PI uI!(1!bI) uI\ #
eI"
"
#u!uI\ #
# u!uI\ #
(36)
if bI((1!bI), that is, 0(bI(0.5, the weighting factor for uI\ becomes higher than the
weighting factor for uI\. For such a condition, the convergence criterion is also met where
eI can be rewritten as follows:
#u!(1!bI) uI\ #
(1
eI:
#u!uI\ #
(37)
In order to ful"l the convergence criterion, aI and bI are de"ned as a function of the rate of
convergence g
aI"C [gI ]K, on ! !!
(38)
bI"C [gI ]K, on ! !!
(39)
In this study, C and m are taken as 1.0 and 0.5, respectively. The rate of convergence in the
overlapped region at the kth step is de"ned by
# eI #
G
gI"
, in ) 5 )
G
# eI\ #
G
Copyright 2000 John Wiley & Sons, Ltd.
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Int. J. Numer. Meth. Engng 2000; 48:1089}1106
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K. PARK AND D. Y. YANG
where subscript i denotes the subdomain number. The rate of convergence is a value between
0 and 1 unless the solution diverges. If it diverges, the rate of convergence is set to 1. As the
solution converges, the rate of convergence decreases continuously and the weighting factors are
updated re#ecting the state of convergence. It is expected that the proposed scheme improves
convergence as well as accuracy of the solution by utilizing the results of the re"ned region for all
the velocity alternating procedures.
5. NUMERICAL EXAMPLES
5.1. Comparison of the velocity alternating schemes
As a numerical example, an analysis of the axisymmetric extrusion process with the mismatching
re"nement is carried out (Figure 2). The workpiece material used in simulation is Al6061 of which
the constitutive relation is given as follows at 400 3C [18]:
pN "53.60 e (MPa)
(41)
The subdomain 1, with a low mesh density, consists of 143 nodes and 120 elements (Figure 2(b))
while the subdomain 2, with a high mesh density, consists of 155 nodes and 120 elements (Figure
2(c)). The analyses of the subdomain 1 and subdomain 2 are carried out iteratively and the result
is compared with that of a full domain analysis which consists of 575 nodes and 520 elements
Figure 2(a)).
Analyses are carried out in order to compare two velocity alternating procedures (Equations
(24)}(27) and (30)}(33)). For each case, the variation of axial velocity components is compared.
The axial velocity components are compared at the center point of the outlet as the iterative
calculation is carried out (Figure 3). The dot line in Figure 3(b) means the velocity of the full
domain analysis without domain decomposition, which is regarded as a reference solution for the
comparison. Comparing the results, the proposed velocity alternating scheme, Equations
(30)}(33), shows more accurate and stable solution than the Schwarz's alternating scheme,
Figure 2. Basic concept of the nonconforming re"nement: (a) full mesh; (b) mesh for subdomain 1; (c) mesh
for subdomain 2.
Copyright 2000 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2000; 48:1089}1106
STEADY-STATE METAL FORMING PROCESS
1097
Figure 3. Axial velocity components at the reference point: (a) position of the reference point; (b) axial
velocity components w.r.t. the number of iterations.
Figure 4. Variation of error measures.
Equations (24)}(27). It appears that the proposed velocity update scheme improves stability and
accuracy of the solution in the case of the mismatching re"nement. In order to check the
convergence of iterative calculation for each subdomain, the variation of error measure, which is
de"ned in Equation (21), is shown in Figure 4 with respect to the number of iterations. It is
concluded that convergence and stability of mismatching re"nement are assured with the
Copyright 2000 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2000; 48:1089}1106
1098
K. PARK AND D. Y. YANG
proposed velocity alternating scheme. The proposed velocity alternating scheme is thus utilized
for the analysis of further numerical examples.
5.2. Analysis of the axisymmetric extrusion process
In order to verify the accuracy and e$ciency of mismatching re"nement, the analysis of the
axisymmetric extrusion process is carried out with a steady-state isothermal assumption at
4003C. The workpiece material is Al6061 of which the constitutive relation is given in Equation
(41). Three mesh structures for the analysis are considered as shown in Figure 5. The mesh 2
(Figure 5(b)) is decomposed into two subdomains with the same mesh density as mesh 1. For the
mismatching re"nement, mesh 3 (Figure 5(c)), the mesh density of subdomain 1 of mesh 3 is
reduced. The ram speed is set as 10 mm/s and the friction factor is assumed to be 0.3.
The iterative calculation is carried out until the error measure is converged below 10\. The
velocity error at the reference point and the strain energy error in the overlapped region are
compared in Table I. All error measures are less than 1.0 per cent in the case of mesh 2 while
a little increase, still less than a 2.0 per cent, occurs in mesh 3. The speed-up ratio relative to the
analysis of the mesh 1 goes up to 1.43 times for mesh 2, and 2.62 times for mesh 3, respectively.
Figure 5. Mesh systems for axisymmetric extrusion: (a) mesh 1; (b) mesh 2; (c) mesh 3.
Table I. Comparison of the results of axisymmetric extrusion.
Subdomain
No. of
nodes
Velocity
error (%)
Energy
error (%)
No. of
iterations
Elapsed
time (sec)
Speed-up
ratio
1
*
575
*
*
*
218.2
1.00
2
1
2
525
155
0.39
0.90
4
151.7
1.43
3
1
2
143
155
1.96
1.24
18
82.9
2.62
Mesh
Copyright 2000 John Wiley & Sons, Ltd.
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STEADY-STATE METAL FORMING PROCESS
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5.3. E+ect of the length of overlapped region
In this section, the e!ect of the length of overlapped region is investigated. Determination of the
length of overlapped region is an important procedure since it may a!ect the convergence as well
as the accuracy of solution. Three mesh structures are considered as shown in Figure 6. Each
mesh has the same structure for subdomain 1, although the length of the overlapped region (l) is
di!erent for subdomain 2. The workpiece material and process parameters are the same as those
of the previous section.
The results with a variation of l are presented in Table II. The convergence limit is 10\, and
the velocity error measure and the speed-up ratio are de"ned with respect to the results of the full
domain analysis. In the case of l"15 and 20 mm, the results show that a better convergence and
an improved speed-up ratio are obtained. The accuracy of solution of both cases is still better
than the result of l"10 mm. Figure 7 shows the state of convergence with the variation of l. In
order to analyse the e!ect of the length of overlapped region, axial velocity component along the
centerline is considered in Figure 8. In this "gure, the result of the subdomain 1 means the
solution after the "rst iteration. The result of subdomain 1 shows almost the same result as that of
the full analysis until the distance from the punch is 45 mm (i.e. l"15 mm), while it shows
a numerical error de"ned as the di!erence from the distance 50 mm (i.e. l"10 mm). It thus
appears that the di!erence of the convergence as shown in Table II is caused by the numerical
Figure 6. Mesh systems for various length of overlapped region: (a) l"10 mm; (b) l"15 mm; (c) l"20 mm.
Table II. Comparison of the results for various
length of overlapped region.
l (mm)
No. of iterations
Velocity error (%)
Elapsed time (s)
Speed-up ratio
Copyright 2000 John Wiley & Sons, Ltd.
10.0
18
1.96
82.9
2.62
15.0
12
1.89
62.2
3.51
20.0
10
2.03
49.4
4.42
Int. J. Numer. Meth. Engng 2000; 48:1089}1106
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K. PARK AND D. Y. YANG
Figure 7. Convergence status with the variation of l.
Figure 8. Axial velocity component along the centerline: (a) de"nition of l and distance; (b) variation
of axial velocity component.
error of the velocity. As a consequence, it is recommended that the length of overlapped region is
determined as considering the result of subdomain 1 at the "rst iteration.
5.4. Analysis for extrusion of a rectangular section
Let us consider three-dimensional extrusion of a rectangular section (Figure 9) in order to
magnify the merit of the proposed method. The ram speed is set as 10 mm/s and the friction factor
Copyright 2000 John Wiley & Sons, Ltd.
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STEADY-STATE METAL FORMING PROCESS
1101
Figure 9. Three-dimensional extrusion process with a rectangular section.
Figure 10. Mesh systems for the rectangular section extrusion: (a) mesh 1; (b) mesh 2; (c) mesh 3.
is taken to be 0.3. The workpiece material used in simulation is Al6061 of which the constitutive
relation is given as follows at 4003C [18]:
pN "53.60 e (MPa)
(42)
Analyses for three mesh systems as shown in Figure 10 are carried out with a steady-state
isothermal assumption at 4003C. The computations are performed on HP 780 workstation with
64MB memory. The axial velocity components and related error measures at the four reference
points (Figure 11) and the strain energy error in the overlapped region are compared in Table III.
Copyright 2000 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2000; 48:1089}1106
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K. PARK AND D. Y. YANG
Figure 11. Position of four reference points.
Figure 12. Schematic description of an
&E'-section extrusion.
Table III. Comparison of velocity components and error measures.
Reference point
Mesh 1
Mesh 2
Mesh 3
1
Velocity (mm/sec)
Error (%)
158.64
*
160.62
1.24
160.61
1.24
2
Velocity (mm/s)
Error (%)
153.43
*
153.96
0.35
154.51
0.70
3
Velocity (mm/s)
Error (%)
135.94
*
133.41
1.89
134.10
1.35
4
Velocity (mm/s)
Error (%)
145.57
*
144.73
0.58
144.97
0.41
0.69
0.91
Energy error (%)
*
All velocity error measures are around 1.0 per cent with a small variation. The problem size and
the computing performance for the three mesh systems are compared in Table IV. The speed-up for
mesh 3 is about twenty times of mesh 1 while about six times for mesh 2. It is concluded that the
proposed mismatching re"nement with domain decomposition guarantees highly e$cient solution
with an allowable accuracy in the "nite element analysis of three-dimensional extrusion process.
5.5. Analysis for extrusion of an *E+-section
Let us consider three-dimensional extrusion of an &E'- section (Figure 12) with a high extrusion
ratio, i.e., thin-walled section extrusion. The ram speed is set as 10 mm/s and the friction factor is
Copyright 2000 John Wiley & Sons, Ltd.
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STEADY-STATE METAL FORMING PROCESS
Table IV. Comparison of problem size and performance.
Subdomain
No. of
nodes
No. of
elements
No. of
iterations
Elapsed time
(h : min)
Speed-up
ratio
1
*
2740
2148
*
37 : 13
1.00
2
1
2
2220
1186
1764
776
4
6 : 04
6.13
3
1
2
711
1186
512
776
8
1 : 53
19.76
Mesh
Figure 13. Full mesh structure (10 886 nodes, 9240 elements).
taken to be 0.3. The workpiece material used in simulation is Al6061 of which the constitutive
relation is given as follows at 5003C [18]:
pN "26.87 e (MPa)
(43)
Numerical analysis is carried out with a steady-state isothermal assumption at 5003C. As
considering the symmetric condition of the geometry, only section of the workpiece is
analysed. The computations are performed on Cray T3E computer with a single CPU and
128MB memory. Figure 13 shows the full mesh structure, which contains 32658 DOFs. The full
analysis, however, is not possible due to the limit of memory capacity. Figure 14 shows mesh
con"guration decomposed into two subdomains with mismatching re"nement. The number of
DOFs is e!ectively reduced as 3630 for subdomain 1 and as 10 167 for the subdomain 2,
respectively.
Copyright 2000 John Wiley & Sons, Ltd.
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K. PARK AND D. Y. YANG
Figure 14. Mesh con"guration for the mismatching re"nement: (a) mesh for subdomain 1 (1210 nodes, 940
elements); (b) mesh for subdomain 2 (3389 nodes, 2352 elements); (c) mesh for the full domain.
Copyright 2000 John Wiley & Sons, Ltd.
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The number of iterations between subdomains is 8 with a convergence limit of 10\. It takes
2947 CPU seconds for the analysis with a given computational condition. It is concluded that the
computation of a large problem size, which is even unable to carry out due to the memory
capacity, can be accomplished within 1 h by employing mismatching re"nement.
6. CONCLUSION
In the present work, mismatching re"nement, a new mesh re"nement scheme, has been proposed
in the analysis of three-dimensional extrusion processes. The domain decomposition method with
overlapping has been utilized in combination with the iterative calculation between several
subdomains. For the convergence of the iterative calculation between subdomains with di!erent
mesh densities, a modi"ed velocity alternating scheme has been introduced and has been
compared with Schwarz alternating procedure. The proposed alternating scheme has shown
better convergence than Schwarz's scheme for the mismatching re"nement. Through several
numerical examples of the extrusion processes, the validity of the proposed method has been
demonstrated. From the results it has been shown that computational e$ciency is highly
increased, especially for the three-dimensional analysis. It is expected that the proposed method
enables a three-dimensional analysis of the industrial extrusion processes with a less computational overhead. In addition, it can be easily implemented on parallel computation which still
more accelerates the computational e$ciency.
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1106
K. PARK AND D. Y. YANG
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