F. Schmidt. Optimization of 3D die extrusion using response surface method

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O ptimization of 3D die extrusion using response surface method.
N. Lebaal1, F.M. Schmidt1, S. Puissant2
1 Ecole des mines d’Albi Carmaux, Laboratoire CROMeP,- Campus Jarlard- Route de Teillet
81013 Albi Cedex 9, France
URL: www.enstimac.fr e-mail: lebaal@enstimac.fr; fabrice.schmidt@enstimac.fr
2 Institut Supérieur d’Ingénierie de la Conception, ERMeP,-27 rue d’Hellieule, 88100 Saint-Dié, France
URL: www.insic.fr e-mail: Stephan.puissant@insic.fr;
ABSTRACT: The primary objective of the geometrical design of extrusion dies in polymer processing is to
obtain a uniform velocity distribution across the die exit. A design procedure for complex coat-hanger die is
presented. While optimizing the exit velocity distribution, geometric constraints are applied. Three
dimensional extrusion simulation software REM3D® is used to simulate the flow in this flat die.
An objective function is defined as the global relative between velocity in exit die and the average exit
velocity. This objective function is minimized by varying the flow channel cross-section. For this
minimization the global response surface method with Kriging interpolation is used.
Key words: Polymer extrusion, Response surface method, Optimization, Kriging Interpolation, REM3D.
1 INTRODUCTION
The design of dies for polymer extrusion often
involves trial and error corrections to the die to
achieve uniform flow at the exit. Manual correction
to die geometry is a time consuming and a costly
procedure. A flat die (Fig.1) is commonly used to
extrude thermoplastics thin sheets [1-3]. If the
channel geometry in a flat die is not designed
properly, the velocity at the exit of the flat die may
not be uniform [4,5]. A non-uniform velocity
distribution at the die exit may lead to a variation in
the sheet thickness across the width of the die. Since
a tight control on thickness is required for a high
quality plastic sheet, the ultimate goal of this work is
to optimize the die channel geometry in a way that a
uniform velocity distribution is obtained at the die
exit [6-8]. Prior works in sheet die optimization have
involved the use of lubrification approximations of
the momentum equations [9-12]. If the geometry is
more complex, a flow channel can be approximated
with simple geometric sections [13]. Others have
used three-dimentional analysis to design die
extrusion [14,15] but they did not take into account
the thermal dependence. Network algorithms have
been developed to optimize die designs [16] but they
are difficult to apply to arbitrary shapes. Michaeli et
al [2] have used a combination of finite-element-
analysis and flow analysis network. To optimize the
die geometry they used respectively the evolution
strategy algorithm and network theory with
isothermal flow. But this geometry is varied
manually and the optimization by evolution strategy
makes a lot of time to converge. Sun et al [18],
Smith [10] have used respectively BFGS and SQP
algorithms to optimize die geometry but the used
algorithms are time consuming and it is not easy to
apply them with different software and to change
different geometries. Reddy et al [6] have used
response surface method with polynomial
approximation, but this approximation is not
accurate and needs a lot of iteration to converge. The
complexity of the polymer rheology further
increases the difficulty of the die optimization
problem [9]. If the polymer rheological behaviour is
not accounted accurately while optimizing the die,
the computed velocity, pressure and temperature
fields are expected to have large errors. There are
many works in the literature that used rheological
behaviours of power law [9, 10, 12, 14, 16];
however, this does not permit to present accurately
the rheological behaviour. Sun et al [18] have
optimized a flat die and they have taken into account
the elongation effect. They obtained optimal
distribution but the pressure drop could not be
decreased. In the present work we have developed
an automatic optimization algorithm based on
response surface method together with Kriging
interpolation. We used REM3D® software to
compute 3D simulation of the flow in extrusion dies.
This software takes into account strain rate and

temperature dependence. The optimal design
procedure is applied to a coat hanger die used to
extrude thermoplastic sheets. The viscosity models
and the values of the various parameters of these
models are summarized in the next section for the
polymer used in the present work.
Fig. 1. Geometry of the Flat die
2 THE OPTIMIZATION BENCHMARK
In this paper, a geometry of a flat die is optimized
for an Acrylonitrile Butadiene Styrene (ABS,
Astalac EPC 10000). The polymer flow is supposed
purely viscous (the viscoelastic behaviour is not
taken into account). According to Sun and Gupta
[17-18] the elongation effect does not influence the
exit velocity distribution at the die exit. The
simulation of extrusion is carried out on a 3D
computation software by finite elements REM3D®
[19]. The behaviour laws used in Rem3D give an
expression of the viscosity in function of the shear
rate and temperature. The thermal and rheological
parameters of the ABS are given in Table 1. Carreau
Yasuda/WLF viscosity model is used to characterize
the temperature and shear rate dependence of
viscosity [20]. It is written as:
()
()
()
()
α
α
τ
γ
ηηη
1
0
2
1
2
1
01exp.
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−+
−
−
−+
−
=
m
ss
s
sref
sref TTA TTA
TTA
TTA & (1)
The thermal conductivity (K), density (ρ) and heat
capacity (Cp) of the polymer used for the flow
simulation were assumed to be constant throughout
the range of temperature in the flat die.
Table1. Margin settings
0
η
Pa.s m no unit 0
α
no unit s
τ
Pa
716.997 0.15862 1 224063
A1 no unit A2 [K] Ts [K] Tref [K]
20.4 101.6 397.7 524.7
2.1 Automation of the finite element model:
In order to save time at the optimization process and
to be able to control effectively and easily the
preprocessing REM3D (GLPre), the solver REM3D
and design routine, it is necessary to use all the
possibilities of automation offered by this code (via
Matlab). Matlab makes it possible to launch the
executable files. This fact enables us to couple
Matlab with REM3D. Fig. 2, shows a diagram of the
principle of operation and possible interactions of
Matlab with REM3D/GLpre, REM3D/Solver and
design routine. It is also possible to automate all the
tasks usually carried out through graphic interface
from the creation of FEM until the recovery of the
results.
Fig. 2. Optimization algorithm.
3 DESIGN VARIABLES AND OBJECTIVE
FUNCTIONS
The variable of optimization is the depth of the
channel of distribution H. Initially, it is fixed at 36.5
mm, but during the processes of optimization it is
limited between 30 and 110 mm (fig.1). It was
mentioned earlier that the goal in design of a flat die
is to minimize the velocity variation across the die
exit. The objective function is:
∑=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
=N
i
i
vvv
N
J1
2
1 (2)
Where N is the total number of nodes at the die exit
in the middle plane, i is the velocity at an exit
node,
v
vis the average exit velocity.
4 OPTIMISATION PROCEDURE
The method of response surface [21] consists in the
Trial Parameters
Pre-Processor
GL-Pre
Operating conditions
Temperature of regulation, flow rate,
rheological parameters and
Mesh
g
eneration
Routine for change channel geometry
D
e
fini
t
i
o
n
o
f
d
i
e
c
h
a
nn
e
l
geo
m
et
r
y
Solver REM3D
Simulation of no-isothermal 3D
Optimization routine
Evaluation of the geometry

construction of an approximate expression of
objective function starting from a limited number of
evaluations of the real function. The main idea is to
approximate the objective function through a
response surface. In order to obtain a good
approximation, we used a Kriging interpolation [22].
In this method, the approximation is computed by
using the evaluation points by composite design of
experiments, the number of evaluation is 5 points
which permits to give a good interpolation. After the
interpolation of the objective function we minimize
it using SQP sequential quadratic programming
method. In addition, to avoid to obtain a local
minimum, we change the initial points of the SQP
method by all the points of the design of experiment
because the evaluation of the interpolation function
does not cause problems and does not take much
time. After we obtain the best minimum of this
interpolation we use the weight function of Gaussian
type which allows to slightly change the
interpolation and gives more importance to the
points which are closer to the minimum and less
importance to the other points. The iterative
procedure stops when the successive points are
superposed with a tolerance ε=10-3.
5 RESULTS AND DISCUSSION
ow conditions, the
optimal value for H is 94.9 mm.
After 2 iterations, the objective function is varying
quite fast at the beginning of the optimization until
the 2nd step. Its final value is about 5.3 10-2.
However the global relative error between exit
velocities in initial and optimal die, Fig. 3, is
minimized for 94.7%. This optimization run
represents 11 hours 23 min of CPU time on a
computer Pentium IV, 3 GHz, 1Go RAM.. Given the
fixed geometry constraints of the die, the rheological
parameters of the ABS and the fl
0100 200 300 400 500
0
0.1
0.2
0.3
0.4
0.5
Width of the die [mm]
Error
Optimal die
Initial die
Fig. 3. Error between th optimal and initial exit
velocity.
e average,
Fig. 4. Velocity distributions of the initial and optimized die.
Velocity distributions at the exit of the optimized die
in Fig. 4-B are more uniform than the exit velocity
in the initial die design in Fig. 4-A, it is evident that
in the initial die design the exit velocity at the border
of the die is significantly higher than the velocity in
the centre of the die. In Fig. 4-B, the exit velocity
distributions for the optimized die design are much
more uniform across the complete width of the half
die. Only in the small regions near the border of this
die, the exit velocity is reduced to zero to satisfy the
no slip condition at the walls.
Fig. 5. Pressure distributions in initial and optimized die
The pressure drop in the initial die design is 4.423
MPa Fig. 5-A. It should be noted that the pressure
drop decreases during the optimization iteration. The
total pressure drop in the optimal die was decreased
by 49.9% or 2.215 MPa, Fig.5-B. With the
optimized geometry, we can obtain the same
pressure as with the initial geometry by increasing
the flow rate by 140%. To summarize, it is possible
to increase the productivity of more than 140% with
the same pressure as for the initial die. To improve
the uniformity of die exit velocity, the width of the
channel cross-section is typically increased from 36
mm to 94.9 mm.
Bad temperature distribution is observed in the
initial die Fig. 6-A. However, it is noted that more
polymer is flowing towards the edge because of the
shear heating. Hence, the temperature is lower near
the middle of the die and higher near the edge. By

comparison, the shear heating in the optimal die and
the temperature distribution are more uniform Fig.
6-B, because of the homogeneous distribution of the
exit velocity.
Fig. 6. Temperature distributions in initial and optimized die.
6 CONCLUSION AND PROSPECTS
We conclude that we obtain good results using the
response surface method with a very fast
convergence (2 iterations), this is due to the good
interpolation obtained with Kriging. The weight
function and the various evaluations of the
interpolation function (which does not take much of
CPU time) permit to give a precise global minimum.
A program for optimizing the geometry of a flat die
for polymer sheet extrusion has been developed. To
save time and to control easily the optimisation
procedure, we have automised this program with
REM3D software and design routine which we have
also developed. The optimization program used the
REM3D software to simulate the polymeric flow in
the flat die. This program optimized successfully the
3D geometry of a flat die such that a uniform
velocity and temperature distribution were obtained
at the die exit with decreasing the pressure drop in
the die. In our perspectives, we envisage to optimise
the temperature of regulation instead of modifying
the geometry while keeping the same objective
function, that is to say homogenizing the exit
velocity in the die.
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