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Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
Available online 30 April 2021
0377-0257/© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Contents lists available at ScienceDirect
Journal of Non-Newtonian Fluid Mechanics
journal homepage: www.elsevier.com/locate/jnnfm
Die shape optimization for extrudate swell using feedback control
M.M.A. Spanjaards a,b, M.A. Hulsen a,∗, P.D. Anderson a
aDepartment of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
bVMI Holland B.V., Gelriaweg 16, 8161 RK Epe, The Netherlands
ARTICLE INFO
Keywords:
Die optimization
Extrudate swell
Viscoelasticity
Feedback control
Free surfaces
Inverse problem
ABSTRACT
In this paper we propose a novel approach to solve the inverse problem of three-dimensional die design for
extrudate swell, using a real-time active control scheme. To this end, we envisioned a feedback connection
between the corner-line finite element method, used to predict the positions of the free surfaces of the
extrudate, and the controller. The corner-line method allows for local mesh refinement and transient flow to
be taken into account (Spanjaards et al., 2019). We show the validity of this method by showing optimization
results for 2D axisymmetric extrusion flows of a viscoelastic fluid for different Weissenberg numbers. In 3D we
first give a proof of concept by showing the results of a Newtonian fluid exiting dies with increasing complexity
in shape. Finally, we show that this method is able to obtain the desired extrudate shape of extrudates of a
viscoelastic fluid for different Weissenberg numbers and different amounts of shear-thinning.
1. Introduction
Extrusion is a common production technique in the polymer pro-
cessing industry to obtain products with a desired cross-section. In
this process a polymer is molten and pushed through a die with a
certain cross-sectional shape, to obtain a product (extrudate) with this
same cross-sectional shape. A common requirement on the extrudate is
dimensional precision. However, the dimensions of the extrudate are
highly influenced by a phenomenon called extrudate swell, where the
extrudate starts to expand due to internal stresses in the polymer once
it leaves the die.
The swelling process involves complex dynamics influenced by
many parameters such as viscoelasticity and temperature. Therefore,
the optimized shape of a die, to obtain an extrudate with desired
dimensions and shape, is now often obtained through trial-and-error.
Especially for complex extrudate shapes, this process becomes time
consuming and therefore inefficient and costly. Hence, there is a need
for a numerical method that can determine the optimal shape of the
die without having to perform extrusion experiments.
Attempts to obtain a desired extrudate by designing the die date
back to the 1950s [1] and 1980s [2,3]. Here, die design was based on a
simple analysis, experience and rules of thumb. The recent review arti-
cle of Pittman [4] shows that there are different approaches to optimize
profile extrusion dies nowadays, using computational methods. Most
often, the mathematical optimization problem is defined by specifying
an objective function, which measures the performance of the die. In
the optimization method this objective function is minimized to obtain
a suitable die.
∗Corresponding author.
E-mail address: m.a.hulsen@tue.nl (M.A. Hulsen).
One frequently used objective function used in the minimization
problem is the difference between point-wise and average extrusion
direction flow velocities on the die exit cross-section. Behr and Elgeti
et al. [5,6] proposed a framework that coupled fluid flow simulations,
using the finite element method, with a mathematical optimization
algorithm where design goals are formulated in terms of an objective
function that is minimized to obtain a homogeneous velocity distribu-
tion at the die exit. The same objective function approach to balance
the flow was also presented by Nobrega et al. [7], coupled with fluid
flow simulations using the finite volume method. Different definitions
of the objective function used in die design are assessed in Szarvagy
et al. [8].
The second approach is design for extrudate swell, where the ‘in-
verse’ problem is solved that answers the question what cross-sectional
shape is required at the die exit to obtain an extrudate with the desired
dimensions and shape. Tran-Cong and Phan-Thien [9] developed an
iterative method using the boundary element method to design a die
that gives the desired extrudate shape. Here, the free surface is defined
by a finite number of particle pathlines. The difference in the shape of
the extrudate and the desired shape is calculated at every iteration, and
all the pathlines are translated by the amount needed to make the final
cross-section of the extrudate equal to the desired cross-section. The
die is then adjusted accordingly. Lee and Ho [10] combined numerical
predictions of the extrudate shape and experiments to design the cross-
section of the die for polystyrene extrudates. Legat and Marchal [11]
developed an implicit finite element model, which can be solved using a
https://doi.org/10.1016/j.jnnfm.2021.104552
Received 29 January 2021; Received in revised form 23 April 2021; Accepted 27 April 2021
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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M.M.A. Spanjaards et al.
Newton–Raphson technique, to find the shape of the extrudate coming
out of a die with a prescribed shape. This method was extended to solve
the inverse problem [12]. Here, the initial condition on the free surface
positions at the die exit are replaced by the constraint that the shape of
the extrudate is prescribed at the extrudate outlet. The Lagrange multi-
plier needed to enforce this constraint is used to adapt the die shape in
such a way that the constraint on the extrudate shape at the extrudate
outlet is satisfied. A disadvantage of this method is that the die exit
and extrudate outlet have to be topologically identical. This means that
mesh refinement near the die exit is not allowed. This inverse problem
is available in the ANSYS Polyflow finite element CFD package and used
by Sharma et al. [13] to design the die for a keyhole shaped extrudate
for non-isothermal extrusion flow. All methods mentioned above are
steady state methods for viscous, or generalized Newtonian fluids.
A novel approach to optimize the die scheme is to solve the inverse
problem making use of a real time active control scheme. Control
is a powerful optimization method, because once you have a stable
controller it optimizes for the desired extrudate shape independent
of what desired shape you choose [14]. An example of using control
in combination with polymeric fluids is used in filament stretching
rheometry [15]. A combination of a feedback/feedforward control
scheme is used in [16] to maintain a constant strain rate of the mid-
filament diameter in a filament stretching rheometer for polymer melts.
This method was incorporated in a finite element method by van Berlo
et al. [17]. If the control parameters were chosen properly, it was
found that this is an effective approach to maintain the desired strain
rate by controlling the radius of the filament. Therefore, in this paper
a similar approach is used for die design for extrudate swell. Instead of
defining an objective function, a feedback interconnection is envisioned
as used in [18]. To this end, the corner line method as described in [19],
is coupled with a feedback control scheme to obtain the desired die
shape.
Although the shape of the extrudate is highly influenced by the
viscoelasticity of the polymer, most published die optimization methods
are based on a viscous, or generalized Newtonian assumptions. There-
fore, the objective of this paper is to develop a 3D method that can
include transient rheological effects and viscoelasticity to be able to
optimize the die shape for rheologically complex fluids. First a problem
description is given in Section 2, followed by a detailed explanation of
the optimization method in Section 3and the numerical method of the
finite element framework in combination with the feedback connection
in Section 4. A proof of concept of this method is given for a 2D
axisymmetric viscoelastic extrusion problem as well as for a viscous
fluid exiting 3D dies with shapes of increasing complexity. Finally, in
Section 53D viscoelastic results are shown for different Weissenberg
numbers and mobility parameters 𝛼in the Giesekus model.
2. Problem description
In this paper we model the optimization of an extrudate exiting an
adjustable die. Only a quarter of the problem is modeled to reduce
computational costs. The problem is schematically depicted in Fig. 1
for an initially rectangular die. The first part of the domain is the fluid
contained in an adjustable die of height 𝐻0and width 𝑊0. A constant
flowrate 𝑄is applied at the inlet 𝛤in of the die. After length 𝐿die the
fluid is exiting the die. The extrudate is modeled for a length 𝐿extr after
the die exit. The corner line of the extrudate, used in the corner-line
method [19], is indicated in red.
2.1. Balance equations
It is assumed that inertia can be neglected, the fluid is incompress-
ible and that there are no external body forces acting on the fluid. Using
these assumptions, the mass- and momentum balance can be written as
follows:
∇⋅𝒖= 0 in 𝛺, (1)
−∇ ⋅𝝈=𝟎in 𝛺, (2)
where, 𝛺is the fluid domain, 𝒖is the fluid velocity and 𝝈is the Cauchy
stress tensor:
𝝈= −𝑝𝑰+ 2𝜂s𝑫+𝝉.(3)
Here, 𝑝is the pressure, 𝑰is the unit tensor, 2𝜂s𝑫is the viscous stress
contribution with the solvent viscosity 𝜂sand the rate of deformation
tensor 𝑫= (∇𝒖+ ∇𝒖𝑇)∕2. The viscoelastic stresses are presented by 𝝉
and are defined in the following section.
2.2. Constitutive equations
The viscoelastic stress tensor can be described using the conforma-
tion tensor 𝒄as follows:
𝝉=𝐺(𝒄−𝑰),(4)
where 𝐺is the polymer modulus. The evolution of the conformation
tensor can be written as:
𝐷𝒄
𝐷𝑡 − (∇𝒖)𝑇⋅𝒄−𝒄⋅∇𝒖+𝒇(𝒄) = 𝟎,(5)
where 𝐷()∕𝐷𝑡 denotes the material derivative and 𝒇(𝒄)depends on the
constitutive model used. In this paper, without loss of generality, the
Giesekus model is used [20]:
𝒇(𝒄) = 1
𝜆𝒄−𝑰+𝛼(𝒄−𝑰)2,(6)
where, 𝜆is the polymer relaxation time and 𝛼is the mobility parameter
that influences shear thinning. The polymer viscosity can now be
defined as 𝜂p=𝐺𝜆 and the ratio between the solvent viscosity and
the total viscosity as 𝛽=𝜂s∕(𝜂s+𝜂p).
2.3. Arbitrary Lagrangian–Eulerian formulation
The problem contains moving boundaries due to the movement of
the free surfaces. Therefore, a body-fitted approach is used to take
the free surface movement into account. To this end, the domain is
described with a mesh that is moving in time in such a way that it
follows the movement of the free surfaces, but not necessarily of the
fluid. Therefore, the governing equations are rewritten in the Arbitrary
Lagrangian–Eulerian (ALE) formulation [21]. The movement of the
mesh has now to be taken into account in equations that contain a
material derivative:
𝐷()
𝐷𝑡 =𝜕()
𝜕𝑡
𝜉+ (𝒖−𝒖m)⋅∇().(7)
Here, 𝜕()∕𝜕𝑡
𝜉denotes the time derivative at a fixed grid point and 𝒖m
is the mesh velocity.
2.4. Corner-line method
The corner-line method is used to obtain the positions of the free
surfaces of the extrudate. In this method, the corner line (indicated in
red in Fig. 1), is described as a material line. In order to obtain the 𝑦-
and 𝑧-positions of this line, the following kinematic equation is solved:
𝜕𝒅
𝜕𝑡 +𝑢𝑥
𝜕𝒅
𝜕𝑥 =𝒖2D,(8)
where 𝒅is the position vector containing the positions 𝑓in 𝑦- and 𝑧-
direction 𝒅= (𝑓𝑦, 𝑓𝑧), and 𝒖2D is the velocity vector containing the
velocities in 𝑦- and 𝑧-direction 𝒖2D = (𝑢𝑦, 𝑢𝑧).
The free surfaces, connected by the corner line, are described using
2D height functions. The domain over which the height functions need
to be applied is not constant, but changes due to the movement of the
corner line. Therefore, this domain is expanded in time, according to
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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M.M.A. Spanjaards et al.
Fig. 1. Schematic depiction of a quarter of the 3D extrusion problem out of an initially rectangular die with the corner line, used in the corner-line method, depicted in red.
the movement of the corner line. This leads to the following equation
to obtain the heights ℎof the free surfaces:
𝜕ℎ
𝜕𝑡
𝜉+𝑢𝑥
𝜕ℎ
𝜕𝑥 + (𝑢𝑧−𝑢m,𝑧 )𝜕ℎ
𝜕𝑧 =𝑢𝑦,(9)
where, 𝜕()∕𝜕𝑡
𝜉denotes the time derivative in a fixed grid point of the
2D grid of the expanding domain, the subscript 𝑧indicates the direction
of the expanding 2D (𝑥,𝑧) domain and 𝑢m,𝑧 the corresponding mesh
velocity. The subscript 𝑦indicates the swell direction of the free surface
𝛤extr,1(see Fig. 1). Two separate 2D surface problems are created for
which Eq. (9) is solved. For surfaces 𝛤extr,2,𝑦and 𝑧are interchanged,
due to the rotation of the surface with respect to 𝛤extr,1.
2.5. Boundary- and initial conditions
A schematic overview of the domain is shown in Fig. 1. To describe
developed inflow conditions at the inlet boundary 𝛤in, a subproblem
of a channel that has the same cross-sectional shape as the die, with
periodic boundaries is solved. A flow rate 𝑄=𝑈avg𝐴is enforced to this
channel as a constraint, using a Lagrange multiplier. Here, 𝑈avg is the
average velocity in the channel and 𝐴is the area of the inlet of the
channel. The resulting velocity 𝒖chan, and conformation tensor solution
𝒄chan of this channel are then prescribed as an essential boundary
condition to the inlet boundary (𝛤in) of the 3D problem. At the walls
of the die (𝛤die) the velocity is the same as the mesh velocity, so
that the fluid follows the movement of the die due to adjustment by
the control-scheme, but fluid cannot penetrate the die wall. the free
surfaces 𝛤free are assumed to be traction free. At the outlet (𝛤out), there
is zero traction in 𝑥-direction. The boundary conditions are given by:
𝒖in =𝒖chan on 𝛤in,
𝒄in =𝒄chan on 𝛤in,
𝒖=𝒖mesh on 𝛤die,
𝑢𝑦= 0 on 𝛤out,
𝑢𝑧= 0 on 𝛤out,
𝑡𝑥= 0 on 𝛤out,
𝒕=𝟎on 𝛤free,
where 𝒖chan and 𝒄chan are obtained from the separate channel problem.
The traction vector on the surface with outwardly directed normal 𝒏is
denoted by 𝒕=𝝈⋅𝒏. To reduce computational costs, only a quarter of the
domain is modeled and the necessary symmetry boundary conditions
are applied to the symmetry planes 𝛤sym, see Fig. 1. Essential boundary
conditions are applied for the material line and the height functions of
the free surfaces such that they stay attached to the die. The initial
conditions for Eqs. (8) and (9) are given by:
𝒅(𝑡= 0) = 𝒅0,(10)
ℎ(𝑡= 0) = 𝐻0,(11)
where, 𝒅0and 𝐻0are equivalent to the initial coordinates of the corner
points of the die and the initial height of the die, respectively.
The initial condition for the conformation tensor 𝒄in Eq. (5) is given
by:
𝒄(𝑡= 0) = 𝑰,(12)
which defines the viscoelastic stress initially to be zero.
3. Optimization
The goal of this paper is not to predict the shape of the extrudate,
but to solve the inverse problem. To this end an adjustable die is
designed of which the shape is defined as a result of an optimization
loop that uses the output of the Finite Element (FE) model as input.
A feedback controller is used to calculate the new die height from
the discrepancy between the real extrudate height obtained with the
FE model and the ideal/desired extrudate height. The optimization
procedure is started with a FE step of the extrudate with an initial shape
as defined by the initial conditions of Eqs. (10) and (11). A schematic
depiction of the optimization scheme is shown in Fig. 2. In this figure
ℎ(𝑖)is the real extrudate height obtained with the FE model at control
step 𝑖. This height is compared to the ideal extrudate height ℎideal. If
they are not the same the difference 𝛿ℎ(𝑖) = ℎideal −ℎ(𝑖)is given as
input to the PI controller, together with the current die height 𝐻(𝑖). The
PI controller calculates a new die height 𝐻(𝑖+ 1) using the following
control equation [18]:
𝐻(𝑖+ 1) = 𝐻(𝑖) + 𝐾𝛿 ℎ(𝑖),(13)
where, 𝐾is a controller parameter that is found through trial-and-error.
3.1. 2D
In 2D the optimization routine is tested on a axisymmetric problem
as schematically shown in Fig. 3. The die has an adjustable die height 𝐻
and the extrudate has height ℎ. A fixed flow rate 𝑄is applied at the inlet
of the die. The feedback controller adjusts the die height 𝐻based on
the difference between the real extrudate height and the ideal extrudate
height 𝛿ℎ. Once the difference between the obtained and desired height
is less or equal to 1% (𝜖=1 − ℎ∕ℎideal≤0.01), the optimization is
stopped and some additional FE steps are performed to make sure the
extrudate is in steady state.
3.2. 3D
In 3D it is no longer sufficient to control one point on the die.
The extrudate now does not only need to have the desired dimensions,
but also the desired shape. To achieve this, multiple controllers are
needed. For the die shapes used in this paper, four controllers are used.
A schematic representation of an extrudate emerging from an initially
rectangular die is shown in Fig. 4(a). Here, the obtained extrudate from
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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M.M.A. Spanjaards et al.
Fig. 2. Schematic representation of the optimization scheme using a feedback controller.
Fig. 3. Schematic representation of a 2D axisymmetric die with adjustable die height 𝐻and fixed applied flowrate 𝑄.
the initial die is shown in blue, whereas the desired extrudate is shown
in black. The height ℎand width 𝑤of the extrudate are compared to
the ideal dimensions. The corner point of the extrudate is controlled
by two controllers via the diagonal length 𝑑and the angle 𝜃. These
are also compared to ideal values. The difference between the obtained
and ideal values are input for four different controllers from the form
of Eq. (13) and give the height 𝐻, width 𝑊, diagonal length 𝐷and
angle 𝛩of the optimized die, as shown in Fig. 4(b). The output of the
four controllers (𝐻,𝑊,𝐷,𝛩) together with the prerequisite that the
die shape is symmetric, give the positions of the red points in Fig. 5.
Sine functions can be constructed through these red points. All nodes
on the die are moved to their new positions according to these sine
functions to obtain the optimized die shape, see Fig. 5.
4. Numerical method
The finite element method is used to solve the governing equa-
tions. The log-conformation representation [22], SUPG [23] and the
G-method [24] are used for stability in solving the constitutive equa-
tion. The corner-line method is used for the description of the free
surfaces [19], here SUPG is also used for stability.
4.1. Spatial discretization
The weak-formulations and spatial discretization are the same as
presented by Spanjaards et al. [25]. For the velocity and the pres-
sure isoparametric, tetrahedral 𝑃2𝑃1(Taylor–Hood) elements are used,
whereas for the conformation tetrahedral 𝑃1elements are used. For the
1D height functions of the corner lines quadratic line elements are used
and for the 2D height functions of the free surfaces quadratic triangular
elements are used. The SUPG parameters are obtained as described by
Spanjaards et al. in [19].
4.2. Time discretization
The numerical procedure of every time step in the finite element
method is explained step by step. The output of the FE scheme serves
as an input for the control scheme as schematically depicted in Fig. 6.
The numerical procedure of the optimization scheme is also presented
in this section.
Finite element method
The numerical procedure for every time step in the finite element
method is explained first. The positions of the free surfaces are obtained
using a predictor–corrector scheme [26]. Since the boundary condition
of the velocity at the die walls is prescribed such that the fluid follows
the movement of the die, there are jumps in time in the velocity after
every optimization step. This means that the solution is not smooth, and
therefore first-order time integration is used. The numerical scheme can
be divided in the following steps:
Step 1: Update the position of the free surfaces, 𝒙free, in the bulk mesh
using a first order prediction of the free surface positions: 𝒙free,pred =
𝒉𝑛
free. Here, 𝒉free are the positions of the free surfaces obtained from
solving the 1D height functions for the corner-lines (8) and the 2D
height functions of the free surfaces (9).
Step 2: Construct the ALE mesh by solving a Poisson equation:
∇2𝒅m=𝟎,(14)
with boundary condition 𝒅m=𝒅free on the free surfaces 𝛤free, where
𝒅free is the displacement of the free surface nodes to the prediction
of the new position. The new coordinates of the mesh nodes 𝒙𝑛+1
mare
calculated with the obtained mesh displacement.
Step 3: The mesh velocities can now be obtained by numerically dif-
ferentiating the mesh displacement using a first-order backward Euler
method:
𝒖𝑛+1
m=𝒙𝑛+1
𝑚−𝒙𝑛
𝑚
𝛥𝑡 ,(15)
where 𝛥𝑡 is the time step used.
Step 4: Compute 𝒖𝑛+1 and 𝑝𝑛+1. The method of D’Avino and Hulsen [27]
for decoupling the momentum balance from the constitutive equation
is applied. This means that at every time step the balance equations are
solved using a prediction for the viscoelastic stress tensor, to find 𝒖𝑛+1
and 𝑝𝑛+1.
Step 5: After solving for the new velocities and pressures, the actual
conformation tensor 𝒄𝑛+1 is found using a first-order Euler combined
backward/forward scheme.
Step 6: Update the position of the material lines by solving Eq. (8). Here,
also first-order time integration is used:
𝒅𝑛+1 −𝒅𝑛
𝛥𝑡 +𝑢𝑛+1
𝑥
𝜕𝒅𝑛+1
𝜕𝑥 =𝒖𝑛+1
2D ,(16)
Step 7: Expand the surface meshes of the height function of the free
surfaces in 𝑧-direction, using the displacement of the material lines. In
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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M.M.A. Spanjaards et al.
Fig. 4. Schematic representation of a cross-section of (a) the desired extrudate (black) and the obtained extrudate (blue) with the four control points of the extrudate and (b) the
original die (black) and the optimized die (blue) with the four control points of the die.
Fig. 5. Schematic representation of a cross-section of the initial die (black) and the optimized die (blue) with the new positions for the height, length, diagonal length and angle
obtained from the controllers (red dots) and the sine functions constructed through these points (dashed blue). The mid-planes of the sine functions are equal to the desired
extrudate height and length (dashed black).
Fig. 6. Schematic representation of the coupling between the FE scheme and the control scheme.
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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M.M.A. Spanjaards et al.
this way it is known how the domain of the height function changes.
Construct the ALE mesh by solving a Poisson equation with Dirichlet
boundary conditions. The new coordinates of the nodes are calculated
with the obtained mesh displacement. The mesh velocities can now be
obtained by numerically differentiating the mesh displacement. Here,
again a backward differencing scheme is used, using the updated mesh
nodes. For the upper free surface in Fig. 1, for example, this gives the
following equation:
𝑢𝑛+1
m,surf =
𝑧𝑛+1
m,surf −𝑧𝑛
m,surf
𝛥𝑡 ,(17)
where 𝑧m,surf is the displacement of the surface mesh in 𝑧-direction.
Step 8: Update the height ℎ𝑛+1 of the free surfaces by solving the
evolution equation of the 2D height function of the free surfaces (9),
using first-order time integration:
ℎ𝑛+1 −ℎ𝑛
𝛥𝑡 +𝑢𝑛+1
𝑥
𝜕ℎ𝑛+1
𝜕𝑥 + (𝑢𝑛+1
𝑧−𝑢𝑛+1
𝑚,𝑧 )𝜕ℎ𝑛+1
𝜕𝑧 =𝑢𝑛+1
𝑦.(18)
For surfaces 𝛤extr,2,𝑦and 𝑧are interchanged, due to the rotation of the
surface with respect to 𝛤extr,1.
Control scheme
The output of the FE scheme after 𝑁cFE steps is used as input
for the control scheme to optimize the die shape. Different choices
for 𝑁cwill be discussed in Section 5. The numerical procedure of the
optimization scheme will now be presented step by step.
Step 1: 𝑁cFE time steps are performed to obtain the extrudate dimen-
sions. The time integration scheme for every FE time step is presented
in the previous paragraph.
Step 2: The extrudate dimensions at the end of the extrudate 𝛤out are
compared to the ideal extrudate dimensions.
Step 3: The difference between the real- and ideal extrudate dimensions
is computed. If this difference is larger than the maximum allowed
difference 𝜖=1 − ℎ∕ℎideal>0.01, new coordinates of the control
points on the die are calculated using feedback controllers according
to Eq. (13). Sine functions are constructed through the new control
points as shown in Fig. 5.
Step 4: All nodes on the die are moved to the positions on these sine
functions by solving the Poisson equation (14) with boundary condition
𝒙displ =𝒅die on the die surfaces 𝛤die, where 𝒅die is the displacement of
the die nodes to the sine functions.
Step 5: 𝑁cFE time steps are performed using the new die shape.
The optimization scheme depicted in Fig. 2 is continued until 𝜖=
1 − ℎ∕ℎideal≤0.01. After that FE time steps are performed with the
final die for a characteristic time corresponding to 25𝜆to make sure the
steady state extrudate with the optimized die has the same dimensions
as the desired extrudate.
4.3. Remeshing and projection
For optimization of viscoelastic extrudates, the change in die shape
can be severe, due to the large amount of swell for viscoelastic fluids.
After a number of optimization steps, the elements get so distorted
that this results in instabilities on the free surfaces and eventually
termination of the numerical algorithm. To avoid this, the initial mesh
is used until it becomes too distorted. When this happens a new mesh,
covering the same domain as the old one, is generated using Gmsh [28]
and the solutions on the old mesh are projected onto the new one.
To this end, a projection problem is solved to project the old solution
variables on the new mesh in the same way as was done by Jaensson
et al. [29].
To quantify the mesh distortion, the change in volume and change
in aspect ratio of the elements are checked:
𝑓1=log(𝑉e∕𝑉e
0),(19)
Table 1
Different meshes used in the convergence study of the 2D swell problem without
optimization.
Mesh #elements ℎelem ℎelem,ref
M0 222 0.4𝐻00.2𝐻0
M1 818 0.2𝐻00.1𝐻0
M2 3 220 0.1𝐻00.05𝐻0
M3 13 034 0.05𝐻00.025𝐻0
𝑓2=log(𝑆e∕𝑆e
0),(20)
where, 𝑉eis the element volume and 𝑉e
0is the volume of the unde-
formed mesh. The aspect ratio is defined is 𝑆e= (𝐿e
max)3∕𝑉e
0, where
𝐿e
max is the maximum length of the sides of an element and 𝑆𝑒
0is
the aspect ratio of an element of the undeformed mesh. Remeshing is
invoked if either 𝑓1>0.4or 𝑓2>0.4which is equivalent to a change
in volume or aspect ratio of 1.5. For the 2D problem, the change in
volume is replaced by the change in area.
5. Results
This section will first show mesh convergence results, followed by
results for 2D optimization for a viscoelastic fluid and 3D optimization
for both viscous and viscoelastic fluids. For all simulations a dimension-
less time step size of 𝛥𝑡𝑈avg∕𝐻0= 1 ⋅10−2 is used, as was shown that
this time step size gave accurate results in our earlier work [19].
5.1. Problem domain and meshes
Detailed convergence studies of the Finite Element Method can
be found in our previous work [19,25]. In this work we are mainly
interested in predicting the extrudate shape. Therefore, meshes are
chosen that can predict the extrudate shape reasonably well, while
limiting the computational costs as much as possible. Swell simulations
are performed for meshes with different element sizes ℎelem compared
to the initial height of the die 𝐻0without optimization steps to find a
suitable element size that can accurately predict the extrudate shape,
but also limits the computational costs of a finite element time step. In
the 2D simulations a Weissenberg number of 2.25 is used, whereas in
the 3D simulations a Weissenberg number of 2 is used, calculated using
the following definition:
Wi0=
𝑈avg
𝐻0
𝜆, (21)
where 𝐻0is the initial height of the adjustable part of the 2D axisym-
metric die and the 3D dies. The subscript 0in Eq. (21) refers to the
initial Weissenberg number of the fluid in the undeformed die.
2D meshes
The 2D meshes are refined near the die exit, where the element
size is ℎelem,ref = 0.5ℎelem with respect to 𝐻0. More information on the
meshes used in this study can be found in Table 1.Fig. 7(a) shows
the steady state free surface profiles for the different meshes listed
in Table 1. The swell ratios of the extrudate height for the different
meshes are listed in Table 2. To limit computational costs but still get
a reasonable prediction of the extrudate shape, mesh M2 is used for the
2D simulations in the remainder of this paper. This mesh is shown in
Fig. 7(b).
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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M.M.A. Spanjaards et al.
Fig. 7. Plots of the free surface profile of the steady state extrudate calculated with different 2D meshes (a) and the chosen mesh M2 to perform the 2D simulations (b).
Fig. 8. Mesh M2 (a) and M2ref (b).
Fig. 9. Plots of the contour of the steady state extrudate calculated with different meshes (a) and a zoomed in version at the corner of the extrudate (b).
Table 2
Swell ratios of the extrudate height for the different 2D meshes.
Mesh ℎ∕𝐻0
M0 1.593
M1 1.606
M2 1.624
M3 1.625
3D meshes
Since the 3D simulations are much more computationally demand-
ing than the 2D simulations, two types of meshes are used: one where
the elements near the die exit are extremely refined to ℎelem,ref =
Table 3
Different meshes used in the convergence study of the 3D swell problem without
optimization.
Mesh #elements ℎelem ℎelem,ref
M0 476 0.4𝐻0–
M1 2191 0.2𝐻0–
M2 15 397 0.1𝐻0–
M3 115 668 0.05𝐻0–
M2ref 149 346 0.1𝐻00.01𝐻0
0.1ℎelem and one where all elements have the same size ℎelem. More
information on the 3D meshes used in this study can be found in
Table 3. Meshes M2 and M2ref are shown in Fig. 8. For this paper we are
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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M.M.A. Spanjaards et al.
Table 4
Swell ratios of the control points for the different meshes.
Mesh ℎ∕𝐻0𝑤∕𝑊0𝑑∕𝐷0𝜃∕𝛩0
M0 1.565 1.432 1.217 0.824
M1 1.461 1.372 1.212 0.831
M2 1.457 1.372 1.236 0.844
M3 1.461 1.370 1.249 0.842
M2ref 1.472 1.376 1.250 0.843
mainly interested in an accurate extrudate shape prediction. Therefore,
a mesh should be used that can accurately predict the extrudate shape
but also limits the computational costs of a FE time step. This means
that we are not looking for a mesh that can accurately predict transient
rheological phenomena, but instead we will compare the swell ratios of
the different control points indicated in Fig. 4 for the different meshes.
The cross-section of the steady state extrudate for the different meshes
is shown in Fig. 9(a) with a zoom in on the corner of the extrudate in
9(b). The swell ratios of the control points for the different meshes are
shown in Table 4. To limit computational costs but still get a reasonable
accurate prediction of the extrudate shape, mesh M2 is used for the
remainder of this paper.
5.2. 2D
This section shows the results of the 2D optimization scheme. First,
two different test cases for the control scheme of the 2D optimization
problem are discussed, followed by the results of the two different test
cases.
2D test cases
With regard to stability of the controller and the FE scheme, two
different cases are tested in this paper. In the first case an optimization
step is done every FE time step. In the second case an optimization
step is performed every 𝑁cFE time steps, where 𝑁csteps correspond
to a characteristic time scale 𝑡c, which can be different for different
fluids. In this case the free surface has time to adjust to the new die
height before the next control step is performed. It was found that this
improves stability of the numerical algorithm.
Case 1: control every FE time step
When a control step was performed every FE time step, it was
found to be hard to keep the controller stable. Relatively small control
parameters 𝐾are needed to obtain a stable controller. The stability of
the controller was tested by performing simulations for different initial
Weissenberg numbers and different control parameters. Fig. 10 shows
the ratio of the error 𝛿ℎ and the ideal height ℎideal as a function of
dimensionless time for different control parameters and different initial
Weissenberg numbers. The results show that for high values of 𝐾the
controller becomes unstable and keeps oscillating around the desired
value of 𝛿ℎ. For too high values of 𝐾the mesh adjustment due to the
control step is too big and the free surface does not have time to adjust
to the new die height and the FE code terminates. For higher initial
Weissenberg numbers the unstable controller behavior was observed
for smaller values of 𝐾. For higher Wi0the swell is more severe, which
leads to a larger value of 𝛿ℎ and therefore a bigger adjustment in
die height given by the controller, for the same 𝐾value. When the
adjustment is too big this leads to unstable behavior for the controller
itself, or the finite element method.
Case 2: control every 𝑁cFE time steps
Since it was hard to keep the controller and the finite element
method stable when a control step was performed every finite element
time step, it is chosen to perform a control step every 𝑁cFE time steps,
where 𝑁ctime steps correspond to a characteristic time typically in the
range of [0.5 − 5]𝜆. The error 𝛿ℎ as a function of dimensionless time
is plotted for different control parameters and different characteristic
times 𝑡cbetween the control steps for Wi0= 4.5in Fig. 11. This figure
shows that for high Wi0and small 𝑡c, the highest control parameter
for which the optimization scheme can be performed is 𝐾= 5 ⋅10−2 .
For higher values of 𝐾the free surface does not have enough time
to adjust to the new die height in between control steps and the FE
scheme becomes unstable and terminates. For larger values of 𝑡cit is
easier to obtain a stable controller and FE scheme, however it takes
longer to reach the desired extrudate height since the time between
control step is larger. Comparing Fig. 11 to the results of Case 1, we
see that stability of the controller is obtained more easily in Case 2,
since oscillations around the desired extrudate height are only observed
for Case 1. Since in 3D it is even more challenging to keep the FE
scheme stable, but fast convergence of the controller is also desired,
a characteristic time of 𝑡c= 2𝜆in between control steps is chosen
throughout the rest of this paper. Fig. 12 shows the results for different
control parameters and different initial Weissenberg numbers when the
characteristic time between control steps is 𝑡c= 2𝜆. This figure shows
that larger 𝐾values lead to a faster convergence of the controller.
What makes these viscoelastic simulations challenging, is the increasing
Weissenberg number in time, due to the adjustment of the die height
by the control scheme. Fig. 13 shows the trace of the conformation
tensor in the fluid domain for Wi0= 4.5and 𝐾= 5 ⋅10−2 , at different
dimensionless times. The black line indicates the desired extrudate
height. The figure shows an increase in the trace of the conformation
tensor when the die height is reduced, indicating that the polymers are
more stretched. Furthermore, it shows that in time, the height of the
extrudate approaches the desired extrudate height.
5.3. 3D
This section shows the results of the 3D optimization scheme. First,
three different cases to construct sine functions through the control
points on the die are discussed. After that, a proof of concept is done
by performing optimization simulations of a Newtonian fluid exiting
dies of increasing complexity. Finally, results are shown for viscoelastic
fluids and an initially rectangular die. The rectangular die in the results
below has an aspect ratio of 2:1. The results of the different cases for the
sine function parametrization of the die are shown for the rectangular
die.
3D test cases
Three different cases are distinguished for constructing sine func-
tions through the control points as schematically depicted in Fig. 14.
Here, the first case is the sine function with a fixed midplane and a free
period, where it is assumed that the mid-plane of the sine equals the
height or width of the desired extrudate, respectively, as presented in
Fig. 5. In the second case the sine no longer has a fixed midplane, but
now has a fixed period of 2𝜋. In the third case, sine functions with a
fixed period are used until the error of all control points is 𝜖≤0.05
after which we change to the fixed midplane approach where the
midplane is the same as the midplane calculated for the sine with the
fixed period. Control parameters 𝐾for the four different controllers are
obtained through trial-and-error. Every controller can have a different
𝐾parameter. In this work, 𝐾values in the range [1.25 − 7.5] ⋅10−3 are
used, where the control parameters for the controllers of the diagonal
length 𝐷and angle 𝛩are generally smaller than the control parameters
for the controllers of the height 𝐻and width 𝑊, respectively. It was
found that stability was harder to achieve for the 3D optimization
scheme compared to the 2D optimization scheme. This is most likely
caused by the more complex die shape. Therefore, a control step is
performed every 𝑁cFE time steps, where 𝑁cagain corresponds to a
characteristic time scale 𝑡cwhich can be different for different fluids. In
an attempt to decouple the controller for 𝐷and 𝛩, they are controlled
every other control step, but not simultaneously.
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Fig. 10. Error in extrudate height as a function of time for different initial Weissenberg numbers and controllers with different control parameters 𝐾when a control step is
performed every FE time step.
Newtonian fluid
To show that our method works for die shapes of increasing com-
plexity, 3D optimization simulations are performed for a Newtonian
fluid exiting an initially square, rectangular and cross-shaped die. To
save computational costs, only a quarter of the problems is modeled.
For the cross-shaped die only one eight of the problem is modeled,
so the problem can still be described with two free surfaces. The
characteristic time between control steps is chosen as 𝑡c=𝑊0∕𝑈avg. The
results are shown in Fig. 15. Here, the blue lines show the optimized
die- and extrudate shapes, whereas the red lines show the initial die
shape and the not-optimized, swollen, extrudate. The figure shows that
the final extrudates that are obtained are in reasonably good agreement
with the desired extrudate shape, indicated by the black dotted line.
For the initially cross-shaped die it was found that a small adjustment
of the die shape description with the sine functions improved the final
extrudate shape. This is shown in Fig. 16.Fig. 16(a) shows the final
extrudate shape when the die shape is described with the sine functions
as discussed in Case I as described in Fig. 14. The control points on the
extrudate now all have an error of 𝜖≤0.01 compared to the desired
extrudate. However, we see that there are points on the red lines of the
extrudate cross-section that have an error that is larger than the error
of the control points. It is now checked on which side of the midplane
on the sine function the maximum error occurs, and the sine function
of the corresponding line on the cross-section of the die, indicated in
red in Fig. 16(b) is now changed to a straight line. The result is that the
sharp corner indicated by the red lines in Fig. 16(a) is better preserved
and the cross-section of the final extrudate does now corresponds better
to the desired extrudate shape. For the remainder of this paper it is
checked if there are maximum errors larger than 𝜖≤0.02 on the lines
of the cross-section of the extrudate in between the control points. If
this is the case, the line-technique is used to reduce this error.
Viscoelastic fluid
Now we demonstrated that the 3D optimization method results in
extrudate shapes that are in good agreement with the desired extrudate
shape for a Newtonian fluid, we can extend the method to optimize the
die for viscoelastic extrudates. Optimization is performed for initially
rectangular shaped dies and for an initial Weissenberg number of Wi0=
1and Wi0= 2. The characteristic time step between control steps is
𝑡c= 2𝜆. Optimization is performed for a Giesekus fluid with a viscosity
ratio 𝛽= 0.4and a mobility parameter 𝛼= 0.1. Furthermore, the
influence of the mobility parameter is studied for a viscosity ratio of
𝛽= 0.1for Wi0= 1.
First we show the optimization results for Wi0= 1 and Wi0= 2
and 𝛽= 0.4in Fig. 17. Here, we use the method as depicted by
Case I in Fig. 14. From this figure it can be observed that while the
description of sine functions with a fixed midplane is sufficient to
obtain the desired extrudate for Wi0= 1, the corner of the extrudate
for the higher initial Weissenberg number is sharper as we would like.
Also it was found to be really challenging to keep the simulation stable
due to the rise of an extreme narrow corner in the die for higher
Weissenberg numbers. Moreover, small wiggles start to appear on the
extrudate surface. Therefore, two different adjustments to the sine
function description are made as shown in Fig. 14. The optimization
result for case II in Fig. 14 is shown in 18(a) for Wi0= 2. Here the
same line-technique as discussed for the cross-shaped die is used to
minimize the final maximum error of the extrudate shape and preserve
the sharp corner. The result for case III in Fig. 14 is shown in Fig. 18(b)
for Wi0= 2. This Figure shows that for higher Wi0the combined
method as described by case III results in the best preservation of the
sharp corners of the extrudate and also leads to more stable simulations
than the results for Case I. However, the differences between the three
cases are small. The Weissenberg number in de 3D simulations does
also increase in time due to the die adjustment by the control scheme.
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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Fig. 11. Error in extrudate height as a function of time for characteristic times in between control steps and controllers with different control parameters 𝐾for Wi0= 4.5.
Fig. 12. Error in extrudate height as a function of time for different initial Weissenberg numbers and controllers with different control parameters 𝐾when the characteristic time
between control steps is 𝑡c= 2𝜆.
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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Fig. 13. Trace of the conformation tensor 𝒄in the fluid domain for Wi0= 4.5and 𝐾= 5 ⋅10−2 at different dimensionless times. The black line indicates the desired extrudate
height.
Fig. 14. (a) Case I: Sine with Fixed midplane of 𝐻0. (b) Case II: Sine with fixed period of 2𝜋. (c) Case III: case II followed by case I (see text).
Fig. 19 shows the Weissenberg number as function of dimensionless
time for the two initial Weissenberg numbers, Wi0= 1 and Wi0= 2
(a) and the trace of the conformation tensor in a cross-section of the
die at a distance 𝐿=𝐻0from the die exit at different moments in
time (b) as indicated in Fig. 19(a) for Wi0= 1. The Weissenberg
number is obtained by calculating the new average velocity 𝑈avg by
dividing the flowrate 𝑄by the area of the die inlet 𝐴die and dividing
it by the new die height 𝐻𝑛+1. The Weissenberg number can than be
calculated using Eq. (21).Fig. 19(b) also shows a clear increase in
the trace of conformation tensor, indicating that the polymers become
more stretched when the die is deformed. Fig. 20 shows the die in dark
gray and the extrudate in light gray at different dimensionless times
𝑡cindicated by the same symbols as used in Fig. 19. The black lines
indicate the initial die and desired extrudate shape. To give a bit more
insight in how the rheology of the fluid influences the final shape of
the die, simulations are performed for Wi0= 1, using a viscosity ratio
𝛽= 0.1and different values for the mobility parameter 𝛼in the Giesekus
model. The shear viscosity 𝜂divided by the zero-shear viscosity 𝜂0as
a function of Weissenberg number is plotted for the different mobility
parameters in Fig. 21(a). Fig. 21(b) shows the final cross-section of the
die for the different 𝛼parameters (left) and the not-optimized (swollen)
extrudate (right). This figure shows that the swell is decreased with
increasing 𝛼, which leads to a slightly less deformed cross-section of
the final die. This effect seems to be more severe for the width of the
die 𝑊, compared to the height of the die 𝐻.
6. Conclusions
In this paper we proposed a novel approach to solve the inverse
problem for three-dimensional die design for extrudate swell. Instead of
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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Fig. 15. Left: not optimized (red) and optimized (blue) die, right: not optimized (red) and optimized (blue) extrudate of a Newtonian fluid exiting a square (a), rectangular (b)
and cross-shaped (c) die. The black dotted line indicates the desired extrudate shape.
Fig. 16. Left: optimized die, right: final extrudate for a die description using only the sine functions (red lines indicate extrudate lines with largest error) (a) and an adjustment
of part of the sine of the red lines to a straight line (b).
Fig. 17. Left: not optimized (red) and optimized (blue) die, right: not optimized (red) and optimized (blue) extrudate for Wi0= 2, using a sine description with a fixed midplane
for an initial Weissenberg number of Wi0= 1 (a) and Wi0= 2 (b). The black dotted line indicates the desired extrudate shape.
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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Fig. 18. Left: not optimized (red) and optimized (blue) die, right: not optimized (red) and optimized (blue) extrudate for Wi0= 2, using a sine description with a fixed period
(Case II) (a) and a combination of a fixed period and a fixed midplane (Case III) (b). The black dotted line indicates the desired extrudate shape.
Fig. 19. The Weissenberg number as function of dimensionless time for initial Weissenberg numbers of Wi0= 1 and Wi0= 2 (a) and the trace of conformation tensor in a
cross-section o the die at a distance 𝐿=𝐻0from the die exit at different times for Wi0= 1.
Fig. 20. Extrudate emerging from the die at different dimensionless times 𝑡c, indicated by the symbols used in Fig. 19, for Wi = 1.0. The die is represented in dark gray and the
black lines indicate the initial die and desired extrudate shape.
following the currently most used approach of minimizing an objective
function, we used a real time active control scheme. To this end
we envisioned a feedback connected between the corner-line finite
element method, used to predict the positions of the free surfaces of the
extrudate, and the control scheme. Advantages of this method are that
there are no restrictions on the position of the elements, which allows
for local mesh refinement, and that transient rheological phenomena
can be taken into account.
The feedback control optimization approach is tested for a 2D
axisymmetric extrusion problem, as well as 3D extrusion problems of
die shapes with increasing complexity. In the 2D axisymmetric case,
one controller is sufficient to control the height of the die in order
to obtain a desired extrudate. For the symmetric 3D dies studied in
this paper, four controllers where used to control the extrudate shape.
Furthermore, sine functions are used to describe the cross-sectional
shape of the 3D dies.
For the 2D axisymmetric problem, two cases where studied; in
the first case a control step was performed every finite element step,
whereas in the second case a control step was performed every 𝑁cFE
steps, where 𝑁ctime steps correspond to a characteristic time 𝑡c. It was
Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104552
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Fig. 21. Dimensionless shear viscosity as a function of the Weissenberg number for different mobility parameters 𝛼in the Giesekus model with viscosity ratio 𝛽= 0.1(a) and the
optimized die shape (left) and the not-optimized extrudate (right) for the different 𝛼parameters (b). Black dotted lines indicate the initial die- and desired extrudate shape.
found that in the second case it was easier to obtain a stable controller
and a characteristic time of 𝑡c= 2𝜆was found to give a good balance
between computation time and stability for the Weissenberg numbers
tested in this paper.
In 3D, optimization simulations are performed for a Newtonian
fluid exiting dies with increasing complexity in shape. To this end, a
square, rectangular and cross-shaped die are used. It was found that the
extrudate shape for all three die shapes were in reasonable agreement
with the desired extrudate shape. Optimization simulations where also
performed for viscoelastic fluids exiting an initially rectangular shaped
die. It was found that these simulations where more challenging due to
the increasing Weissenberg number in time because of the adjustment
of the die shape by the controller. Furthermore, it was found that
for high Weissenberg numbers there were slight differences in the
final extrudate shape for different sine function descriptions. For more
complex, asymmetric die shapes a description of the die shape using
sine functions might be too limited. In this case splines might be a more
suitable approach to describe the cross-section of the die.
Finally 3D viscoelastic simulations were performed for different
amounts of shear-thinning by varying the mobility parameter 𝛼in the
Giesekus model. These results show that for increasing shear-thinning
the extrudate will swell less and hence the adjustment of the die was
found to be less severe for larger mobility parameters.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgments
The results of this study have been obtained through the FLEX-
Pro project, which was in part funded by the European Funding for
Regional Development (EFRO PROJ-00679). The research is performed
in collaboration with VMI Holland B.V.
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