modeling of optimization strategies in the incremental cnc_2004
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Modeling of Optimization Strategies in the Incremental CNCSheet Metal Forming Process
M. Bambach, G. Hirt, J. Ames
Institute of Materials Technology/Precision Forming (LWP), Saarland University, Germany
Abstract. Incremental CNC sheet forming (ISF) is a relatively new sheet metal forming process for small batch productionand prototyping. In ISF, a blank is shaped by the CNC movements of a simple tool in combination with a simplified die. Thestandard forming strategies in ISF entail two major drawbacks: (i) the inherent forming kinematics set limits on the maximumwall angle that can be formed with ISF. (ii) since elastic parts of the imposed deformation can currently not be accounted forin CNC code generation, the standard strategies can lead to undesired deviations between the target and the sample geometry.
Several enhancements have recently been put forward to overcome the above limitations, among them a multistage formingstrategy to manufacture steep flanges, and a correction algorithm to improve the geometric accuracy. Both strategies have beensuccessful in improving the forming of simple parts. However, the high experimental effort to empirically optimize the tool
paths motivates the use of process modeling techniques.This paper deals with finite element modeling of the ISF process. In particular, the outcome of different multistage
strategies is modeled and compared to collated experimental results regarding aspects such as sheet thickness and the onsetof wrinkling. Moreover, the feasibility of modeling the geometry of a part is investigated as this is of major importance withrespect to optimizing the geometric accuracy. Experimental validation is achieved by optical deformation measurement thatgives the local displacements and strains of the sheet during forming as benchmark quantities for the simulation.
INTRODUCTION
The incremental CNC sheet forming process as de-
scribed in [1-5] has been developed to meet the demands
of small batch sheet metal forming and rapid prototyp-
ing. Recent experimental work has revealed the need for
non–conventionalforming strategies to overcome currentlimitations, i.e. strategies that help optimize the tool path
to (i) produce steep flanges and (ii) reduce deviations
from the target geometry [6]. Currently, these strategies
are based on trial-and-error optimization of the tool path.
The experimental effort inherent in empirical tool path
optimization could in principle be reduced by process
modeling. The present paper provides first results of the
FE modeling of non–conventional ISF strategies.
PROCESS TECHNOLOGY
Process description
In ISF, a metal blank which is clamped into a rect-
angular blank holder is shaped by the continuous move-
ment of a simple ball-headed forming tool. The tool path
is prescribed by NC data that is generated from a CAD
model of the component to be formed. The conventional
forming strategy consists of a single forming stage where
the tool traces along a sequence of contour lines with a
vertical feed in between. Generally, a distinction is made
between "single point forming", where the bottom con-
tour of the part is supported by a rig and "two-point form-
ing", where full or partial positive dies support critical
surface areas of the part (Figure 1).
blank tool partial die
post
two-point ISFsingle point ISF
blankholder
FIGURE 1. Process variants in ISF
Process limit: sheet thinning
In conventional ISF, sheet thinning depends strongly
on the wall angle. Presuming volume constancy and a
deformation mode close to plane strain conditions, theso–called sine law
t 1
t 0
sin¡
90¢ £
α ¥
(1)
relates the initial (t 0
) and actual (t 1
) sheet thickness for
a given wall angle α . At wall angles of approximately
60¢
for Al99.5 and mild steel, a localization of the plas-
tic deformation can be observed [6]. Accordingly, wall
© 2004 American Institute of Physics 0-7354-0188-8/04/$22.00edited by S. Ghosh, J. C. Castro, and J. K. Lee
CP712, Materials Processing and Design: Modeling, Simulation and Applications, NUMIFORM 2004,
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angles of about 60-65¢
can be considered the maximum
for conventional ISF with sheets of 1–1.5 mm thickness.
This limitation on the maximum wall angle restricts the
potential scope of shapes and applications.
Process limit: geometric accuracy
A second process limit stems from the elastic portion
of the deformation. The tool path is generated exclu-
sively according to geometric information specified by
the CAD model of the desired part. Since the elastic por-
tion of the deformation (including backstresses induced
by the cyclic loading history) are neglected, experimental
work shows undesired deviations from the target geome-
try (see Figure 2).
mild steel (t0=0.8mm)
>3.00
2.25 – 3.00
1.50 – 2.25
0.75 – 1.50
0.00 – 0.75
-0.75 – 0.00
-1.50 – -0.75
-2.25 – -1.50
-3.00 – -2.25
< -3.00
deviation [mm]
stainless steel (t0=1.0mm)
FIGURE 2. Deviations from the target geometry for ademonstator part
OPTIMIZATON STRATEGIES
Multistage forming
Inspired by the ideas for multistage forming strategies
for axisymmetric components [9], a modified multistage
forming strategy (Figure 3) has been developed for non-
axisymmetric parts.
The multistage strategy can be described as follows:
• In the first "preforming stage" (Figure 3a) a preform
with a shallow wall angle (45¢
in this example)
is produced by using the conventional two–point
forming.
• Then, a number of stages follow in which the pitch
motion of the forming tool alternates from upward
(Figure 3b) to downward (Figure 3c).
• From one stage to the next the tool path is generally
designed with an increase in angle of 3¢
or 5¢
. This
means that 7 to 12 stages are needed to produce
components with an angle of about 80¢
.
The described forming strategy introduces a number of
new process variables such as the shape of the preform,
tool
partial die
blankholder
before forming during forming stage
down
α
preforming
stage
a
fixture
up upward
stage
b
down downward
stage
c
FIGURE 3. Multistage forming strategy
and the shape and number of intermediate stages. The
number of intermediate stages should be as low as pos-
sible to avoid the occurrence of surface wear [7] and to
reduce the process time. It should be noted that reducing
the number of intermediate shapes increases the risk of
sheet rupture and wrinkling (Figure 4).
31
2
(3) wear
(1) wrinkling
(2) rupture
FIGURE 4. Multistage forming strategy
Correction algorithm
Reducing geometric deviations can be considered an
optimization problem of finding a tool path that yields
the desired part with a specified geometric tolerance. Dueto the fact that the ISF process is very reproducible, a
general correction algorithm has been developed (Fig-
ure 5). First, a part is produced based on the uncorrected
tool path. Then, "deviation vectors" pointing from a set
of target points to the corresponding points on the actual
geometry are determined using a coordinate measuring
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actual position
targetcontour
actualcontour
target point
corrected point
correctedcontour
ci
¦
ti
¦
ai
¦
c⋅di
¦
di
¦
x
z
y
targetcontour
part
actualcontour
x
z
y
FIGURE 5. correction algorithm
machine. These vectors are inverted and scaled by a cor-
retion factor c, yielding a new trial tool path to produce a
further part. The correction module can be considered a
proportional controller. It can be applied repeatedly until
a specified tolerance is met. Figure 6 shows the reduced
deviations after applying the correction algorithm once.
deviation [mm]
>2.52.5 – 2.02.0 – 1.51.5 – 2.01.0 – 1.50.5 – 1.00.0 – 0.5-0.5 – -1.0-1.0 – -1.5-1.5 – -2.0-2.0 – -2.5<-2.5
uncorrected component
corrected component (c=0.7)
FIGURE 6. Comparison between uncorrected and correctedpart geometry
PROCESS MODELING
In earlier work ([8],[9]), a modeling framework in
ABAQUS/Explicit has been successfully used to model
aspects in conventional ISF, e.g. sheet thinning, stress
and strain fields under the action of the tool as well as
damage evolution during forming. The present paper fo-
cuses on finite element modeling of the following aspects
of non–conventional ISF strategies:
• For multistage forming it is important to find a com-
bination of process parameters that avoids the limi-tations shown in Figure 4. Here, we will restrict our-
selves to the prediction of wrinkling and compare
two different variants to produce the same compo-
nent by multistage ISF.
• Since the correction algorithm presented above acts
as a proportional controller, convergence and thus
the experimental effort involved depend crucially on
the correction factor c. In order to reduce the experi-
mental effort by simulation, the finite element mod-
eling must predict the part geometry as accurately
as possible, i.e. at least as good as the desired ge-
ometric tolerance. This will be investigated later in
this paper by comparing modeling results to relatedoptical deformation measurement.
Finite elemet modeling of multistage
forming
Problem definition
We consider two different variants of multistage ISF
for the forming of a four–sided pyramid with a flange
angle of 81¢
(Figure 7).
final shape α=81°
preforming stages
(α=45°)
variant 1 variant 2
FIGURE 7. Multistage forming of variants 1 and 2
The different variants can be described as follows:
1. In variant 1, a constant corner radius of 15 mm isused throughout all forming stages.
2. For variant 2, a variable corner radius is used. The
preform has a bottom radius of 60 mm which de-
creases continously to a top radius of 15 mm. The
corner radii are gradually reduced to yield a con-
stant edge radius of 15 mm (from top to bottom)
after the final stage.
For both variants, the wall angle on the flat side walls
is increased by 3¢
per stage (yielding 12 stages after the
preform). Under the described conditions, both variants
should produce the same final shape, using different
shapes for the preform and the intermediate stages.
Finite element model
Modeling of ISF is computationally very intensive,
mainly due to the fact that the ISF process has a time
range of minutes or hours in reality. Since the tool path
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consists of a large number of points (104£
106) even
for demonstrator components, the FE model has to deal
with a huge number of contact situations. In the case
of multistage forming, the process duration is further
increased. With 12 stages in the example to be presented,
the process time is increased by a factor 12 with respect
to the conventional process.Consequently, in order to reduce model complexity
and calculation time, only a quarter of the pyramid is
considered (Figure 8), with symmetry boundary condi-
tions applied to the cutting edges. Considering a quarter
pyramid introduces disturbances in the stress and strain
fields as compared to a full pyramid, but test calculations
have shown that these deviations are small and restricted
to the immediate vicinity of the cutting edges, i.e. far
away from the region where wrinkling occurs.
sheet
tool
partial
die
tool path
1 2
3
symmetry b.c.
in 2-directionsymmetry b.c.
in 1-direction
FIGURE 8. FE model for multistage forming of a pyramid
The partial die used for the experiments has a length
of 72 mm, a width of 52 mm and a 9 mm edge radius,
yielding a quarter die with edge lengths of 36 mm and
of 26 mm, respectively, for the simulation. The sheet
has a size of 130x140 mm and is meshed with 2912
shell elements. In the experiments, 1.5 mm A1050-H14
aluminum sheets have been used. Since previous results
have shown that anisotropy has a negligible effect on
the outcome of the forming operation [9], isotropic J 2
plasticity with isotropic hardening has been used.
Results and discussion
The outcome of the simulation of the different mul-
tistage ISF variants is given in Figure 9 (both parts are
shown from the inside). For variant 1, the simulationpredicts excessive wrinkling in the corner region of the
pyramid at stage 9 of 12 stages. This corresponds well
to experimental results (Figure 10). On the contary, the
simulation predicts that variant 2 enables forming with-
out wrinkling. This is also found in the corresponding
experiment, where a pyramid with a final wall angle of
81¢
has been successfully formed (Figure 10).
thickness
[mm]
variant 1 variant 2
wrinkling no wrinkling2.2
1.9
1.6
1.3
1.0
0.7
0.4
FIGURE 9. FE modeling of forming variants 1 and 2
wrinkling
wrinkling successful strategy
FIGURE 10. Pyramid with a wall angle of 81 §
Two factors are crucial for wrinkling:
• In order to allow for steep flanges to be formed,
the perimeter of the part at an arbitrary z–level¡
0¨
z© £
60mm¥
is gradually reduced, and addi-
tional material from the preform is included into the
forming of the part. The reduction of the perimeter
causes compressive stresses in circumferential di-
rection that can entail wrinkling.
• At a fixed stage, the considered variants differ in the
shape of the corner region. The constant corner ra-
dius of variant 1 leads to a smaller inclination of the
side wall compared to the inclination of variant 2
for the preform. Thus, for all stages, the sheet vol-
ume included in the forming of the corner region is
bigger for variant 1, yielding a larger sheet thick-
ness and thus stiffness than variant 2. Since the flat
side walls have the same wall angle for both variants
and consequently the same sheet thickness, variant
1 shows a steeper increase in stiffness at the junc-
tion between side wall and corner region and is thus
more susceptible to wrinkling.
Modeling the part geometry
Due to the presence of elastic waves, the explicit FEmethod is generally considered ill–suited to predict the
shape of the sheet after springback (i.e. the geometric
accuracy). The implicit method, on the contrary, is well–
suited for springback analyzes, but a small time step has
to be chosen in order to deal with the huge number of
contact situations in simulations of the ISF process. This
leads at present to an unacceptable computational effort.
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In the following, a benchmark example is considered to
investigate the feasibility of our explicit FE model for
springback analysis:
We consider single–point forming of a four–sided
pyramid made of 1.5 mm aluminum sheet A1050-H14.
The pyramid has a square bottom contour of 200
200
mm with a 15 mm radius at the corners, a side wall angleof 50
¢
and a height of 60 mm. The corresponding finite
element model is given in Figure 11.
tool
rig
sheet
2 0 0 m
m2 0 0 m m
1 0 0 m
m 1 0 0 m m
vertical pitch/
break point
observed
area
FIGURE 11. FE model for comparison with optical defor-mation measurement
During forming, the part geometry is analyzed by means
of an optical deformation measurement system consist-
ing of two calibrated CCD cameras. The cameras record
the deformation on the underside of the sheet. Since
bulging occurs primarily on the unsupported side walls
of the pyramid, a square of 100 mm edge length that
completely contains one side wall has been observed by
deformation measurement.
In order to reduce the number of cycles to be recorded,
a tool path with a relatively coarse vertical pitch of 1 mm
has been chosen. The G–code file for the CNC control is
designed in such a way that the machine is halted aftereach cycle of the tool just before the vertical feed motion
is carried out so that the deformation after each cycle can
be evaluated. In order to synchronize the FE simulation
with the experiment, the tool path information specified
in the G–code file has been translated into input data
for ABAQUS/Explicit. By maintaining the break points
set in the experiments, we generate field outputs of the
calculated displacements and strains for exactly the same
time points as in the experiment.
The finite element mesh is depicted in Figure 12 for
a forming depth of 50 mm. The mesh consists of 6,400
shell elements with 5 integration points over the sheet
thickness. The mesh region that corresponds to the area
observed with optical deformation measurement is high-lighted. The results given next compare the FE model
with experimental data along the depicted centered sec-
tion, where the maximum amount of bulging occurs.
The comparison between the results obtained by sim-
ulation and deformation measurement is shown in Figure
13 for six stages from 10 mm to 60 mm forming depth.
mesh region corresponding to
the area observed by
deformation measurement
centered section for
the comparison with
deformation measurement
FIGURE 12. FE model for comparison with optical defor-mation measurement
In almost all cases, side wall bulging is underestimated
in the finite element simulation. For the final stage (z =
-60 mm), a maximum deviation of 4 mm between defor-
mation analysis and simulation has been found.
0 20 40 60−30
−20
−10
0z = −10 mm
z
0 20 40 60−30
−20
−10
0z = −20 mm
0 20 40 60−40
−20
0z = −30 mm
z
0 20 40 60−40
−20
0z = −40 mm
0 20 40 60−60
−40
−20
0z = −50 mm
y
z
0 20 40 60−60
−40
−20
0z = −60 mm
y
simulationexperiment
FIGURE 13. Comparison between the geometry predictedby FEM and related deformation measurement
Since the misfit to the experimental results is partially
due to the the presence of elastic waves, the analysis
has been restarted and a viscous pressure load has been
applied to the surface of the shell elements to damp out
transient wave effects. Quasi–static equilibrium can be
reached quickly by calculating one additional second of
process time, which is a negligible effort compared to
the 300 seconds of process time that have been taken for
the simulation of whole process. Figure 14 compares the
geometry of the part along the centered section for the
final stage (z=-60mm).
With damping the misfit has been reduced to a maxi-mum deviation of 1.5 mm. While this can be considered
a good conformance, it is still a considerable misfit as
measured by the demands of the correction module. It
is worth mentioning that the influence of the discretiza-
tion and shell element formulation has been carefully
checked in a series of test calculations.
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y
z
experimentsimulation as issimulation with viscous pressure load
0
-10
-20
-30
-40
-50
-600 10 20 30 40 50 60
FIGURE 14. Comparison between the geometry predictedby FEM and related deformation measurement
Reasons for the remaining deviations can be:
• The strong dependence of the results on the elas-tic modulus. Since we assume isotropic elastic be-
haviour and since the elastic modulus of the unde-
formed sheet can only be determined with limited
accuracy, this could have a considerable influence
on the result. Furthermore, we do not account for
the decrease of the elastic modulus due to damage
evolution in the sheet. Since decreasing the elastic
modulus increases the elastic part of the deforma-
tion, accounting for this effect could improve the
prediction of geometric accuracy.
• The constitutive framework used in this work does
not account for kinematic hardening and the build–
up of backstresses. Accounting for kinematic hard-
ening should provide a better prediction of spring-
back than the isotropic hardening law used here.
• The blank holder is at present modeled by constrain-
ing the corresponding translational and rotational
degrees of freedom along the edges of the sheet. A
realistic modeling of the clamping conditions could
further improve the results.
SUMMARY AND OUTLOOK
In this paper finite element calculations for non–
conventional forming strategies in incremental sheet
forming have been presented. The enclosed non–conventional forming strategies aim at optimizing the
tool path to enable the production of steep flanges and to
reduce geometric deviations. Since these strategies are
at present based on trial–and–error optimization, finite
element calculations could in principle help reduce the
experimental effort. In particular, two different variants
for the multistage forming of a pyramid shape have been
compared. The occurrence of wrinkling in one of the
variants has been modeled successfully. Furthermore, the
evolution of the geometry of a part during forming has
been tracked using optical deformation measurement.
The related finite element calculation with an explicit
code could describe side wall bulging fairly well afterthe elastic waves have been damped out using a viscous
pressure load. However, the remaining maximum devia-
tion of 1.5 mm between model and experiment is still too
large to allow for an optimization of the part geometry.
Future work will focus on the role of the elasic modulus
and the build–up of internal stresses in order to improve
the prediction of the part geometry.
ACKNOWLEDGMENTS
This research is supported by the German Research
Foundation (DFG) in the framework of SPP 1146: Mod-
eling of incremental forming operations.
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