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This is the accepted version of a paper published in Composites. Part A, Applied science and
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Citation for the original published paper (version of record):
Kaufmann, M., Zenkert, D., Åkermo, M. (2010)
Cost/weight optimization of composite prepreg structures for best draping strategy.
Composites. Part A, Applied science and manufacturing, 41(4): 464-472
http://dx.doi.org/10.1016/j.compositesa.2009.11.012
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Cost/weight optimization of composite prepreg
structures for best draping strategy
Markus Kaufmann, Dan Zenkert and Malin Åkermo
Kungliga Tekniska Högskolan (KTH), 10044 Stockholm, Sweden
May 2009
The application of hand-laid carbon fiber prepreg is very expensive from
a labor perspective. Therefore the manufacturing cost should be included
in the design process. In this work, we propose a novel optimization
framework which contains a draping simulation in combination with a
detailed cost estimation package and the calculation of the structural
performance based on FE. We suggest applying the methodology in
two steps. First, a draping knowledge database is generated in which
combinations of seed points and reference angles are evaluated in terms of
fiber angle deviation, scrap, ultrasonic cuts and material shear. Second,
a cost/weight optimization framework picks the best sets of plies during
the subsequent optimization. The methodology is tested by means of a
curved C-spar which is designed using plain weave and unidirectional
prepreg. Different objectives in the generation of the draping database
lead to different design solutions.
Keywords: Cost/Weight Optimization, Finite element analysis,
Prepreg, Lay-up (manual/automated)
1
Introduction
Aircraft manufacturers aim to design airliners with the lowest possible direct operating cost (DOC). This direct operating cost includes cost for depreciation, insurance,
landing fees and not least fuel consumption. As the latter is a substantial part of
the DOC (20-40%, depending on the operator), one can understand why a lot of
efforts have been undertaken to lower the fuel consumption. One possible strategy
to save fuel is to lower the structural weight.
1
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M. Kaufmann, D. Zenkert and M. Åkermo
During the last 20 years, carbon fiber reinforced plastics (CFRP) were carefully
introduced into primary aircraft structures in order to lower the structural weight.
The drawback of CFRP, however, is the increased manufacturing cost. Gutowski et
al. described the composite’s disadvantages in terms of cost already back in 1991 [1].
In particular, the authors spotted the high labor cost as the largest single contributor
to direct cost – in spite of the higher material cost.
Some years later, Rais-Rohani et al. identified the risk that the high material,
fabrication and tooling costs may make the use of high-performance materials
cost-ineffective. As a consequence, the costs of raw materials, tooling, fabrication,
assembly, scrap, repair, certification and environmental factors should be included
in the design of composite structures [2].
Since that time, a lot of research and development has been performed in
the field of composite structures. For example, new material systems evolved,
and automatic tape laying or fiber placement machines were installed. A lot of
parts, however, were still manufactured with rather expensive hand-laid carbon
epoxy prepreg systems which were cured in an autoclave. In addition, the stacking
sequences generally remained the same, using a quasi-isotropic layup with 0, 90
and ±45◦ angles. Soutis pointed out that early composite designs were ”replicas
of those that employed metallic materials”, and that ”the high material cost and
man-hour-intensive laminate production jeopardized their acceptance” [3].
Authors saw the lack of sophisticated cost models as the reason behind the absence
of cost optimization frameworks for composite components, see SobieszczanskiSobieski and Haftka [4]. Some costing information was integrated into the design
process by Geiger et al., though [5]. There, the authors presented a conceptual
model for an automated design-to-cost approach by providing a framework for
the decision-making process. Heinmuller and Dilts adapted this framework to the
aerospace industry [6], and Kassapoglou published a series of articles on combined
cost/weight optimization of aircraft structures [7–9].
Not an optimization per se, but a methodology to compare different material
systems on the basis of cost/performance efficiency was proposed by Bader [10].
Among else, he mentioned the high amount of waste when cutting plies from
standard prepreg. Park et al. pursued the optimization of components manufactured
by resin-transfer-molding. In particular, the stacking sequence of a composite plate
was optimized in order to maximize the stiffness while minimizing the mold filling
time [11].
Curran et al. combined the manufacturing cost and the weight and used a
simplified form of the above-mentioned direct operating cost as the objective function
[12, 13]. A similar approach of weighted sums was implemented in the work of Iqbal
et al. who applied it on the optimization of a truss structure [14]. The work of
these authors was resumed by Kaufmann et al. who used the DOC of composite
components in combination with FE, a feature-based parametric manufacturing
cost model [15] and a non-destructive testing cost model [16]. This framework
incorporated the manufacturing cost of a component and the influence of its weight
into one objective function.
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DOC
DOCº
Manufacturing
process adjustments
DOC*
Design
solutions
Figure 1: Parameter adjustments can lead to a better solution DOC ∗ , while
constant manufacturing parameters converge to DOC ◦ . Note that this
is only a schematic drawing.
In Kaufmann et al. [17], the use of a knowledge database for the sub-optimization
of manufacturing cost was explored. A center wing box rear spar was optimized
in terms of cost and weight. It could be shown that the initial settings of the
manufacturing cost model should not remain constant during the optimization of
the geometry of spar and stiffeners. Hence, a framework was proposed in which the
machining cost of the aluminum stiffeners was sub-optimized in each iteration of
the superordinate shape optimization. This idea is illustrated in Figure 1. There,
it is shown that the global optimum DOC ∗ cannot be reached when the process
parameters are not adjusted. Instead, the optimization with constant machining
parameters (such as cutter diameter or cutter material) would lead to a local
optimum DOC ◦ .
In all the work performed by Kaufmann et al., only nominal fiber angles with 0,
90 and ±45◦ were used. In reality, particularly when manufacturing double-curved
components, the manufactured fiber angles cannot match these nominal fiber angles
in every point of the component due to the limitations in material shear. For the
optimization of composite structures, it would therefore be advantageous to use
the draping-corrected fiber angles. As a consequence, a more detailed evaluation of
the desired fiber angles would allow for tailoring the layup more efficiently, and the
structural response would comply closer with reality.
Recent developments in design software included tools such as more elaborate
cost estimation systems and draping simulations. The question was now how these
tools allowed for the more advanced assessment of a design solution in terms of cost
versus structural performance than done previously. In this work, it is investigated
whether and in what form a draping model could be used for the cost/weight
optimization of composite aircraft components.
Looking at available draping simulations, we can distinguish between two different
types of modeling approaches. On the one hand, mechanical algorithms can be
applied which use the materials equilibrium equations to drape the material with
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M. Kaufmann, D. Zenkert and M. Åkermo
the help of an external force. For instance, Boisse et al. [18] and Badel et al. [19]
developed finite elements for a woven unit cell and compared the simulation of a
draped hemisphere with experiments. Further, Nguyen et al. [20] modeled the fiber
rotation, shear and slip of plain weave by means of a truss model. Commercial
software packages are available (e.g. AniForm1 or PAM-FORM2 ). As all mechanical
models, they are based on computationally rather heavy explicit FE code.
On the other hand, kinematic algorithms (also known as mapping or fishnet
algorithms) map a unit square of fabric or tape onto a geometric surface. Kinematic
algorithms normally do not account for boundary conditions, loads or friction
(although some developments included the effects of the blank-holder forces, see
Wiggers [21]). These restrictions reduce their application mainly to the draping of
dry weave and the hand layup of prepreg. The advantages, however, are the greatly
reduced computation time and the reduced model complexity – which is reflected
in the reduced amount of input parameters needed. Examples for commercially
available packages using a kinematic approach are PAM-QUIKFORM2 , FiberSIM3 ,
Interactive Drape4 or Composite Modeler5 .
Most of the kinematic codes are based on the work performed by Mack and
Taylor who proposed the pin-jointed net idealization [22]. Wang et al. compared
the results of a pin-jointed net simulation with experiments for a series of articles,
such as a rudder tip, D and nose ribs and flap panels [23]. They concluded that
the simulations agreed well with the experiments, although the choice of the initial
contact point (seed point) ”may result in considerably different draped patterns”.
According to Guillermin [24], the use of draping tools should not only allow for a
more detailed assessment of the draped fabric, but also for an optimization in terms
of total fabric shear deformation, prescribed fiber orientations at control points,
unavoidable darts and splices at noncritical areas of the part, and the optimization
of the best layup start point. Some drape optimization of woven composites was
performed by Skordos et al. [25] who modeled and optimized a hemisphere by means
of FE and a genetic algorithm. Thus, wrinkling was minimized by optimizing the
holding force distribution.
2
Aims and scope
In this work we resumed Guillermin’s idea of drape optimizing composite parts
by the use of a kinematic draping software. Hence, we adapted the framework
proposed in [17] for the use with a draping simulation. By accounting for the
material consumption and process time, it is tried to find tradeoffs between the
structural performance of a component and its manufacturing cost.
1 http://www.aniform.com
2 http://www.esi-group.com
3 http://www.vistagy.com
4 http://www.interprot.com
5 http://www.simulayt.com
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The methodology is applied in two steps. First, a draping knowledge database is
generated. There, combinations of seed points (location of initial contact between
the ply stack and the next layer) and reference angles (the angle between the 1-axis
of the prepreg ply and a reference coordinate system) are evaluated in terms of fiber
angle deviation, material consumption, cuts and material shear (see Section 3.2).
For an explanation of seed point, reference angle, reference coordinate system and
fiber angle deviation it is referred to Figure 2. Second, the cost/weight optimization
framework picks the best sets of plies during the subsequent optimization. It is tried
to find combinations of seed points and reference angles that are optimal from a
structural and a cost perspective.
seed point
mold
surface
1-axis of the draped ply
φ
reference angle
draped 0°
direction
fiber angle deviation
at control points
reference
coordinate
system
nominal 0°
direction
Figure 2: Definition of seed point, reference coordinate system, reference angle φ,
nominal fiber direction, draped fiber direction and fiber angle deviation
at control points.
The methodology is aimed to be used in the preliminary design phase; thus, the
output of the optimization is neither a detailed manufacturing documentation nor a
ply book. The introduction of a draping model in this phase is rather a way to (a)
provide the cost model with more detailed manufacturing knowledge, (b) to perform
a structural analysis with fiber angles closer to reality and producibility constraints,
and (c) to show that the choice of the draping strategy can influence the cost and
weight balance of a component.
3
Method
The proposed optimization framework consists of two parts. First, there is a global
cost/weight optimization routine which includes the calculation of the weight, cost
and structural performance. Second, this routine is extended by a draping knowledge
database which supports the cost estimation of the composite parts.
3.1
The cost/weight optimization framework
The cost/weight optimization framework as illustrated in Figure 3 contains the
necessary modules to calculate the objective function and constraints as well as the
solver.
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M. Kaufmann, D. Zenkert and M. Åkermo
design
draping database
weight
best ply
+
cost
NDT
DOC
dk
objective function
solver
constraints
FE
Figure 3: The cost/weight optimization framework with the novel draping
knowledge database within dashed lines.
As described in the introduction, we decided to use a simplified form of the
direct operating cost DOC as the objective function. This DOC was defined as
DOC = Cman + p · W
(1)
where Cman represents the manufacturing costs (in e), p is a weight penalty (in
e/kg), and W is the structural weight (in kg) of the component. The global
optimization problem was formulated as
min
subject to
DOC of an aircraft component
structural requirements
xi < xi < xi ,
(2)
i = 1 . . . n.
with the variables xi and their lower and upper boundaries xi and xi , respectively.
The problem given in Equation (2) was transformed to its mathematical formulation
min f (x),
x
x ∈ Rn
gj (x) ≤ 0,
j = 1, . . . , m
xi ≤ xi ≤ xi ,
i = 1, . . . , n.
(3)
In this case, the cost/weight objective function f (x) and the constraint functions
gj (x) could not be expressed by explicit formulae, since both objective function
and constraints were dependent on the output of separate modules. Thus, only
the results of the weight and cost estimation modules and the calculation of the
structural performance were fed back to the solver.
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The cost model seen in Figure 3 was based on the code SEER-MFG by Galorath
Inc.6 . In Table 1, an example for the output of that code is given. Note the level
of detail obtained for the different cost contributors. The advantage of using a
commercial code was the reliability of the estimation and the broad acceptance in
the aircraft industry. The same applied to the structural model. Unlike similar
work in this field which relied on closed-form solutions provided by the Engineering
Sciences Data Unit (see Curran et al. [26]), a commercial FE package was used for
the calculation of the structural performance. Despite the higher computational
effort, we believe that the use of ABAQUS/CAE provides greater flexibility in terms
of geometry and load cases, especially when dealing with composite structures. It
was decided not to include the NDT module described in Kaufmann et al. [16].
Table 1: Example of a manufacturing cost estimation as obtained from SEERMFG. During the optimization, four decimal places were used for the
calculation of exact finite differences.
LABOR TOTAL
Manufacturing Labor Total
Setup
Direct
Inspection
Rework
Assembly Labor Contribution
ADDITIONAL COST
Material
Tooling
Other
TOTAL COST
ADDITIONAL DATA
Finished Weight (kg)
Material Placement (min)
Cutting Time (min)
Layup Time (min)
Tool Closing (min)
Cure Cycle (min)
Remove Part (min)
Part Finish (min)
Tool Cleaning (min)
6 see
http://www.galorath.com
min/Unit
Cost/Unit
1774.6
1774.6
0.6
1774.0
0.0
0.0
4436.0
4436.0
1.0
4435.0
0.0
0.0
0.0
956.3
711.0
0.0
245.3
5392.3
8.1
339.5
20.8
318.7
950.7
285.9
168.8
16.4
12.0
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M. Kaufmann, D. Zenkert and M. Åkermo
A solver was chosen that would process the output of these models, not be
too sensitive to disturbances in the form of non-smooth objective and constraint
functions, and thus lead to a good convergence rate. Otherwise, the choice of solver
was arbitrary. For a detailed description of the optimization framework it is referred
to Kaufmann et al. [15–17].
3.2
Draping knowledge database
It would be ideal to use a multi-level optimization framework as shown in Kaufmann
et al. [17] to find the best draping strategy for each solution in the geometrical
optimization. This would, however, involve a very large number of draping calculations with many redundant initial conditions among them. Therefore, it was
decided to generate a draping knowledge database before the actual optimization
was performed. Thus, we sought a set of best plies which conformed to the 0, 90 and
±45◦ directions, fulfilled certain criteria and which could be used as the input to
the global optimization of the component’s geometry. For that purpose Simulayt’s
Composite Modeler was used.
Composite Modeler is an add-on to ABAQUS/CAE which can map unidirectional
and woven composites onto a previously modeled shell surface. Input parameters to
Composite Modeler were the type of the material (weave or tape), the extension type
(which governs the draping kinematics e.g. by minimizing shear strain deformation
energy), seed points, reference angles and material parameters (such as maximum
strain, warp/weft angles and warp/weft ratios). The procedure in ABAQUS/CAE
was the following: mesh the part, call the draping plug-in, define layup, material
and draping properties, export the flat pattern, and map the simulated composite
properties back onto ABAQUS sections. All these steps could be performed in batch
mode. Further, a recently implemented functionality allowed the export of a range
of properties to a text file. In that sense, access to the draping strain, ply thickness,
and ply angle in every element was gained before they were mapped back onto
sections. This text file was then used for the creation of the knowledge database.
The generation of the knowledge database was done in three steps. First, a ply
set with nominal fiber angles (i.e. nominal 0, 90 and ±45◦ plies) was created. Second,
sets of deviating reference angles were seeded in various locations of the component.
Third, these sets were scanned for combinations of good seed points and reference
angles which formed the knowledge database. This scan can be described as follows:
(i) Consider only feasible sets of seed points and reference angles by limiting the
shear angle γ during draping. The latter is depending on the maximum shear
angle γmax which is a material property.
(ii) Select the combinations of seed point and reference angle which conform best
to the four nominal plies. The selection is based on an objective function F .
Figure 4 shows a schematic drawing of two flat patterns generated using different
seed points. Clearly, it can be seen that the material consumption of the left pattern
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1
1
2
2
Figure 4: Two flat patterns as exported from Composite Modeler. Note the
differences in material consumption versus average deviation from a 45◦
ply.
is higher than the one on the right. On the other hand, the left pattern complies
better with a nominal 45◦ ply than the one on the right. Depending on the objective
function F , the solver would choose the left (for minimum fiber angle deviation) or
the right pattern (for minimum material consumption).
Obviously, one has different alternatives to define the objective function. One
possibility is to use the error between the seeded and the nominal fiber angle
distribution, the so-called fiber angle deviation. This norm is given as
Fθ = kθseed − θnom k
(4)
where θseed is a vector containing the simulated fiber angle distribution at the
current seed point and reference angle, and θnom is a vector containing the nominal
fiber angle distribution. It might not be necessary to closely follow the nominal
fiber angle distribution in all points of the ply. Therefore, control points, a control
line or a control surface can be selected in which the norm Fθ is calculated. Hence,
a set of best plies which minimize Fθ leads to a solution closer to the nominal fiber
angles and to more quasi-isotropic fiber angles. Note that quasi-isotropic layups
do not have to be optimal to meet the stiffness criteria. It is, however, common in
aircraft design to use 0, 90 and ±45◦ fiber angles, as non-conform designs would
need additional efforts in material characterization and certification.
Similarly, one could define an objective function Fγ as the norm of all in-plane
shear components γ in a ply. This norm is given as
Fγ = kγk.
(5)
A set of best plies which minimizes Fγ leads to a better producible solution in terms
of drapability, as the risk for wrinkles and likewise problems is smaller.
A third way to define the objective function is to reduce the material consumption,
thus minimizing the amount of raw material needed by using
FA = A.
(6)
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M. Kaufmann, D. Zenkert and M. Åkermo
Parameter A is the total material consumption as shown in Figure 4. The authors
are aware of the fact that FA might overestimate the amount of material needed and
create an upper bound for the material consumption; in fact, tools are available which
try to reduce the scrap rate by means of nesting algorithms [27]. The implementation
of such a nesting algorithm in the current framework is promising and should be
considered for future research activities.
A way to reduce the time of ultrasonic cutting is to minimize the perimeter. In
this case, the objective function for the generation of the knowledge database is
defined as
FC = C
(7)
with C being the perimeter of the flat pattern. The draping tool also allows for the
introduction of splits and splices. The splits enhance the drapability and minimize
the effects of fiber angle deviations. If the ply is not split, however, the cut length is
basically constant and its optimization would not make sense.
Thus, the knowledge database consists of draped plies with different seed angles
and reference angles, some of the plies being marked with a tag best ply and the
governing objective function (4)-(7).
4
Case study
It was decided to use a curved C-spar as a test case. This component had a length
of 1.2 m, a sweeping web height of 120-150 mm and a flange width of 100 mm. As
shown in Figure 5, the load case involved a uniformly distributed pressure on the
outer flange corresponding to the cabin pressure of an airliner. Simply supported
flanges formed the boundary conditions while the spar had to withstand a maximum
strain criterion and a maximum displacement at mid-length.
The component
cabin
pressure
Simply
supported
max displacement
Figure 5: Geometry and load case of the C-spar used in the case study
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First, a ”weak” ply was seeded along a curve corresponding to the spar’s center
line in order to generate the nominal 0◦ ply. For the generation of the draping
knowledge database, a series of seed points was chosen, see Figure 6. The knowledge
database generator then swept the reference angle from −90◦ to +90◦ in each of the
seed points and saved the results in terms of total material consumption, cut length,
shear angle and fiber angle distribution.
S4
S2
S6
S7
S3
S5
S1
S8
Figure 6: Eight seed points used in this case study
Two material systems were used. On the one hand, a plain weave prepreg was
chosen to provide a high drapability in spite of moderate structural properties. A
range of maximum shear angles could be found in the literature; we adopted a value
of 30◦ based on Lin et al. [28]. On the other hand, a UD prepreg would provide
high stiffness and strength at a rather low drapability. Thus, a maximum shear
angle of 10◦ was assumed. During the generation of the knowledge database, all
plies with shear angles exceeding these values were considered as unfeasible. The
material properties are compiled in Table 2.
The best plies of this knowledge database were selected using objective functions
Fθ and FA . We decided not to use the cut length FC as an objective, as these
lengths remained the same unless splits were introduced.
The total framework was implemented by means of the scripting language Python.
The main advantage of Python was the seamless integration of ABAQUS/ CAE
and Composite Modeler commands. In addition, Python controlled the execution
of the cost estimation, the variable handling and not least the draping knowledge
database in the form of a Python ”shelf database”.
SEER-MFG was used for the estimation of the manufacturing cost. This tool
had to be adapted in order to estimate the cost of a C-spar based on the inputs
from the draping module. In particular, one had to implement and parameterize the
cost model in terms of material consumption for all four main directions (0◦ , 90◦
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M. Kaufmann, D. Zenkert and M. Åkermo
Table 2: Material properties. PW and UD refer to plain weave prepreg
and unidirectional prepreg, respectively.
Young’s modulus [GPa]
Shear modulus [GPa]
Poisson ratio [-]
Density [kg/m3 ]
maximum shear [◦ ]
ply thickness [mm]
E1
E2
G12
G23 = G13
ν12
ρ
γmax
t
UD
PW
151
8.5
4.2
4.2
0.30
1500
10
0.25
75
75
4.2
4.2
0.037
1500
30
0.25
and ±45◦ ), a functionality which was not part of the original SEER distribution.
These four ply directions were modeled as following.
- First, the four ply directions were handled separately. Hence, four separate
batches of raw material were cut, each rounded off by a (discrete) number of
layers. All four batches provided an estimation of the material cost and the
cutting time.
- Second, the batches were lumped together for the estimation of layup time,
tool closing time, cure cycle duration, part removal and part finish duration,
and tool cleaning time. Depending on the complexity of the part, these time
estimations were adjusted by a so-called part complexity factor which varied
between 0.9 for very simple laminates and 4.3 for very complex components.
- Third, these labor hours underwent a learning curve based on 100 manufactured
units.
- Finally, the time-corrected labor hours were converted into manufacturing
costs, as can be seen in Table 1. The labor rate of e150/h included overhead
cost such as autoclaves and other equipment.
Once the geometry and thus the complexity of the part were set, the layup rate
was assumed to be independent of the draping strategy. This was a simplification
and shortcoming of the cost and draping tools used. Ward et al. indeed showed that
the time to drape a ply can depend on the material, the seed point and the draping
sequence [29] by using Hancock’s Virtual Fabric Placement software [30]. A material
placement model which provides the functionality to calculate the time to drape a
ply would therefore be an interesting enhancement for this optimization framework.
In order to reduce the computational workload, a gradient-based method was
chosen for the optimization. The method of moving asymptotes (MMA) was developed by Svanberg [31] for the purpose of structural optimization and further
Paper D
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enhanced to improve the global convergence [32]. As the overall methodology was
intended to be used in the preliminary design phase, the time to solve the problem
was crucial for its applicability. Here, we saw the advantage of a gradient-based
method over evolutionary algorithms. In addition, MMA existed as a tool called
Xopt7 . Optimization trials were performed as follows:
The selection of the best plies for minimum fiber angle deviation Fθ or lowest
material consumption FA was done directly after the generation of the draping
knowledge database. This was done in order to reduce the number of FE calculations in this stage of the design chain. Obviously, one could also use all possible
combinations of plies during the optimization. The latter was implemented using a
genetic algorithm, but not used for this study.
The thickness of the 0◦ , the 90◦ and the ±45◦ direction were used as the three
continuous variables throughout the optimization of the UD prepreg. Similarly,
the optimization of the plain weave prepreg contained two variables, one for 0/90◦
and one for ±45◦ . The solver first disturbed the first variable by a small value
and calculated the finite difference of the objective function and the constraints.
Then the second variable was disturbed, followed by the third. The final step of
an iteration was the calculation of a new solution based on the values of the three
foregoing finite difference calculations. 20-40 iterations were necessary to reach
convergence.
A series of different optimizations runs were performed and analyzed. An
overview is given in Table 3. For both material systems, a minimum cost and a
minimum weight solution were sought; the two objective functions corresponded to
a weight penalty of 0 and ∞, respectively.
Table 3: Optimization runs performed for the case study. PW and UD refer to
plain weave prepreg and unidirectional prepreg, respectively.
7 see
number
material
KB
objective function
1
2
3
4
5
6
7
8
UD
UD
UD
UD
PW
PW
PW
PW
Fθ
Fθ
FA
FA
Fθ
Fθ
FA
FA
min
min
min
min
min
min
min
min
http://www.alfgam.se
weight
cost
weight
cost
weight
cost
weight
cost
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5
M. Kaufmann, D. Zenkert and M. Åkermo
Results
In Table 4, the cost and the weight optimization of a unidirectional prepreg with
minimum fiber angle deviations are shown. As can be seen, the results were very
similar in terms of cost and weight. This is not surprising, as the C-spar was
modeled using only additive processes, such as hand-laid prepreg systems. Some
differences, however, could be found in the thickness of the 90◦ and the ±45◦ layers.
In particular, the latter was reduced in the case of cost optimization as a direct
consequence of the higher material consumption of the 45◦ layer. Note that the
layup time and the tool closing time were the same, as the total thickness was
rounded up to the same number of plies.
The optimization results of unidirectional prepreg with minimum material consumption FA were even more similar, see Table 5. While the direct labor cost was
exactly the same, the minor difference in material cost was a result of the slightly
different thicknesses of the 90◦ and the ±45◦ layers.
In Table 6, the results of the cost and weight optimization of a plain weave
prepreg with minimum fiber angle deviations are shown. Differences in terms of cost
and weight can be seen. This was a consequence of the much thicker 0/90◦ layer in
the cost-optimized layup, which led to a significant reduction of material cost. By
contrast, the direct labor cost was the same for both optimization runs.
Finally, the biggest differences could be seen in Table 7. The minimum weight
solution showed a ±45◦ governed fiber layup, whereas the minimum cost solution
was 0/90◦ dominated. This was a consequence of the cost saving potential of this
draping set whose plies were selected according to minimum material consumption
FA . Note the differences in material cost for these two solutions, as well as the
differences in material cost compared to Table 6.
6
Discussion
As expected, the minimum weight solution always resulted in higher manufacturing
cost. For both material systems, the lowest cost was achieved by using the best set
of plies corresponding to a minimum material consumption criterion. Similarly, the
ply set with minimum fiber deviation accounted for the minimum weight solution.
These differences were not too big in this case study. One could imagine aircraft
components where the potential for cost reductions could considerably influence the
choice of the draping strategy, though. Examples for such parts are:
(i) Shape optimized parts with tradeoffs between the thickness and the geometry.
Examples are stiffened panels, where the stringer shape and stringer pitch
is responsible for the cost/weight balance of the overall part, see Curran et
al. [13] and Kaufmann et al. [15, 16].
(ii) Subtractive processes, such as machined aluminium parts, see Kaufmann et
al. [15].
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-28.8° (0°)
-59.6° (90°)
44.9° (45°)
-1.69° (0/90°)
-48.68° (±45°)
-43.0° (-45°)
-43.0° (-45°)
-48.68° (±45°)
-1.69° (0/90°)
44.9° (45°)
-59.6° (90°)
-28.8° (0°)
Figure 7: Plain weave and UD stacking sequence for a control point in the center
of the spar web (depicted as point S3 in Figure 6). Note the differences
in fiber angles compared to the nominal angles (given in brackets).
Second, the weight and the cost of the UD prepreg solution were higher than the
PW prepreg solution. This was not obvious at first sight, as the optimization of UD
prepreg involved more independent variables than the optimization of PW prepreg.
A closer look to the local fiber angles of the two material systems, however, gave
an indication for a possible reason. Figure 7 shows the fiber angles at the center
of the spar web (depicted as S3 in Figure 6). In particular, the fiber angles of the
plain weave and the UD prepreg are shown for optimization runs 1 and 5 (weight
optimized Fθ ply sets). As can be seen, the plain weave had only a minimal fiber
angle deviation compared to the nominal layup, whereas the UD prepreg layup
showed significant deviations. This is most obvious for the 90◦ layers which locally
deviate up to thirty degrees as a consequence of the reduced shear deformability.
Hence, the big differences in fiber angles compared with the nominal layup resulted
in thicker 0◦ and 90◦ plies which in turn worsened the cost and weight balance at
the disadvantage of the UD prepreg. Note that in reality, this disadvantage would
be accounted for by splitting the layers appropriately, which is why a direct PW/UD
comparison is not possible here.
The reduced deformability of UD prepreg was also noticed when the knowledge
database was generated. Many solutions were unfeasible due to the tight constraint
of 10◦ maximum shear, and the ”best” plies with rather high fiber angle deviations
were chosen from a highly reduced range of solutions.
For two particular cases, the layup sequence was very much dominated by one
fiber direction, i.e. by the ±45◦ direction of the minimum weight solution in Table
6 and by the 0/90◦ of the minimum cost solution in Table 7. There, common
lamination rules of at least 10% in each direction were not fulfilled anymore. As
a consequence, these rules were implemented in the optimization routine and the
results rerun. As expected the solutions became more quasi-isotropic; otherwise no
relevant changes could be observed.
e 5392.35
e 710.99
e 4435.04
e 5394.29
e 711.95
e 4436.01
21.23 min
318.68 min
950.65 min
0.50 mm
0.90 mm
2.54 mm
total cost
material cost
direct labor
cutting time
layup time
closing time
0◦ layer
90◦ layer
±45◦ layer
0.50 mm
1.34 mm
2.37 mm
20.84 min
318.68 min
950.65 min
7.63 kg
7.62 kg
min Cost
weight
min Weight
Table 4: Optimization run 1&2 - UD - min Fθ
0◦ layer
90◦ layer
±45◦ layer
cutting time
layup time
closing time
total cost
material cost
direct labor
weight
0.50 mm
1.50 mm
2.30 mm
21.71 min
318.68 min
954.38 min
e 5360.06
e 664.35
e 4449.38
7.65 kg
min Weight
0.50 mm
1.34 mm
2.38 mm
21.71 min
318.68 min
954.38 min
e 5359.73
e 664.02
e 4449.38
7.65 kg
min Cost
Table 5: Optimization run 3&4 - UD - min FA
D 16
M. Kaufmann, D. Zenkert and M. Åkermo
e 4830.62
e 536.43
e 4063.15
e 4905.04
e 611.29
e 4062.70
19.17 min
294.63 min
870.74 min
0.70 mm
5.30 mm
total cost
material cost
direct labor
cutting time
layup time
closing time
0/90◦ layer
±45◦ layer
2.23 mm
3.89 mm
19.62 min
294.63 min
870.74 min
7.10 kg
6.96 kg
min Cost
weight
min Weight
Table 6: Optimization run 5&6 - PW - min Fθ
0/90◦ layer
±45◦ layer
cutting time
layup time
closing time
total cost
material cost
direct labor
weight
1.29 mm
4.87 mm
19.62 min
300.65 min
871.56 min
e 4863.51
e 551.37
e 4081.10
7.14 kg
min Weight
5.72 mm
0.50 mm
20.20 min
300.65 min
871.56 min
e 4645.43
e 333.87
e 4080.52
7.22 kg
min Cost
Table 7: Optimization run 7&8 - PW - min FA
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M. Kaufmann, D. Zenkert and M. Åkermo
Finally, the merit of a solution in terms of direct operating cost DOC was
computed. Hence, each cost/weight solution given in Tables 4-7 was recalculated by
using Equation (1) and three different weight penalties. In Table 8, the result of
this calculation is shown. As can be seen, the best draping strategy depended on
the value of the weight penalty. In particular, the best strategy for p =e150/kg was
a cost optimized C-spar whose plies were chosen with regard to minimum material
consumption. For p =e1500/kg and p =e15000/kg, the choice would shift to a
weight-minimized spar with minimum fiber angle deviation. There, the benefits of
the lower weight outweighed the slightly higher manufacturing cost compared to a
cost optimized solution.
Table 8: The solutions of Tables 4-7 compiled in terms of Cman , W and DOC =
Cman + pW . Three weight penalties p were applied. The lowest DOC
for each weight penalty is marked in bold.
(a) Results for the plain weave prepreg
PW - min Fθ
minW
minC
PW - min FA
minW
minC
Cman [e]
W [kg]
4905
6.96
4831
7.10
4864
7.14
4645
7.22
DOC [e], p =e150/kg
DOC [e], p =e1500/kg
DOC [e], p =e15000/kg
5949
15345
109305
5896
15481
111331
5935
15574
111964
5728
15475
112945
(b) Results for the unidirectional prepreg
UD - min FA
minW
minC
UD - min Fθ
minW
minC
Cman [e]
W [kg]
5360
7.65
5360
7.65
5394
7.62
5392
7.63
DOC [e], p =e150/kg
DOC [e], p =e1500/kg
DOC [e], p =e15000/kg
6508
16835
120110
6507
16835
120110
6537
16824
119694
6537
16837
119842
In order to examine the sensitivity of the touch labor cost with respect to the
fabric type and mass/unit area, the manufacturing cost of the spar (i.e. the result
of optimization run 5) was remodeled with twice the ply thickness. Obviously, the
total weight did not change, as the cross-section’s thickness remained the same. The
number of plies and the cutting time, however, were reduced by 50%; the layup and
tool closing times decreased by 49% and 43%, respectively, and the part removal
time (which included debagging after each debulking step) decreased by 41%. As
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D 19
expected, the total cost did not halve if twice the ply thickness was used; it dropped
from e4905 to e3334. Thus, 32% manufacturing cost could be saved.
The use of one general hourly labor rate might not be adequate throughout all
manufacturing processes (i.e. ply cutting, tool preparation, layup, debulking, curing,
part removal and inspection), since manual processes are discriminated in terms
of cost. In our opinion, however, SEER-MFG provided the best functionality and
cost sensitivity along with a strong acceptance in the aircraft industry. Thus, we
refrained from replacing the SEER-MFG algorithms by an own implementation. In
addition, every other definition of labor rates would be company-specific and not as
general in its use.
A more detailed study is presented in Kaufmann et al. [33], where five different
material systems and the effect of changes in labor rates, production quantities and
material costs are examined.
7
Conclusions
An existing cost/weight optimization framework for composite structures was extended by the implementation of a draping simulation tool. Thus, an optimal draping
strategy in terms of seed point and reference angle was selected in each iteration of
the geometric cost/weight optimization. These optimal draping strategies which led
to best plies could be evaluated with different objectives, such as minimum material
consumption, minimum cut length, or minimum fiber angle deviation examined at
a predefined set of control points. The manufacturing cost was calculated by an
established cost estimation program. These cost estimates were modified in a way
that inputs such as material consumption and cut length were directly imported
from the draping simulation. Hence, the merit of a draping strategy could precisely
be evaluated in terms of manufacturing cost and structural performance.
In a case study, a curved C-spar built of plain weave prepreg or unidirectional
prepreg was cost and weight optimized while keeping the set of best plies constant.
For both material systems, the cost-optimal solution was a set of best plies which
was chosen according to minimum material consumption objectives. Similarly, the
weight-optimal solution was achieved with a set of best plies which had the least
fiber angle deviation compared to a nominal layup. It was concluded that the
plain weave prepreg component performed better than the unidirectional prepreg
component. This was a consequence of the reduced drapability of the UD material.
Hence, the material systems were not quite comparable. In practice, one would
introduce a number of splits in the UD plies in order to minimize the effects of fiber
angle deviations.
A certain comparison was done anyway on the level of direct operating cost.
It could be seen that the optimal draping strategy shifted when different weight
penalties were applied. For a high weight penalty, for example, the plain weave
prepreg with a ply set of minimum fiber angle deviation was found to perform the
best. For that particular case, a stacking sequence close to the nominal fiber angles
D 20
M. Kaufmann, D. Zenkert and M. Åkermo
was favorable, whereas the effects of the reduced material consumption played a
minor role for the value of DOC.
The benefits of the integration of this draping tool were as follows. First, drapecorrected fiber angles were used instead of nominal 0/90/+45/-45◦ . This fiber angle
deviation had a substantial impact on the structural performance of the component
and should therefore not be neglected. Second, the use of a feasible draping strategy
leads to producible design solutions already in an early design phase. Thus, draping
tools such as Composite Modeler are on their way to become a standard in the design
process of aircraft manufacturers and component suppliers, and should therefore
also be included in an optimization framework. Some drawbacks exist, though. The
draping database, for example, can only be generated when the overall geometry of
the component is frozen. Hence, the framework would appear in a second step of
the preliminary design phase – after the completion of a less detailed cost/weight
optimization as presented in previous articles by the same authors. Further, the time
to place a ply is independent of the draping strategy itself. This is a shortcoming of
the combination Composite Modeler and SEER-MFG used in this work.
Two areas of future work were identified: First, it would be interesting to implement a nesting algorithm in order to estimate scrap rates and material consumptions
closer to reality. Second, a draping model could be used which estimates the exact
time of placing the material based on the draping route. Thus, the labor cost could
be calculated more accurately by taking into account the intrinsic effect of the
draping strategy.
Acknowledgements
This work is part of the European Framework Program 6, project ALCAS, AIP4-CT2003-516092, and the Swedish Flight Research Program 4, project KEKS. Special
thanks go to Simulayt Inc. for the use of Composite Modeler, to Alfgam AB for the
use of Xopt and to Galorath International for the use of the cost estimation package
SEER-MFG.
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