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Genetic Programming for Fiber-Threading for Fiber-Reinforced Plastics

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Genetic Programming for Fiber-Threading for Fiber-Reinforced Plastics

Abstract and Figures

Setting up fiber-threading for a pultrusion line is tedious, error prone and takes a long time. Between 100 and 1000 fibers have to be arranged into a two-dimensional shape, which have to be threaded between several support plates without causing crossovers. When manually planning this process based on intuition, it is hard to keep track of the complexity. This slows the process down to where it can take several hours or several days, and shortening this duration reduces the cost considerably. As planning the setup takes up a large chunk of time, we are proposing a simulation and an algorithm to automatically calculate how the fiber bundles need to be threaded from the creels through the support plates to result in the desired shape. Using a three-dimensional simulation for collision detection in conjunction with a genetic algorithm, we are able to shorten the planning of the fibers to around 10 minutes on a modern 8-core personal computer. Based on this data, further work can be done to further improve, visualize or permanently store the data in a digitized company.
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Genetic Programming for Fiber-Threading for
Fiber-Reinforced Plastics
Jonas Wilfert, Simon Stieber, Frederik Wilhelmand Wolfgang Reif
,Institute for Software & Systems Engineering—University of Augsburg, Germany
jonas.wilfert@uni-a.de {stieber; reif}@isse.de
Fraunhofer Institute for Casting, Composite and Processing Technology IGCV—Augsburg, Germany
frederik.wilhelm@igcv.fraunhofer.de
Abstract—Setting up fiber-threading for a pultrusion line is
tedious, error prone and takes a long time. Between 100 and
1000 fibers have to be arranged into a two-dimensional shape,
which have to be threaded between several support plates without
causing crossovers. When manually planning this process based
on intuition, it is hard to keep track of the complexity. This slows
the process down to where it can take several hours or several
days, and shortening this duration reduces the cost considerably.
As planning the setup takes up a large chunk of time, we
are proposing a simulation and an algorithm to automatically
calculate how the fiber bundles need to be threaded from the
creels through the support plates to result in the desired shape.
Using a three-dimensional simulation for collision detection in
conjunction with a genetic algorithm, we are able to shorten the
planning of the fibers to around 10 minutes on a modern 8-core
personal computer. Based on this data, further work can be done
to further improve, visualize or permanently store the data in a
digitized company.
Index Terms—pultrusion, fiber-threading, genetic algorithm,
fiber-reinforced plastic
I. INTRODUCTION
Long set-up times delay production and increase the cost
of the final product, and the pultrusion of fiber-reinforced
profiles is no exception. Neither the pultrusion line nor the
staff can be used profitably during this time. Before the
fibers can be inserted into the heated die, the individual
rovings (fiber bundles) must be arranged in a shape vaguely
representing the final profile without intersecting. Preparing
the plan manually takes from several hours up to an entire
business day, preventing the employee from getting other work
done. This uneconomical ratio is even worse where frequent
changes of the profile type are required due to small lot
sizes. A considerable improvement can be achieved by an
algorithmic solution.
A. Pultrusion
The pultrusion process is an industrial manufacturing pro-
cess in which fiber-reinforced plastic profiles are produced.
Continuous fibers, mostly glass or carbon fibers, are used
in this process. For this purpose, the prepared fiber package
is impregnated with different resins, which are subsequently
cured to obtain a solid composite material. Figure 1 provides
an overview of a pultrusion line, the individual steps of which
are explained in the following. Before the fibers are drawn
through the resin bath (2), they must be arranged in the
© Fraunhofer-Einrichtung für Gießerei-, Composite-und Verarbeitungstechnik IGCV
(1)
(2)
(3)
(4)
(5)
(a) (b) (c)
Fig. 1: Structure of a pultrusion line
shape of the desired profile. To achieve this, the fiber bundles
are threaded from the racks through several support plates
(a) to (c) in the fiber-threading area (1). Between each pair
of support plates several rovings are combined to achieve
the desired density and arrangement. If the profile is to be
reinforced by textile semi-finished products, such as glass fiber
mats, these are also introduced here. Planning this part is
the focus of our work and will be discussed more closely
in the next section. Afterward, the previously arranged fiber
package is pulled through the resin bath (2) to impregnate
it with resin. The impregnated fiber package is then passed
through a heated molding die (3) to allow the resin to cure
into the final shape. Finally, the cooled profile is sawed off
(5) to the desired length. The continuous movement of the
fibers is ensured by the pulling unit (4). This moves the
fiber composite through the line at constant speed. Due to
the high degree of automation of a pultrusion line, profiles
with similar mechanical properties can be produced more
cost effectively than with other manufacturing processes for
fiber-reinforced composites. Products from this manufacturing
process are used, for example, in lightweight construction or
in areas subject to high corrosion as a replacement for steel
beams. [1], [2]
B. Fiber-threading
In this work, the focus is set on the fiber routing framed in
red in Fig. 1. The number of rovings required depends on the
area of the profile and thus scales quadratically in the worst
case when the profile is enlarged. Depending on the shape
and complexity of the profile, between 100 and several 1000
rovings are usually combined. Usually, up to 10 rovings are
taken from the racks and combined into one joint strand in
the subsequent plate. The most import thing is that the fibers
are not crossed so that they are protected from abrasion. Such
defects could lead to defects in the finished product and must
therefore be avoided at all costs. To make the routing process
easier for the line worker, the joined spools should lie next to
each other on the rack. Rovings that are combined into one
strand are also referred to as a cluster in the following.
Often the fiber routing is planned in advance to make
execution at the machine as trouble-free as possible. Currently,
this is mostly done by hand using either an Excel spreadsheet
or pencil and paper. This process is very tedious and error
prone. Since two dimensions are insufficient for realistically
representing the positions relative to each other, overlaps can
only be estimated. This results in a error rate of up to 10%,
which are undetectable until the fibers are beging set up at the
plant. Manually creating a plan can take several hours up to a
several business days. In this work, we present an algorithmic
solution for solving these mentioned problems and to speed
up the process preparation.
C. Requirements
To be able to ensure the quality of a fiber-threading plan,
the requirements have to be defined. They have to cover both
the feasibility in reality as well as ensuring the comfort of the
worker at the plant. The following list is given in descending
priority.
1) Prevention of overlapping fibers
When fibers overlap, abrasion occurs, which can lead
to incalculable weakening of the final product and must
therefore be avoided at all costs.
2) Prohibiting punctured clusters
Even if the individual rovings do not overlap, it is
possible for another roving to puncture the cluster,
like intertwined fingers. In addition to increasing the
difficulty of the execution, this increases the probability
of abrasion in sagging fibers and should therefore be
avoided as well.
3) Ensuring close proximity of rovings within a cluster
By placing fibers in close proximity that belong to the
same cluster, they can be grabbed together and routed
to the next plate, resulting in a simplified execution at
the plant.
4) Ensuring short fiber distances between support
plates
Short fiber runs ensure that fibers stay in a the same
area on the target plate as they did on the source plate.
This means that fibers are less likely to run across other
clusters, reducing the likelihood of collisions.
D. Abstraction
To simplify the pultrusion line, we represent the fiber-
threading area with simple geometrical shapes. As such, we
view each support plate as a two-dimensional plane, referred
to as container in the following. The slots in the support
plate are simplified to a set of points located on the plane,
called the elements. Fibers connecting two plates can be
abstracted to a line-segment in three-dimensional space. Using
this abstraction, we can focus solely on the necessary data
required to prepare a fiber-threading plan.
II. RE LATE D WORK
No previous attempts at automatically creating a fiber-
threading plan were found in the literature. However, the
problem can be broken into smaller parts where significant
work has been published. It boils down to finding clusters of
a defined size in a set of points while the physical limitations
of the rovings have to be considered.
Therefore, the solution can consist of the following parts: Fiber
clustering, collision detection and an algorithm to combine
these two.
A. Clustering
Clustering the fibers as described in Section I-C is required
to easily replicate the plan at the pultrusion line. A commonly
used algorithm to combine a arbitrary set of points into
clusters is k-means [3]. Using an iterative approach k-means
separates the point-set into k different clusters of different
sizes. To improve the performance and increase the ability
to respect further factors multiple publications have been
published which use a genetic algorithm to enhance the k-
means algorithm [4], [5]. This can be used as a basis to the
clustering part of our problem.
B. Collision Detection
Using a physics simulation as a basis for an algorithm is
commonly used in actual practice. Since most ready-to-use
simulation tools or game engines are generally optimized for
dynamically moving objects, they cause a undesired overhead
in our context [6]. Looking at a lower level there are further
libraries implementing collision detection like ncollide1on a
more basic level which can be enhanced with state of the
art triangle intersection algorithms like the M¨
oller-Trumbore-
Algorithm [7] or the recently published algorithm by Baldwin
and Weber [8].
C. Algorithm Selection
Selecting the algorithm is a critical step that influences
the result as well as the duration the solution needs to be
computed. One might think about using a clustering algorithm
like k-means, but it is unable to deal with collisions in the
three-dimensional space without sufficient modifications.
Another failed approach consisted of using a kd-tree [9]
to split the rovings into equal sectors. Choosing the closest
sector with enough free rovings ensures the correct clustering
of the coherent rovings. However, it is unable to respect
three-dimensional collisions nor can it deal with distances e.g.
between multiple rovings.
Albeit loosing the fast performance from the solutions above
an iterative approach allows for more control. Optimizations
1Sebastien Crozet, ncollide: ncollide is a 2 and 3-dimensional collision-
detection library written with the rust programming language. Available:
https://ncollide.org
can lead to a higher degree of freedom by using different
fitness function to incrementally improve the solution. To
assist in the selection of a fitting type of algorithm Haupt et
al. [10] provides an auxiliary metric called epistasis. Epistasis
looks at how interdependent the variables are. It describes how
much a Variable Bwill change when a different variable A
changes. At low values, a minimum-seeking algorithm such
as gradient descent is recommended. At the higher range of
spectrum, random search is the only feasible solution. The
middle range is where the genetic algorithm shines. Each fiber
position can be viewed as an independent variable. This leads
to the suggestion that a genetic algorithm would be the most
fitting.
In conclusion, we used physics simulation in conjunction
with a genetic algorithm [10]–[12] inspired by biological evo-
lution. Operations such as selection, crossover and mutation
are adapted from nature to solve an optimization problem
using a high number of variables. To start off with, an initial
population is randomly generated. Using a fitness function, the
generated chromosomes are now evaluated and assigned scalar
values for comparison. During the selection stage, suitable
parents for the crossover operation are selected, which are
combined into a new child generation. Additionally, those
new children are randomly mutated to cover a larger range of
possible solution. This process is repeated until a predefined
termination condition is met, e.g., no more improvements in
the fitness function. Our implementation of the operations are
described more closely in Section VI.
III. DATA STRUC TU RE S
As described in Section I-D the fiber routing is subdivided
into the individual plates, which are called container (cf.
Fig. 2). Each of these containers stores its position and rotation
using a global offset and a normal vector pointing in the pull-
off direction. The containers are arranged in such a way that
the pull-off direction is in positive z-direction. A uniform axis
orientation ensures consistency between different imports from
CAD files. To be able to refer to each container unambigu-
ously, each one is given an integer identification number. The
containers are further divided into elements to represent the
slots in the support plates or roving creels. Storing the position
of the element in a local two-dimensional coordinate system
relative to the respective container allows the global position
to be determined by coordinate transformations.
When preparation an experiment, the initial setup of the
fiber routing is mirrored in a digital twin. Each element stores
either how many fibers are provided (out-fibers), consumed
(in-fibers) or can be forwarded (capacity). These values only
describe the initial configuration of the fiber routing and not
how the fibers are ultimately routed, i.e., the capacity is only
an upper boundary of fibers and does not necessarily require
a fiber to pass through that element.
Since it’s frequently the case that it can only be determined
during the ongoing test how many fibers are required at
which position, the concept of excess fibers was introduced.
The algorithm should treat these teh same way as in-fibers,
but are only held in reserve during system preparation when
they are not yet threaded together with the other fibers. By
already reserving rovings by the algorithm for specific slots,
subsequent additions of excess fibers are simplified.
Rovings that run between containers are represented as a
path between two elements. Here, each connection between
two containers is considered a single segment. These line
segments can be used to determine whether the fiber routing
plan is feasible by performing collision checks. Additionally,
checks for collisions with the frame of the pultrusion line could
be checked with several bounding boxes, such as Oriented
Bounding Boxes (OBB) [13]. This representation is well suited
for storing the complete fiber routing line, however the genetic
algorithm needs additional information to work with. There-
fore, an extended high-performance data structure is presented
that focuses exclusively on the interaction between two fiber
guide plates. In the following, this structure will be referred to
as graph and will be used as the chromosome of the genetic
algorithm.A visual representation can be seen in Figure 2.
From the source plate, each element, i.e. each slot, is
converted into a target. This stores the position of the element
and the number of rovings that can be provided. In the case
of a source, such as a fiber rack, this number is usually 1 as
noted in the out-fibers. In the case of a router, such as a support
plate, the maximum capacity of simultaneous rovings that can
be fed through one of the slots is stored. These elements do not
have to be completely filled, but must not exceed the maximum
number. A cluster is assigned each element on the target plate.
A cluster consists of any number of nodes and one fixpoint.
The fixpoint stores the the coordinate of the element on the
target plate to which the nodes should converge. Nodes are
not static and do not have a fixed position. However, they can
reserve a target and inherit its position. Only one node can be
assigned per target at a time. Dynamic nodes are necessary to
allow continuous changes during the calculation of the fiber
routing.
cluster
nodes of a cluster
each connected to
a target
fixpoints
(elements)
rovings
targets
(elements)
inner distance
between nodes
outer distance
to the fixpoint
Fig. 2: Multiple cluster displayed in 3D
Like the targets, the number of nodes in a cluster depends
on the type of the source plate. If it is a router, the necessary
number of rovings is stored entirely in a single node, as they
are threaded like a strand. In the case of a source, one node
is created per consumed roving, so that different targets can
be assigned to each roving. In general, this corresponds to
the minimum of the sum of the capacity and the out-fibers.
More vividly, it can be explained as follows (cf. Fig. 2):
The elements of the target plate (fixpoint) are connected with
multiple rovings (nodes) to the source plate (target) and are
supposed to be spatially close to each other (cluster). This
entity forms a graph. The data structure presented here only
considers the interaction of two containers, but the number of
containers connected in series is arbitrary. To overcome this
limitation, several of these graphs can be computed one after
the other. Here, we start at the last target plate directly in
front of the die. The distribution of the nodes on the targets of
the original plate can be used as the target plate for the next
container, so that the container connections are calculated in
pairs, against the direction of pull-off, until finally the fiber
source is reached.
IV. DEG RE ES O F FRE ED OM
If a graph is created from two containers, they are scaled to
the same size along the XY plane. It was observed empirically
that only the position and angles of intersections change, but
the intersections are not resolved, so that it becomes clearer
how the fibers are routed. Furthermore, this allows perfor-
mance optimizations using a bounding volume hierarchy on
non overlapping bounding boxes [9], [14]. Allowed operations
in this case are scaling and translating the fiber guide plates,
but not rotating (Figure 3).
Base form Scaling Translation Scaling
+
Translation
Rotation
Fig. 3: Scaling and translation of a support plate in 2D
V. FI TN ES S FUNCTIONS
As described in Section II-C, a genetic algorithm uses a
scalar fitness score to compare multiple different genomes.
However, one fitness function is not enough to ensure that
all requirements are respected in the final result. We present
4 individual cost functions, which are summed up after being
multiplied by a weight parameter to form the combined fitness
function. These weights represent the importance of each
cost function and balance out differences in the units of
measurement used. In this work, we refer to cost as the inverse
of the fitness. While a higher fitness is better, the costs have
to be decreased to improve the result.
A. Inner Distance
To be able to guarantee the proximity of the nodes of
a cluster (requirement 3 from Section I-C), the Euclidean
distances between all nodes are calculated. In Figure 2, these
connecting lines are shown in green. Before summing the
individual values, they are potentiated by a configurable factor
to weight short distances exponentially more. The result is
normalized using the triangle number to prevent the calculated
cost from being dependent on the number of nodes. Through
this number series, the number of interactions between the
nodes can be determined. This number can be directly calcu-
lated by the Gaussian summation. By starting the inner sum
at i+ 1 we can avoid measuring the interaction between two
nodes twice as well as skipping self interactions. The final
formula for the cost of a cluster Ccorresponds to:
costinner(C) = Pn
i=1 Pn
k=i+1 d(Ni, Nk)p
n·(n+1)
2
(1)
Where Niare the nodes of the cluster and pis an exponen-
tial factor.
B. Outer Distance
To keep fiber distances as short as possible and to ensure
that nodes in a cluster are as close as possible to their fixpoint,
the distance is also taken into account in the weighting. For
this purpose, the Euclidean distances between the fixpoint and
the positions of the respective nodes are measured. These
connections are shown in purple in Figure 2. Like the previous
section, this value is also normalized by the number of nodes.
Leading to the following equation:
costouter(C) = Pn
i=1 d(Ni, F )p
n(2)
Where Niis the nodes of a cluster, Fis the fixpoint, and
pis the exponent.
C. 2D Angle
If no overlaps occur between two rovings, it is never-
theless disadvantageous if they cross in different directions.
By projecting the fibers from three-dimensional space onto a
two-dimensional plane, the angles of the now crossing fibers
can be checked. Visually speaking, the rovings are viewed
from a bird’s eye view (XZ plane) as well as a frontal view
(XY plane). The angles between the two vectors a, b in the
respective planes can be calculated by the following formula:
αXY (a, b) = arccos ax·bx+ay·by
qa2
x+a2
y·qb2
x+b2
y
(3)
If this angle exceeds a configurable threshold simultane-
ously with a collision of the vectors in the respective pro-
jection, the total cost is increased. When disregarding this
condition the rovings would have to be pulled over or under
already threaded fiber packages.
D. Triangle Check
Checking for triangular collisions is the most relevant metric
for implementing threading at the plant. This part of the fitness
function is strongly aligned with the problem formulated
in Requirement 2: preventing punctured clusters. It can be
represented by a mathematical cost function with triangle-
line collisions, where the interior of the cluster is treated as
a triangle and the rovings as line segments. With the target
positions of two rovings as well as the fixpoint of the cluster
a triangle can be spanned. The edges of the triangle thus
form the two rovings leading to the two nodes as well as
an imaginary connection between these two nodes (cf. Fig 4).
To find out between which nodes a triangle has to span, a
collision
spanned
triangle
fixpoint
roving
connection
between nodes
Fig. 4: Spanning of a triangle between the fixpoint and two
nodes with a collision of a roving
hull is formed around the cluster. Thus, only the triangles of a
simplified shape must be calculated. The Graham scan yields
an algorithm that can compute the convex hull in a worst case
runtime of O(n·log n)by iterative angle checking over all
nodes [15]. Unfortunately, the algorithm is only defined for 2-
dimensional space. However, by projecting the position vectors
of the nodes on to a hyperplane orthogonal to the normal
vector, the the application of the algorithm is possible under
the assumption of a flat support plate. Often, the simple case
occurs where the support plate is parallel to the XY plane,
i.e., along the standard orientation without rotation. In this
case, the projection simplifies to a vector with only the x
and ycomponents. The algorithm forms the outline of the
cluster along of which edges the triangles can be spanned. All
collisions with edges of inner nodes excluded by the Graham
scan are also covered by the connections of the contour, which
means that n+ 1 triangles will have to be checked in a worst-
case, instead of n·(n+1)
2if pair-wise interactions had been
employed.
E. Clustering
In many cases, the inner distance is insufficient for deter-
mining whether a fiber packet is arranged as a cluster. The
change of two directly adjacent nodes only leads to a very
small change in total cost, which can be overshadowed by
a simultaneous degradation of another cost function, such as
the outer distance. To support this metric, we additionally
determine how well it is clustered by counting the nearby
nodes of foreign clusters. Here the opposite direction of well-
known clustering algorithms as k-means is considered: check-
ing clusters that have already been formed instead of finding
related clusters in a set of points. In addition, restrictions to
prevent collisions must be observed here, which are difficult
to integrate directly into k-means. Despite the limitations, the
algorithm presented here strongly resembles k-means [16] in
terms of dividing the space into segments.
To quantify clustering, as shown in Fig. 5, a node Ncentral
is selected around which the space is divided into 4 segments,
each with 90° opening angle. Now, individually for each
segment, the nodes are sorted in ascending order by Euclidean
distance to the central node. Additional costs are imposed if a
node with a foreign or without a cluster occurs in this sorted
list before all nodes of the own cluster have been processed.
Said procedure is repeated for all nodes in each cluster and
the costs are added up. This can be implemented efficiently
using an additional variable for each segment as described
in Algorithm 1. The variable acts as a memory mithat gets
committed to the final cost when coming across a node of
the own cluster. Thus, if the total cost of this cost function
reaches zero, according to this metric, all nodes have arranged
themselves in clusters.
6542
3
1
69
7
1235
4
enclosed node of the
green cluster
(causes additional cost)
ascending order using
euclidean distance to
the central node
enclosed node without
cluster
(causes low additional cost)
central node
Fig. 5: Quantification of the clustering using 4 segments with
each 90° opening angle
Algorithm 1: Clustering Cost-Function
Result: cost
1cost 0;
2foreach Ncentral nodes do
3mn0n∈ {0,1,2,3};
4foreach Nother in nodes sorted by distance to Ncentral do
5sindex of sector of Nother;
6if cluster(Ncentral ) = cluster(Nother )then
7cost cost +ms;
8ms0;
9else
10 msms+ 1;
VI. GE NE TI C ALGORITHM
In the following sections, we want to present our imple-
mentations of elemental operations of the genetic algorithm
as described in II-C.
A. Selection
Based on fitness, it is possible to compare graphs pairwise.
However, to choose which graphs to use for the crossover,
a selection procedure is needed that selects two elements
from the weighted set as parents. There are different ways
to accomplish this task. E.g. Weighted Order selects parents
weighted by position in the list sorted by fitness. Similarly
Upper Half selects a random element in the upper half of the
sorted list while ignoring the fitness score. Weighting/Roulette
Wheel assigns each element a probability based on the share
the fitness of the element accounts of in the total sum.
Comparable to a roulette wheel, the element is now selected
at random based on the probability distribution. When using
Tournament Selection nelements of the unsorted list are
selected at random of which only the best element is kept.
In a comparison between several selection methods by Shukla
et al. [17] Tournament Selection turned out to be the best
method, both in terms of performance as well as convergence
behavior.
In this work, a combination of both Upper Half and
Tournament Selection methods was used. It was decided to
include only the top 50% of the population in the Tournament
Selection, as a single mutation can cause very large changes in
fitness, leading to the possibility of getting very poor results.
The calculation time of the fitness function is several orders
of magnitude higher than the duration spent in selection, there
the performance of the selection method is negligible.
B. Crossover
During crossover, two chromosomes, in this case graphs, are
merged into one child that adheres to the following procedure.
The number of clusters and nodes is already fixed from the
beginning and cannot change during runtime. This allows both
graphs to be traversed in parallel and is guaranteed to have a
partner for each node.
The process begins with an additional empty graph, the child
graph. The positions of each node are transferred to it during
the crossover. For each pair of nodes that are processed, a
random number is generated that is converted to a boolean
value. These are used to determine from which of the two
parents the position of the current nodes should be carried over.
We use quasi-random number sequences instead of pseudo-
random numbers to both increase the probability of a better
solution as well as increasing the performance, as described
by Shuhei and Matsumura [18], [19]. Now, if one of the two
nodes of the parents has been selected, it must be checked
whether the position on the third, the child graph, is already
occupied. By different assignment of the parents nodes, it is
possible that the position of a target is requested twice. In this
case, the nearest possible free fiber target is selected. This
guarantees that no erroneous graphs are created where targets
are assigned more than once. By selecting a nearby target,
a degree of clustering can still be retained. This process is
repeated until all nodes of both parents have been transferred
to the child.
C. Mutation
Random mutations are used to introduce certain changes
with a low probability into the graphs. Mutations can either
affect a node or a whole cluster. Nodes can either be swapped
with an empty target nearby or swapped in place with a
different node. Clusters are treated similarly and are either
moved to a new free location in the graph or swapped with
an existing cluster. If the clusters have a different number of
nodes, the remaining ones are distributed to nearby free targets,
similar to the crossover.
The cluster mutations can cause movements of entire clus-
ters without violating requirement 3 (proximity of nodes
within a cluster). In node-based operations, scattering nodes
of a cluster often occurs, which could cause a temporary
degradation of the overall fitness due to the cost functions
inner distance and clustering. This makes it highly likely that
this mutation will not be carried over to the next generation.
To optimize the clusters, nodes must be preferentially swapped
with nodes from clusters nearby, and a procedure similar to
Tournament Selection is used to do this. nnodes or clusters
are randomly selected, but only the closest of these are used
for swapping. Thus, the probability for a resulting change is
indirectly proportional to the distance.
D. Termination Condition
In order to determine when the computation of the algorithm
is complete, a primary termination condition must be defined.
As a genetic algorithm converges to some value the change in
fitness between iterations can be used to determine at which
part of the convergence curve the algorithm is located. If
the difference falls below a certain threshold for an extended
period of time, the algorithm is declared complete and can be
terminated.
However, this is not the only termination condition present
in the proposed algorithm. Since some errors are fatal for
the functionality of the fiber-threading plan, parts of the
fitness function can additionally be considered individually.
Specifically, the cost function triangle check is relevant here.
If errors of this type are still present in a plan, it cannot be
executed reasonably in actual practice. It has been observed
that errors in the triangle checks are strongly dependent on
the initial population, which is why the concept of restart was
introduced here. If triangle collisions are still present after
a defined number of iterations, here 10000, the algorithm
is restarted. In this case the complete population is deleted
and randomly generated again. The number of iterations after
which the restart occurs must be smaller than that of the
primary termination condition. Therefore, it is impossible to
complete the algorithm successfully with triangular intersec-
tions.
VII. RES ULTS A ND DISCUSSION
A. Hyperparameter Optimization
The developed algorithm provides a large number of pa-
rameters. These parameters can affect the runtime as well as
the result of the algorithm, so they must be chosen wisely.
Meaningful values can be achieved using hyperparameter
optimization. A sufficiently complex tube-profile with over
500 fibers was selected for the hyperparameter optimization,
as these offer the greatest challenge for the algorithm while
still being computable in a reasonable amount of time (see
Section VII-B).
a) Population Size: Population size is one of the two
most important factors of the genetic algorithm along with
the mutation chance. It determines how many chromosomes
are created and evaluated in each generation. Since a genetic
algorithm is based on randomness, the probability of obtaining
a good mutation or crossover is higher with a higher popu-
lation size. Furthermore, the algorithm starts with a higher
variance in the initial population, which allows for a better
result. However, execution time increases drastically as the
computational effort increases proportionally to the population
size. The effect of the population size on the computation time
as well as on the total cost value is shown in the graph 6b.
b) Mutation chance: The second of the two most impor-
tant parameters of a genetic algorithm is the mutation chance,
which is the probability at which a random change is induced
in the chromosome. The mutation chance is not as flexibly
selectable as the population size, but ranges within a certain
interval. Graph 6a shows that a lower mutation chance leads
to a faster convergence with a lower final cost. Above a value
of 2% it was no longer possible for the algorithm to complete
successfully, resulting in termination after 100,000 iterations.
Below a value of 0.025%, no result could be generated either,
since the algorithm was trapped in an infinite loop due to the
termination condition formulated in Section VI-D for triangle
collisions. These could not be solved in time with such a low
mutation chance before a restart was triggered.
The way the probability is applied explains this progression.
Instead of triggering a single mutation somewhere on the
whole graph with a high probability, a mutation is triggered
per node with a very low probability. Thus, if the percentage
is too high, multiple mutations may occur at the same time. In
this case, an improvement in the fitness can be immediately
overshadowed by a bad change. Unfortunately, the chance
of mutation is consequently dependent on the number of
fibers to be threaded. Hence, choosing a very small value
is necessary for complex problems, but results in a slightly
increased computation time for simple plans with few fibers.
B. Profiles
To ensure that the presented solution supports a broad
spectrum of profiles, 3 different profiles are tested here. The
algorithm is run with the following profiles and the result is
then evaluated, with the type of profiles becoming increasingly
complex.
duration [min]
(a)
2
12
22
32
42
52
62
0
100
200
300
400
500
600
700
800
0 500 1000
duration [min]
cost
population
(b)
Fig. 6: Impact of the mutation chance (a) and population size
(b) on the runtime duration and the final cost
Type Duration for 500 Rovings Stage 1 Stage 2
Flat 2:49 min 11% 89%
U 5:28 min 6% 94%
Tube 7:15 min 21% 79%
TABLE I: Durations for the different types of profiles
Flat Profile (50 Elements each with 10 Rovings)
U-Profile (56 Elements each with 9 Rovings)
Tube-Profile (100 Elements each with 5 Rovings)
Creating the fiber-threading plan is divided into two steps.
First, the final plate is mapped onto the support plate. Here, the
fiber packages are not yet divided into the individual rovings
but are threaded as a strand. The data created here is used for
the second stage. There, the occupied elements of the support
plate are threaded onto the racks. The fiber bundles are now
divided into the individual rovings so that they can be taken
individually from the racks. Thus, the second step is more
computationally intensive and error prone due to the higher
number of separate rovings.
C. Results
In the following, the calculations of the previous profiles are
evaluated. Table I shows the duration the individual profiles
took to complete both stages while running on an 8-core
Ryzen 3700X. Figures 7 and 8 show selected excerpts of
the fiber guidance for the tube profile. In the graphs, the Xs
correspond to the source and the circles to the target. The
elements connected by rovings are marked in color.
Reducing the planning time from multiple hours down to a
few minutes on consumer hardware is a major improvement
between one and two orders of magnitude. If the user is not
satisfied with the provided result, the algorithm can simply
be rerun while still maintaining a speedup to the manual
approach.
Figure 7a shows the left part of the first stage. Due to the
low density in the center compared to the border, the fibers
extend in all directions while preventing collisions. As can be
seen in the projected image, the algorithm prevents overlap
of fibers even on complex shapes. This is hard to replicate
using a manual approach and simplifies the work with non-
basic shapes. Figure 7b shows an excerpt from the second
stage. Different colors represent different clusters. It shows
how the algorithm attempts to place nodes of clusters in close
proximity, which in this case means mostly on one row in the
rack. Where possible, the cluster is placed near the fixpoint.
However, if it would result in a collision between different
rovings, the whole cluster is moved farther away. This ensures
that the plan can be implemented without having to thread
fibers between other clusters while also preventing abrasion.
(a) (b)
Fig. 7: Excerpt from the first (a) and second stage (b) of the
round profile
However, the algorithm is only an heuristic. In rare cases,
the results are suboptimal with regard to the ease of implemen-
tation. In our experiments, in 95% of the cases, the algorithm
provided a sensible fiber-threading plan. This percentage fell
to 90% for complex plans with several stages and many
fibers. However, by inspecting the projected images after the
algorithm is complete, the user can easily identify deficiencies
in the fiber-threading plan. In contrast to Figure 7 which shows
a good plan we can see such a defect in Figure 8. The two
opposed clusters (1) and (2) should be swapped for a simpler
executability. This could be done by a manual postprocessing
step or by rerunning the algorithm with the same input data.
D. Limitations and Future Work
A functional algorithm, which greatly simplifies the work
and shortens the preparation time, was created. Nevertheless,
the approach is not without limitations, which should be
considered when implementing the software in production.
Most obvious is the long runtime of the algorithm. For
large plans, it can take up to 30 minutes to produce a
meaningful result. For further acceleration, more profound
measures than software optimization must be used, resulting
in fewer iterations of the algorithm. By focusing on only one
container and its direct successor at a time, the algorithm
has fewer possibilities that need to be checked, improving its
performance. A better overall result could be achieved if the
algorithm processed the complete plan from the fiber rack to
the die in parallel. In the current implementation, only a locally
good result is targeted in each stage, but without checking
whether this fiber distribution is advantageous for the next
stages. A simultaneous calculation would also allow the fiber
routing plan to be calculated without any pre-subdivision.
(1)
(2)
Fig. 8: Two opposed clusters (1) and (2) which should be
swapped
VIII. CONCLUSION
In this paper, we presented an algorithm that calculates
with high probability a correct fiber-threading plan that can
be implemented in reality from several connected fiber support
plates. For this purpose, a simulation and a genetic algorithm
with four different cost functions were used. The simulation
provides a model that uses collision detection for checking
whether the routing plan is correct. The fitness function
is used to create an objective evaluation based on firmly
defined requirements. With the help of these so-called fitness
scores, the genetic algorithm can select, cross and mutate the
chromosomes in order to iteratively obtain a better result,
however the best solution cannot be guaranteed.
Through various optimization approaches, an average run-
time of between 5 and 15 minutes was achieved. Since the
algorithm allows many parameters to be adjusted, the most
important settings were explained as well as a reasonable
configuration was determined in a parameter comparison.
The intended benefit was to reduce the preparation time
as well as to verify the fiber-threading plan ahead of time
to prevent errors when realizing the fiber-threading at the
pultrusion line. Both goals were achieved with a calculation
time of less than one hour and collision detection in three-
dimensional space.
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