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Citation: Babiˇc, M.; Marinkovi´c, D. A
New Approach to Determining the
Network Fractality with Application
to Robot-Laser-Hardened Surfaces of
Materials. Fractal Fract. 2023,7, 710.
https://doi.org/10.3390/
fractalfract7100710
Academic Editor: Erick Ogam
Received: 9 May 2023
Revised: 20 September 2023
Accepted: 22 September 2023
Published: 27 September 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
fractal and fractional
Article
A New Approach to Determining the Network Fractality with
Application to Robot-Laser-Hardened Surfaces of Materials
Matej Babiˇc 1and Dragan Marinkovi´c 2, *
1Complex Systems and Data Science Laboratory, Faculty of Information Studies, 8000 Novo Mesto, Slovenia;
matej.babic@fis.unm.si or babicster@gmail.com
2
Department of Structural Mechanics and Analysis, Faculty of Mechanical Engineering and Transport Systems,
Technical University Berlin, 10623 Berlin, Germany
*Correspondence: dragan.marinkovic@tu-berlin.de
Abstract:
A new method to determine a fractal network in chaotic systems is presented together with
its application to the microstructure recognition of robot-laser-hardened (RLH) steels under various
angles of a laser beam. The method is based on fractal geometry. An experimental investigation
was conducted by investigating the effect of several process parameters on the final microstructures
of material that has been heat-treated. The influences of the surface temperature, laser speed, and
different orientation angles of the laser beam on the microstructural geometry of the treated surfaces
were considered. The fractal network of the microstructures of robot-laser-hardened specimens
was used to describe how the geometry was changed during the heat treatment of materials. In
order to predict the fractal network of robot-laser-hardened specimens, we used a method based on
intelligent systems, namely genetic programming (GP) and a convolutional neural network (CNN).
The proposed GP model achieved a prediction accuracy of 98.4%, while the proposed CNN model
reached 96.5%. The performed analyses demonstrate that the angles of the robot laser cell have a
noticeable effect on the final microstructures. The specimen laser-hardened under the conditions
of 4 mm/s, 1000
◦
C, and an impact angle of the laser beam equal to 75
◦
presented the maximum
fractal network. The minimum fractal network was observed for the specimen before the robot-laser-
hardening process.
Keywords: fractal networks; modeling; robot laser hardening
1. Introduction
The term “fractal” was proposed for the first time by Mandelbrot [
1
], derived from
the Latin root “fract” (lat. “Fractare”—break, fraction; “fractus”—dismembered, broken;
English “fractal”—fractional). According to Mandelbrot’s definition, a set is denoted
as ‘fractal’ if its Hausdorff–Besikovich dimension is strictly greater than its topological
dimension. Simply put, a fractal set is a set of dimensions different from what is usually
called a topological dimension, represented by a number as a whole. Mandelbrot [
1
] also
provided another definition: “a fractal is a structure made up of parts that resemble a
whole in some way”. However, there is no strict and specific definition of a fractal today.
A fractal structure is created by any infinitely repeating (iterating) initial shape, say H,
in decreasing (or increasing) proportions, in a specific algorithm. This relatively simple
feedback process provides a surprisingly diverse morphogenesis that often resembles
the emergence of natural forms. Fractals are therefore characterized by self-similarity or
large-scale invariance. Traditional geometric objects have integer dimensions: lines are
one-dimensional, planes are two-dimensional, and spheres are three-dimensional. A fractal
object is characterized by a fractal dimension. Such dimensions were introduced by F.
Hausdorff. If a smooth Euclidean line just fills one-dimensional space, then the fractal line
goes beyond it and partially fills two-dimensional space—its dimension is broken, and
Fractal Fract. 2023,7, 710. https://doi.org/10.3390/fractalfract7100710 https://www.mdpi.com/journal/fractalfract
Fractal Fract. 2023,7, 710 2 of 13
it lies between the original dimension of the line and the two-dimensional space of the
fractal form Happened yes. The motivation of our research is presenting a new method for
determining network fractality.
For example, fractal coastlines have dimensions between 1 and 2; fractal surfaces
(mountains, clouds) have dimensions between 2 and 3. The fractal dimension [
2
] is a
topological invariant of every fractal structure, which is a special symmetry. So, as men-
tioned earlier, fractal lines go beyond one-dimensional space into two-dimensional space.
Their dimensions are broken, and lie between the original dimension of lines and the
two-dimensional space of morphogenesis—fractal genesis. Likewise, fractal planes extend
partially into three-dimensional space; theoretically, we also believe that the output of a
three-dimensional surface is the result of its fractalization into higher-dimensional spaces.
For example, coastlines and many other natural structures and processes are fractal (or,
more accurately, quasi-fractal): rivers and their tributaries, lightning, mountain surfaces,
cloud cover, the distribution of galaxies, solar activity. The natural landscape around us is
created by the dynamic chaos of natural processes. The fractal properties of natural objects
demonstrate the ability to construct very realistic computer landscapes, virtual worlds, in
relatively simple fractal programs, where the resemblance to reality is achieved through
randomization and a certain degree of irregularity achieved through random number man-
agement. However, fractals can also be indistinguishable, such as scaly, granular, fibrous
fractals, etc. The self-similarity of fractal structures as a result of functional iteration with
reverse communication (self-referential feedback) determines near (local) connections to
distant (global) orders and enables concise mathematical descriptions of structures and
processes. These structures and processes are hardly suitable for mathematical descrip-
tion and mathematical understanding. Simple mathematical rules generate self-similar
structures with respect to nonlinear transformations, thus enabling fairly complex shapes.
This means that simple laws are at the very heart of complex systems and processes. The
application of fractal structures in the field of mechanical engineering is most developed
in fractal systems. Brittle iron is relatively weak, Mandelbrot also said. Fractal geometry
has been successfully used for the detailed analysis of various mechanical properties of
materials such as molybdenum, stone, concrete, steel, and composite materials of various
lengths. The values obtained from the analyzed fracture sites confirm the concept of a
global fracture measurement, independent of the fracture analysis obtained by scientific
researchers for this type of material. This approach also facilitates product integration.
Robot laser hardening (RLH) belongs to the group of modern and highly effective
methods for hardening the surface layers of materials [
3
]. Thermal hardening of metals
and alloys occurs locally when a portion of the radiation surface is heated followed by the
fast cooling of this surface portion due to the removal of heat to the internal cold metal
layers. Important advantages of laser hardening also include the reduction in the additional
processing of the material, the eliminated need for a separate cooling process, and the ability
to process inhomogeneous 3D blanks and complex geometries. Robot systems that provide
not only heat treatment, but also positioning, the automatic feeding of workpieces, results
measurement, etc., are becoming more relevant. The complex control system analyzes this
data stream in real-time and, if necessary, corrects the parameters of laser radiation. This is
important in the conditions of practical work when the workpiece warms up in the process
of heat hardening. If the radiation power is not proportionally adjusted, overheating will
occur, causing deterioration in the results of the heat-hardening process. The control system
provides automatic calibration in the framework of the service tasks and makes appropriate
corrections in the technological operations performed.
Heat-treatment processes, including the hardening and tempering of tool steels to
achieve a desired microstructure with improved mechanical properties, is one of the last
production steps which determines the final bulk residual stresses within the workpiece [
4
].
Babiˇc et al. [
5
] presented a procedure based on the calculation of the FD complexity
index. It was developed to assess the complex geometry of RLH materials’ microstructures
Fractal Fract. 2023,7, 710 3 of 13
upon changing the process parameters, and the obtained results were in agreement with
experimental data.
Modeling of thermo-mechanical–metallurgical interactions during heat treatment is
quite important in order to predict the evolution of internal stresses and distortions as a
consequence of the process [6].
Babiˇc et al. [
7
] introduced a new method for quantifying the complexity of networks
based on the network nodes given in Cartesian coordinates. The nodes were mapped
to polar coordinates, and fractal dimensions were computed using the ReScaled Ranged
(R/S) method. This work revealed various challenges, thus calling for further research in
this field.
The aim of this manuscript is to propose a new approach to determining network
fractality, with an application to robot-laser-hardened surfaces of materials, and to mod-
eling the fractal networks of RLH specimens’ microstructures depending on the process
parameters of the robot laser cell.
Section 2presents the preparation of materials, methodology and data mining, and
Section 3gives the results and discussion, while conclusions are given in the last section.
2. Material Preparation, Methodology and Data Mining
2.1. Material Preparation
In the present study, steel standard tools [
8
] were hardened using a laser cell, equipped
with a robot RV60-40 by Reis Robotics Company from Obernburg, Germany (technical
details: manufacturer: Reis Robotics; model: HRV60; category: industrial robots; weight:
1200 kg; height: 100 com; length: 200 cm; width: 100 cm). The robot controls the laser
head position and the laser beam orientation (fiber coupled 10 kW diode, laser Laserline
LDF10000, 1.5
µ
m fiber). The laser can deliver a nominal maximum power equal to 1500 W.
The manufacturing cell is also equipped with a thermal scanner that measures the surface
temperature during the heat treatments.
During the experiments, the following process parameters were controlled:
◦Laser beam speed v ∈[2 mm/s, 5 mm/s];
◦Surface temperature T ∈[850 ◦C, 1400 ◦C];
◦Impact angles ϕ∈[45◦, 60◦, 75◦], ϕ∈[105◦, 120◦, 135◦].
Figure 1provides schematic views of the laser hardening performed by the robot,
including different options to modify the laser beam angle, either in the plane defined by
the laser motion or in the plane perpendicular to it. Each robot-laser-hardened specimen
was etched (selective removal of material from a sample using chemical or physical means)
and polished (an abrasive material was used to finish the metal surface), which was carried
out at the Institute of Metals and Technology (IMT) in Ljubljana, Slovenia. Afterwards
the samples were studied under a microscope at the Jožef Stefan Institute IJS in Ljubl-
jana, Slovenia. The image in Figure 2was obtained by field emission scanning electron
microscopy using a JMS-7600F from JEOL (Tokyo, Japan) and represents the complexity of
the microstructure.
FractalFract.2023,7,xFORPEERREVIEW3of13
uponchangingtheprocessparameters,andtheobtainedresultswereinagreementwith
experimentaldata.
Modelingofthermo-mechanical–metallurgicalinteractionsduringheattreatmentis
quiteimportantinordertopredicttheevolutionofinternalstressesanddistortionsasa
consequenceoftheprocess[6].
Babičetal.[7]introducedanewmethodforquantifyingthecomplexityofnetworks
basedonthenetworknodesgiveninCartesiancoordinates.Thenodesweremappedto
polarcoordinates,andfractaldimensionswerecomputedusingtheReScaledRanged
(R/S)method.Thisworkrevealedvariouschallenges,thuscallingforfurtherresearchin
thisfield.
Theaimofthismanuscriptistoproposeanewapproachtodeterminingnetwork
fractality,withanapplicationtorobot-laser-hardenedsurfacesofmaterials,andtomod-
elingthefractalnetworksofRLHspecimens’microstructuresdependingontheprocess
parametersoftherobotlasercell.
Section2presentsthepreparationofmaterials,methodologyanddatamining,and
Section3givestheresultsanddiscussion,whileconclusionsaregiveninthelastsection.
2. MaterialPreparation,MethodologyandDataMining
2.1.MaterialPreparation
Inthepresentstudy,steelstandardtools[8]werehardenedusingalasercell,
equippedwitharobotRV60-40byReisRoboticsCompanyfromObernburg,Germany
(technicaldetails:manufacturer:ReisRobotics;model:HRV60;category:industrialrobots;
weight:1200kg;height:100com;length:200cm;width:100cm).Therobotcontrolsthe
laserheadpositionandthelaserbeamorientation(fibercoupled10kWdiode,laserLa-
serlineLDF10000,1.5µmfiber).Thelasercandeliveranominalmaximumpowerequal
to1500W.Themanufacturingcellisalsoequippedwithathermalscannerthatmeasures
thesurfacetemperatureduringtheheattreatments.
Duringtheexperiments,thefollowingprocessparameterswerecontrolled:
oLaserbeamspeedv∈[2mm/s,5mm/s];
oSurfacetemperatureT∈[850°C,1400°C];
oImpactanglesφ∈[45°,60°,75°],φ∈[105°,120°,135°].
Figure1providesschematicviewsofthelaserhardeningperformedbytherobot,
includingdifferentoptionstomodifythelaserbeamangle,eitherintheplanedefinedby
thelasermotionorintheplaneperpendiculartoit.Eachrobot-laser-hardenedspecimen
wasetched(selectiveremovalofmaterialfromasampleusingchemicalorphysical
means)andpolished(anabrasivematerialwasusedtofinishthemetalsurface),which
wascarriedoutattheInstituteofMetalsandTechnology(IMT)inLjubljana,Slovenia.After-
wardsthesampleswerestudiedunderamicroscopeattheJožefStefanInstituteIJSin
Ljubljana,Slovenia.TheimageinFigure2wasobtainedbyfieldemissionscanningelectron
microscopyusingaJMS-7600FfromJEOL(Tokyo,Japan)andrepresentsthecomplexityof
themicrostructure.
(a)(b)
Figure 1. Cont.
Fractal Fract. 2023,7, 710 4 of 13
FractalFract.2023,7,xFORPEERREVIEW4of13
(c)(d)
(e)(f)
Figure1.Schematicviewofthelaserhardeningviarobot:(a)thelaserbeamimpactsthetreated
surfaceperpendicularly(90°);(b)theimpactanglemodifiedtotherightorleftside:φ∈[45°,60°,
75°]and[105°,120°,135°],respectively;(c)theimpactanglemodifiedforward:φ∈[45°,60°,75°];
(d)theimpactangleismodifieddownward:φ∈[105°,120°,135°];(e)robotlaserhardeningwith
overlapping;(f)pointrobotlaserhardening(thelaserbeamdoesnotmove).
Figure2depictsthemicrostructureofalaser-hardenedspecimen,withhardening
performedbyarobot.ThemagnificationusedforSEMis2000×,andtheimageresolution
is250×200pixels.
Figure2.Microstructureofalaser-hardenedspecimenusingarobot.
2.2.Methodology
First,weapplyanetworkofinterconnectednodes[9]todescribethemicrostructure.
Anodeisoneofthetwomostdistinctiveelementsofsuchadiagram,oftenunderstood
asanapex.Itrepresentsanintersection.Thiscouldbeaparticularposition,property,etc.
Itistypicallyenvisagedasasphere.Anothermostcardinalcharacteristicofthediagram
isthelinks.Alinkisanassociationbetweentwonodes.Typically,itisshownasaline
connectingtwonodes.Figure3illustratessuchanetwork.
Figure 1.
Schematic view of the laser hardening via robot: (
a
) the laser beam impacts the treated
surface perpendicularly (90
◦
); (
b
) the impact angle modified to the right or left side:
ϕ∈
[45
◦
, 60
◦
,
75
◦
] and [105
◦
, 120
◦
, 135
◦
], respectively; (
c
) the impact angle modified forward:
ϕ∈
[45
◦
, 60
◦
, 75
◦
];
(
d
) the impact angle is modified downward:
ϕ∈
[105
◦
, 120
◦
, 135
◦
]; (
e
) robot laser hardening with
overlapping; (f) point robot laser hardening (the laser beam does not move).
FractalFract.2023,7,xFORPEERREVIEW4of13
(c)(d)
(e)(f)
Figure1.Schematicviewofthelaserhardeningviarobot:(a)thelaserbeamimpactsthetreated
surfaceperpendicularly(90°);(b)theimpactanglemodifiedtotherightorleftside:φ∈[45°,60°,
75°]and[105°,120°,135°],respectively;(c)theimpactanglemodifiedforward:φ∈[45°,60°,75°];
(d)theimpactangleismodifieddownward:φ∈[105°,120°,135°];(e)robotlaserhardeningwith
overlapping;(f)pointrobotlaserhardening(thelaserbeamdoesnotmove).
Figure2depictsthemicrostructureofalaser-hardenedspecimen,withhardening
performedbyarobot.ThemagnificationusedforSEMis2000×,andtheimageresolution
is250×200pixels.
Figure2.Microstructureofalaser-hardenedspecimenusingarobot.
2.2.Methodology
First,weapplyanetworkofinterconnectednodes[9]todescribethemicrostructure.
Anodeisoneofthetwomostdistinctiveelementsofsuchadiagram,oftenunderstood
asanapex.Itrepresentsanintersection.Thiscouldbeaparticularposition,property,etc.
Itistypicallyenvisagedasasphere.Anothermostcardinalcharacteristicofthediagram
isthelinks.Alinkisanassociationbetweentwonodes.Typically,itisshownasaline
connectingtwonodes.Figure3illustratessuchanetwork.
Figure 2. Microstructure of a laser-hardened specimen using a robot.
Figure 2depicts the microstructure of a laser-hardened specimen, with hardening
performed by a robot. The magnification used for SEM is 2000
×
, and the image resolution
is 250 ×200 pixels.
2.2. Methodology
First, we apply a network of interconnected nodes [
9
] to describe the microstructure.
A node is one of the two most distinctive elements of such a diagram, often understood as
an apex. It represents an intersection. This could be a particular position, property, etc. It is
typically envisaged as a sphere. Another most cardinal characteristic of the diagram is the
links. A link is an association between two nodes. Typically, it is shown as a line connecting
two nodes. Figure 3illustrates such a network.
Fractal Fract. 2023,7, 710 5 of 13
FractalFract.2023,7,xFORPEERREVIEW5of13
Figure3.Exampleofgraph/network.
Furthermore,weapplyfractalgeometrytodescribethecomplexityoftheinvestigated
geometry.Dimensionsofself-similarfractalsarethesameateachenlargement,whichisnot
anissueforstatisticalunmixed-relativefractals[10].Figure4depictsself-similarandself-af-
finetransformations.Self-affinetransformationsarealsotiltingtransformations.
Figure4.Self-similarandself-affinetransformations.
Itmaybeemphasizedthattheself-similarityspaceisasubsetofself-affinity.Self-
affinefractals—includingself-similargeometry—canbedividedintoexactlyself-affine,
quasi-self-affineandstatisticallyself-affine,asshowninFigure5.
(a)(b)(c)
Figure5.Threetypesofself-similarfractalobjects:(a)self-similargeometricobject;(b)quasi-self-
similarnaturalobject;(c)statisticallyself-affinemicrostructureofrobot-laser-hardenedspecimen.
Thetwo-dimensionalrepresentationoftheRLHspecimen’smicrostructureisquite
complicatedanddefinitelynotperfectlyalikebutcanstillbeexaminedstatisticallyusing
thementionedapproach.CanonicalEuclideangeometrycannotbeappliedtothisprob-
lem.Thus,fractalgeometrywasselectedforthispurpose.Wedevelopedanewmethod
toestimatefractalpropertiesofthenetwork,anditispresentedbelow.
First,weconvertSEMimagesintobinaryimages(blackandwhitesquares).Then,
weuseonlyblacksquares,i.e.,weremovethewhiteones.Theblacksquarespresentthe
nodesofthenetwork.Weconvertalltheseblacknodesintoapolarcoordinatesystem
Figure 3. Example of graph/network.
Furthermore, we apply fractal geometry to describe the complexity of the investigated
geometry. Dimensions of self-similar fractals are the same at each enlargement, which is
not an issue for statistical unmixed-relative fractals [
10
]. Figure 4depicts self-similar and
self-affine transformations. Self-affine transformations are also tilting transformations.
FractalFract.2023,7,xFORPEERREVIEW5of13
Figure3.Exampleofgraph/network.
Furthermore,weapplyfractalgeometrytodescribethecomplexityoftheinvestigated
geometry.Dimensionsofself-similarfractalsarethesameateachenlargement,whichisnot
anissueforstatisticalunmixed-relativefractals[10].Figure4depictsself-similarandself-af-
finetransformations.Self-affinetransformationsarealsotiltingtransformations.
Figure4.Self-similarandself-affinetransformations.
Itmaybeemphasizedthattheself-similarityspaceisasubsetofself-affinity.Self-
affinefractals—includingself-similargeometry—canbedividedintoexactlyself-affine,
quasi-self-affineandstatisticallyself-affine,asshowninFigure5.
(a)(b)(c)
Figure5.Threetypesofself-similarfractalobjects:(a)self-similargeometricobject;(b)quasi-self-
similarnaturalobject;(c)statisticallyself-affinemicrostructureofrobot-laser-hardenedspecimen.
Thetwo-dimensionalrepresentationoftheRLHspecimen’smicrostructureisquite
complicatedanddefinitelynotperfectlyalikebutcanstillbeexaminedstatisticallyusing
thementionedapproach.CanonicalEuclideangeometrycannotbeappliedtothisprob-
lem.Thus,fractalgeometrywasselectedforthispurpose.Wedevelopedanewmethod
toestimatefractalpropertiesofthenetwork,anditispresentedbelow.
First,weconvertSEMimagesintobinaryimages(blackandwhitesquares).Then,
weuseonlyblacksquares,i.e.,weremovethewhiteones.Theblacksquarespresentthe
nodesofthenetwork.Weconvertalltheseblacknodesintoapolarcoordinatesystem
Figure 4. Self-similar and self-affine transformations.
It may be emphasized that the self-similarity space is a subset of self-affinity. Self-
affine fractals—including self-similar geometry—can be divided into exactly self-affine,
quasi-self-affine and statistically self-affine, as shown in Figure 5.
FractalFract.2023,7,xFORPEERREVIEW5of13
Figure3.Exampleofgraph/network.
Furthermore,weapplyfractalgeometrytodescribethecomplexityoftheinvestigated
geometry.Dimensionsofself-similarfractalsarethesameateachenlargement,whichisnot
anissueforstatisticalunmixed-relativefractals[10].Figure4depictsself-similarandself-af-
finetransformations.Self-affinetransformationsarealsotiltingtransformations.
Figure4.Self-similarandself-affinetransformations.
Itmaybeemphasizedthattheself-similarityspaceisasubsetofself-affinity.Self-
affinefractals—includingself-similargeometry—canbedividedintoexactlyself-affine,
quasi-self-affineandstatisticallyself-affine,asshowninFigure5.
(a)(b)(c)
Figure5.Threetypesofself-similarfractalobjects:(a)self-similargeometricobject;(b)quasi-self-
similarnaturalobject;(c)statisticallyself-affinemicrostructureofrobot-laser-hardenedspecimen.
Thetwo-dimensionalrepresentationoftheRLHspecimen’smicrostructureisquite
complicatedanddefinitelynotperfectlyalikebutcanstillbeexaminedstatisticallyusing
thementionedapproach.CanonicalEuclideangeometrycannotbeappliedtothisprob-
lem.Thus,fractalgeometrywasselectedforthispurpose.Wedevelopedanewmethod
toestimatefractalpropertiesofthenetwork,anditispresentedbelow.
First,weconvertSEMimagesintobinaryimages(blackandwhitesquares).Then,
weuseonlyblacksquares,i.e.,weremovethewhiteones.Theblacksquarespresentthe
nodesofthenetwork.Weconvertalltheseblacknodesintoapolarcoordinatesystem
Figure 5.
Three types of self-similar fractal objects: (
a
) self-similar geometric object; (
b
) quasi-self-
similar natural object; (c) statistically self-affine microstructure of robot-laser-hardened specimen.
The two-dimensional representation of the RLH specimen’s microstructure is quite
complicated and definitely not perfectly alike but can still be examined statistically using
the mentioned approach. Canonical Euclidean geometry cannot be applied to this problem.
Thus, fractal geometry was selected for this purpose. We developed a new method to
estimate fractal properties of the network, and it is presented below.
First, we convert SEM images into binary images (black and white squares). Then,
we use only black squares, i.e., we remove the white ones. The black squares present the
nodes of the network. We convert all these black nodes into a polar coordinate system
Fractal Fract. 2023,7, 710 6 of 13
with coordinates (r,
ϕr
). For each line
ϕi
, we check how many of the nodes lie on it (n). So,
ϕi= n.
We determine all pairs (
ϕi
,n), which results in a linear graph
Γ
(
ϕ
,n). In addition,
the linear graph
Γ
(
ϕ
,n) includes nodes T1(
ϕ
1, n1), T2(
ϕ
2, n2)
. . .
Tk(
ϕ
k, nk). For this graph,
we estimate the Hurst exponent H [
11
]. After this, we can calculate the fractal dimension
using the equation D = 2
−
H. The Hurst exponent has a value in the interval H
∈
(0, 1);
thus, the fractal dimension of complex networks is D
∈
(1, 2). Figure 6shows the procedure
of calculating the fractal dimension of the network.
FractalFract.2023,7,xFORPEERREVIEW6of13
withcoordinates(r,
r).Foreachline
i,wecheckhowmanyofthenodeslieonit(n).So,
i=n.Wedetermineallpairs(
i,n),whichresultsinalineargraph(
,n).Inaddition,
thelineargraph(
,n)includesnodesT1(
1,n1),T2(
2,n2)…Tk(
k,nk).Forthisgraph,
weestimatetheHurstexponentH[11].Afterthis,wecancalculatethefractaldimension
usingtheequationD=2−H.TheHurstexponenthasavalueintheintervalH∈(0,1);
thus,thefractaldimensionofcomplexnetworksisD∈(1,2).Figure6showstheprocedure
ofcalculatingthefractaldimensionofthenetwork.
Figure6.Procedureofcalculatingfractaldimensionofthenetwork.
2.3.DataMining
Weusemethodsofintelligentsystems[12]topredictthecomplexityofthemicro-
structuresofrobot-laser-hardenedspecimens—weuseCNN,GPandMR.
Geneticprogramming(GP)[11,13]isanapproachbasedonthecreativepowerinherent
innaturalevolutionfortheautomaticdevelopmentofcomputerprograms.Ittriestoimitate
mechanismsofnaturalevolutioninordertoautomaticallygenerateprogramsthatsolvea
givenproblem.GPhasbeenusedsuccessfullyinanumberofapplicationsforbothsolving
purelymathematicalproblemsandsolvingreal-worldproblems.Weusedthefollowingat-
tributesoftheGP:sizeofthepopulationoforganisms:600;maximumnumberofgenerations:
100;reproductionprobability:0.5;crossoverprobability:0.6;maximumpermissibledepthin
Figure 6. Procedure of calculating fractal dimension of the network.
2.3. Data Mining
We use methods of intelligent systems [
12
] to predict the complexity of the microstruc-
tures of robot-laser-hardened specimens—we use CNN, GP and MR.
Genetic programming (GP) [
11
,
13
] is an approach based on the creative power inherent
in natural evolution for the automatic development of computer programs. It tries to imitate
mechanisms of natural evolution in order to automatically generate programs that solve
a given problem. GP has been used successfully in a number of applications for both
solving purely mathematical problems and solving real-world problems. We used the
following attributes of the GP: size of the population of organisms: 600; maximum number
of generations: 100; reproduction probability: 0.5; crossover probability: 0.6; maximum
permissible depth in the creation of the population: 8; maximum permissible depth after the
operation of the crossover of two organisms: 10; smallest permissible depth of organisms
Fractal Fract. 2023,7, 710 7 of 13
in generating new organisms: 4; and tournament size used for the selection of organisms:
7. Parameters are given in Table 1.
Table 1. Attributes of the GP.
Attributes #
size of the population of organisms 600
maximum number of generations 100
reproduction probability 0.5
crossover probability 0.6
maximum permissible depth in the creation of the population 8
maximum permissible depth after the operation of crossover of two organisms 10
smallest permissible depth of organisms in generating new organisms 4
tournament size used for selection of organisms 7
Deep learning neural networks have a particular subset known as convolutional
neural networks, sometimes known as CNNs or ConvNets [
14
]. The inputs to a CNN are
often pictures, and it uses one or more convolutional layers that typically conduct a 2D
convolution. CNNs have gained a lot of popularity recently and have produced some
truly remarkable image recognition results that have had a significant impact on the field
of computer vision. John von Neumann first suggested the concept of cellular automata
(CAs) [
15
]. Although it is evident that the speed and temperature of the robot laser have a
significant impact on the fractal dimension of the network of the microstructure of RLH
specimens, we were curious as to whether this prediction could be strengthened by looking
at specimen photos. Our objective was to determine whether the fractal dimension of
the network of the microstructure of RLH specimens can be predicted from the specimen
photos, and if so, whether this prediction is much more accurate than the one that relies
just on the speed and temperature. We applied two settings:
1.
To forecast the fractal dimension of the network of the microstructure of RLH speci-
mens without photos, only non-image input parameters were used.
2.
In addition to other input characteristics, images were also employed to forecast the
fractality of the network of the microstructure of RLH specimens.
In the first instance, the network fractality of RLH specimens’ microstructures was
predicted using two inputs—temperature and speed—based on a standard multilayer
perceptron. In the second instance, as seen in Figure 7, a convolutional neural network
(CNN) was used to forecast the network fractality of the microstructure of RLH specimens.
FractalFract.2023,7,xFORPEERREVIEW7of13
thecreationofthepopulation:8;maximumpermissibledepthaftertheoperationofthecross-
overoftwoorganisms:10;smallestpermissibledepthoforganismsingeneratingneworgan-
isms:4;andtournamentsizeusedfortheselectionoforganisms:7.Parametersaregivenin
Tabl e1.
Tab l e1.AttributesoftheGP.
Attributes#
sizeofthepopulationoforganisms600
maximumnumberofgenerations100
reproductionprobability0.5
crossoverprobability0.6
maximumpermissibledepthinthecreationofthepopulation8
maximumpermissibledepthaftertheoperationofcrossoveroftwoorganisms10
smallestpermissibledepthoforganismsingeneratingneworganisms4
tournamentsizeusedforselectionoforganisms7
Deeplearningneuralnetworkshaveaparticularsubsetknownasconvolutionalneural
networks,sometimesknownasCNNsorConvNets[14].TheinputstoaCNNareoftenpic-
tures,anditusesoneormoreconvolutionallayersthattypicallyconducta2Dconvolution.
CNNshavegainedalotofpopularityrecentlyandhaveproducedsometrulyremarkable
imagerecognitionresultsthathavehadasignificantimpactonthefieldofcomputervision.
JohnvonNeumannfirstsuggestedtheconceptofcellularautomata(CAs)[15].Althoughitis
evidentthatthespeedandtemperatureoftherobotlaserhaveasignificantimpactonthe
fractaldimensionofthenetworkofthemicrostructureofRLHspecimens,wewerecuriousas
towhetherthispredictioncouldbestrengthenedbylookingatspecimenphotos.Ourobjective
wastodeterminewhetherthefractaldimensionofthenetworkofthemicrostructureofRLH
specimenscanbepredictedfromthespecimenphotos,andifso,whetherthispredictionis
muchmoreaccuratethantheonethatreliesjustonthespeedandtemperature.Weapplied
twosettings:
1. ToforecastthefractaldimensionofthenetworkofthemicrostructureofRLHspeci-
menswithoutphotos,onlynon-imageinputparameterswereused.
2. Inadditiontootherinputcharacteristics,imageswerealsoemployedtoforecastthe
fractalityofthenetworkofthemicrostructureofRLHspecimens.
Inthefirstinstance,thenetworkfractalityofRLHspecimens’microstructureswas
predictedusingtwoinputs—temperatureandspeed—basedonastandardmultilayer
perceptron.Inthesecondinstance,asseeninFigure7,aconvolutionalneuralnetwork
(CNN)wasusedtoforecastthenetworkfractalityofthemicrostructureofRLHspeci-
mens.
Figure7.Convolutionalneuralnetwork.Inthefirstsetting(withoutimages),theglobalaveragepooling
layerisnotconnectedtofullyconnectedlayers,whileinthesecond(withimages),itisconnected.
Figure 7.
Convolutional neural network. In the first setting (without images), the global average
pooling layer is not connected to fully connected layers, while in the second (with images), it
is connected.
3. Results, Discussion, GP and CNN Approach
In Table 2, the parameters and the network fractality of RLH specimens are presented.
We mark specimens from S1 to S22. The parameter X1 represents the temperature [
◦
C], X2
Fractal Fract. 2023,7, 710 8 of 13
is the speed of hardening (mm/s), X3 is the angle (
◦
) of hardening, X4 is the number of
nodes of the fractal network and Y represents the network fractality of the RLH specimens.
S1–S4 are the specimens hardened using the process presented in Figure 1. Specimens
S5–S8 represent the material next to the hardened zone. S9–S11 represent the results
of the robot laser hardening under the following conditions: the variation in the angle
ϕ∈(45, 60, 75)◦
between the left side of the laser beam and the material surface. S12–S14
represent the results of the robot laser hardening under these conditions: the variation
in the incidence angle
ϕ∈
(45, 60, 75)
◦
between the right side of the laser beam and the
material surface. S15–S17 represent the results of the robot laser hardening under another
set of conditions: the angle was changed in the plane defined by the direction of the laser
beam
ϕ∈
[45
◦
, 60
◦
, 75
◦
]. S18–S20 represent the results of robot laser hardening under
the final set of variable conditions: the angle was changed in the plane perpendicular to
the direction of the laser beam
ϕ∈
[105
◦
, 120
◦
, 135
◦
]. S21 represents the results of the
point-robot-laser-hardened specimens. Finally, specimen S22 represents the material before
using the process of hardening. Table 3gives the measured and predicted network fractality
of RLH specimens, while Figure 8gives the predicted network fractality of RLH specimens
by using GP and CNN.
Table 2. Parameters of RLH specimens.
S X1 X2 X3 X4 Y
S1 1000 2 90 120,515 1.615
S2 1000 3 90 125,579 1.623
S3 1000 4 90 123,695 1.618
S4 1000 5 90 124,585 1.655
S5 1400 2 90 124,378 1.589
S6 1400 3 90 132,292 1.682
S7 1400 4 90 126,714 1.631
S8 1400 5 90 130,751 1.625
S9 1000 2 45 123,345 1.612
S10 1000 3 60 126,147 1.526
S11 1000 4 75 124,048 1.715
S12 1000 2 45 123,581 1.608
S13 1000 3 60 127,895 1.539
S14 1000 4 75 122,152 1.533
S15 1000 2 45 120,370 1.713
S16 1000 3 60 116,564 1.543
S17 1000 4 75 132,783 1.519
S18 1400 0 105 130,226 1.663
S19 1000 0 120 130,795 1.641
S20 950 0 135 106,468 1.585
S21 850 0 90 120,565 1.559
S22 0 0 0 94,842 1.374
The laser heat treatment technique for robots is straightforward, but this study reveals
that the evolution of the microstructure during the laser-hardening process is an important
aspect that needs to be addressed. It shows that the changing laser beam angles affect the
intricate networks that make up the microstructure of specimens that were laser-hardened
by means of robots. The proposed technique allows identifying the complicated network
Fractal Fract. 2023,7, 710 9 of 13
structure discovered in the microstructure of laser-hardened specimens at various angles.
The maximum complexity specification is obtained for specimen S11: 1.715. This specimen
was hardened at a rate of 4 mm/s at 1000
◦
C with the variation in the incidence angle
ϕ∈
75
◦
between the left side of the laser beam and the material surface. The minimal
complexity was observed for the specimen S22, which represents the material before
the robot-laser-hardening process. Hence, the complexity of the microstructure of the
robot-laser-hardened specimens indicates how the heat treatment of a material alters its
microstructure’s geometry.
Table 3. The measured and predicted network fractality of RLH specimens.
S Y P GP P CNN
S1 1.615 1.58072 1.608
S2 1.623 1.62035 1.585
S3 1.618 1.61631 1.655
S4 1.655 1.61513 1.631
S5 1.589 1.61024 1.663
S6 1.682 1.69476 1.519
S7 1.631 1.61795 1.663
S8 1.625 1.61576 1.663
S9 1.612 1.60483 1.585
S10 1.526 1.53524 1.533
S11 1.715 1.61746 1.608
S12 1.608 1.60483 1.612
S13 1.539 1.53524 1.526
S14 1.533 1.61746 1.615
S15 1.713 1.60483 1.608
S16 1.543 1.53524 1.608
S17 1.519 1.61746 1.539
S18 1.663 1.65291 1.641
S19 1.641 1.64808 1.663
S20 1.585 1.58919 1.655
S21 1.559 1.55913 1.585
S22 1.374 1.37364 1.608
Furthermore, the network complexity of microstructural materials hardened at various
speeds, temperatures, and laser beam angles is modeled using genetic programming (GP).
The measured data and the genetic programming model data yield a high agreement of
98.4%. As a result, the complexity network parameter X3 (angle) has a greater effect on
the model. In order to create 100 different predictions based on the various parameters
employed in the robot laser cell, the internal GP system was executed 100 times. On an
I7 Intel processor (Intel Corporation, Santa Clara, CA, USA) with 8 GB of RAM, each run
lasted roughly 2.5 h. Using an internal genetic programming system written in AutoLISP
and connected with AutoCAD (v24.2), we used 100 separate models for forecasting network
fractality of RLH specimens. The mean square of errors from the monitored data was used
to measure the model’s fitness. It is given by
∆=Σ∆i2/n,
Fractal Fract. 2023,7, 710 10 of 13
where iis the square of the error of a single sample of data, and nis the size of the
monitored data.
FractalFract.2023,7,xFORPEERREVIEW9of13
S31.6181.616311.655
S41.6551.615131.631
S51.5891.610241.663
S61.6821.694761.519
S71.6311.617951.663
S81.6251.615761.663
S91.6121.604831.585
S101.5261.535241.533
S111.7151.617461.608
S121.6081.604831.612
S131.5391.535241.526
S141.5331.617461.615
S151.7131.604831.608
S161.5431.535241.608
S171.5191.617461.539
S181.6631.652911.641
S191.6411.648081.663
S201.5851.589191.655
S211.5591.559131.585
S221.3741.373641.608
Figure8.ThecalculatedandpredictednetworkfractalityofRLHspecimensbasedonGPandCNN.
Thelaserheattreatmenttechniqueforrobotsisstraightforward,butthisstudyre-
vealsthattheevolutionofthemicrostructureduringthelaser-hardeningprocessisan
importantaspectthatneedstobeaddressed.Itshowsthatthechanginglaserbeamangles
affecttheintricatenetworksthatmakeupthemicrostructureofspecimensthatwerelaser-
hardenedbymeansofrobots.Theproposedtechniqueallowsidentifyingthecomplicated
networkstructurediscoveredinthemicrostructureoflaser-hardenedspecimensatvari-
ousangles.ThemaximumcomplexityspecificationisobtainedforspecimenS11:1.715.
Thisspecimenwashardenedatarateof4mm/sat1000°Cwiththevariationintheinci-
denceangle75°betweentheleftsideofthelaserbeamandthematerialsurface.The
minimalcomplexitywasobservedforthespecimenS22,whichrepresentsthematerial
beforetherobot-laser-hardeningprocess.Hence,thecomplexityofthemicrostructureof
therobot-laser-hardenedspecimensindicateshowtheheattreatmentofamaterialalters
itsmicrostructure’sgeometry.
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1.75
0 1 2 3 4 5 6 7 8 9 10111213141516171819202122
Fractalitynetwork
Specimens
Y
PGP
PCNN
Figure 8.
The calculated and predicted network fractality of RLH specimens based on GP and CNN.
Simply put, an individual organism’s single sample data variance is
∆i = Ei −Gi
where Ei and Gi are the projected and actual scrap fractions, respectively, and they only
depend on the surface flaws.
The developed optimal model of genetic programming is given as follows:
Y=1.83992 −0.118464 ×(1.83992 + (8.50567 ×(0.454976×X1
X3−X1
(−2.19792+0.312703×X1
X3)+
2×X2+−2.19792+X3−5.28948×(−2.19792−X1
X4)
4.39584−1
X2−X2−0.059118×X3+X1
X2(X1
X3+X3)
−−2.19792+X3−5.28948×(−0.454976×X1
X3–X3
X4)
4.39584−1
X2−X2−0.059118×X3+X1
X2(X1
X3+X3)
))/
(X3
2X2−
−2.19792+X3−
0.189055×
X3−
−2.19792+X3−0.189055×X3
X4
−2.19792−0.312703×X1
X3
−2.19792−X1
X4
−4.81742−1+X2
X3+X1
X2(X3+X1
X3×X3)
))
The specimen photos were reduced to a standard of 300
×
300 pixels. There was just
one channel needed because they were given as grayscale. The values were scaled from the
[0, 255] interval to the [0, 1] interval. To reduce the image size further, a two-dimensional
max-pooling (3
×
3) layer was applied after a two-dimensional convolution layer with
a filter size of 3
×
3 and four kernels. The 2D conv-layer and max-poling combination
were used once more to obtain more compression. The third convolutional layer and an
average pooling layer, whose output was flattened so that it could be combined with the
non-image data, were then applied. Figure 9presents the microstructure of one example of
22 specimens characterized by speed treatments of 2 mm/s (S1), 3 mm/s (S2), 4 mm/s (S3),
and 5 mm/s (S4) at 1000 ◦C.
The RMSprop learning method was used to train the CNNs over 1200 iterations at a
learning rate of 0.001 error in the mean square (MSE). Both examined cases, those without
photographs and those with images, had their mean squared error (MSE) recorded. Eight
training configurations were tested for each setting, including two and three hidden layers,
two different layer sizes (i.e., the number of artificial neurons), and two different activation
modes for the neurons (sigmoid vs. relu). The output neuron’s activation function was
always sigmoid, and it was utilized to predict the surface roughness. Table 4represents
Fractal Fract. 2023,7, 710 11 of 13
the findings. The average of 10 runs, along with the standard deviation, is the outcome
for each round of training. For each parameter set (speed, temperature), two of the three
photos were utilized for training, and the third one was used for testing. Therefore, we also
have test results in the “image” setting. Because testing would produce identical results
for the training in the “no-image” mode, only the training’s findings are provided. The
bold numbers in Table 4are the best results obtained by means of CNN. The CNN model
delivers a precision of 96.5%.
FractalFract.2023,7,xFORPEERREVIEW10of13
Furthermore,thenetworkcomplexityofmicrostructuralmaterialshardenedatvari-
ousspeeds,temperatures,andlaserbeamanglesismodeledusinggeneticprogramming
(GP).Themeasureddataandthegeneticprogrammingmodeldatayieldahighagree-
mentof98.4%.Asaresult,thecomplexitynetworkparameterX3(angle)hasagreater
effectonthemodel.Inordertocreate100differentpredictionsbasedonthevariouspa-
rametersemployedintherobotlasercell,theinternalGPsystemwasexecuted100times.
OnanI7Intelprocessor(IntelCorporation,SantaClara,CA,USA)with8GBofRAM,
eachrunlastedroughly2.5h.Usinganinternalgeneticprogrammingsystemwrittenin
AutoLISPandconnectedwithAutoCAD(v24.2),weused100separatemodelsforfore-
castingnetworkfractalityofRLHspecimens.Themeansquareoferrorsfromthemoni-
toreddatawasusedtomeasurethemodel’sfitness.Itisgivenby
Δ=ΣΔi2/n,
whereiisthesquareoftheerrorofasinglesampleofdata,andnisthesizeofthemoni-
toreddata.
Simplyput,anindividualorganism’ssinglesampledatavarianceis
Δi=Ei−Gi
whereEiandGiaretheprojectedandactualscrapfractions,respectively,andtheyonly
dependonthesurfaceflaws.
Thedevelopedoptimalmodelofgeneticprogrammingisgivenasfollows:
Y=1.83992−0.118464×(1.83992+(8.50567×(.
. .
2𝑋2. ..
.
.
. ..
–
.
.
/
𝑋3
⎝
⎜
⎜
⎛
2𝑋2.
...
..
.
.
⎠
⎟
⎟
⎞
Thespecimenphotoswerereducedtoastandardof300×300pixels.Therewasjust
onechannelneededbecausetheyweregivenasgrayscale.Thevalueswerescaledfrom
the[0,255]intervaltothe[0,1]interval.Toreducetheimagesizefurther,atwo-dimen-
sionalmax-pooling(3×3)layerwasappliedafteratwo-dimensionalconvolutionlayer
withafiltersizeof3×3andfourkernels.The2Dconv-layerandmax-polingcombination
wereusedoncemoretoobtainmorecompression.Thethirdconvolutionallayerandan
averagepoolinglayer,whoseoutputwasflattenedsothatitcouldbecombinedwiththe
non-imagedata,werethenapplied.Figure9presentsthemicrostructureofoneexample
of22specimenscharacterizedbyspeedtreatmentsof2mm/s(S1),3mm/s(S2),4mm/s
(S3),and5mm/s(S4)at1000°C.
Figure 9.
Microstructure of specimens characterized by a speed treatment of (from
left
to
right
)
2 mm/s (S1), 3 mm/s (S2), 4 mm/s (S3), and 5 mm/s (S4) at 1000 ◦C.
Table 4.
Roughness prediction errors (the mean squared error, MSE) in both settings: without and
with images.
Number of
Hidden Layers
Size of
Layers
Activation
Function
Without Images With Images
Train (=Test) Train Test
Mean Std Mean Std Mean Std
2 hidden layers 20, 10 sigmoid 0.022 ±0.001 0.021 ±0.002 0.022 ±0.002
relu 0.024 ±0.002 0.018 ±0.003 0.017 ±0.003
10, 5 sigmoid 0.027 ±0.001 0.028 ±0.004 0.028 ±0.004
relu 0.024 ±0.003 0.021 ±0.004 0.022 ±0.004
3 hidden layers 20, 20, 10 sigmoid 0.022 ±0.001 0.019 ±0.002 0.021 ±0.002
relu 0.021 ±0.002 0.015 ±0.001 0.015 ±0.001
10, 10, 5 sigmoid 0.026 ±0.001 0.026 ±0.003 0.027 ±0.002
relu 0.025 ±0.004 0.021 ±0.005 0.021 ±0.004
4. Conclusions
With the aim of analyzing the invariance property of the network fractality of real
microstructure images (and fractography), it was necessary to develop a new method for
determining the object complexity that can be seen in the microstructures of RLH specimens.
In order to predict the complexity of robot-laser-hardened specimens, we applied an AI
method, namely genetic programming. The genetic programming delivered an accuracy of
above 98%.
The highest complexity specification was found for the specimen S11: 1.715. This
specimen was hardened at a rate of 4 mm/s, at 1000
◦
C, with the variation in the incidence
angle of
ϕ∈
75
◦
between the left side of the laser beam and the material surface. The
minimal network fractality was found in the specimen before the RLH process. Finally,
because the microstructure of RLH specimens is very chaotic and complex, it is clear
that for the use of fractal-based analysis for the evaluation of the microstructure of RLH
materials, careful calibrations are required to develop a technological tool for real-life
application. There are a number of aspects that remain to be investigated to obtain a
complete understanding of the fractal behavior of microstructures of RLH specimens using
images and in order to implement its application for the evaluation of materials.
Furthermore, we successfully applied a convolutional neural network (CNN) to predict
the surface roughness also using the photos of the materials’ surfaces. The mean squared
error was in the range from 0.021
±
0.002 to 0.015
±
0.001 when images were also used. A
t-test was applied to compare the means, yielding t= 0.023/0.017 = 1.35 at 16 degrees of
Fractal Fract. 2023,7, 710 12 of 13
freedom. This implies that the p-value is equal to 0.18. Hence, the prediction using images
noticeably improved compared to the case when images were not used. The CNN model
delivered a precision of 96.5%.
Making sprockets extremely hard throughout the hardening process is a very signif-
icant issue. In this study, we described a novel technique for hardening materials under
various laser beam angles using a robot laser cell to address the issue of sprocket hard-
ening (the idea of two laser-beam RLH). We split the laser beam into two parts using a
prism. In the future, we intend to investigate the effects of different laser beam angles and
characteristics on the topography of various materials using two laser beams for RLH.
An important advantage of the proposed method is its applicability in a wide range
of fields such as in medicine, in the analysis of biochemical interactions, in the analysis of
biomedical images in technical science, in the analysis of the influence of process parameters
on a material’s mechanical properties, and even in social sciences. Of course, until more
knowledge is acquired about its application in various fields, the method is to be used with
special care.
Author Contributions:
Conceptualization, M.B. and D.M.; methodology, M.B.; software, M.B.; val-
idation, M.B. and D.M.; formal analysis, M.B. and D.M.; investigation, M.B. and D.M.; resources,
M.B. and D.M.; data curation, M.B. and D.M.; writing—original draft preparation, M.B. and D.M.;
writing—review and editing, M.B. and D.M.; visualization, M.B. and D.M.; supervision, M.B. and
D.M.; project administration, M.B. and D.M.; funding acquisition, M.B. and D.M. All authors have
read and agreed to the published version of the manuscript.
Funding:
We acknowledge support by the German Research Foundation and the Open Access
Publication Fund of TU Berlin.
Data Availability Statement: Not applicable.
Acknowledgments:
We acknowledge support by the German Research Foundation and the Open
Access Publication Fund of TU Berlin.
Conflicts of Interest: The authors declare no conflict of interest.
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