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Fractal Fract
Special Issues
The Materials Structure and Fractal Nature
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The Materials Structure and Fractal Nature

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 14297

Special Issue Editors

Prof. Dr. Cristina Serpa

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Guest Editor
ISEL - Instituto Superior de Engenharia de Lisboa, CMAFcIO - Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal
Interests: fractals; dynamical systems; functional analysis; fractal regression and Hausdorff dimension
Prof. Dr. Hans-Jörg Fecht

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Guest Editor
Institute of Functional Nanosystems, Ulm University, Albert-Einstein-Allee 47, 89081 Ulm, Germany
Interests: advanced nanomaterials; nanotechnology; interface engineering
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Branislav Randjelovic

E-Mail Website
Guest Editor
1. Faculty of Electronic Engineering, University of Niš, Niš 18106, Serbia
2. Faculty of Teachers’ Education, University of Priština Kosovska Mitrovica, Leposavić 38218, Serbia
Interests: graph theory; fractals; neural networks; ceramics; numerical mathematics; applied algebra
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Vojislav V. Mitic †
*
E-Mail Website
Guest Editor
1. Faculty of Electronic Engineering, University of Nis, 18000 Nis, Serbia
2. Institute of Technical Sciences, Serbian Academy of Sciences and Arts, University of Belgrade, 11000 Belgrade, Serbia
Interests: electronic ceramics; microstructure; neural networks; graphs and fractals; sintering
* We dedicate the memory of the editor, Prof. Vojislav Mitic, who passed away during this special issue period.

Special Issue Information

Dear Colleagues,

The material sciences, with fractal nature analysis, open a new frontier within total matter.

Chaos defines a phenomenon that is in apparent disorder. Its complex behavior can be difficult to understand. It may appear everywhere and with different mathematical structure laws. The high sensitivity to initial conditions brings complexity to an analytical perspective. Fractals are a form of chaos, where order is ruled by self-similarity, i.e., where parts of the whole are repeated in a smaller scale in some portions of the total structure. A pure fractal is characterized by a chain of an infinite number of levels of self-similarity. In applied studies, when an object or a phenomenon present a complex structure, one way to analyze it is to work with fractals, by doing a fractal reconstruction (through fractal regression or interpolation), or by estimating its Hausdorff dimension, through geometric or numerical methods. Fractal coefficients (or contraction factors, either vertical factors) and the Hausdorff dimension are key indicators of the fractal characterization of real data. Bigger coefficients mean higher fractal structure oscillations. Fractalization may be understood as a process of approximating a given data set to a fractal function. Fractal regression is a method that includes a finite chain of fractal levels, and the estimates of fractal coefficients are obtained numerically.

One special case of fractals is so-called random fractals. Several approaches to the study of Brownian random functions are possible, such as fractional calculus. In this context, since the behavior of Brownian motion is intrinsically irregular and does not suit a traditional differential perspective, it is conceivable to establish relations between fractional calculus and fractals in this topic. This is one fact that may foresee some relationship between fractional calculus and fractals.

Both fractals and fractional mathematics integrate and open up new insights that can help toward a more complete understanding of the total nature of matter, including biophysical and technical sciences systems in the frame of overall reality.

Prof. Dr. Cristina Serpa
Prof. Dr. Hans-Jörg Fecht
Prof. Dr. Branislav Randjelovic
Prof. Dr. Vojislav V. Mitic
Guest Editors

During work on this Special Issue, Prof. Vojislav Mitic (1955–2021) lost his fight against COVID.

He was president of the Serbian Ceramic Society, and member of the European Academy of Sciences and Arts, World Ceramic Academy, Pan-European Union Serbia, Honorary Senate of Europe, International Ceramic Federation, Japanese Society for Materials Testing, Australian Ceramic Society and IEEE. He was a participant in many scientific research projects in the field of consolidation of BaTiO3 ceramics, and a member of the organizing and scientific committees of several scientific conferences. He published over 500 scientific papers and the monograph "Structure and electrical properties of BaTiO3-ceramics". From 1989 to 1992, he served as the president of the Executive Council of the city of Nis. From 1997 to 2006, he was the Chairman of the Board of Directors of EI Corporation. He leaves remarkable and significant contributuons to science, education, politics, the economy, and international relations and in his local community.

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Fractal
  • Fractal reconstruction
  • Chaos
  • Hausdorff dimension
  • Fractalization
  • Fractional

Published Papers (7 papers)

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Research

23 pages, 12535 KiB  
Article
Lateral Fractal Formation by Crystallographic Silicon Micromachining
by Lucas Johannes Kooijman, Yasser Pordeli, Johan Willem Berenschot and Niels Roelof Tas
Fractal Fract. 2023, 7(2), 202; https://doi.org/10.3390/fractalfract7020202 - 18 Feb 2023
Viewed by 1109
Abstract
A novel wafer-scale silicon fractal fabrication method is presented here for forming pyramids only in the lateral direction using the crystal orientation of silicon. Fractals are fabricated in silicon by masking only the corners (corner lithography) of a cavity in silicon with silicon [...] Read more.
A novel wafer-scale silicon fractal fabrication method is presented here for forming pyramids only in the lateral direction using the crystal orientation of silicon. Fractals are fabricated in silicon by masking only the corners (corner lithography) of a cavity in silicon with silicon nitride, where the shape is determined by the crystal {111} planes of the silicon. The octahedral cavity shaped by the {111} planes was previously only used for forming octahedral fractals in all directions, but by using a planar silicon dioxide hard-mask on a silicon (100) wafer, the silicon octahedral cavity is “cut in half”. This creates a pyramid with sharper edges and vertices at its base than those determined by just the {111} planes. This allows selective corner lithography patterning at the vertices of the base while leaving the apex unpatterned, leading to lateral growing of pyramidal fractals. This selective patterning is shown mathematically and then demonstrated by creating a fractal of four generations, with the initial pyramid being 8 µm and the two final generations being of submicron size. Full article
(This article belongs to the Special Issue The Materials Structure and Fractal Nature)
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12 pages, 3637 KiB  
Article
Microstructure of Epoxy-Based Composites: Fractal Nature Analysis
by Ivana Stajcic, Aleksandar Stajcic, Cristina Serpa, Dana Vasiljevic-Radovic, Branislav Randjelovic, Vesna Radojevic and Hans Fecht
Fractal Fract. 2022, 6(12), 741; https://doi.org/10.3390/fractalfract6120741 - 15 Dec 2022
Cited by 2 | Viewed by 1668
Abstract
Polymers and polymer matrix composites are commonly used materials with applications extending from packaging materials to delicate electronic devices. Epoxy resins and fiber-reinforced epoxy-based composites have been used as adhesives and construction parts. Fractal analysis has been recognized in materials science as a [...] Read more.
Polymers and polymer matrix composites are commonly used materials with applications extending from packaging materials to delicate electronic devices. Epoxy resins and fiber-reinforced epoxy-based composites have been used as adhesives and construction parts. Fractal analysis has been recognized in materials science as a valuable tool for the microstructural characterization of composites by connecting fractal characteristics with composites’ functional properties. In this study, fractal reconstructions of different microstructural shapes in an epoxy-based composite were performed on field emission scanning electron microscopy (FESEM) images. These images were of glass fiber reinforced epoxy as well as a hybrid composite containing both glass and electrospun polystyrene fibers in an epoxy matrix. Fractal reconstruction enables the identification of self-similarity in the fractal structure, which represents a novelty in analyzing the fractal properties of materials. Fractal Real Finder software, based on the mathematical affine fractal regression model, was employed to reconstruct different microstructure shapes and calculate fractal dimensions to develop a method of predicting the optimal structure–property relations in composite materials in the future. Full article
(This article belongs to the Special Issue The Materials Structure and Fractal Nature)
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15 pages, 4330 KiB  
Article
Fractal Geometry and Convolutional Neural Networks for the Characterization of Thermal Shock Resistances of Ultra-High Temperature Ceramics
by Shanxiang Wang, Zailiang Chen, Fei Qi, Chenghai Xu, Chunju Wang, Tao Chen and Hao Guo
Fractal Fract. 2022, 6(10), 605; https://doi.org/10.3390/fractalfract6100605 - 17 Oct 2022
Cited by 1 | Viewed by 1361
Abstract
The accurate characterization of the surface microstructure of ultra-high temperature ceramics after thermal shocks is of great practical significance for evaluating their thermal resistance properties. In this paper, a fractal reconstruction method for the surface image of Ultra-high temperature ceramics after repeated thermal [...] Read more.
The accurate characterization of the surface microstructure of ultra-high temperature ceramics after thermal shocks is of great practical significance for evaluating their thermal resistance properties. In this paper, a fractal reconstruction method for the surface image of Ultra-high temperature ceramics after repeated thermal shocks is proposed. The nonlinearity and spatial distribution characteristics of the oxidized surfaces of ceramics were extracted. A fractal convolutional neural network model based on deep learning was established to realize automatic recognition of the classification of thermal shock cycles of ultra-high temperature ceramics, obtaining a recognition accuracy of 93.74%. It provides a novel quantitative method for evaluating the surface character of ultra-high temperature ceramics, which contributes to understanding the influence of oxidation after thermal shocks. Full article
(This article belongs to the Special Issue The Materials Structure and Fractal Nature)
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13 pages, 4115 KiB  
Article
Characterization of Ceramic Thermal Shock Cracks Based on the Multifractal Spectrum
by Changxu Shao, Hao Guo, Songhe Meng, Yingfeng Shao, Shanxiang Wang, Shangjian Xie and Fei Qi
Fractal Fract. 2022, 6(10), 539; https://doi.org/10.3390/fractalfract6100539 - 24 Sep 2022
Cited by 2 | Viewed by 1467
Abstract
Ceramics are commonly used as high-temperature structural materials which are easy to fracture because of the propagation of thermal shock cracks. Characterizing and controlling crack propagation are significant for the improvement of the thermal shock resistance of ceramics. However, observing crack morphology, based [...] Read more.
Ceramics are commonly used as high-temperature structural materials which are easy to fracture because of the propagation of thermal shock cracks. Characterizing and controlling crack propagation are significant for the improvement of the thermal shock resistance of ceramics. However, observing crack morphology, based on macro and SEM images, costs much time and potentially includes subjective factors. In addition, complex cracks cannot be counted and will be simplified or omitted. Fractals are suitable to describe complex and inhomogeneous structures, and the multifractal spectrum describes this complexity and heterogeneity in more detail. This paper proposes a crack characterization method based on the multifractal spectrum. After thermal shocks, the multifractal spectrum of alumina ceramics was obtained, and the crack fractal features were extracted. Then, a deep learning method was employed to extract features and automatically classify ceramic crack materials with different strengths, with a recognition accuracy of 87.5%. Full article
(This article belongs to the Special Issue The Materials Structure and Fractal Nature)
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11 pages, 2890 KiB  
Article
Fractal Nature Bridge between Neural Networks and Graph Theory Approach within Material Structure Characterization
by Branislav M. Randjelovic, Vojislav V. Mitic, Srdjan Ribar, Dusan M. Milosevic, Goran Lazovic, Hans J. Fecht and Branislav Vlahovic
Fractal Fract. 2022, 6(3), 134; https://doi.org/10.3390/fractalfract6030134 - 28 Feb 2022
Viewed by 2453
Abstract
Many recently published research papers examine the representation of nanostructures and biomimetic materials, especially using mathematical methods. For this purpose, it is important that the mathematical method is simple and powerful. Theory of fractals, artificial neural networks and graph theory are most commonly [...] Read more.
Many recently published research papers examine the representation of nanostructures and biomimetic materials, especially using mathematical methods. For this purpose, it is important that the mathematical method is simple and powerful. Theory of fractals, artificial neural networks and graph theory are most commonly used in such papers. These methods are useful tools for applying mathematics in nanostructures, especially given the diversity of the methods, as well as their compatibility and complementarity. The purpose of this paper is to provide an overview of existing results in the field of electrochemical and magnetic nanostructures parameter modeling by applying the three methods that are “easy to use”: theory of fractals, artificial neural networks and graph theory. We also give some new conclusions about applicability, advantages and disadvantages in various different circumstances. Full article
(This article belongs to the Special Issue The Materials Structure and Fractal Nature)
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10 pages, 2738 KiB  
Article
Complexity-Based Analysis of the Effect of Forming Parameters on the Surface Finish of Workpiece in Single Point Incremental Forming (SPIF)
by Ali Akhavan Farid, Shin Shen Foong, Ondrej Krejcar and Hamidreza Namazi
Fractal Fract. 2021, 5(4), 241; https://doi.org/10.3390/fractalfract5040241 - 24 Nov 2021
Cited by 9 | Viewed by 1591
Abstract
Nowadays, the manufacturing industry is focused on newer modern manufacturing methods, such as single point incremental forming (SPIF). The popularity of the SPIF process in the manufacturing industry is increasing due to its capability for rapid prototyping, forming complex geometry with simple steps, [...] Read more.
Nowadays, the manufacturing industry is focused on newer modern manufacturing methods, such as single point incremental forming (SPIF). The popularity of the SPIF process in the manufacturing industry is increasing due to its capability for rapid prototyping, forming complex geometry with simple steps, and customizing products for customers. This study investigates the effect of forming parameters (feed rate and step size) on the surface structure of the aluminum AA6061 sheet. We employ fractal theory to investigate the complexity of deformed surfaces. Accordingly, we study the relationship between the complexity and roughness of the deformed surface. The results show that the complexity and roughness of the deformed surface vary due to the changes in forming parameters. Fractal analysis can be further employed in other manufacturing processes to investigate the relation between the complexity and roughness of processed surfaces. Full article
(This article belongs to the Special Issue The Materials Structure and Fractal Nature)
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25 pages, 460 KiB  
Article
A q-Gradient Descent Algorithm with Quasi-Fejér Convergence for Unconstrained Optimization Problems
by Shashi Kant Mishra, Predrag Rajković, Mohammad Esmael Samei, Suvra Kanti Chakraborty, Bhagwat Ram and Mohammed K. A. Kaabar
Fractal Fract. 2021, 5(3), 110; https://doi.org/10.3390/fractalfract5030110 - 03 Sep 2021
Cited by 11 | Viewed by 2396
Abstract
We present an algorithm for solving unconstrained optimization problems based on the q-gradient vector. The main idea used in the algorithm construction is the approximation of the classical gradient by a q-gradient vector. For a convex objective function, the quasi-Fejér convergence [...] Read more.
We present an algorithm for solving unconstrained optimization problems based on the q-gradient vector. The main idea used in the algorithm construction is the approximation of the classical gradient by a q-gradient vector. For a convex objective function, the quasi-Fejér convergence of the algorithm is proved. The proposed method does not require the boundedness assumption on any level set. Further, numerical experiments are reported to show the performance of the proposed method. Full article
(This article belongs to the Special Issue The Materials Structure and Fractal Nature)
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