Relevant bibliographies by topics / Fractal Koch geometry

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Fractal Koch geometry'

Author: Grafiati
Published: 4 June 2025
Last updated: 15 July 2025

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Journal articles on the topic "Fractal Koch geometry"

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Camp, Dane R. "A Fractal Excursion." Mathematics Teacher 84, no. 4 (1991): 265–75. http://dx.doi.org/10.5951/mt.84.4.0265.

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Recently, chaos theory and the related topic of fractal geometry have blossomed as creative fields of study in mathematics and physics. Fractals are shapes containing self-similarity on arbitrary magnification. One such object, the Koch curve, is generated by simple recursion on an equilateral triangle. The process used to produce the curve is a great way to introduce students to some concepts of fractal geometry.
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Jamil, Atif, Muhammad Rauf, Abdul Sami, Arsalan Ansari, and Muhammad Dawood Idrees. "A Wideband Hybrid Fractal Ring Antenna for WLAN Applications." International Journal of Antennas and Propagation 2022 (February 21, 2022): 1–8. http://dx.doi.org/10.1155/2022/6136916.

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We propose the design of a novel fractal antenna that is both unique and performance-driven. Two important antenna design features, miniaturization and wideband operation, are combined in this work. A ring-shaped antenna is designed using the well-known fractal geometry. This hybrid geometry is a fusion of meander and Koch curve shapes. The geometrical construction of the proposed antenna is compared to the standard Koch curve geometry. It is shown that combining the meander and Koch curve shapes increases the effective electrical length. The wider bandwidth is achieved by bringing the higher modes together. The overall dimensions of proposed meander Koch curve fractal ring antenna are 45 × 25 × 1.6 mm3. The resonance frequency of the antenna is between 4.94 and 6.12 GHz (% BW = 21.83), which covers the entire 5 GHz WLAN band. The prototype has been fabricated and experimentally verified.
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Jaffri, Zain Ul Abidin, Zeeshan Ahmad, Asif Kabir, and Syed Sabahat Hussain Bukhari. "A novel miniaturized Koch-Minkowski hybrid fractal antenna." Microelectronics International 39, no. 1 (2021): 22–37. http://dx.doi.org/10.1108/mi-07-2021-0069.

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Purpose Antenna miniaturization, multiband operation and wider operational bandwidth are vital to achieve optimal design for modern wireless communication devices. Using fractal geometries is recognized as one of the most promising solutions to attain these characteristics. The purpose of this paper is to present a unique structure of patch antenna using hybrid fractal technique to enhance the performance characteristics for various wireless applications and to achieve better miniaturization. Design/methodology/approach In this paper, the authors propose a novel hybrid fractal antenna by combining Koch and Minkowski (K-M) fractal geometries. A microstrip patch antenna (MPA) operating at 1.8 GHz is incorporated with a novel K-M hybrid fractal geometry. The proposed fractal antenna is designed and simulated in CST Microwave studio and compared with existing Koch fractal geometry. The prototype for the third iteration of the K-M fractal antenna is then fabricated on FR-4 substrate and tested through vector network analyzer for operating band/voltage standing wave ratio. Findings The third iteration of the proposed K-M fractal geometry results in achieving a 20% size reduction as compared to an ordinary MPA for the same resonant frequency with impedance bandwidth of 16.25 MHz and a directional gain of 6.48 dB, respectively. The operating frequency of MPA also lowers down to 1.44 GHz. Originality/value Further testing for the radiation patterns in an anechoic chamber shows good agreement to those of simulated results.
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TATOM, FRANK B. "THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS." Fractals 03, no. 01 (1995): 217–29. http://dx.doi.org/10.1142/s0218348x95000175.

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The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).
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Mathur, Vinita, and Dr Manisha Gupta. "Morphology of Koch Fractal Antenna." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 13, no. 2 (2014): 4157–63. http://dx.doi.org/10.24297/ijct.v13i2.2902.

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Antenna is paramount element for the radio communication entity using radio frequency and microwaves. In twenty-first century wireless communication systems, there is a demand for wider bandwidth, multiband and low profile antennas for both commercial and military purposes. This has initiated antenna analysis in various directions; one of them is using fractal shaped antenna elements. Fractal concepts have emerged and advanced as a unique technique for designing compact UWB antennas, because of the self-similarityand space-filling attributes. In a fractal antenna, the multiple frequencies of operation depend on the total dimensions of the design and the scale factor. There are varied fractal geometries that have been found to be favorable in developing novel and new models for antennas. In this paper we will be discussing one such type of fractal antennas i.e. the Koch structure. Koch antenna can be designed with a triangle, rectangle and pentagon as its initiator. Their geometry and parameters have been analyzed in this paper.Â
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Zhao, Shanrong, Jin Tan, Jiyang Wang, Xiaohong Xu, and Hong Liu. "A Dendrite with "Sierpinski Gasket" Fractal Morphology in Matt Glaze of LiAlSiO4-SiO2 System." Fractals 11, no. 03 (2003): 271–76. http://dx.doi.org/10.1142/s0218348x03001525.

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In this paper, we introduce a dendritic crystal, formed in matt glaze of LiAlSiO 4- SiO 2, having "Sierpinski gasket" fractal morphology. The crystal structure of this "Sierpinski gasket" dendrite is β-quartz. β-quartz can grow two kinds of fractal patterns: snow-shaped dendrite and "Sierpinski gasket" dendrite, depending on different supercooling conditions. These two kinds of fractals can develop together in one dendritic crystal. The evolution of the boundary morphologies between these two kinds of fractal dendrites can be described by another fractal — Koch curve. The "Sierpinski gasket" dendrite is a rather new fractal growth pattern which can introduce new opportunities to fractal growth research of nonlinear sciences.
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Purnomo, Kosala Dwidja, Siti Fatimah, and Bagus Juliyanto. "Generation of Fractal Objects with Iterated Function System on the Developments of Trellis Ornament Designs." BERKALA SAINSTEK 13, no. 1 (2025): 1–7. https://doi.org/10.19184/bst.v13i1.25656.

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Fractals are one of a mathematical concept that provides artistic value and is therefore widely used to design various kinds of objects. The purpose of this study is to obtain various trellis ornament designs generated from fractal objects. Some fractal objects that will be used are Koch Snowflake (m,n,c), Koch Anti-Snowflake (m,n,c) and dragon curve. The basic trellis pattern is built from basic geometry, namely line segments, rhombuses and elliptical curved lines with certain sizes. In this study, the generation of fractal objects was carried out using the Iterated Function Systems (IFS) method. In this case, IFS is carried out by utilizing Affine transformations, namely dilation, rotation and reflection. Related to the generation of the Koch Snowflake curve (m,n,c), an m-sided polygon with 3≤m≤5 is used and the side looping form uses an n-sided polygon with 3≤n≤5. The c value or the middle segment divisor used is 0.3; 0.2; and 0.19. The dilation scale on the dragon curve is 0.6≤k≤9.8 and the angle θ=90°. The iteration used to generate the Koch curve is 2 iterations while the dragon curve is 15 iterations. By taking several parameters, a trellis ornament design consisting of 5 patterns is obtained and each pattern has 3 variations of trellis motifs.
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RAMÍREZ, JOSÉ L., GUSTAVO N. RUBIANO, and BORUT JURČIČ ZLOBEC. "GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION." Fractals 23, no. 04 (2015): 1550047. http://dx.doi.org/10.1142/s0218348x15500474.

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In this paper, we introduce the [Formula: see text]-circle inversion which generalizes the classical inversion with respect to a circle ([Formula: see text]) and the taxicab inversion [Formula: see text]. We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the [Formula: see text]-circle inversion.
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LI, JIE, YONGPING HUANG, CHENGBIN ZHANG, and XIANGDONG LIU. "NUMERICAL STUDY ON THE SOLIDIFICATION PERFORMANCE OF A LATENT HEAT STORAGE UNIT WITH KOCH-FRACTAL FIN." Fractals 27, no. 07 (2019): 1950108. http://dx.doi.org/10.1142/s0218348x19501081.

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Inspired by the snowflake structure, an innovative Koch-fractal fin is proposed to optimize the fin geometry of the latent heat thermal energy storage (LHTES) units. A model of unsteady heat transfer accompanied with phase change is developed and numerically analyzed to investigate the effect of fin structure on the discharging process of a LHTES unit. The dynamic response of heat release, the dynamic temperature response and solidification front evolution of a LHTES unit with Koch-fractal fins are discussed and compared with the corresponding radial fins. Furthermore, a comprehensive evaluation of thermal performance of LHTES units is conducted in terms of the TES capacity, TES rate and solidification time. The results indicate that the heat release rate of a LHTES unit with Koch-fractal fins is faster than that with radial fins. Moreover, because the Koch-fractal fins have advantages of higher specific surface area, faster heat flow path from point to surface and smaller thermal resistance arising from the reasonable spatial layout, the evolution of solidification front is faster and the temperature distribution is more uniform. The results of quantitative evaluation show that although the TES capacity is identical, the TES rate of a LHTES unit with Koch-fractal fins is six times that with radial fins.
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ZHANG, SHUAI, XUEYE CHEN, ZHONGLI WU, and YUE ZHENG. "NUMERICAL STUDY ON KOCH FRACTAL BAFFLE MICROMIXER." Fractals 27, no. 03 (2019): 1950026. http://dx.doi.org/10.1142/s0218348x19500269.

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This paper is mainly to study the application of Koch fractal baffle to passive micromixers. It can be determined that the mixing efficiency of secondary Koch fractal baffle (SKFB) micromixer is better than that of primary Koch fractal baffle (PKFB). We compare and analyze the mixing efficiency when the angle between the baffle and the microchannel is [Formula: see text], [Formula: see text] and [Formula: see text] with the height 100[Formula: see text][Formula: see text]m. With the changing of the angle, it contributes to enhance the chaotic convection of the micromixer. Especially at the angle of [Formula: see text], the vortex caused by the Koch fractal baffle structure is more obvious, the mixing efficiency of micromixer is more than 95% at Re [Formula: see text] 0.05 and 100. When the height of Koch fractal baffle is 50, 75 and [Formula: see text]m, the mixing efficiency of the micromixer gradually increases. The whirling and spiral phenomenon of the streamlines increases the chaotic convection and promotes the improvement of the mixing efficiency. In the direction of microchannel, nine sections which have a significant effect on the mixing efficiency are investigated. The encircling and split phenomenon affected by the chaotic convection is shown in nine sections at Re [Formula: see text] 0.05, 10 and 100.
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Dissertations / Theses on the topic "Fractal Koch geometry"

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Almeida, Filho Valdez Arag?o de. "Arranjos Log-Peri?dicos Compactos em Microfita com Elementos Fractais de Koch." Universidade Federal do Rio Grande do Norte, 2010. http://repositorio.ufrn.br:8080/jspui/handle/123456789/15315.

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Made available in DSpace on 2014-12-17T14:55:43Z (GMT). No. of bitstreams: 1 ValdezAAFi_DISSERT.pdf: 1620850 bytes, checksum: c1208b8ca13742d0d9cd3ac88c864f60 (MD5) Previous issue date: 2010-06-14<br>Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior<br>This work aims to present how the application of fractal geometry to the elements of a log-periodic array can become a good alternative when one wants to reduce the size of the array. Two types of log-periodic arrays were proposed: one with fed by microstrip line and other fed by electromagnetic coupling. To the elements of these arrays were applied fractal Koch contours, at two levels. In order to validate the results obtained some prototypes were built, which were measured on a vector network analyzer and simulated in a software, for comparison. The results presented reductions of 60% in the total area of the arrays, for both types. By analyzing the graphs of return loss, it was observed that the application of fractal contours made different resonant frequencies appear in the arrays. Furthermore, a good agreement was observed between simulated and measured results. The array with feeding by electromagnetic coupling presented, after application of fractal contours, radiation pattern with more smooth forms than the array with feeding by microstrip line<br>Este trabalho tem como objetivo apresentar como a aplica??o de contornos fractais aos elementos de um arranjo log-peri?dico se torna uma alternativa bastante interessante quando se deseja reduzir as dimens?es do arranjo. Foram propostos dois tipos de arranjos log-peri?dicos: um com alimenta??o por linha de microfita e outro com alimenta??o por acoplamento eletromagn?tico. Aos elementos desses arranjos foram aplicados contornos fractais de Koch, em dois n?veis. Com a finalidade de validar os resultados obtidos foram constru?dos prot?tipos, que foram caracterizados experimentalmente em um analisador de rede vetorial e simulados em software, para compara??o. Os resultados mostraram redu??es de 60% nas dimens?es dos arranjos, para ambos os tipos. Atrav?s da an?lise dos gr?ficos da perda de retorno, p?de-se observar que a aplica??o dos contornos fractais fez com que aparecessem diferentes frequ?ncias de resson?ncia nos arranjos. Al?m disso, observa-se uma boa concord?ncia entre os resultados medidos e simulados. O arranjo com alimenta??o por acoplamento eletromagn?tico apresentou, ap?s aplica??o dos contornos fractais, menores valores de diretividade do que o arranjo com alimenta??o por linha de microfita
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Book chapters on the topic "Fractal Koch geometry"

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T. Abed, Amer, Mahmood J. Abu-AlShaer, and Aqeel M. Jawad. "Fractal Antennas for Wireless Communications." In Modern Printed-Circuit Antennas. IntechOpen, 2020. http://dx.doi.org/10.5772/intechopen.90332.

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When the length of the antenna is less than a quarter of the wavelength of the operating frequency, good radiation properties are difficult to obtain. However, size limitations can be overcome in this case using a fractal geometry antenna. The shape is repeated in a limited size such that the total length of the antenna is increased to match, for example, half of the wavelength of the corresponding desired frequency. Many fractal geometries, e.g., the tree, Koch, Minkowski, and Hilbert fractals, are available. This chapter describes the details of designing, simulations, and experimental measurements of fractal antennas. Based on dimensional geometry in terms of desired frequency bands, the characteristics of each iteration are studied carefully to improve the process of designing the antennas. In depth, the surface current distribution is investigated and analyzed to enhance the circular polarization radiation and axial ratio bandwidth (ARBW). Both, simulation and experimental, results are discussed and compared. Two types of fractal antennas are proposed. The first proposed fractal antenna has a new structure configured via a five-stage process. The second proposed fractal antenna has a low profile, wherein the configuration of the antenna was based on three iterations.
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Gonzales Palacios, Orlando Francois, Ricardo Erick Diaz Vargas, Patrick H. Stakem, and Carlos Enrique Arellano Ramirez. "Koch Snowflake Fractal Antenna Design in the Deep Space Bands for a Constellation of Cubesat Explorers." In Proceedings of CECNet 2021. IOS Press, 2021. http://dx.doi.org/10.3233/faia210419.

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This paper presents the design and simulation of a Koch curve fractal antenna, developed according to the second iteration of the Koch snowflake fractal for S-band, C-band, X-band and Ku-band. The simulated antenna shows good performance for the operating frequencies and desirable gain, bandwidth and VSWR parameters. Being a compact antenna, it has a size, geometry and characteristics that go in accord with the CubeSat’s structure standards. The antenna was fabricated on a 1.5 mm thick FR-4 substrate. The VSWR achieved values are lower than 1.4 for the frequencies used (2.1 GHz to 2.4 GHz and 7.4 GHz to 8.9 GHz) with a simulated omnidirectional radiation pattern. A maximum gain of 6.8 dBi was achieved. As this antenna works optimally in the S, C and X bands, it is adequate for deep space applications, especially in low-power consumption systems. This approach would be ideal for constellations of Cubesat explorers.
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Bassingthwaighte, James B., Larry S. Liebovitch, and Bruce J. West. "Generating Fractals." In Fractal Physiology. Oxford University PressNew York, NY, 1994. http://dx.doi.org/10.1093/oso/9780195080131.003.0009.

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Abstract Examples provided earlier as introductions to fractal ideas, such as the Koch snowflake, fall into the class of geometric fractals. They are simple, beautiful, and powerful. They startle us: so much diversity is captured in such simple beginnings. The power is not so much in the “beginning,” but in the process of recursion. A simple act, repeated sufficiently often, creates extraordinary, often unsuspected results. Playing with recursive operations on the computer is the key to the revelation; reading the book spoils the story when the result is before you.
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Conference papers on the topic "Fractal Koch geometry"

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Minal, Nitin Dhama, R. B. Patel, and B. P. Singh. "Antenna Miniaturization Using Koch Snowflake Fractal Geometry." In INTERNATIONAL CONFERENCE ON METHODS AND MODELS IN SCIENCE AND TECHNOLOGY (ICM2ST-10). AIP, 2010. http://dx.doi.org/10.1063/1.3526256.

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Monsef Khoshhesab, Mona, and Yaning Li. "The Strength of Dissimilar Fractal Joints." In ASME 2016 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/imece2016-66830.

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In this investigation, the influences of fractal geometry and material properties on the strength of dissimilar joints were studied. The fractal geometry explored was an iterative Koch curve. The interfacial layer joining two different materials was designed to be a Koch layer with three different numbers of iteration. The mechanical behaviors of the fractal dissimilar joints under both normal tensile traction and shear traction were simulated via finite element (FE) method. In the three-phase FE models, isotropic elasto-perfect-plastic material models with different stiffness and yielding strength were used for all three phases. By varying the stiffness and strength ratio of the Koch layer and the dissimilar materials, fractal dissimilar joints with both perfect bonding and imperfect bonding were simulated and compared. It was found that the fractal geometry plays a very important role in enhancing both the tensile and shearing strength of dissimilar joints, especially for the cases with imperfect bonding.
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Vieira Ferreira, Gustavo, Weliton Dal Pizzol Maria, and Adriano Rodrigues de Melo. "Introdução à Geometria Fractal no Ensino Médio Técnico: Uma Abordagem com Programação Python." In Computer on the Beach. Universidade do Vale do Itajaí, 2021. http://dx.doi.org/10.14210/cotb.v12.p543-546.

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This work is inserted in the context of technical high school andit aimed to analyze the integration between the branches of FractalGeometry, Analytical Geometry and Computer Programming.For this purpose, we carried out a bibliographic search about whatcharacterizes and distinguishes Fractal Geometry from EuclideanGeometry, we also seek in our readings to list the most famousfractals. Then, we developed (in python language) several fractalgeneration programs. It was possible to work with amazing andeasily programmable fractal shapes, such as the Cantor Set, theHilbert Curve and Sierpinski Triangle. We also built two new familiesof fractal shapes from a generalization of the Koch Curve. Weconclude that programming fractals in the context of technical highschool is productive and challenging, as it requires many changesin the representations of fractal patterns.
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Warren, Peter, Sandip Haldar, Seetha Raghavan, and Ranajay Ghosh. "Modeling Thermally Grown Oxides in Thermal Barrier Coatings Using Koch Fractal." In ASME Turbo Expo 2019: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/gt2019-91722.

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Abstract Growth of the Thermally Grown Oxide (TGO) between the bond coat and thermal barrier coating (TBC) during service is one of the most common causes of failure within thermal barrier coating (TBC) systems. Initially this oxide will provide protection from oxidation for the substrate, but stress build up will contribute to delamination of the topcoat. Research has been carried out over the stresses caused by this TGO growth, and how to best mitigate these induced stresses. The interface topography plays a critical role for air plasma sprayed (APS) TBCs in development of stress profiles across the TGO/TBC interface [1, 2]. The APS TBCs fail by cracking in the TBC close to the TGO-TBC interface. Most models treat TGO as a sinusoidal wavelength interface. However, most TGO surfaces have been experimentally observed to have fractal like patterns at the interfacial region of the bondcoat and topcoat. Fractals provide us a better understanding of interactions at rough interfaces between two materials adhered to one another. In this work, we model the topography of the TGO using a Koch fractal. We find the geometry selected to model the TGO layer has a direct effect on the stress generation and creep strain during simulation.
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Rahman, N. A., and M. F. Jamlos. "A Wideband Log Periodic Antenna with Fractal Koch Geometry and Rectangular Stub." In 2016 International Conference on Computer and Communication Engineering (ICCCE). IEEE, 2016. http://dx.doi.org/10.1109/iccce.2016.15.

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Jibrael, Fawwaz J., and Shahad D. Sateaa. "Multiband characteristics and fractal dimension of dipole antenna with square koch curve geometry." In 2010 2nd International Conference on Education Technology and Computer (ICETC). IEEE, 2010. http://dx.doi.org/10.1109/icetc.2010.5529951.

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Ahmed, Hayder S., Zahraa Salah Ahmed, Rafid S. Zamel, and Taha Ahmed Elwi. "Compact MIMO Antenna Array for 5G Applications based Novel Hayder-Koch Fractal Geometry." In 2022 International Telecommunications Conference (ITC-Egypt). IEEE, 2022. http://dx.doi.org/10.1109/itc-egypt55520.2022.9855710.

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Gurgel, Nathan, Idalmir Queiroz, Humberto Andrade, and Tagleorge Silveira. "Miniaturization of microstrip patch antennas using Koch fractal geometry on the ground plane." In 2021 SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference (IMOC). IEEE, 2021. http://dx.doi.org/10.1109/imoc53012.2021.9624798.

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Adelpour, Zahra, Farzad Mohajeri, and Mojtaba Sadeghi. "Dual-frequency microstrip patch antenna with modified Koch fractal geometry based on genetic algorithm." In Propagation Conference (LAPC). IEEE, 2010. http://dx.doi.org/10.1109/lapc.2010.5666299.

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Gvozdarev, Aleksey S., and Tatiana K. Artemova. "An analysis of the multiband non-planar koch-type fractal dipole with steerable geometry." In 2018 Moscow Workshop on Electronic and Networking Technologies (MWENT). IEEE, 2018. http://dx.doi.org/10.1109/mwent.2018.8337283.

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