The fractal nature of isolines in a field

The fractal nature of isolines in a field. Relief and Fraktal

Michel Felgenhauer, Berlin

When evaluating image data from flow analyzes, the task emerges to quantify the quality of currents that exist in the afterlife of a interference contour. The image data comes from laboratory experiments on the wind tunnel or are the calculation result from computer simulations. The intervention names a procedure of digital image processing, which is suitable for detection in -idy densities in speed fields of regular flow fields.

Contour lines, or isolinia (Greek: isos = equal) are lines that connect points of the same values within a field. In order to visualize measurement data or calculation variables from numerical simulations, isolinias are preferably shown in axis - compliant cuts of the three -dimensional field. In this smart essay, Isoline fields in selected cutting levels primarily treat representations of speed fields. Isotachen are called isolinia over speeds. In Isolinia, the viewer is also aware of gradient in a natural way, because an accumulation or absence of curves of the same values (isolinia) indicates a more or less large change in the parameter shown.¹

The nature of isoline visualization is in the advancement of isolinia, which is now being evaluated via the plan, which must now be evaluated. The task of detecting a gradient immediately understands the human eye. In digital image processing, the so -called fractal nature of graphic representations is occasionally argued.

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The significant analysis parameter is the fractal dimension FR, which is based on the non-supplementary, broken Hausdorff-Number used in chaos theory². The concept introduced by Felix Hausdorff assigns rational numbers to any metric rooms. For simple geometric objects, their value agrees with that of the ordinary dimension concept. Unlike the Euclidean dimensions theory, the Hausdorff­Number assumes that the dimension of a structure can be broken and has a mantis, i.e. are not rational numbers. This made the Benoit Mandelbrot³to shape the term "fractal" and to define as follows:

According to definition, a fractal is a lot whose Hausdorff-Besicovitch dimension really exceeds the topological dimension.

Ein Fraktal ist nach Definition eine Menge, deren Hausdorff-Besicovitch- Dimension echt die topologische Dimensionübersteigt .

The complexity parameter FR listed in this essay has already been used successfully to solve classification tasks that are related to the correlations of physical properties in and the homotopes of vertebral structures. The code of the algorithm used here for determining the fractal dimension FR is based on statistical methods of image analysis and is very efficient and fast. He uses the box-count method for regular matrices that come from RGB images. The algorithm filters an RGB image into a (quasibinarian) matrix A and invert it. The new matrize B now only contains elements Hi = 256 (RGB-White) and LO = 0 (RGB-Black).

This also applies to very complex facts, such as the detection of priority in a speed field, which itself is again a vector.

Michel Felgenhauer, Berlin

Michel Felgenhauer is the pseudonym of the engineer Michael Dienst from Wiesbaden, Germany. I live and work in Berlin, I am spokesman for the Bionic Research Unit and I have been a lecturer in Bionic Engineering at the Berlin University of the Arts and the Industrial Design Institute of the Magdeburg, University of Applied Sciences.

Martha Felgenhauer died in 1943 as a young woman in Ziegenhals, Silesia. Those who knew her say we are kindred spirits. So occasionally I tell my grandmother stories of friendly science.

Berlin, 2023

BIBLIOGRAPHIE, Quellen und weiterführende Literatur

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[Die09-4] Dienst, Mi.(2009) Physical Modelling driven Bionics. GRIN-Verlag München.

[DUB-95] Dubbel, Handbuch des Maschinenbaus, Springer Verlag Berlin, 15.Auflage 1995.

[Fel - 19-2] Felgenhauer, Mi. (2019) BIONIK UND DIGITALE BILDVERARBEITUNG Laterale Inhibition und Aktivierung. Grin Verlag München, ISBN(e-Book) 9783668874541, ISBN(Buch): 9783668874558

[Fel -19-1] Felgenhauer, Mi. (2019) MATRIZENVERFAHREN ZUR DIGITALEN BILDVERARBEITUNG. Facettenaugen als Vorbild schneller Algorithmen in der Bildsynthese. Grin Verlag München

[Fel 20-1] Felgenhauer, Mi. (2020). Die Verteilung von Induktionswirkungen Lagrange Kohärenter Objekte. Zur Topographie und Kondition von Geschwindig­keitsfeldern. GRIN-Verlag GmbH München, ISBN (e-Book): 783346285904, ISBN(Buch):783346142146, VNR:535307

[Fel 20-2] Felgenhauer, Mi. (2020) About artificial Lagrangian Coherent

Structures. GRIN-Verlag GmbH München, PDF-Version (pdf), ISBN: 9783346285904, ISBN (Buch): 9783346285911 Katalognummer. v913092

[Fel 21-5] Felgenhauer, Mi. (2021). Implicite Coherent Fluid Systems. Fluid within a Fluid. GRIN-Verlag GmbH München, ISBN(e-Book): 9783346407030. ISBN (Buch): 9783346407047, VNR: v1012490

[Fren-94] French, M.: Invention and Evolution: design in nature and engineering. Cambridge University Press. Cambridge 1994.

[Fren-99] French, M.: Conceptual Design for Engineers. Berlin, Heidelberg, New York, London, Paris, Tokio: Springer: 1999

[Guen-98] Günther, B., Morgado, E. (1998) Dimensional analysis and allometric equations concerning Cope's rule.RevistaChilena de Historia Natural 71: 1989

[Gör-75] Görtler, H. Diemensionsanalyse. Berlin Springer 1975

[Gorr-17] Edgar Gorrell, S. Martin: Aerofoils and Aerofoil Structural Combinations. In: NACA Technical Report. Nr. 18, 1917.

[Guen-66] Günther, B., Leon, B. (1966) Theorie of biological Similarities, nondimensional Parameters and invariant Numbers. Bulletin ofMathematicalBiophysics Volume 28, 1966.

[Hüt-07] Hütte, 2007, 33. Auflage, Springer Verlag. S.E147

[Kra-86] Krasny, R. (1986) Desingularization of Periodic Vortex Sheet Roll-up. Courant lnstirute oJ' Mathematical Sciences, New York Unioersity, 251Mercer Street, Nen, York, New York 10012, received November 15, 1981; revised July 25, 1985

[Kra 91] Krasny, R. (1991) Vortex Sheet Computations: Roll-Up, Wakes, Separation. In: Lectures in Applied Mathematics, Vol.: 28. (1991)

[Lun-82] T. S. Lundgren, T.S. (1982) Strained spiral vortex model for turbulent fine structure, The Physics of Fluids 25, 2193 (1982); https://doi.org/10.1063/1.863957

[Man 87] Mandelbrot, B.B. (1987) Die fraktale Geometrie der Natur. Birkhäuser Verlag. Basel, Boston, Berlin

[Mial-05] B. Mialon, M. Hepperle: "Flying Wing Aerodynamics Studies at ONERA and DLR", CEAS/KATnet Conference on Key Aerodynamic Technologies, 20.-22. Juni 2005, Bremen.

[PaBe-93] Pahl. G.; Beitz, W.: Konstruktionslehre, 3.Auflage. Berlin- Heidelberg-New York-London-Paris-Tokio: Springer 1993

[Pei 88] Peitgen, H.O. (1988) Fraktale: Computerexperimente entzaubern

komplexe Strukturen. In: 115. Verhandlungen der Gesellschaft Deutscher Naturforscher und Ärzte 17. Bis 20.9 1988. S. 123ff.

[Pflu-96] Pflumm, W. (1996) Biologie der Säugetiere. Berlin: Blackwell Wissenschaftsverlag.

[Sun-16] Sun,P.N., Colagrossi, A. Marrone, S. , Zhang, A.M, (2016) Detection of Lagrangian Coherent Structures in the SPH framework, College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China; CNR-INSEAN, Marine Technology Research Institute, Rome, Italy; Ecole Centrale Nantes, LHEEA Lab. (UMR CNRS), Nantes, France.

[Tho-59] Thompson, D'Arcy, W. (1959) On Growth and Form. London: Cambridge University Press. (Neuauflage der Originalschrift 1907)

[Tho-92] Thompson, D W., (1992). On Growth and Form. Dover reprint of 1942 2nd ed. (1st ed., 1917). ISBN 0-486-67135-6

[Zie - 72] Zierep, J. (1972) Ähnlichkeitsgesetze und Modellregeln der Strömungslehre.

Bildmaterial

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https://commons.wikimedia.Org/wiki/File:Relief-with-contour-lines.jpg

MikeRun, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons

<a title="MikeRun, CC BY-SA 4.0 &lt;https://creativecommons.org/licenses/by-sa/4.0&gt;, via

Wikimedia Commons" href="https://commons.wikimedia.org/wiki/File:Relief-with-contour- lines.jpg"><img width="512" alt="Relief-with-contour-lines" src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Relief-with-contour- lines.jpg/512px-Relief-with-contour-lines.jpg"></a>

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¹ Aus: MikeRun, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons, https://commons.wikimedia.Org/wiki/File:Relief-with-contour-lines.jpg

² Die Chaosforschung oder Chaostheorie bezeichnet ein nicht klar umgrenztes Teilgebiet der nichtlinearen Dynamik bzw. der dynamischen Systeme, welches der mathematischen Physik oder angewandten Mathematik zugeordnet ist. In den 80er Jahren dvJ. beschäftigt sich die CT mit Ordnungen in speziellen dynamischen Systemen, deren zeitliche Entwicklung unvorhersagbar erscheint, obwohl die zugrundeliegenden Gleichungen deterministisch sind.

³Benoit B. Mandelbrot (* 20. November 1924 in Warschau; t 14. Oktober 2010 in Cambridge, Massachusetts) war ein französisch-US-amerikanischer Mathematiker. Er leistete Beiträge zu einem breiten Spektrum mathematischer Probleme, einschließlich der theoretischen Physik, der Finanzmathematik und der Chaosforschung. Am bekanntesten aber wurde er als Vater der fraktalen Geometrie (wikipedia).

Benoit B. Mandelbrot ((1987) Die fraktale Geomerie der Natur. Birkhäuser Verlag Basel, S.27.