МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ МЕТАЛОГРАФІЧНОГО ЗОБРАЖЕННЯ ЗА ДОПОМОГОЮ РІШЕННЯ ГРАНИЧНИХ ЗАДАЧ ДЛЯ РІВНЯННЯ ЛАПЛАСУ
Journal Title: Математичне моделювання - Year 2018, Vol 1, Issue 1
Abstract
MATHEMATICAL MODELING OF THE METALLOGRAPHIC IMAGE BY SOLUTVING THE LAPLACE EQUATION BOUNDARY PROBLEMS Klochko O. Yu. Abstract In the present paper we describe the new approaches for construction of computer models of metallographic heterogeneous structure using mathematical modeling and solving Laplace equation boundary problems, which allow one to obtain the high-accuracy images of structures. In our research we take the diffusion processes that occur during structure formation into account. Our research has been carried out by modeling the structure formation of the complex heterophase alloys (e.g., high-chromium cast iron rolls) affected by a large number of complex processes similar to the hydrodynamic ones. The modeling has been performed with applicain of a range of boundary conditions in order to obtain and compare the solutions of Laplace equation. In the process of modeling we have studied the interior regions specified by the microstructure image with the finite-difference iteration method used. As a result, the values of the conventional colors and their distribution have been found, while making a comparison of the original structure metallographic image with the distribution of the conventional colors. The described method helps one to achieve the minimum deviation in the results References [1] Ladyzhenskaja O.A. [Boundary-value problems of mathematical physics] Kraevye zadachi matematicheskoj fiziki. M.:“Nauka”, 1973, 407 s. [2] Bazhal I.G., Kurilenko O.D. [Recrystallization in disperse systems] Perekristallizacija v dispersnyh sistemah. Kiev, Naukova dumka, 1975, 216 s. [3] Brailsford A.D., Bullough R. The theory of sink strengths. Philos. Trans. R. Soc. London, Ser. A, 1981, v. 302, p. 87–137. [4] Calef D.F., Deutch J.M. Diffusion- controlled reactions. Ann. Rev. Phys. Chem., 1983, v. 34, p. 493–524. [5] Zgaevskij V.Je., Janovskij Ju.G. [Calculation of the effective viscosity of concentrated suspensions of rigid particles on the basis of the crystal model] Vychislenie jeffektivnoj vjazkosti koncentrirovannyh suspenzij zhestkih chastic na osnove kristallicheskoj modeli. Mehanika kompozicionnyh materialov i konstrukcij, 1996, t. 2, № 1, s. 137–167. [6] Traitak S.D. Methods for solution of boundary value problems for regions with multiply connected boundaries // J.Composite Mech. Design, Vol. 9, № 4 (2003), Р. 495–521. [7] Patrick J. Roache. Fundamentals of Computational Fluid Dynamics. Publisher: Hermosa Pub, 1998, 648 p. [8] T.S. Skoblo, E.L. Belkin, O.Yu. Klochko. [Application justification of the concepts of fluid dynamics Navier-Stokes equations for the analysis of metallographic images]. Obosnovanie primenenij ponyatij uravnenij gidrodinamiki Navje-Stocksa dlya analiza mettalographicheskih izobrazhenij /Materiały VII Mięzdynarodowej naukowi-praktycznej konferencji. Przemyśl: 2011. Vol. 21. P. 94–96. [in Russian] [9] Pikulin V.P., Pohozhaev S.I. [Practical course on equations of mathematical physics] Prakticheskij kurs po uravnenijam matematicheskoj fiziki. – Piksel, 2013. – 208 s. [10] Bekkenbah Je.F. [Modern mathematics for engineers] Sovremennaja matematika dlja inzhenerov. Ripol Klassik, 2013, 506 s. [11] Vasidzu K. [Variational methods in the theory of elasticity and plasticity] Variacionnye metody v teorii uprugosti i plastichnosti. M: Mir, 1987, 542 s. [12] Skoblo T.S., Klochko O.Yu., Belkin E.L. Structure of high-chromium cast iron //Steel in Translation. – March 2012, Volume 42, Issue 3, pp 261–268. [13] [Method for modeling the structure of metals by permuting pixel images] Metodika modelirovanija struktury metallov s pomoshh'ju perestanovki pikselej izobrazhenija / T.S.Skoblo, O.Ju. Klochko, E.L. Belkin // Vіsnik HNTUSG. – Vipusk 115, 2011. – S. 10–21. [14] [Mathematical analysis of the estimation of the dispersion of the structure of alloyed cast irons] Matematicheskij analiz ocenki dispersnosti struktury legirovannyh chugunov / T.S. Skoblo, O.Ju. Klochko, E.L. Belkin, A.I. Sidashenko // Stal'. – 2017. – № 2. – S. 51–54.
Authors and Affiliations
О. Ю. Клочко
Keywords
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- EP ID EP297125
- DOI 10.31319/2519-8106.1(38)2018.129040
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