Ways of regularization of materials science ill-posed problems

Yu. I. Dubrov, V. M. Volchuk


Introduction. The ill-posedness of the problems is mainly caused by the incompleteness of their formal axiomatics, which is a consequence of the incompleteness of our knowledge of the object of identification. This fact demonstrates the formal axiomatic incompleteness, arising in description of the metal structure elements by means of traditional configurations of Euclidean geometry, which triggers the need to use other promising approaches to the structure assessment. The aim of the work is to establish solutions to some conditionally incorrect materials science problems. Statement of basic materials. On the basis of the analysis of quantitative and qualitative assessment of the real structures of many iron carbon alloys, there are observed some significant, for practical purposes, divergences between results of direct experiments and results of their prediction. Basic ways of regularization of identification problems of multi-parameter objects, criteria selection for multi-criteria technology optimization, ranging of multi-parameter technologies quality criteria, material quality estimation are considered. It is shown that regularization of materials science conditionally ill-posed problems is acceptable with application of specific algorithms for every specific case. Conclusions. It is shown that some of the conditionally incorrect problems are solved by applying a language of a higher level - the language of fractal geometry.


ill-posed problems; multiparametric technology; the area of self-similarity; materials; the regularization

Full Text:



Tikhonov A.N., Arsenin V.Yu. Solutions of Ill-Posed Problems. New York : Winston, 1977, 258 p.

Gödel K. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 1931, vol. 38, no. 1, рр. 173-198. (in Germany).

Bolshakov Vad.I., Bolshakov V.I., Volchuk V.N. and Dubrov Yu.I. Chastkova kompensatsiya nepovnoty formal’noyi aksiomatyky pry identyfikatsiyi struktury metalu [The partial compensation of incompleteness of formal axiomatics in the identification of the metal structure]. Visnyk akademiyi nauk Ukrayiny [Bulletin of the National Academy of Sciences of Ukraine]. 2014, no. 12, pp. 45−48. (in Ukrainian). – Available at:

Bir S. Kibernetika i upravleniye proizvodstvom [Cybernetics and production management]. Moscow: Nauka, 1963, 276 p. (in Russian).

Mandelbrot B.B. The Fractal Geometry of Nature : monograph. New-York, San Francisco: Freeman, 1982, 480 p. – Available at:

Volchuk V., Klymenko I., Kroviakov S. and Orešković M. Method of material quality estimation with usage of multifractal formalism. Tehnički glasnik − Technical Journal, 2018, vol. 12, no. 2, рр. 93−97. – Available at:

Calcagni G., Giuseppe Nardelli G. and Rodríguez-Fernández D. Particle-physics constraints on multifractal spacetimes. Physical Review D. 2016, vol. 93, no. 2, рр. 025005. – Available at:

Zhuravel' I.M. Computer Analysis of the Distribution of Grain Sizes in the Structure of 12Kh1MF Steel After Operation. Materials Science. 2019, vol. 55, no 2, рp. 187−192.

Mishutin A.V., Kroviakov S.O., Mishutin N.V. and Bogutsky V.L. Modified expanded clay lightweight concretes for thin-walled floating structures. Proceeding of the Second International Conference on Concrete Sustainability (ICCS16). Madrid, Spain on 13−15 June, 2016. – Barcelona, Spain: International Center for Numerical Method in Engineering, 2016, pp. 743−749.

Ivanova V.S., Balankin A.S., Bunin I.Zh. and Oksogoev A.A. Sinergetika i fraktaly v materialovedenii [Synergetics and fractals in materials science]. Moscow : Nauka, 1994, 383 p. (in Russian).

Dubrov Yu., Bolshakov V. and Volchuk V. Puti identifikatsii periodicheskikh mnogokriterial'nykh tekhnologiy [Road periodic identification of multi-criteria Technology]. Saarbrucken : Palmarium Academic Publishing, 2015, 236 p. (in Russian).

Bolshakov V.I., Volchuk V.N. and Dubrov Yu.I. O prognozirovanii kachestva tselevogo produkta v periodicheskikh tekhnologiyakh [Predicting the quality of a desired product in periodic technologies]. Dopovidi Natsionalnoi akademii nauk Ukrainy [Reports of the National Academy of Sciences of Ukraine]. 2014, no. 11, pp. 7781. (in Russian). – Available at:

Bolshakov V.I., Volchuk V.N. and Dubrov Yu.I. Etapy identyfikatsiyi bahatoparametrychnykh tekhnolohiy ta shlyakhy yikh realizatsiyi [Stages multiparameter identification technologies and ways of their implementation]. Visnyk Natsionalʹnoyi akademiyi nauk Ukrayiny [Bulletin of the National Academy of Sciences of Ukraine]. 2013, no. 8, pp. 66–72. (in Ukrainian). – Available at:

Bolshakov V.I., Volchuk V.M. and Dubrov Yu.I. Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy. Metallofizika i Noveishie Tekhnologii, 2018, vol. 40, no. 9, pр. 11651171. – Available at:

DOI: 10.15407/mfint.40.09.1165

Volchuk V.M. K primeneniyu fraktal'nogo formalizma pri ranzhirovanii kriteriyev kachestva mnogoparametricheskikh tekhnologiy [On the Application of Fractal Formalism for Ranging Criteria of Quality of Multiparametric Technologies]. Metallofizika I noveyshiye tekhnologii [Metal Physics and Advanced Technologies]. 2017, vol. 39, no 3, рp. 949−957. (in Russian). – Available at:

Kroviakov S., Zavoloka M., Dudnik L. and Kryzhanovskyi V. Comparison of strength and durability of concretes made with sulfate-resistant portland cement and portland cement with pozzolana additive. Electronic Journal of the Faculty of Civil Engineering Osijek-e-GFOS, 2019, vol. 10, no. 19, pр. 8186. – Available at:

Bolshakov V.I. and Volchuk V.N. Materialovedcheskiye aspekty primeneniya veyvletno-mul'tifraktal'nogo podkhoda dlya otsenki struktury i svoystv malouglerodistoy stali [Material science aspects of the use of wavelet and multifractal approach for assessing of the structure and properties of low-carbon steel]. Metallofizika i noveyshiye tekhnologii [Metal Physics and Advanced Technologies]. 2011, vol. 33, no. 3, рp. 347−360. (in Russian).

Bolshakov V., Volchuk V. and Dubrov Yu. Fractals and properties of materials. Saarbrucken: Lambert Academic Publishing, 2016, 140 p.

Bolshakov V.I., Volchuk V.M. and Dubrov Yu.I. Osnovy organizacii fraktal'nogo modelirovaniya [Fundamentals of fractal modeling]. Kyiv, Ukraine : PH "Akademperiodyka" National Academy of Sciences of Ukraine, 2017, 170 p. (in Russian).

Blum A.S., Soto C.M., Wilson C.D., Cole J.D., Kim M., Gnade B., Chatterji A., Ochoa W.F., Lin T. W., Johnson J.E. and et al. An engineered virus as a scaffold for three-dimensional self-assembly on the nanoscale. Small, 2005, no. 1, pp. 702–706. – Available at: PMID 1719 3509

GOST Style Citations