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Direct and First Inverse Problems of Genetic Identification of Materials at a Fixed Smoothness Class



Author(s)

P. A. Belov¹ and Elliot Weiss²

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DOI:10.17265/1934-7359/2026.08.002

1. Institute of Applied Mechanics, Russian Academy of Sciences, Moscow 125040, Russia 2. Independent Researcher, USA

Within the framework of the proposed genetics of elastoplastic materials, a mathematical formulation of the genetic identification of materials is developed on the basis of the correspondence between the experimental deformation curve, the governing differential operator, and the space of its fundamental solutions. A fixed smoothness class Cᵏ uniquely determines the order of the governing operator, the number of second-order factors, and the dimension of the space of fundamental solutions. A canonical interpolation curve is introduced as the unique geometric reference associated with the experimental sample and the selected smoothness class; it is independent of the mechanical parameters of the material. Two complementary bounds are established. First, every proper mechanical model curve that does not coincide with the canonical interpolation curve has a strictly nonzero approximation error relative to it. Second, the generalized Reinsch construction implies that a one-segment canonical interpolant of class Cᵏ is a polynomial of degree at most 2k+1; therefore, any finite sample of N points admits one-segment canonical interpolation for a sufficiently high class satisfying k ≥ N-22. A third result establishes a class-dependent upper bound on the number of identifiable segments: for a fixed sample size, the continuous segmentation capacity decreases strictly with smoothness class, whereas the maximum integer number of segments decreases stepwise. These results separate the class-dependent geometric limits of representation from measurement accuracy, which enters only at the stage of inverse identification. The direct and first inverse problems are then formulated for a fixed smoothness class and segment architecture.

Genetics of materials, genetic identification, direct problem, first inverse problem, canonical interpolation curve, smoothness class, factorization of a differential operator, fundamental solutions.

P. A. Belov and Elliot Weiss . (2026).Direct and First Inverse Problems of Genetic Identification of Materials at a Fixed Smoothness Class,  Journal of Civil Engineering and Architecture, August 2026, Vol. 20, No. 8, 322-339