This article is cited in 1 paper

Mathematical Modeling, Numerical Methods and Software Complexes

Triply periodic surface description using Laplace–Beltrami operator and a statistical machine learning model

M. I. Smolkov

Samara State Technical University, Samara, 443086, Russian Federation

Abstract: Triply periodic surfaces (TPS) and their minimal analogs (TPMS) are currently widely used in various fields, including mechanics, biomechanics, aerodynamics, hydrodynamics, and radiophysics. In this context, the problem of establishing correlations between the topological and geometric properties of surfaces and their physical characteristics arises. To address this problem, it is necessary to introduce a measure of similarity between surfaces with different topological and geometric features. This work focuses on describing TPS and TPMS in terms of a specific metric space of descriptors. The problem is solved using the mathematical framework of image recognition theory. A descriptor is constructed based on a set of eigenvectors and eigenvalues of the Beltrami–Laplace operator and a joint Bayesian model. A metric based on a probabilistic measure of surface similarity is introduced in the descriptor space. The effectiveness of the method developed in this work has been tested on 51 surfaces of class P. The accuracy of predicting the surface type is 92.8 %. The developed machine learning model enables the determination of whether a given surface belongs to the class of P-surfaces.

Keywords: topological structure, discrete analog of the Laplace–Beltrami equation, eigenvectors, eigenvalues, Bayesian probabilities, probabilistic similarity measure

UDC: 517.9+514.7+519.6

MSC: 68T05, 58J50, 53A10

Received: July 24, 2024
Revised: February 19, 2025
Accepted: February 21, 2025
First online: March 10, 2025

DOI: 10.14498/vsgtu2105

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