chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2025-26-4-370-382cheb-2092Research ArticleСтатьиArticleСпектральный метод оценки эффективных свойств керна на основе быстрого преобразования Фурье: алгоритм и сравнение с методом конечных элементовSpectral method for evaluating effective rock sample properties based on the fast Fourier transform: algorithm and comparison with the finite element methodЦыбаковМихаил НиколаевичTsybakovMikhail Nikolaevichmihailtsybakov@yandex.ruВершининАнатолий ВикторовичVershininAnatoly Viktorovich

доктор физико-математических наук

doctor of physical and mathematical sciences

versh1984@mail.ruМосковский государственный университет им. М. В. ЛомоносоваРоссияLomonosov Moscow State UniversityRussian Federation202529122025264370382Copyright © Цыбаков М.Н., Вершинин А.В., 20252025Цыбаков М.Н., Вершинин А.В.Tsybakov M.N., Vershinin A.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/2092

В статье рассматривается спектральный метод оценки эффективных свойств композитов и кернов на основе быстрого преобразования Фурье (FFT). Имея ограничения наприкладываемые граничные условия, метод предлагает высокую эффективность анализа напряжений и деформаций в периодической ячейке за счет асимптотики FFT по времени 𝑂(𝑛 · 𝑙𝑜𝑔(𝑛)). Проведённое сравнение результатов гомогенизации с Методом Конечных Элементов показывает высокую точность спектрального метода при оценке свойств керна. Представлен анализ эффективности и сходимости алгоритма для различных периодических ячеек и материалов.

The article considers a spectral method for evaluating the effective properties of composites and rock samples based on the Fast Fourier Transform (FFT). Having limitations on the applied boundary conditions, the method offers high efficiency of stress and strain analysis in a periodic cell due to the asymptotics of FFT in time 𝑂(𝑛 · 𝑙𝑜𝑔(𝑛)). The comparison of the results of homogenization with the Finite Element Method shows the high accuracy of the spectral method in estimating the properties of the rocks. An analysis of the efficiency and convergence of the algorithm for various periodic cells and materials is presented.

эффективные свойстваспектральные методыкомпьютерная томографиябыстрое преобразование Фурьеанализ напряжений и деформацийметод конечных элементов.effective propertiesspectral methodsComputed tomographyFast Fourier Transformstress-strain analysisfinite element method.ReferencesMoulinec H., Suquet P. A fast numerical method for computing the linear and nonlinear mechanical properties of composites. // Comptes Rendus de l’Acad´emie des sciences. S´erie II. M´ecanique, physique, chimie, astronomie, 1994. hal-03019226.Moulinec, H. & Suquet, P., 1994. “A fast numerical method for computing the linear and nonlinear mechanical properties of composites,” Comptes Rendus de l’Acad´emie des sciences. S´erie II. M´ecanique, physique, chimie, astronomie. hal-03019226.Levin V.A., Vershinin A.V., Yakovlev M.Y. et al. Computed Tomography Based Stress-Strain Analysis of Heterogeneous Models of Rocks and Biological Tissues Using Unstructured Meshes. // Russian Physics Journal, 2024. v.67, pp. 140–146. https://doi.org/10.1007/s11182-024-03100-9.Levin, V.A., Vershinin, A.V., Yakovlev, M.Y. et al., 2024. “Computed Tomography Based Stress-Strain Analysis of Heterogeneous Models of Rocks and Biological Tissues Using Unstructured Meshes,” Russian Physics Journal, vol. 67, pp. 140–146. https://doi.org/10.1007/s11182-024-03100-9.Moulinec H., Suquet P. A numerical method for computing the overall response of nonlinear composites with complex microstructure. // Computer Methods in Applied Mechanics and Engineering, 1998. v.157, n.1-2, pp. 69–94. doi:10.1016/s0045-7825(97)00218-1.Moulinec, H. & Suquet, P., 1998. “A numerical method for computing the overall response of nonlinear composites with complex microstructure,” Computer Methods in Applied Mechanics and Engineering, vol. 157, no. 1-2, pp. 69–94. doi:10.1016/s0045-7825(97)00218-1.Cooley J.W., Tukey J.W. An algorithm for the machine calculation of complex Fourier series. // Mathematics of Computation, 1965. v.19, n.90, pp. 297-301. https://doi.org/10.1090/S0025-5718-1965-0178586-1.Cooley, J.W. & Tukey, J.W., 1965. “An algorithm for the machine calculation of complex Fourier series,” Mathematics of Computation, vol. 19, no. 90, pp. 297-301. https://doi.org/10.1090/S0025-5718-1965-0178586-1.Willot F., Abdallah B., Pellegrini Y.P. Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields. // International Journal for Numerical Methods in Engineering, 2014. v.98, n.7, pp. 518-533. doi: 10.1002/nme.4641.Willot, F., Abdallah, B. & Pellegrini, Y.P., 2014. “Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields,” International Journal for Numerical Methods in Engineering, vol. 98, no. 7, pp. 518-533. doi: 10.1002/nme.4641.Kaptchouang N.B.N., G´el´ebart L. Multiscale coupling of FFT-based simulations with the LDC approach. // Computer Methods in Applied Mechanics and Engineering, 2022. v.394, Article 114921. doi: 10.1016/j.cma.2022.114921.Kaptchouang, N.B.N. & G´el´ebart, L., 2022. “Multiscale coupling of FFT-based simulations with the LDC approach,” Computer Methods in Applied Mechanics and Engineering, vol. 394, Article 114921. doi: 10.1016/j.cma.2022.114921.Vondˇrejc J., Zeman J., Marek I. An FFT-based Galerkin Method for Homogenization of Periodic Media. // Computers and Mathematics with Applications, 2014. v.68, n.3, pp. 156-173. doi: 10.1016/j.camwa.2014.07.015.Vondˇrejc, J., Zeman, J. & Marek, I., 2014. “An FFT-based Galerkin Method for Homogenization of Periodic Media,” Computers and Mathematics with Applications, vol. 68, no. 3, pp. 156-173. doi: 10.1016/j.camwa.2014.07.015.Nagra J.S., Brahme A., Lebensohn R.A., Inal K. Efficient fast Fourier transform-based numerical implementation to simulate large strain behavior of polycrystalline materials. // International Journal of Plasticity, 2017. v.98, pp. 65-82. doi: 10.1016/j.ijplas.2017.07.001.Nagra, J.S., Brahme, A., Lebensohn, R.A. & Inal, K., 2017. “Efficient fast Fourier transformbased numerical implementation to simulate large strain behavior of polycrystalline materials,” International Journal of Plasticity, vol. 98, pp. 65-82. doi: 10.1016/j.ijplas.2017.07.001.Eloh K.S., Jacques A., Berbenni S. Development of a new consistent discrete Green operator for FFT-based methods to solve heterogeneous problems with eigenstrains. // International Journal of Plasticity, 2019. v.116, pp. 1-23. doi: 10.1016/j.ijplas.2018.10.011.Eloh, K.S., Jacques, A. & Berbenni, S., 2019. “Development of a new consistent discrete Green operator for FFT-based methods to solve heterogeneous problems with eigenstrains,” International Journal of Plasticity, vol. 116, pp. 1-23. doi: 10.1016/j.ijplas.2018.10.011.Cao Y.J., Shen W.Q., Shao J.F., Wang W. A novel FFT-based phase field model for damage and cracking behavior of heterogeneous materials. // International Journal of Plasticity, 2020. Article 102786. doi: 10.1016/j.ijplas.2020.102786.Cao, Y.J., Shen, W.Q., Shao, J.F. & Wang, W., 2020. “A novel FFT-based phase field model for damage and cracking behavior of heterogeneous materials,” International Journal of Plasticity, vol. 2020, Article 102786. doi: 10.1016/j.ijplas.2020.102786.

The authors declare that there are no conflicts of interest present.