рус eng
«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2023. Vol 45

Triangulation Method for Approximate Solving of Variational Problems in Nonlinear Elasticity

Author(s)
Vladimir V. Klyachin1,2, Vladislav V. Kuzmin1,2, Ekaterina V. Khizhnyakova1,2

1Volgograd State University, Volgograd, Russian Federation

2Novosibirsk State University, Novosibirsk, Russian Federation

Abstract
A variational problem for the minimum of the stored energy functional is considered in the framework of the nonlinear theory of elasticity, taking into account admissible deformations. An algorithm for solving this problem is proposed, based on the use of a polygonal partition of the computational domain by the Delaunay triangulation method. Conditions for the convergence of the method to a local minimum in the class of piecewise affine mappings are found.
About the Authors

Vladimir A. Klyachin, Dr. Sci. (Phys.–Math.), Prof., Volgograd State University, Volgograd, 400062, Russian Federation, klchnv@mail.ru

Vladislav V. Kuzmin, Postgraduate, Volgograd State University, Volgograd, 400062, Russian Federation, vlad329@yandex.ru

Ekaterina V. Khizhnyakova, Senior Lecturer, Volgograd State University, Volgograd, 400062, Russian Federation, kate1995yakovleva@yandex.ru

For citation
Klyachin V. A., Kuzmin V. V., Khizhnyakova E. V. Triangulation Method for Approximate Solving of Variational Problems in Nonlinear Elasticity. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 45, pp. 54–72. (in Russian) https://doi.org/10.26516/1997-7670.2023.45.54
Keywords
stored energy functional, variational problem, gradient descent method, Delaunay triangulation, finite element method
UDC
517.97
MSC
49J35, 65K10
DOI
https://doi.org/10.26516/1997-7670.2023.45.54
References
  1. Boluchevskaya A.V. On the Quasiisometric Mapping Preserving Simplex Orientation. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, no. 2, pp. 20–23. https://doi.org/10.18500/1816-9791-2013-13-1-2-20-23 (in Russian)
  2. Vodop’yanov S.K., Molchanova A.O. Variational problems of nonlinear elasticity in certain classes of mappings with finite distortion. Doklady Mathematics, 2015, vol. 92, no. 3, pp. 739–742. https://doi.org/10.7868/S086956521535008X (in Russian)
  3. Glowinski R., Lions J.-L., Tremolieres R. Numerical analysis of variational inequalities. Moscow, Mir Publ., 1979, 576 p. (in Russian)
  4. Delone B.N. Geometry of positive quadratic forms. Uspekhi Mat. Nauk, 1937, no. 3, pp. 16–62. (in Russian)
  5. Klyachin V.A., Shirokii A.A. The Delaunay triangulation for multidimensional surfaces and its approximative properties. Russian Mathematics, 2012, vol. 56, pp. 27–34. https://doi.org/10.3103/S1066369X12010045 (in Russian)
  6. Klyachin V.A., Pabat E.A. The 𝐶1-approximation of the level surfaces of functions defined on irregular meshes. Sibirskii Zhurnal Industrial’noi Matematiki, 2010, vol. 13, no. 2, pp. 69–78. (in Russian)
  7. Klyachin V.A., Chebanenko N.A. About linear preimages of continuous maps, that preserve orientation of triangles. Mathematical Physics and Computer Simulation, 2014, no. 3, pp. 56–60. (in Russian)
  8. Klyachin V.A. On a multidimensional analogue of the Schwarz example. Izvestiya: Mathematics, 2012, vol. 76, no. 4, pp. 681–687. https://doi.org/10.4213/im6845 (in Russian)
  9. Klyachin V.A. Modified Delaunay empty sphere condition in the problem of approximation of the gradient. Izvestiya: Mathematics, 2016. vol. 80, no. 3, pp. 549–556. https://doi.org/10.4213/im8350 (in Russian)
  10. Molchanova A. A variational approximation scheme for the elastodynamic problems in the new class of admissible mappings. Siberian Journal of Pure and Applied Mathematics, 2016, vol. 16, no. 3. pp. 55–60.https://doi.org/10.17377/PAM.2016.16.305 (in Russian)
  11. Prokhorova M.F. Problems of homeomorphism arising in the theory of grid generation. Proceedings of the Steklov Institute of Mathematics, 2008, vol. 261, no. 1, pp. 165–182. https://doi.org/10.1134/S0081543808050155 (in Russian)
  12. Ciarlet Ph. Mathematical Elasticity. Moscow, Mir Publ., 1992, 472 p.
  13. Ball J.M. Convexity conditions and existence theorems in nonlinear elasticity. Arch.Ration. Mech. Anal, 1977, no. 63, pp. 337–403.
  14. Delaunay B.N. Sur la sphere vide. A la memoire de Georges Voronoi. Izvestiya: Mathematics, 1934, no. 6, pp. 793–800.
  15. Ortigosa R., Martinez-Frutos J., Mora-Corral C., Pedregal P., Periago F. Optimal control of soft materials using a Hausdorff distance functional. SIAM Journal on Control and Optimization, 2021, vol. 59, no. 1, pp. 393–416.https://doi.org/10.13140/RG.2.2.25255.50084
  16. Vodopyanov S.K., Molchanova A. Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity. Calculus of Variations and PDE, 2020, vol. 59, pp. 1–25. https://doi.org/10.1007/s00526-019-1671-4

Full text (russian)