Geodesic
Triangulation method for approximate solving of variational problems in nonlinear elasticity
The Bulletin of Irkutsk State University. Series Mathematics, Tome 45 (2023), pp. 54-72 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: stored energy functional, variational problem, finite element method.
Mots-clés : gradient descent method, Delaunay triangulation 
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Vladimir A. Klyachin; Vladislav V. Kuzmin; Ekaterina V. Khizhnyakova. Triangulation method for approximate solving of variational problems in nonlinear elasticity. The Bulletin of Irkutsk State University. Series Mathematics, Tome 45 (2023), pp. 54-72. http://geodesic.mathdoc.fr/item/IIGUM_2023_45_a3/

[1] Boluchevskaya A.V., “On the Quasiisometric Mapping Preserving Simplex Orientation”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 13:2 (2013), 20–23 (in Russian) | DOI | Zbl

[2] Vodop'yanov S.K., Molchanova A.O., “Variational problems of nonlinear elasticity in certain classes of mappings with finite distortion”, Doklady Mathematics, 92:3 (2015), 739–742 | DOI | DOI | MR | Zbl

[3] Glowinski R., Lions J.-L., Tremolieres R., Numerical analysis of variational inequalities, Mir Publ, M., 1979, 576 pp. (in Russian)

[4] Delone B.N., “Geometry of positive quadratic forms”, Uspekhi Mat. Nauk, 1937, no. 3, 16–62 (in Russian)

[5] Klyachin V.A., Shirokii A.A., “The Delaunay triangulation for multidimensional surfaces and its approximative properties”, Russian Mathematics, 56 (2012), 27–34 | DOI | MR | Zbl

[6] Klyachin V.A., Pabat E.A., “The $C^1$-approximation of the level surfaces of functions defined on irregular meshes”, Sibirskii Zhurnal Industrial'noi Matematiki, 13:2 (2010), 69–78 (in Russian) | Zbl

[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, 56–60 (in Russian)

[8] Klyachin V.A., “On a multidimensional analogue of the Schwarz example”, Izvestiya: Mathematics, 76:4 (2012), 681–687 | DOI | DOI | MR | Zbl

[9] Klyachin V.A., “Modified Delaunay empty sphere condition in the problem of approximation of the gradient”, Izvestiya: Mathematics, 80:3 (2016), 549–556 | DOI | DOI | MR | Zbl

[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, 16:3 (2016), 55–60 (in Russian) | DOI | MR | Zbl

[11] Prokhorova M.F., “Problems of homeomorphism arising in the theory of grid generation”, Proceedings of the Steklov Institute of Mathematics, 261:1 (2008), 165–182 | DOI | MR

[12] Ciarlet Ph., Mathematical Elasticity, Mir Publ, M., 1992, 472 pp.

[13] Ball J. M., “Convexity conditions and existence theorems in nonlinear elasticity”, Arch. Ration. Mech. Anal., 1977, no. 63, 337–403 | MR | Zbl

[14] Delaunay B.N., “Sur la sphere vide. A la memoire de Georges Voronoi”, Izvestiya: Mathematics, 1934, no. 6, 793–800 | Zbl

[15] R. Ortigosa, J. Martinez-Frutos, C. Mora-Corral, P. Pedregal, F. Periago, “Optimal control of soft materials using a Hausdorff distance functional”, SIAM Journal on Control and Optimization, 59:1 (2021), 393–416 | DOI | MR | Zbl

[16] Vodopyanov S. K., Molchanova A., “Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity”, Calculus of Variations and PDE, 59 (2020), 1–25 | DOI | MR | Zbl