Implicit rare mesh scheme for solving 3D elasticity problems
https://doi.org/10.26907/2541-7746.2025.1.169-180
Abstract
Anew implicit scheme for solving 3D dynamic elasticity problems was considered. To approximate the elasticity equations by spatial coordinates, a rare mesh FEM scheme based on a four-node finite element with a linear approximation of displacements within the element was employed. The finite elements are located in the centers of hexahedral cells, with each cell containing a single element. As a result, for meshes with the same element size, this scheme uses five times fewer finite elements and half as many nodes as traditional schemes utilizing four-node linear finite elements, which makes it highly efficient. The equations were approximated in time based on the implicit unconditionally stable Crank–Nicolson numerical scheme (trapezoidal rule). The applicability of the scheme was discussed, with a focus on the class of problems for which it outperforms the explicit scheme. An example of a test model problem solved using this scheme was provided.
Keywords
implicit scheme,
rare mesh scheme,
3D elasticity problem,
finite element method,
Crank–Nicolson scheme
Supporting agencies: This study was supported by the Russian Science Foundation (project no. 24 29-00422).
About the Authors
D. T. Chekmarev
Lobachevsky State University of Nizhny Novgorod
Russian Federation
Russian Federation
Dmitry T. Chekmarev, Dr. Sci. (Physics and Mathematics), Associate Professor, Leading Researcher, Research Institute of Mechanics
Nizhny Novgorod
E. G. Glazova
Lobachevsky State University of Nizhny Novgorod
Russian Federation
Russian Federation
Elena G. Glazova, Cand. Sci. (Physics and Mathematics), Senior Researcher, Research Institute of Mechanics
Nizhny Novgorod
E. G. Glazova
Lobachevsky State University of Nizhny Novgorod
Russian Federation
Russian Federation
Elena G. Glazova, Junior Researcher, Research Institute of Mechanics
Nizhny Novgorod
References
1. Zhidkov A.V., Zefirov S.V., Kastalskaya K.A., Spirin S.V., Chekmarev D.T. Rare mesh scheme for solution of three-dimensional dynamic problems of elasticity and plasticity. Vestn. NNGU, 2011, no. 4 (4), pp. 1480–1482. (In Russian)
2. Krutova K.A. Numerical solution of three-dimensional dynamic problems of the elasticity and plasticity theory based on a rare variational-difference scheme. Extended Abstract of Cand. Sci. (Physics and Mathematics) Diss. Nizhny Novgorod, 2015. (In Russian)
3. Zhidkov A.V., Spirin S.V., Chekmarev D.T. A rare mesh finite element scheme for solving static elasticity problems. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2012, vol. 154, no. 4, pp. 26–32. (In Russian)
4. Lakes R. Elastic and viscoelastic behavior of chiral materials. Int. J. Mech. Sci., 2001, vol. 43, no. 7, pp. 1579–1589. https://doi.org/10.1016/S0020-7403(00)00100-4.
5. Coombs W.M., Charlton T.J., Cortis M., Augarde Ch.E. Overcoming volumetric locking in material point methods. Comput. Methods Appl. Mech. Eng., 2018, vol. 333, pp. 1–21. https://doi.org/10.1016/j.cma.2018.01.010.
6. Courant R., Friedrichs K., Lewy G. On difference equations of mathematical physics. Usp. Mat. Nauk, 1941, no. 8, pp. 125–160. (In Russian)
7. Burago N.G., Nikitin I.S. Matrix-free conjugate gradient implementation of implicit schemes. Comput. Math. Math. Phys., 2018, vol. 58, no. 8, pp. 1247–1258. https://doi.org/10.1134/S0965542518080043.
8. Tovbis E., Krutikov V., Stanimirovi´c P., Meshechkin V., Popov A., Kazakovtsev L. A family of multi-step subgradient minimization methods. Mathematics, 2023, vol. 11, no. 10, art. 2264. https://doi.org/10.3390/math11102264.
9. Carson A.M., Banks J.W., Henshaw W.D., Schwendeman D.W. High-order accurate implicit explicit time-stepping schemes for wave equations on overset grids. https://doi.org/10.48550/arXiv.2404.14592.
10. Golubev V.I., Nikitin I.S., Mi X. Numerical schemes of higher approximation orders for dynamic problems of elastoviscoplastic media. Zh. SFU. Ser. Mat. Fiz., 2024, vol. 17, no. 1, pp. 8–17. (In Russian)
11. Golovanov A.I., Berezhnoi D.V. Metod konechnykh elementov v mekhanike deformiruemykh tverdykh tel [Finite Element Method in Mechanics of Deformable Solids]. Kazan, DAS, 2001. 300 p. (In Russian)
12. Zienkiewicz O. Metod konechnykh elementov v tekhnike [The Finite Element Method in Engineering Science]. Moscow, Mir, 1975. 541 p. (In Russian)
13. Bathe K., Wilson E. Chislennye metody analiza i metod konechnykh elementov [Numerical Methods in Finite Element Analysis]. Alekseev A.S. et al. (Trans.). Smirnov A.F. (Ed.). Moscow, Stroiizdat, 1982. 448 p. (In Russian)
14. Samarskii A.A., Gulin A.V. Chislennye metody. Ucheb. posobie dlya vuzov [Numerical Methods: A Textbook for Universities]. Moscow, Nauka, Gl. Red. Fiz.-Mat. Lit., 1989. 432 p. (In Russian)
15. Danilin A.N., Markov A.V. Modeling the dynamics of flexible rod systems unfolding with various initial geometry transformations. Mater. VIII Mezhdunar. simpoz. “Dinamicheskie i tekhnologicheskie problemy mekhaniki konstruktsii i sploshnykh sred” (Yaropolets, 11–15 fevr. 2002 g.) [Proc. VIII Int. Symp. “Dynamic and Technological Problems of Structural and Continuous Media Mechanics” (Yaropolets, February 11–15, 2002)]. Moscow, 2002, p. 61. (In Russian)
16. Shalashilin V.I., Kuznetsov E.B. Metod prodolzheniya resheniya po parametru i nailuchshaya parametrizatsiya v prikladnoi matematike i mekhanike [Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics]. Moscow, Ed. URSS, 1999. 224 p. (In Russian)
17. Bazhenov V.G., Lomunov V.K. Methodology for calculating the dynamic deformation of geometrically variable flat rod systems. Probl. Prochn. Plast., 2002, no. 64, pp. 55–63. (In Russian)
Review
For citations:
Chekmarev D.T., Glazova E.G., Glazova E.G. Implicit rare mesh scheme for solving 3D elasticity problems. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2025;167(1):169-180. (In Russ.) https://doi.org/10.26907/2541-7746.2025.1.169-180
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