L.S. Bryndin, V.A. Belyaev. Collocation methods with fourth degree polynomials on triangular grids and their application to the calculation of bending of round plates with holes ... P. 43-60

A new collocation method ($h$-CM$_4$) is developed for the numerical solution of two-dimensional elliptic problems with second-order highest derivatives. Fourth-degree polynomials on triangular cells of a grid generated by Gmsh are used as an approximation. Unknown coefficients of the polynomial decomposition are determined from the solution of a system of linear algebraic equations (SLAE) consisting of collocation equations, matching conditions, and boundary conditions. In the $h$-CM$_4$, the SLAE is quadratic in contrast to published versions of the least-squares collocation method, where similar equations are written, but the SLAE is overdetermined. This leads to an increase in computation time and the need to search for special values of the weight coefficients multiplying the equations of the approximate problem. The fourth order of convergence of the $h$-CM$_4$ is established numerically on smooth test solutions of the Poisson's equation and of a system of partial differential equations (PDEs) arising in the calculation of bending within the Reissner-Mindlin plate theory (RMPT). The possibility of calculation of the stress-strain state (SSS) of sufficiently thin plates in the RMPT is demonstrated. It is shown that in order to solve the PDE system describing the plate bending within the Kirchhoff-Love plate theory (KLPT) in a mixed formulation, it is necessary to increase the number of equations of the approximate problem in the $h$-CM$_4$. Thus, the approximation is reduced to the construction of a new version of the least-squares collocation method ($h$-LSCM$_4$), whose convergence order is no worse than the third. The SSS of round plates with holes is analyzed depending on the thickness of a plate in the RMPT and KLPT as well as on eccentricity in the case of one hole. Adaptive grids are used to improve accuracy in problems with large gradients and limited smoothness of the solution, which resulted in improving the order of convergence in the latter case. The application of adaptive grids expands the capabilities of the $h$-CM$_4$ and $h$-LSCM$_4$ compared to previous versions of the least-squares collocation method, which is confirmed by numerical examples.

Keywords: collocation method, Poisson’s equation, Reissner–Mindlin theory, Kirchhoff–Love theory, plate bending

Received October 23, 2023

Revised November 20, 2023

Accepted November 27, 2023

Funding Agency: This research was carried out within a state task to the Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences.

Luka Sergeevich Bryndin, doctoral student, Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia; Novosibirsk State University, Novosibirsk, 630090 Russia, e-mail: l.bryndin@g.nsu.ru

Vasilii Alexeyevich Belyaev, Cand. Sci. (Phys.-Math.), Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia, e-mail: belyaevasily@mail.ru

REFERENCES

1.   Belyaev V.A. On the effective implementation and capabilities of the least-squares collocation method for solving second-order elliptic equations. Vychisl. Metody i Progr., 2021, vol. 22, no. 3, pp. 211–229 (in Russian). doi: 10.26089/NumMet.v22r313

2.   Belyaev V.A. Solving a Poisson equation with singularities by the least-squares collocation method // Numer. Anal. Appl., 2020, vol. 13, no. 3, pp. 207–218. doi: 10.1134/S1995423920030027

3.   Reddy J. N. Mechanics of laminated composite plates and shells: theory and analysis, 2nd edn. Boca Raton, London, NY, Washington, D.C.: CRC Press, 2004, 858 p. doi: 10.1201/b12409

4.   Timoshenko S., Woinowsky-Krieger S. Theory of plates and shells, Engineering Societies Monographs, 2nd Ed., NY, Toronto: McGraw-Hill Book Comp., 1959, 580 p. ISBN 0-07-064779-8. Translated to Russian under the title Plastiny i obolochki, Moscow, Fizmatgiz Publ., 1966, 625 p.

5.   Lee W.M., Chen J.T. Free vibration analysis of a circular plate with multiple circular holes by using indirect BIEM and addition theorem. J. Appl. Mech., 2011, vol. 78, no. 1, art. no. 011015, pp. 1–10. doi: 10.1115/1.4001993

6.   Schillinger D., Evans J. A., Reali A., Scott M. A., Hughes T. J. R. Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput. Methods Appl. Mech. Eng., 2013, vol. 267, pp. 170–232. doi: 10.1016/j.cma.2013.07.017

7.   Kireev V.A. Hermite bicubic collocation method in domain with curvilinear boundary. Vestnik Sibirskogo Gosudarstvennogo Aerokosmicheskogo Universiteta imeni Akademika M.F. Reshetneva, 2014, no. 3 (55), pp. 73–77 (in Russian).

8.   Shao W., Wu X., Chen S. Chebyshev tau eshless method based on the integration-differentiation for biharmonic-type equations on irregular domain. Eng. Anal. Bound. Elem., 2012, vol. 36, no. 12, pp. 1787–1798. doi: 10.1016/j.enganabound.2012.06.005

9.   Belyaev V.A., Bryndin L.S., Golushko S.K., Semisalov B.V., Shapeev V.P. H-, p-, and hp-versions of the least-squares collocation method for solving boundary value problems for biharmonic equation in irregular domains and their applications. Comput. Math. Math. Phys., 2022, vol. 62, no. 4, pp. 517–537. doi: 10.1134/S0965542522040029

10.   Mai-Duy N., See H., Tran-Cong T. A spectral collocation technique based on integrated Chebyshev polynomials for biharmonic problems in irregular domains. Appl. Math. Model., 2009, vol. 33, no. 1, pp. 284–299. doi: 10.1016/j.apm.2007.11.002

11.   Semisalov B.V. A fast nonlocal algorithm for solving Neumann–Dirichlet boundary value problems with error control. Vychisl. Metody i Progr., 2016, vol. 17, no. 4, pp. 500–522 (in Russian).

12.   Katsikadelis J.T. Boundary elements: theory and applications, Amsterdam, London, NY, Elsevier, 2002, 448 p. ISBN: 9780080528243 . Translated to Russian under the title Granichnye elementy: teoriya i prilozheniya, Moscow, Izd. Assots. Stroitel’nykh Vuzov, 2007, 348 p. ISBN 978-5-93093-473-1 .

13.   Drozdov G. M., Shapeev V. P. CAS application to the construction of high-order difference schemes for solving Poisson equation. Lect. Notes Comput. Sci., 2014, vol. 8660, pp. 99–110. doi: 10.1007/978-3-319-10515-4_8

14.   Shao W., Wu X. An effective Chebyshev tau meshless domain decomposition method based on the integration–differentiation for solving fourth order equations. Appl. Math. Model., 2015, vol. 39, no. 9, pp. 2554–2569. doi: 10.1016/j.apm.2014.10.048

15.   Sleptsov A. G., Shokin Yu. I. An adaptive grid-projection method for elliptic problems. Comput. Math. Math. Phys., 1997, vol. 37, no. 5, pp. 558–571.

16.   Belyaev V.V., Shapeev V.P. The collocation and least squares method on adaptive grids in a domain with a curvilinear boundary. Vychislit. Tekhnologii, 2000, vol. 5, no. 4, pp. 13–21 (in Russian).

17.   Isaev V.I., Shapeev V.P., Eremin S.A. An investigation of the collocation and the least squares method for solution of boundary value problems for the Navier–Stokes and Poisson equations. Vychislit. Tekhnologii, 2007, vol. 12, no. 3, pp. 53–70 (in Russian).

18.   Golushko S.K., Idimeshev S.V., Shapeev V.P. Application of collocations and least residuals method to problems of the isotropic plates theory. Vychislit. Tekhnologii, 2013, vol. 18, no. 6, pp. 31–43.

19.   Idimeshev S. V. Modificirovannyj metod kollokacij i naimen’shih nevyazok i ego prilozhenie v mekhanike mnogoslojnyh kompozitnyh balok i plastin [Modified method of collocations and least residuals and its application in the mechanics of multilayer composite beams and plates]. Dissertation, Cand. Sci. (Phys.–Math.), Novosibirsk, 2016, 179 p. (in Russian).

20.   Garcia O. Fancello E.A., de Barcellos C.S., Duarte C.A. Hp-clouds in Mindlin’s thick plate model. Int. J. Numer. Methods Eng., 2000, vol. 47, no. 8, pp. 1381–1400. doi: 10.1002/(SICI)1097-0207(20000320)47:8<1381::AID-NME833>3.0.CO;2-9

21.   Tiago C.,  Leitão V.M.A. Eliminating shear-locking in meshless methods: a critical overview and a new framework for structural theories. Advances in meshfree techniques. Computational methods in applied sciences, 2007, vol. 5, pp. 123–147. doi: 10.1007/978-1-4020-6095-3_7

22.   Ike C. C. Mathematical solutions for the flexural analysis of Mindlin’s first order shear deformable circular plates. Mathematical Models in Engineering, 2018, vol. 4, no. 2, pp. 50–72. doi: 10.21595/mme.2018.19825

23.   Davis T.A. Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization ACM Trans. Math. Softw., 2011, vol. 38, no. 1, pp. 8:1–8:22. doi: 10.1145/2049662.2049670

24.   Shapeev V. P., Shapeev A. V. Solutions of the elliptic problems with singularities using finite difference schemes with high order of approximation. Vychislit. Tekhnologii, 2006, vol. 11, part 2, special iss., pp. 84–91 (in Russian).

Cite this article as: L.S. Bryndin, V.A. Belyaev. Collocation methods with fourth degree polynomials on triangular grids and their application to the calculation of bending of round plates with holes. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 43–60.