Geodesic
Cubic version of the least-squares collocation method and its application to plate bending analysis
Sibirskij žurnal industrialʹnoj matematiki, Tome 27 (2024) no. 3, pp. 36-56 Cet article a éte moissonné depuis la source Math-Net.Ru

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A new cubic version of the least-squares collocation method based on adaptive grids is developed. The approximate values of the solution and its first derivatives at the vertices of quadrangular cells are the unknowns. This approach has made it possible to eliminate the matching conditions from the global overdetermined system of linear algebraic equations consisting of collocation equations and boundary conditions. The preconditioned system is solved using the SuiteSparse library by the orthogonal method with the CUDA parallel programming technology. We consider the Reissner—Mindlin plate problem in a mixed setting. A higher accuracy of deflections and rotations of the transverse normal in comparison with the isogeometric collocation method as well as the uniform convergence of shear forces in the case of a thin plate are shown in the proposed method. Bending of an annular plate and round plates with an off-center hole is analyzed. An increase in the shear force gradient in the vicinity of the hole is shown both with a decrease in the plate thickness and with an increase in the eccentricity. The second order of convergence of the developed method is shown numerically. The results obtained using the Reissner—Mindlin theory are compared with the ones in the Kirchhoff—Love theory and three-dimensional finite element simulation.
Keywords: least-squares collocation method, automatic solution continuity, adaptive grid, Reissner—Mindlin plate theory, off-center hole. 
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S. K. Golushko; L. S. Bryndin; V. A. Belyaev; A. G. Gorynin. Cubic version of the least-squares collocation method and its application to plate bending analysis. Sibirskij žurnal industrialʹnoj matematiki, Tome 27 (2024) no. 3, pp. 36-56. http://geodesic.mathdoc.fr/item/SJIM_2024_27_3_a3/

[1] Reddy J. N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edn., CRC Press, Boca Raton—London—N. Y.—Washington, 2004 | DOI

[2] S. K. Golushko and Yu. V. Nemirovskii, Direct and Inverse Problems of Mechanics of Elastic Composite Plates and Shells of Revolution, Fizmatlit, M., 2008 (in Russian)

[3] Ya. M. Grigorenko and A. M. Timonin, “Solution of problems on bending of plates of complex shape in orthogonal curvilinear coordinates”, Dokl. Akad. Nauk. Ukr. SSR. Ser. A. Fiz.-Mat. Tekh. Nauki, 1987, no. 2, 51–54 (in Russian) | Zbl

[4] Ascher U. M., Mattheij R. M. M., Russell R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, 1995 | DOI | MR | Zbl

[5] S. K. Golushko, V. V. Gorshkov, and A. V. Yurchenko, “On two numerical methods for solving multipoint nonlinear boundary value problems”, Vychisl. Tekhnol., 7:2 (2002), 24–33 (in Russian) | MR

[6] S. K. Golushko, E. V. Morozova, and A. V. Yurchenko, “On the numerical solution of boundary value problems for stiff systems of differential equations”, Vestn. KazNU. Ser. Mat. Mekh. Inf., 10:2 (2005), 12–26 (in Russian)

[7] Luo Y., Shear Locking in Finite Elements, licentiate thesis, Kungliga Tekniska högskolan, Stockholm, 1997

[8] S. A. Ambartsumyan, Theory of Anisotropic Plates: Strength, Stability, and Vibrations, Nauka, M., 1987 (in Russian) | MR

[9] S. V. Idimeshev, A modified method of collocations and least residuals and its application in mechanics of multilayer composite beams and plates, Cand. of. Sci. (Phys.-Math.) Dissertation, IVT SO RAN, Novosibirsk, 2016 (in Russian)

[10] 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., 47:8 (2000), 1381–1400 | 3.0.CO;2-9 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | Zbl

[11] Kiendl J., Auricchio F., Beirão da Veiga L., Lovadina C., Reali A., “Isogeometric collocation methods for the Reissner—Mindlin plate problem”, Comput. Methods Appl. Mech. Eng., 284:12 (2015), 489–507 | DOI | MR | Zbl

[12] Ben-Artzi M., Chorev I., Croisille J.-P., Fishelov D., “A compact difference scheme for the biharmonic equation in planar irregular domains”, SIAM J. Numer. Anal., 47:4 (2009), 3087–3108 | DOI | MR | Zbl

[13] Shao W., Wu X., Chen S., “Chebyshev tau meshless method based on the integration-differentiation for biharmonic-type equations on irregular domain”, Eng. Anal. Bound. Elem., 36:12 (2012), 1787–1798 | DOI | MR | Zbl

[14] S. K. Golushko, S. V. Idimeshev, and V. P. Shapeev, “Method of collocations and least residuals in application to problems of mechanics of isotropic plates”, Vychisl. Tekhnol., 18:6 (2013), 31–43 (in Russian)

[15] V. A. Belyaev, L. S. Bryndin, S. K. Golushko, B. V. Semisalov, and V. P. Shapeev, “h-, p-, and hpversions of the least-squares collocation method for solving boundary value problems for biharmonic equation in irregular domains and their applications”, Comput Math. Math. Phys., 62:4 (2022), 517–537 | DOI | DOI | MR | Zbl

[16] V. A. Belyaev and V. P. Shapeev, “Variants of the collocation and least residual methods for solving problems of mathematical physics in convex quadrangular domains”, Model. Anal. Inf. Sist, 24:5 (2017), 629–648 (in Russian) | MR

[17] B. D. Annin and Y. M. Volchkov, “Nonclassical models of the theory of plates and shells”, J. Appl. Mech. Tech. Phys., 57:5 (2016), 769–776 | DOI | DOI | MR | Zbl

[18] Drozdov G. M., Shapeev V. P., “CAS application to the construction of high-order difference schemes for solving Poisson equation”, Lect. Notes Comput. Sci., 8660, 2014, 99–110 | DOI | Zbl

[19] V. A. Belyaev, “Solving a Poisson equation with singularities by the least-squares collocation method”, Numer. Anal. Appl., 13:3 (2020), 207–218 | DOI | DOI | MR | Zbl

[20] A. G. Sleptsov and Yu. I. Shokin, “An adaptive grid-projection method for elliptic problems”, Comput. Math. Math. Phys., 37:5 (1997), 558–571 | MR | MR | Zbl

[21] Semisalov B. V., Belyaev V. A., Bryndin L. S., Gorynin A. G., Blokhin A. M., Golushko S. K., Shapeev V. P., “Verified simulation of the stationary polymer fluid flows in the channel with elliptical cross-section”, Appl. Math. Comput., 430 (2022), 127294, 1–25 | DOI | MR

[22] Katsikadelis J. T., Boundary Elements: Theory and Applications, Elsevier, Amsterdam—London—New York—Oxford—Paris—Tokyo—Boston—San Diego—San Francisco—Singapore—Sydney, 2002

[23] 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., 267 (2013), 170–232 | DOI | MR | Zbl

[24] V. I. Isaev, V. P. Shapeev, and S. A. Eremin, “Study of the properties of the collocation and least squares method for solving boundary value problems for the Poisson equation and the Navier—Stokes equations”, Vychisl. Tekhnol., 12:3 (2007), 53–70 (in Russian) | Zbl

[25] Zienkiewicz O. C., Taylor R. L., Zhu J. Z., The Finite Element Method: Its Basis and Fundamentals, eth edn, Elsevier, Amsterdam—Boston—Heidelberg—London—New York—Oxford—Paris—San Diego—San Francisco—Singapore—Sydney—Tokyo, 2013 | MR | Zbl

[26] Cho J. Y., Atluri S. N., “Analysis of shear flexible beams, using the meshless local Petrov—Galerkin method, based on a locking-free formulation”, Eng. Comput., 18:1/2 (2001), 215–240 | DOI | MR | Zbl

[27] V. A. Nesterov, “Finite element calculation of a cylindrical shell pliable under transverse shear”, Vestn. SibGU im. akad. M. F. Reshetneva, 2013, no. 2, 64–70 (in Russian)

[28] S. K. Golushko, S. V. Idimeshev, and V. P. Shapeev, “Development and application of the collocation and least residuals method to solving problems of mechanics of anisotropic layered plates”, Komp'ut. Tekh., 19:5 (2014), 24–36 (in Russian) | Zbl

[29] Reberol M., Georgiadis C., Remacle J.-F., Quasi-structured quadrilateral meshing in Gmsh — a robust pipeline for complex CAD models, 2021 | DOI

[30] V. A. Kireev, “Collocation method with bicubic Hermitian basis in a domain with a curvilinear boundary”, Vestn. SibGU im. akad. M. F. Reshetneva, 2014, no. 3, 73–77 (in Russian)

[31] J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997 | DOI | MR | Zbl

[32] Ramšak M., Škerget L., “A subdomain boundary element method for high-Reynolds laminar flow using stream function-vorticity formulation”, Int. J. Numer. Meth. Fluids, 46:8 (2004), 815–847 | DOI | Zbl

[33] Davis T. A., “Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization”, ACM Trans. Math. Softw., 38:1 (2011), 1–22 | DOI | MR | Zbl

[34] SuiteSparse, https://github.com/DrTimothyAldenDavis/SuiteSparse/blob/dev/SPQR/Demo/qrdemo_gpu.cpp

[35] Ike C. C., “Mathematical solutions for the flexural analysis of Mindlin's first order shear deformable circular plates”, Math. Models Eng., 4:2 (2018), 50–72 | DOI

[36] Dhondt G., CalculiX crunchix user's manual version 2.12 https://www.dhondt.de/ccx_2.12.pdf