Numerical solution of unsteady elastic equations with C-Bézier basis functions

Lanyin Sun; Kunkun Pang; Lanyin Sun; Kunkun Pang

doi:10.3934/math.2024036

AIMS Mathematics

2024, Volume 9, Issue 1: 702-722. doi: 10.3934/math.2024036

Previous Article Next Article

Research article

Numerical solution of unsteady elastic equations with C-Bézier basis functions

Lanyin Sun ^,,
Kunkun Pang

School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

Received: 16 August 2023 Revised: 23 October 2023 Accepted: 13 November 2023 Published: 04 December 2023
MSC : 65D18, 65M60

In this paper, the finite element method is applied to solve the unsteady elastic equations, C-Bézier basis functions are used to construct the shape function spaces, the semi-discrete scheme of the unsteady elastic equations is obtained by Galerkin finite element method and then the fully discretized Galerkin method is obtained by further discretizing the time variable with $ \theta $-scheme finite difference. Furthermore, for several numerical examples, the accuracy of approximate solutions are improved by 1–3 order-of magnitudes compared with the Lagrange basis function in $ L^\infty $ norm, $ L^2 $ norm and $ H^1 $ semi-norm, and the numerical examples show that the method proposed possesses a faster convergence rate. It is fully demonstrated that the C-Bézier basis functions have a better approximation effect in simulating unsteady elastic equations.
- unsteady elastic equation,
- finite element method,
- C-Bézier basis functions
Citation: Lanyin Sun, Kunkun Pang. Numerical solution of unsteady elastic equations with C-Bézier basis functions[J]. AIMS Mathematics, 2024, 9(1): 702-722. doi: 10.3934/math.2024036

Related Papers:

Abstract

In this paper, the finite element method is applied to solve the unsteady elastic equations, C-Bézier basis functions are used to construct the shape function spaces, the semi-discrete scheme of the unsteady elastic equations is obtained by Galerkin finite element method and then the fully discretized Galerkin method is obtained by further discretizing the time variable with $ \theta $-scheme finite difference. Furthermore, for several numerical examples, the accuracy of approximate solutions are improved by 1–3 order-of magnitudes compared with the Lagrange basis function in $ L^\infty $ norm, $ L^2 $ norm and $ H^1 $ semi-norm, and the numerical examples show that the method proposed possesses a faster convergence rate. It is fully demonstrated that the C-Bézier basis functions have a better approximation effect in simulating unsteady elastic equations.

References

[1]	S. Patnaik, S. Sidhardh, F. Semperlotti, A Ritz-based finite element method for a fractional-order boundary value problem of nonlocal elasticity, Int. J. Solids Struct., 202 (2020), 398–417. https://doi.org/10.1016/j.ijsolstr.2020.05.034 doi: 10.1016/j.ijsolstr.2020.05.034
[2]	O. A. González-Estrada, S. Natarajan, J. J. Ródenas, S. P. A. Bordas, Error estimation for the polygonal finite element method for smooth and singular linear elasticity, Comput. Math. Appl., 92 (2021), 109–119. https://doi.org/10.1016/j.camwa.2021.03.017 doi: 10.1016/j.camwa.2021.03.017
[3]	A. M. Vargas, Finite difference method for solving fractional differential equations at irregular meshes, Math. Comput. Simulat., 193 (2022), 204–216. https://doi.org/10.1016/j.matcom.2021.10.010 doi: 10.1016/j.matcom.2021.10.010
[4]	T. A. Bullo, G. A. Degla, G. F. Duressa, Parameter-uniform finite difference method for singularly perturbed parabolic problem with two small parameters, Int. J. Comput. Methods Eng. Sci. Mech., 23 (2022), 210–218. https://doi.org/10.1080/15502287.2021.1948148 doi: 10.1080/15502287.2021.1948148
[5]	J. Jeon, J. Lee, S. J. Kim, Finite volume method network for the acceleration of unsteady computational fluid dynamics: Non‐reacting and reacting flows, Int. J. Energy Res., 46 (2022), 10770–10795. https://doi.org/10.1002/er.7879 doi: 10.1002/er.7879
[6]	U. S. Fjordholm, M. Musch, N. H. Risebro, Well-posedness and convergence of a finite volume method for conservation laws on networks, SIAM J. Numer. Anal., 60 (2022), 606–630. https://doi.org/10.1137/21M145001X doi: 10.1137/21M145001X
[7]	S. Sengupta, N. A. Sreejith, P. Mohanamuraly, G. Staffelbach, L. Gicquel, Global spectral analysis of the Lax-Wendroff-central difference scheme applied to convection-diffusion equation, Comput. Fluids, 242 (2022), 105508. https://doi.org/10.1016/j.compfluid.2022.105508 doi: 10.1016/j.compfluid.2022.105508
[8]	R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., 49 (1943), 1–23.
[9]	R. W. Clough, Y. Rashid, Finite element analysis of axi-symmetric solids, J. Eng. Mech. Div., 91 (1965), 71–85. https://doi.org/10.1061/JMCEA3.0000585 doi: 10.1061/JMCEA3.0000585
[10]	K. Feng, Difference scheme based on variational principle, Appl. Math. Comput., 2 (1965), 238–262.
[11]	M. I. Ivanov, I. A. Kremer, Y. M. Laevsky, Solving the pure Neumann problem by a mixed finite element method, Numer. Anal. Appl., 15 (2022), 316–330. https://doi.org/10.1134/S1995423922040048 doi: 10.1134/S1995423922040048
[12]	H. D. Gao, W. W. Sun, Optimal analysis of non-uniform Galerkin-mixed finite element approximations to the Ginzburg-Landau equations in superconductivity, SIAM J. Numer. Anal., 61 (2023), 929–951. https://doi.org/10.1137/22M1483670 doi: 10.1137/22M1483670
[13]	X. L. Wang, X. L. Meng, S. Y. Zhang, H. F. Zhou, A modified weak Galerkin finite element method for the linear elasticity problem in mixed form, J. Comput. Appl. Math., 420 (2023), 114743. https://doi.org/10.1016/j.cam.2022.114743 doi: 10.1016/j.cam.2022.114743
[14]	B. Deka, N. Kumar, A systematic study on weak Galerkin finite element method for second‐order parabolic problems, Numer. Methods Partial Differ. Equ., 39 (2023), 2444–2474. https://doi.org/10.1002/num.22973 doi: 10.1002/num.22973
[15]	E. Chung, Y. Efendiev, Y. B. Li, Q. Li, Generalized multiscale finite element method for the steady state linear Boltzmann equation, Multiscale Model. Simul., 18 (2020), 475–501. https://doi.org/10.1137/19M1256282 doi: 10.1137/19M1256282
[16]	J. H. Yue, G. R. Liu, M. Li, R. P. Niu, A cell-based smoothed finite element method for multi-body contact analysis using linear complementarity formulation, Int. J. Solids Struct., 141–142 (2018), 110–126. https://doi.org/10.1016/j.ijsolstr.2018.02.016 doi: 10.1016/j.ijsolstr.2018.02.016
[17]	Y. Cheng, Q. Zhang, Local analysis of the fully discrete local discontinuous Galerkin method for the time-dependent singularly perturbed problem, J. Comput. Math., 35 (2017), 265–288. https://doi.org/10.4208/jcm.1605-m2015-0398 doi: 10.4208/jcm.1605-m2015-0398
[18]	J. B. Lin, H. Li, Z. M. Dong, Z. H. Zhao, Error estimations of SUPC stabilized space-time finite element approximations for convection-diffusion-reaction equations (Chinese), Math. Appl., 33 (2020), 275–294. https://doi.org/10.13642/j.cnki.42-1184/o1.2020.02.002 doi: 10.13642/j.cnki.42-1184/o1.2020.02.002
[19]	V. D. Varma, S. K. Nadupuri, N. Chamakuri, A posteriori error estimates and an adaptive finite element solution for the system of unsteady convection-diffusion-reaction equations in fluidized beds, Appl. Numer. Math., 163 (2021), 108–125. https://doi.org/10.1016/j.apnum.2021.01.012 doi: 10.1016/j.apnum.2021.01.012
[20]	Z. C. Shi, On spline finite element method, Math. Numer. Sin., 1 (1979), 50–72. https://doi.org/10.12286/jssx.1979.1.50 doi: 10.12286/jssx.1979.1.50
[21]	T. J. R. Hughes, J. A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng., 194 (2005), 4135–4195. https://doi.org/10.1016/j.cma.2004.10.008 doi: 10.1016/j.cma.2004.10.008
[22]	X. Li, F. Chen, On the instability in the dimension of splines spaces over T-meshes, Comput. Aided Geom. D., 28 (2011), 420–426. https://doi.org/10.1016/j.cagd.2011.08.001 doi: 10.1016/j.cagd.2011.08.001
[23]	X. Peng, H. J. Lian, Z. W. Ma, C. Zheng, Intrinsic extended isogeometric analysis with emphasis on capturing high gradients or singularities, Eng. Anal. Bound. Elem., 134 (2022), 231–240. https://doi.org/10.1016/j.enganabound.2021.09.022 doi: 10.1016/j.enganabound.2021.09.022
[24]	M. J. Peake, J. Trevelyan, G. Coates, Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems, Comput. Methods Appl. Mech. Eng., 284 (2015), 762–780. https://doi.org/10.1016/j.cma.2014.10.039 doi: 10.1016/j.cma.2014.10.039
[25]	Y. P. Zhu, X. L. Han, New cubic rational basis with tension shape parameters, Appl. Math. J. Chin. Univ., 30 (2015), 273–298. https://doi.org/10.1007/s11766-015-3232-8 doi: 10.1007/s11766-015-3232-8
[26]	Q. Y. Chen, G. Z. Wang, A class of Bézier-like curves, Comput. Aided Geom. D., 20 (2003), 29–39. https://doi.org/10.1016/S0167-8396(03)00003-7 doi: 10.1016/S0167-8396(03)00003-7
[27]	C. Y. Li, C. Zhu, Designing developable C-Bézier surface with shape parameters, Mathematics, 8 (2020), 1–21. https://doi.org/10.3390/math8030402 doi: 10.3390/math8030402
[28]	L. Y. Sun, F. M. Su, Application of C-Bézier and H-Bézier basis functions to numerical solution of convection-diffusion equations, Bound. Value. Probl., 2022 (2022), 66. https://doi.org/10.1186/s13661-022-01647-5 doi: 10.1186/s13661-022-01647-5

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)