Semi-discrete Weak Galerkin Finite Element Method of Viscoelastic Wave Propagation
Department of Mathematics of Jinjiang College, Sichuan University, Pengshan 620860, China
| Abstract: | This paper primarily proposes a weak Galerkin finite element of arbitrary order for the wave propagation equation of the Maxwell viscoelasticity model. The quasi-static viscoelastic equations of the Maxwell model serve as the foundation for this approach. By introducing the velocity variable in the model problem, the weak variational form of the velocity-stress model is established. The spatial variables are then discretized using the weak Galerkin finite element method, resulting in a semi-discrete format for the velocity-stress model. In the discretization process of the weak Galerkin finite element method, piecewise polynomials of degree k (k≥1) and k+1 are used to approximate the stress and velocity within each element, and piecewise polynomials of degree k are employed to approximate the trace of the velocity on the element boundaries. By analyzing and proving the method, this paper demonstrates the stability of the semi-discrete format's solution using Gronwall's inequality, and finally establishes the optimal error estimation with respect to the spatial mesh size for this method. The research findings of this paper indicate that by introducing the velocity variable and utilizing the weak Galerkin finite element method, the wave propagation equation of the viscoelasticity based on the Maxwell model can be effectively solved. |
| Keywords: | Viscoelastic Wave Propagation; Weak Galerkin Finite Element; Error Estimate |
| DOI: | 10.57237/j.wjms.2024.02.002 |
View:
Downloads:
Received: 12 September 2024
Accepted: 19 November 2024
Published Online: 23 December 2024
References
| [1] | BLAND D. The theory of viscoelasticity [M]. Oxford: Pergamon Press, 1960. |
| [2] | CHRISTENSEN R M. Theory of viscoelasticity: An introduction [M]. Academic press, 2007. |
| [3] | DILL E H. Continuum mechanics: Elasticity, Plasticity, Viscoelasticity [M]. CRC Press, 2007. |
| [4] | DROZDOV A D. Mechanics of viscoelastic solids [M]. Wiley, 1998. |
| [5] | GURTIN M E, Sternberg E. On the linear theory of viscoelasticity [J]. Archive for Rational Mechanics and Analysis, 1962, 11(1): 291–356. |
| [6] | CARCIONE J, KOSLOFF D, KOSLOFF R. Wave propagation simulation in a linear viscoacoustic medium [J]. Geophysical Journal International, 1988, 93: 393–407. |
| [7] | CARCIONE J, KOSLOFF D, KOSLOFF R. Wave propagation simulation in a linear viscoelastic medium [J]. Geophysical Journal International, 1988, 95: 597–611. |
| [8] | Janovsky V, Shaw S, Warby M K, et. al. Numerical methods for treating problems of viscoelastic isotropic solid deformation [J]. Journal of Computational Applied Mathematics, 63(1-3): 1–107, 1995. |
| [9] | HA T, SANTOS J, SHEEN D. Nonconforming finite element methods for the simulation of waves in viscoelastic solids [J]. Computer Methods in Applied Mechanics and Engineering. 2002, 191: 5647–5670. |
| [10] | ROGNES M, WINTHER R. Mixed finite element methods for linear viscoelasticity using weak symmetry [J], Mathematical Models & Methods in Applied Sciences. 2010, 20: 955–985. |
| [11] | LEE J. Mixed Methods with Weak Symmetry for Time Dependent Problems of Elasticity and Viscoelasticity [D], PhD Thesis, University of Minnesota, 2012. |
| [12] | 王少杰, 谢小平. 基于Maxwell模型的线性粘弹性波传导问题的半离散/全离散杂交应力有限元法 [J]. 高等学校计算数学学报, 2021, 43(1): 28-58. |
| [13] | YUAN H, XIE X P. Mixed finite element discretization for Maxwell viscoelastic model of wave propagation [J]. Advances in Applied Mathematics and Mechanics. 2022, 14(2): 344–364. |
| [14] | Xiao J H, Zhu Z M, Xie X P. Semi-discrete and fully discrete weak Galerkin finite element methods for a quasistatic Maxwell viscoelastic model [J]. Numerical Mathematics: Theory, Methods and Applications. 2023, 16(1): 79-110. |
| [15] | WANG J P, YE X. A weak Galerkin finite element method for second order elliptic problems [J]. J. Comput. Appl Math, 2013, 241: 103-115. |
| [16] | WANG J P, YE X. A Weak Galerkin mixed finite element method for second-order elliptic problems [J]. Mathematics of Computation, 2014, 83: 2101-2126. |
| [17] | CHEN G, FENG M F, XIE X P, A class of robust WG finite element method for convection diffusion-reaction equations, Journal of Computational Applied Mathematics. 315 (2017), 107–125. |
| [18] | LIN R C, YE X, ZHANG S Y, et. al. A weak Galerkin finite element method for singularly perturbed convection- diffusion-reaction problems [J]. SIAM Journal of Numerical Analysis, 2018, 56: 1482–1497. |
| [19] | CHEN G, XIE X P. A robust weak Galerkin finite element for linear elasticity with strong symmetric stresses [J], Computational Methods in Applied Mathematics, 2016, 16: 389–408. |
| [20] | 张然. 弱有限元方法在线弹性问题中的应用 [J]. 计算数学, 2020, 42(1): 1-17. |
| [21] | ZHAI Q L, ZHANG R, WANG X. A hybridized weak Galerkin finite element scheme for the Stokes equations [J]. Science. China Mathematics, 2015, 58: 2455-2472. |
| [22] | 王军平, 叶秀, 张然. 弱有限元方法简论 [J]. 计算数学, 2016, 38(3): 289-308. |
| [23] | 王军平, 王春梅. 椭圆型偏微分方程的弱有限元方法 [J]. 中国科学: 数学, 2015, 45(7): 1061-1092. |
| [24] | Wang X P, Gao F Z, Sun Z J. Weak Galerkin finite element method for viscoelastic wave equations [J]. Journal of Computational Applied Mathematics, 2020, 375, 112816. |
Join the Editorial Board
We invite active, qualified and high profile scientists and researchers to join as Editorial Board Members.
Join UsBecome a Reviewer
Scholars with a strong interest in reviewing are invited to join the reviewer panel to ensure the quality of the research to be published.
Join Us