Multiscale Element–Free Galerkin Method with Penalty for 2D Burgers’ Equation
DOI:
https://doi.org/10.11113/jt.v62.1888Keywords:
Burgers’ equation, multiscale method, meshless method, penalty method, critical areaAbstract
In this paper, a new numerical method which is based on the coupling between multiscale method and meshless method with penalty is developed for 2D Burgers’ equation. The advantage of meshless method over the finite element method (FEM) is that remeshing process is not required. This is because the meshless method approximation is constructed entirely in terms of a set of nodes. Since the moving least squares (MLS) shape function does not satisfy the Kronecker delta property, so penalty method is adopted to enforce the essential boundary conditions in this paper. In order to obtain the fine scale approximation, the local enrichment basis is applied. The local enrichment basis may adopt the polynomial basis functions or any other analytical basis functions. Here, the polynomial basis functions are chosen as local enrichment basis. This multiscale meshless method with penalty will provide a more accurate result especially in the critical region which requires higher accuracy. It is believed that this proposed method is an attractive approach for solving more general problems which involve large deformation.References
Bateman, H. 1915. Some Recent Researches on the Motion of Fluids. MonWeather Rev. 43: 163–70.
Burgers, J. M. 1948. A Mathematical Model Illustrating the Theory of Turbulence. Advanced in Applied Mechanics. 1: 171–199.
Bahadir, A. R. 2003. A Fully Implicit Finite-difference Scheme for Two Dimensional Burgers’ Equation. Applied Mathematics and Computation. 137: 131–137.
Kutluay, S., A. R. Bahadir, A. Ozdes. 1999. Numerical Solution of One Dimensional Burgers Equation: Explicit and Exact-explicit Finite Difference Methods. Journal of Compuational and Applied Math. 103: 251–261.
Froncioni, A. M., P. Labbe, A. Garon, and R. Camarero. 1997. Interpolation-free Space-time Remeshing for the Burgers Equation. Communications in Numerical Methods Engineering. 13: 875–884.
Dogan, A. 2004. A Galerkin Finite Element Approach to Burgers’ Equation. Apply Math. Comput. 157: 331–346.
Chino, E., N. Tosaka. 1998. Dual Reciprocity Boundary Element Analysis of Time-independent Burgers’ Equation. Engineering Analysis with Boundary Elements. 21: 261–270.
Kakuda, K., N. Tosaka. 1990. The Generalized Boundary Element Approach to Burgers’ Equation. Int J Numer Methods Eng. 29: 245– 261.
Belytschko, T., Y. Y. Lu, and L. Gu. 1994. Element-free Galerkin Methods. Int. J. Numer. Methods Eng. 37(2): 229–256.
Belytschko, T., Y. Krongauz, D. Organ, and M. Fleming. 1996. Meshless Method: An Over View and Recent Developments. Comput Methods Appl Mech Eng. 139: 3–47.
Liu, G. R., Y. T. Gu. 2005. An Introduction to Meshfree Methods and Their Programming. Berlin and NewYork: Springer.
Ouyang, J., X. H. Zhang, and L. Zhang. 2009. Element-free Characteristic Galerkin Method for Burgers’ Equation. Engineering Analysis with Boundary Elements. 33: 356–362.
Young, D. L., C. M. Fan, S. P. Hu, and S. N. Atluri. 2008. The Eulerian–Lagrangian Method of Fundamental Solutions for Two-Dimensional Unsteady Burgers’ Equations. Engineering Analysis with Boundary Elements. 32: 395–412.
Xie, S. S., S. Heo, S. Kim, G. Woo, and S. Yi. 2008. Numerical Solution of One-dimensional Burgers’ Equation Using Reproducing Kernel Function. Journal of Computational and Applied Mathematics. 214(2): 417–434.
Zhang, L., J. Ouyang, X. X. Wang, and H. H. Zhang. 2010. Variational Multiscale Element-free Galerkin Method for 2D Burgers’ Equation. Journal of Computational Physics. 229: 7147–7161.
Hughes, T. J. R. 1995. Multiscale Phenomena: Green’s Function, the Dirichlet-to-Neumann Formulation, Subgrid Scale Models, Bubbles and the Origins of Stabilized Methods. Comput. Methods Appl. Mech. Eng. 127: 387–401.
Hughes, T. J. R, G. R. Feijoo, M. Luca, and Q. Jean-Baptiste. 1998. The Variational Multiscale Method-A Paradigm for Computational Mechanics. Comput. Mehtods Appl. Mech. Eng. 166(1–2): 3–24.
Masud, A. 2002. On a stabilized finite element formulation for incompressible Navier-Stokes equations. Proceedings of the Fourth US–Japan Conference on Computational Fluid Dynamics. 28–30.
Masud, A., T. J. R. Hughes. 2002. A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Eng. 191(39–40): 4341–4370.
Masud, A., R. A. Khurram. 2004. A Multiscale/Stabilized Finite Element Method for the Advection-diffusion Equation. Comput. Methods Appl. Mech. Eng. 193(21–22): 1997–2018.
Zhang, L., J. Ouyang, T. Jiang, and C. Ruan. 2011. Variational Multiscale Element Free Galerkin Method for the Water Wave Problems. Journal of Computational Physics. 230(2011): 5045–5060.
Zhang, L., J. Ouyang, X. H. Zhang, and W. B. Zhang. 2008. On a Multi-scale Element-free Galerkin Method for the Stokes Problem. Applied Mathematics and Computation. 203: 745–753.
Jeoung, H. Y., K. Y. Sung. 2008. Variational Multiscale Analysis of Elastoplastic Deformation Using Meshfree Approximation. International Journal of Solids and Structures. 45: 4709–4724.
Lu, Y. Y., T. Belytschko, and L. Gu. 1994. A New Implementation of the Element-free Galerkin Method. Comput. Meth. Appl. Mech. Engrg. 113: 397–414.
Chu, Y. A., B. Moran. 1995. A Computational Model for Nucleation of the Solid-Solid Phase Transformations, Modell. Simul. Mater. Sci. Engrg. 3: 455–471.
Mukherjee, Y. X., S. Mukherjee. 1997. On Boundary Conditions in the Element Free Galerkin Method. Comput. Mech. 19(4): 264–270.
Fan, S. C., X. Liu, and C. K. Lee. 2004. Enriched Partition of Unity Finite Element Method for Stress Intensity Factors at Crack Tips. Computers & Structures. 82(4–5): 445–461.
Hazard, L., P. Bouillard. 2007. Structural Dynamics of Viscoelastic Sandwich Plates by the Partition of Unity Finite Element Method. Comput. Methods in Appl. Mechanics and Eng. 196(41–44): 4101–4116.
Li, S. C., S. C. Li, Y. M. Cheng. 2005. Enriched Meshless Manifold Method for Two-Dimensional Crack Modelling. Theor. Appl. Fract. Mec. 44(3): 234–248.
Melenk, J. M., I. Babuška. 1996. The Partition of Unity Finite Element Method: Basic Theory And Applications. Comput. Methods Appl. Eng. 139: 289–314.
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