Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)
Abstract
Summary The finite element methods (FEMs) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES‐FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial basis functions , which we construct using local weighted least‐squares approximations. The method preserves the theoretical framework of FEM and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES‐FEM can use higher‐degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES‐FEM and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time‐independent convection–diffusion equation. The numerical results demonstrate that AES‐FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM overmore »
- Authors:
-
[1];
- Department of Applied Mathematics and Statistics Stony Brook University Stony Brook 11794 NY USA
- Publication Date:
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1401188
- Resource Type:
- Publisher's Accepted Manuscript
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Additional Journal Information:
- Journal Name: International Journal for Numerical Methods in Engineering Journal Volume: 108 Journal Issue: 9; Journal ID: ISSN 0029-5981
- Publisher:
- Wiley Blackwell (John Wiley & Sons)
- Country of Publication:
- United Kingdom
- Language:
- English
Citation Formats
Conley, Rebecca, Delaney, Tristan J., and Jiao, Xiangmin. Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM). United Kingdom: N. p., 2016.
Web. doi:10.1002/nme.5246.
Conley, Rebecca, Delaney, Tristan J., & Jiao, Xiangmin. Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM). United Kingdom. https://doi.org/10.1002/nme.5246
Conley, Rebecca, Delaney, Tristan J., and Jiao, Xiangmin. Wed .
"Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)". United Kingdom. https://doi.org/10.1002/nme.5246.
@article{osti_1401188,
title = {Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)},
author = {Conley, Rebecca and Delaney, Tristan J. and Jiao, Xiangmin},
abstractNote = {Summary The finite element methods (FEMs) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES‐FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial basis functions , which we construct using local weighted least‐squares approximations. The method preserves the theoretical framework of FEM and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES‐FEM can use higher‐degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES‐FEM and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time‐independent convection–diffusion equation. The numerical results demonstrate that AES‐FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor‐quality meshes. Copyright © 2016 John Wiley & Sons, Ltd.},
doi = {10.1002/nme.5246},
journal = {International Journal for Numerical Methods in Engineering},
number = 9,
volume = 108,
place = {United Kingdom},
year = {Wed Mar 23 00:00:00 EDT 2016},
month = {Wed Mar 23 00:00:00 EDT 2016}
}
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Works referenced in this record:
On some convergence results for FDM with irregular mesh
journal, March 1984
- Demkowicz, Leszek; Karafiat, Andrzej; Liszka, Tadeusz
- Computer Methods in Applied Mechanics and Engineering, Vol. 42, Issue 3
The Mathematical Theory of Finite Element Methods
book, January 2008
- Brenner, Susanne C.; Scott, L. Ridgway
- Texts in Applied Mathematics
A review of extended/generalized finite element methods for material modeling
journal, April 2009
- Belytschko, Ted; Gracie, Robert; Ventura, Giulio
- Modelling and Simulation in Materials Science and Engineering, Vol. 17, Issue 4
Generalizing the finite element method: Diffuse approximation and diffuse elements
journal, January 1992
- Nayroles, B.; Touzot, G.; Villon, P.
- Computational Mechanics, Vol. 10, Issue 5
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
journal, January 2002
- Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo
- SIAM Journal on Numerical Analysis, Vol. 39, Issue 5
Generalized finite element method for modeling nearly incompressible bimaterial hyperelastic solids
journal, October 2008
- Srinivasan, K. R.; Matouš, K.; Geubelle, P. H.
- Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 51-52
The finite difference method at arbitrary irregular grids and its application in applied mechanics
journal, February 1980
- Liszka, T.; Orkisz, J.
- Computers & Structures, Vol. 11, Issue 1-2
An h-p adaptive method using clouds
journal, December 1996
- Duarte, C. Armando; Oden, J. Tinsley
- Computer Methods in Applied Mechanics and Engineering, Vol. 139, Issue 1-4
TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator
journal, February 2015
- Si, Hang
- ACM Transactions on Mathematical Software, Vol. 41, Issue 2
Selected computational aspects of the meshless finite difference method
journal, July 2012
- Milewski, Sławomir
- Numerical Algorithms, Vol. 63, Issue 1
Estimating differential quantities using polynomial fitting of osculating jets
journal, February 2005
- Cazals, F.; Pouget, M.
- Computer Aided Geometric Design, Vol. 22, Issue 2
A new mesh generation scheme for arbitrary planar domains
journal, August 1985
- Lo, S. H.
- International Journal for Numerical Methods in Engineering, Vol. 21, Issue 8
Generalized finite element methods for three-dimensional structural mechanics problems
journal, June 2000
- Duarte, C. A.; Babuška, I.; Oden, J. T.
- Computers & Structures, Vol. 77, Issue 2
Solving parabolic and hyperbolic equations by the generalized finite difference method
journal, December 2007
- Benito, J. J.; Ureña, F.; Gavete, L.
- Journal of Computational and Applied Mathematics, Vol. 209, Issue 2
On the Angle Condition in the Finite Element Method
journal, April 1976
- Babuška, I.; Aziz, A. K.
- SIAM Journal on Numerical Analysis, Vol. 13, Issue 2
The partition of unity finite element method: Basic theory and applications
journal, December 1996
- Melenk, J. M.; Babuška, I.
- Computer Methods in Applied Mechanics and Engineering, Vol. 139, Issue 1-4
A general finite difference method for arbitrary meshes
journal, April 1975
- Perrone, Nicholas; Kao, Robert
- Computers & Structures, Vol. 5, Issue 1
Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator
book, January 1996
- Shewchuk, Jonathan Richard
- Applied Computational Geometry Towards Geometric Engineering, p. 203-222
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986
- Saad, Youcef; Schultz, Martin H.
- SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3
A Survey of Condition Number Estimation for Triangular Matrices
journal, December 1987
- Higham, Nicholas J.
- SIAM Review, Vol. 29, Issue 4
Application of the generalized finite difference method to solve the advection–diffusion equation
journal, February 2011
- Prieto, Francisco Ureña; Benito Muñoz, Juan José; Corvinos, Luis Gavete
- Journal of Computational and Applied Mathematics, Vol. 235, Issue 7
Delaunay refinement algorithms for triangular mesh generation
journal, May 2002
- Shewchuk, Jonathan Richard
- Computational Geometry, Vol. 22, Issue 1-3
A spectral element method for fluid dynamics: Laminar flow in a channel expansion
journal, June 1984
- Patera, Anthony T.
- Journal of Computational Physics, Vol. 54, Issue 3
Finite difference techniques for variable grids
journal, February 1972
- Jensen, Paul S.
- Computers & Structures, Vol. 2, Issue 1-2
Automatic three-dimensional mesh generation by the finite octree technique
journal, September 1991
- Shephard, Mark S.; Georges, Marcel K.
- International Journal for Numerical Methods in Engineering, Vol. 32, Issue 4
The extended/generalized finite element method: An overview of the method and its applications
journal, January 2010
- Fries, Thomas-Peter; Belytschko, Ted
- International Journal for Numerical Methods in Engineering
Influence of several factors in the generalized finite difference method
journal, December 2001
- Benito, J. J.; Ureña, F.; Gavete, L.
- Applied Mathematical Modelling, Vol. 25, Issue 12
Consistent computation of first- and second-order differential quantities for surface meshes
conference, January 2008
- Jiao, Xiangmin; Zha, Hongyuan
- Proceedings of the 2008 ACM symposium on Solid and physical modeling - SPM '08
An analysis and comparison of parameterization-based computation of differential quantities for discrete surfaces
journal, June 2009
- Wang, Duo; Clark, Bryan; Jiao, Xiangmin
- Computer Aided Geometric Design, Vol. 26, Issue 5
Element-free Galerkin methods
journal, January 1994
- Belytschko, T.; Lu, Y. Y.; Gu, L.
- International Journal for Numerical Methods in Engineering, Vol. 37, Issue 2
Meshless Finite Difference Method with Higher Order Approximation—Applications in Mechanics
journal, February 2012
- Milewski, Sławomir
- Archives of Computational Methods in Engineering, Vol. 19, Issue 1
Finite Element Methods of Least-Squares Type
journal, January 1998
- Bochev, Pavel B.; Gunzburger, Max D.
- SIAM Review, Vol. 40, Issue 4
Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement
journal, October 2005
- Hughes, T. J. R.; Cottrell, J. A.; Bazilevs, Y.
- Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 39-41
Surfaces generated by moving least squares methods
journal, September 1981
- Lancaster, P.; Salkauskas, K.
- Mathematics of Computation, Vol. 37, Issue 155