Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- Part I Fundamental concepts
- Part II Code verification
- Part III Solution verification
- 7 Solution verification
- 8 Discretization error
- 9 Solution adaptation
- Part IV Model validation and prediction
- Part V Planning, management, and implementation issues
- Appendix Programming practices
- Index
- Plate Section
- References
9 - Solution adaptation
from Part III - Solution verification
Published online by Cambridge University Press: 05 March 2013
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- Part I Fundamental concepts
- Part II Code verification
- Part III Solution verification
- 7 Solution verification
- 8 Discretization error
- 9 Solution adaptation
- Part IV Model validation and prediction
- Part V Planning, management, and implementation issues
- Appendix Programming practices
- Index
- Plate Section
- References
Summary
The previous chapter focused on the estimation of numerical errors and uncertainties due to the discretization. In addition to estimating the discretization error, we also desire methods for reducing it when either it is found to be too large or when the solutions are not yet in the asymptotic range and therefore the error estimates are not reliable. Applying systematic mesh refinement, although required for assessing the reliability of all discretization error estimation approaches, is not the most efficient method for reducing the discretization error. Since systematic refinement, by definition, refines by the same factor over the entire domain, it generally results in meshes with highly-refined cells or elements in regions where they are not needed. Recall that for 3-D scientific computing applications, each time the mesh is refined using grid halving (a refinement factor of two), the number of cells/elements increases by a factor of eight. Thus systematic refinement for reducing discretization error can be prohibitively expensive.
Targeted, local solution adaptation is a much better strategy for reducing the discretization error. After a discussion of factors affecting the discretization error, this chapter then addresses the two main aspects of solution adaptation:
methods for determining which regions should be adapted and
methods for accomplishing the adaption.
- Type
- Chapter
- Information
- Verification and Validation in Scientific Computing , pp. 343 - 368Publisher: Cambridge University PressPrint publication year: 2010
References
and (2000). A Posteriori Error Estimation in Finite Element Analysis, New York, Wiley Interscience.CrossRefGoogle Scholar
(1986). Accuracy Estimates and Adaptive Refinements in Finite Element Computations, New York, Wiley.Google Scholar
and (1978a). A posteriori error estimates for the finite element method, International Journal for Numerical Methods in Engineering. 12, 1597–1615.CrossRefGoogle Scholar
and (1978b). Error estimates for adaptive finite element computations, SIAM Journal of Numerical Analysis. 15(4), 736–754.CrossRefGoogle Scholar
, , , and (1997). Pollution error in the h-version of the finite element method and local quality of the recovered derivatives, Computer Methods in Applied Mechanics and Engineering. 140, 1–37.CrossRefGoogle Scholar
(1997). Mesh adaptation strategies for problems in fluid dynamics, Finite Elements in Analysis and Design, 25, 243–273.CrossRefGoogle Scholar
(2005). Adaptive modification of time evolving meshes, Computer Methods in Applied Mechanics and Engineering. 194, 4977–5001.CrossRefGoogle Scholar
(1996). Hierarchical bases and the finite element method, Acta Numerica. 5, 1–45.CrossRefGoogle Scholar
, , and (2008). On sub-linear convergence for linearly degenerate waves in capturing schemes, Journal of Computational Physics. 227, 6985–7002.CrossRefGoogle Scholar
and (1972). A table of solutions of the one-dimensional Burgers' equation, Quarterly of Applied Mathematics. 30, 195–212.CrossRefGoogle Scholar
and (1985). Automatic adaptive grid refinement for the Euler equations. AIAA Journal, 23(4), 561–568.CrossRefGoogle Scholar
(2006). A new adaptive remeshing scheme based on the sensitivity analysis of the SPR point wise error estimation, Computer Methods in Applied Mechanics and Engineering. 195(4–6), 462–478.CrossRefGoogle Scholar
(2006). Analytic study of 2D and 3D grid motion using modified Laplacian, International Journal for Numerical Methods in Fluids. 52, 163–197.CrossRefGoogle Scholar
(2008). Heuristic a posteriori estimation of error due to dissipation in finite volume schemes and application to mesh adaptation, Journal of Computational Physics. 227(5), 2845–2863.CrossRefGoogle Scholar
(1982). A Vectorized, Finite-Volume, Adaptive Grid Algorithm Applied to Planetary Entry Problems, AIAA Paper 1982–1018.
and (2001). Differentiating between Source and Location of Error for Solution-Adaptive Mesh Refinement, AIAA Paper 2001–2660.
(1990). Numerical Computation of Internal and External Flows: Volume 2, Computational Methods for Inviscid and Viscous Flows, Chichester, Wiley.Google Scholar
, , and (1998). Adaptive finite element solution of compressible turbulent flows, AIAA Journal. 36(12), 2187–2194.CrossRefGoogle Scholar
and (1992). Adaptive finite element methods in computational mechanics, Computer Methods in Applied Mechanics and Engineering. 101(1–3), 143–181.CrossRefGoogle Scholar
(1997). Solver-Independent r-Refinement Adaptation for Dynamic Numerical Simulations, Doctoral Thesis, North Carolina State University.Google Scholar
(1999). Truncation error on structured grids, in Handbook of Grid Generation, , , and , eds., Boca Raton, CRC Press.Google Scholar
(2000). r-refinement grid adaptation algorithms and issues, Computer Methods in Applied Mechanics and Engineering. 189, 1161–1182.CrossRefGoogle Scholar
and (1999). Dynamic grid adaption and grid quality, in Handbook of Grid Generation, , , and , eds., Boca Raton, FL, CRC Press, 34-1–34-33.Google Scholar
and (1997). Computational techniques for adaptive hp finite element methods, Finite Elements in Analysis and Design. 25(1–2), 27–39.CrossRefGoogle Scholar
, , , and (1987). Adaptive remeshing for compressible flow computations, Journal of Computational Physics. 72(2), 449–466.CrossRefGoogle Scholar
and (1997). A feed-back approach to error control in finite element methods: application to linear elasticity, Computational Mechanics. 19(5), 434–446.CrossRefGoogle Scholar
(2003). Grid convergence error analysis for mixed-order numerical schemes, AIAA Journal. 41(4), 595–604.CrossRefGoogle Scholar
(2009). Strategies for Driving Mesh Adaptation in CFD, AIAA Paper 2009–1302.
and (1996). A posteriori error estimation and adaptive finite element computation of the Helmholtz equation in exterior domains, Finite Elements in Analysis and Design. 22(1), 15–24.CrossRefGoogle Scholar
, , and (1985). Numerical Grid Generation: Foundations and Applications, New York, Elsevier ().Google Scholar
and (2003). Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows, Journal of Computational Physics. 187, 22–46.CrossRefGoogle Scholar
(1999). A review of a posteriori error estimation techniques for elasticity problems, Computer Methods in Applied Mechanics and Engineering. 176(1–4), 419–440.CrossRefGoogle Scholar
and (1999). On Multi-dimensional Unstructured Mesh Adaption, AIAA Paper 1999–3254.
and (1992a). The superconvergent patch recovery and a posteriori error estimates, Part 2: Error estimates and adaptivity, International Journal for Numerical Methods in Engineering. 33, 1365–1382.CrossRefGoogle Scholar
and (1992b). Superconvergent patch recovery (SPR) and adaptive finite element refinement, Computer Methods in Applied Mechanics and Engineering. 101(1–3), 207–224.CrossRefGoogle Scholar