Adaptive mesh refinement for high-resolution finite element schemes

International Journal for Numerical Methods in Fluids

Volumn 52, Issue 5, 2006, Pages 545-569

  Möller, Matthias ^a   Kuzmin, D ^a  

^a TU DORTMUND UNIVERSITY   (Germany)

Author keywords

Adaptive mesh refinement; Convection dominated problems; Error estimation; Gradient recovery; High resolution schemes

Indexed keywords

ADAPTIVE ALGORITHMS; COMPRESSIBLE FLOW; COMPUTATIONAL METHODS; COMPUTER SIMULATION; ERROR DETECTION; FINITE ELEMENT METHOD; SLOPE STABILITY;

ADAPTIVE MESH REFINEMENT; ERROR ESTIMATION; HIGH-RESOLUTION SCHEME;

COMPUTATIONAL FLUID DYNAMICS;

ADAPTIVE ALGORITHMS; COMPRESSIBLE FLOW; COMPUTATIONAL FLUID DYNAMICS; COMPUTATIONAL METHODS; COMPUTER SIMULATION; ERROR DETECTION; FINITE ELEMENT METHOD; SLOPE STABILITY;

EID: 33749180346     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.1183     Document Type: Article

References (54)

1. Babuška I, Rheinboldt W. A posteriori error estimates for the finite element method. International Journal for Numerical Methods in Engineering 1978; 12:1597-1615.
2. Ainsworth M, Oden JT. A Posteriori Error Estimation in Finite Element Analysis. Wiley: New York, 2000.
3. Babuška I, Szabo B. Finite Element Analysis. Wiley: New York, 1991.
4. Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley/Teubner: New York/Stuttgart, 1996.
5. Kuzmin D, Turek S. Flux correction tools for finite elements. Journal of Computational Physics 2002; 175:525-558.
6. Kuzmin D, Möller M, Turek S. High-resolution FEM-FCT schemes for multidimensional conservation laws. Computer Methods in Applied Mechanics and Engineering 2004; 193:4915-4946.
7. Kuzmin D, Turek S. High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. Journal of Computational Physics 2004; 198:131-158.
8. Kuzmin D, Möller M. Algebraic flux correction I. Scalar conservation laws. In Flux-Corrected Transport: Principles, Algorithms, and Applications, Kuzmin D, Löhner R, Turek S (eds). Springer: Berlin, 2005; 155-206.
9. Kuzmin D, Möller M. Algebraic flux correction II. Compressible Euler equations. In Flux-Corrected Transport: Principles, Algorithms, and Applications, Kuzmin D, Löhner R, Turek S (eds). Springer: Berlin, 2005; 207-250.
10. Berger MJ, Oliger J. Adaptive mesh refinement for hyperbolic partial differential equations. Journal of Computational Physics 1984; 53:484-512.
11. Berger MJ, Colella P. Local adaptive mesh refinement for shock hydrodynamics. Journal of Computational Physics 1989; 82:64-84.
12. Berger MJ, LeVeque R. Adaptive mesh refinement using wave propagation algorithms for hyperbolic systems. SIAM Journal on Numerical Analysis 1998; 35:1439-1461.
13. Roache PJ. Perspective: a method for uniform reporting of grid refinement studies. Journal of Fluids Engineering 1994; 116:405-413.
14. Roy CJ. Grid convergence error analysis for mixed-order numerical schemes. AIAA Journal 2003; 41:595-604.
15. Geppert J. Adaptive Gitterverfeinerung bei Flux-Limiter-Verfahren. Ph.D. Thesis, University of Hamburg, 1996.
16. Zienkiewicz OC, Zhu JZ. A simple error estimator and adaptive procedure for practical engineering analysis. International Journal for Numerical Methods in Engineering 1987; 24:337-357.
17. Ainsworth M, Zhu JZ, Craig AW, Zienkiewicz OC. Analysis of the Zienkiewicz-Zhu a posteriori error estimator in the finite element method. International Journal for Numerical Methods in Engineering 1989; 28:2161-2174.
18. Zienkiewicz OC, Zhu JZ. The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery techniques. International Journal for Numerical Methods in Engineering 1992; 33:1331-1364.
19. Zienkiewicz OC, Zhu JZ. The superconvergent patch recovery and a posteriori error estimates. Part 2: error estimates and adaptivity. International Journal for Numerical Methods in Engineering 1992; 33:1365-1382.
20. Prudhomme S, Oden JT. Computable error estimators. In: Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Barth TJ, Deconinck H (eds). Springer: Berlin, 2003.
21. Prudhomme S. Adaptive finite element simulation of a supersonic underexpanded jet. Master Thesis, University of Virginia, 1992.
22. Cockburn B, Shu CW. TVD Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework. Mathematics of Computation 1989; 52:411-435.
23. Jameson A. Analysis and design of numerical schemes for gas dynamics 1. Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. International Journal of Computational Fluid Dynamics 1995; 4:171-218.
24. Oden JT, Brauchli HJ. On the calculation of consistent stress distribution in finite element applications. International Journal for Numerical Methods in Engineering 1971; 3:317-325.
25. Oden JT, Reddy JN. Note on an approximate method for computing consistent conjugate stresses in elastic finite elements. International Journal for Numerical Methods in Engineering 1973; 6:55-61.
26. Hinton E, Campbell J. Local and global smoothing of discontinuous finite element functions using a least squares method. International Journal for Numerical Methods in Engineering 1974; 8:461-480.
27. Cantin G, Loubignac C, Touzot C. An iterative scheme to build continuous stress and displacement solutions. International Journal for Numerical Methods in Engineering 1978; 12:1493-1506.
28. Lyra PRM. Unstructured grid adaptive algorithms for fluid dynamics and heat conduction. Ph.D. Thesis, University of Wales, Swansea, 1994.
29. Zhang Z, Zhu JZ. Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (I). Computer Methods in Applied Mechanics and Engineering 1995; 123:173-187.
30. Zhang Z, Zhu JZ. Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (II). Computer Methods in Applied Mechanics and Engineering 1998; 163:159-170.
31. Zhang Z. Ultraconvergence of the patch recovery technique II. Mathematics of Computation 2000; 69:141-158.
32. Naga A, Zhang Z. A posteriori error estimates based on polynomial preserving recovery. SIAM Journal on Numerical Analysis 2004; 42:1780-1800.
33. Zhang Z, Naga A. A new finite element gradient recovery method: superconvergence property. SIAM Journal on Scientific Computing 2005; 26:1192-1213.
34. GiD - The personal pre and postprocessor. http://gid.cimne.uOpc.es/
35. Brandt A. Multilevel adaptive computations in fluid dynamics. AIAA Journal 1980; 18:1165-1172.
36. Hackbusch W. Iterative Solution of Large Sparse Systems of Equations. Applied Mathematical Sciences, vol. 95. Springer: Berlin, 1994.
37. Brandt A. Multi-level adaptive solutions to boundary value problems. Mathematics of Computation 1977; 31:333-390.
38. Feistauer M, Felcman J, Straškraba I. Mathematical and Computational Methods for Compressible Flow. Oxford University Press: Oxford, 2003.
39. Shapiro RA. Adaptive Finite Element Solution Algorithm for the Euler Equations. Notes on Numerical Fluid Mechanics, vol. 32. Vieweg: Braunschweig, 1991.
40. Hansbo P. A free-Lagrange finite element method using space-time elements. Computer Methods in Applied Mechanics and Engineering 2000; 188:347-361.
41. Bank RE, Sherman AH, Weiser A. Refinement algorithms and data structures for regular local mesh refinement. Scientific Computing (Applications of Mathematics and Computing to the Physical Sciences). North-Holland: Amsterdam, 1983; 3-17.
42. Babuška I, Rheinboldt WA. Reliable error estimation and mesh adaptation for the finite element method. In Computational Methods in Nonlinear Mechanics. North-Holland: Amsterdam, 1980; 67-108.
43. Rivara MC. Design and data structure of fully adaptive multigrid finite element software. ACM Transactions on Mathematical Software 1984; 10:242-264.
44. Rivara MC. New mathematical tools and techniques for the refinement and/ or improvement of unstructured triangulations. Proceedings of the 5th International Meshing Roundtable 96, Pittsburgh, 1996; 77-86.
45. Rivara MC. Using longest-side bisection techniques for the automatic refinement of delaunay triangulations. International Journal for Numerical Methods in Engineering 1997; 40:581-597.
46. De Cougny HL, Shephard MS. Parallel refinement and coarsening of tetrahedral meshes. International Journal for Numerical Methods in Engineering 1999; 46:1101-1125.
47. Kobbelt L, Schroder P. A multiresolution framework for variational subdivision. Technical Report CS-TR-97-05, Computer Graphics Research Group, Computer Science Department, California Institute of Technology, 1997.
48. Walter MAT, Abdu AAQ, da Silva LFF, Azevedo JLF. Evaluation of adaptive mesh refinement and coarsening for the computation of compressible flows on unstructured meshes. International Journal for Numerical Methods in Fluids 2005; 49:999-1014.
49. Bank RE, Smith RK. Mesh smoothing using a posteriori error estimates. SIAM Journal on Numerical Analysis 1997; 34:979-997.
50. Forsyth PA. A control volume finite element approach to NAPL groundwater contamination. SIAM Journal on Scientific and Statistical Computing 1991; 12:1029-1057.
51. Freitag LA. On combining Laplacian and optimization-based mesh smoothing techniques. Trends in unstructured mesh generation. ASME Applied Mechanics Division 1997; 220:37-44.
52. Peraire J, Vahdati M, Morgan K, Zienkiewicz OC. Adaptive remeshing for compressible flow computations. Journal of Computational Physics 1987; 72:449-466.
53. Oran ES, Boris JP. Numerical Simulation of Reactive Flow. Cambridge University Press: Cambridge, MA, 2001.
54. NPARC Alliance. Computational fluid dynamics (CFD) verification and validation web site: http://www.grc.nasa.gov/WWW/wind/valid/