POD and CVT-based reduced-order modeling of Navier-Stokes flows

Computer Methods in Applied Mechanics and Engineering

Volumn 196, Issue 1-3, 2006, Pages 337-355

  Burkardt, John ^a   Gunzburger, Max ^a   Lee, Hyung Chun ^b  

^a FLORIDA STATE UNIVERSITY   (United States)

^b Ajou University   (South Korea)

Author keywords

Centroidal Voronoi tessellations; Navier Stokes equations; Proper orthogonal decomposition; Reduced order modeling

Indexed keywords

COMPUTER SIMULATION; MATHEMATICAL MODELS; NAVIER STOKES EQUATIONS;

CENTROIDAL VORONOI TESSELLATIONS; CONCRETE INCOMPRESSIBLE FLOW; NAVIER-STOKES FLOWS; PROPER ORTHOGONAL DECOMPOSITION; REDUCED-ORDER MODELING;

INCOMPRESSIBLE FLOW;

COMPUTER SIMULATION; INCOMPRESSIBLE FLOW; MATHEMATICAL MODELS; NAVIER STOKES EQUATIONS;

EID: 33846479940     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2006.04.004     Document Type: Article

References (51)

1. Aubry N., Lian W., and Titi E. Preserving symmetries in the proper orthogonal decomposition. SIAM J. Sci. Comput. 14 (1993) 483-505
2. Berkooz G., Holmes P., and Lumley J. The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech. 25 (1993) 539-575
3. Berkooz G., and Titi E. Galerkin projections and the proper orthogonal decomposition for equivariant equations. Phys. Lett. A 174 (1993) 94-102
4. Christensen E., Brons M., and Sorensen J. Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Sci. Comput. 21 (2000) 1419-1434
5. Deane A., Kevrekidis I., Karniadakis G., and Orszag S. Low-dimensional models for complex geometry flows: applications to grooved channels and circular cylinders. Phys. Fluids A 3 (1991) 2337-2354
6. Du Q., Faber V., and Gunzburger M. Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41 (1999) 637-676
7. Q. Du, M. Gunzburger, Model reduction by proper orthogonal decomposition coupled with centroidal Voronoi tessellation, in: Proc. Fluids Engineering Division Summer Meeting, FEDSM2002-31051, ASME, 2002.
8. Du Q., and Gunzburger M. Grid generation and optimization based on centroidal Voronoi tessellations. Appl. Math. Comput. 133 (2002) 591-607
9. Du Q., and Gunzburger M. Centroidal Voronoi tessellation based proper orthogonal decomposition analysis. Proc. 8th Conference on Control of Distributed Parameter Systems (2002), Birkhauser, Basel 137-150
10. Du Q., Gunzburger M., and Ju L. Meshfree, probabilistic determination of points sets and support regions for meshless computing. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1349-1366
11. Du Q., Gunzburger M., and Ju L. Constrained CVTs on general surfaces. SIAM J. Sci. Comput. 24 (2003) 1488-1506
12. Du Q., Gunzburger M., and Ju L. Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere. Comput. Methods Appl. Mech. Engrg. 192 (2003) 3933-3957
13. A. Faulds, Centroidal Voronoi Decompositions, Algorithms and Applications, M.S. Thesis, Department of Mathematics, Penn State University, 2002.
14. Faulds A., and King B. Sensor location in feedback control of partial differential equation systems. Proc. 2000 IEEE CCA/CACSD (2000), IEEE, Washington 536-541
15. Raviart P.-A., and Girault V. Finite Element Methods for Navier-Stokes Equations (1986), Springer, Berlin
16. Graham M., and Kevrekidis I. Pattern analysis and model reduction: some alternative approaches to the Karhunen-Lóeve decomposition. Comput. Chem. Engrg. 20 (1996) 495-506
17. Golub G., and van Loan C. Matrix Computations (1996), Johns Hopkins University, Baltimore
18. Gunzburger M. Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice and Algorithms (1989), Academic, Boston
19. Hartigan J. Clustering Algorithms (1975), Wiley
20. Hartigan J., and Wong M. Algorithm AS 136: A K-Means Clustering Algorithm. Appl. Stat. 28 (1979) 100-108
21. Holmes P., Lumley J., and Berkooz G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry (1996), Cambridge University, Cambridge
22. Holmes P., Lumley J., Berkooz G., Mattingly J., and Wittenberg R. Low-dimensional models of coherent structures in turbulence. Phys. Rep. 287 (1997) 337-384
23. Ju L., Du Q., and Gunzburger M. Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations. J. Parallel Comput. 28 (2002) 1477-1500
24. Kunisch K., and Volkwein S. Control of Burger's equation by a reduced order approach using proper orthogonal decomposition. JOTA 102 (1999) 345-371
25. K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Spezialforschungsbereich F003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 153, Graz, 1999.
26. Lumley J. Stochastic Tools in Turbulence (1971), Academic, New York
27. Martinez W., and Martinez A. Computational Statistics Handbook with MATLAB (2002), Chapman and Hall/CRC
28. Nagy D. Modal representation of geometrically nonlinear behavior by the finite element method. Comput. Struct. 10 (1979) 693
29. Noor A. Recent advances in reduction methods for nonlinear problems. Comput. Struct. 13 (1981) 31-44
30. Noor A., Anderson C., and Peters J. Reduced basis technique for collapse analysis of shells. AAIA J. 19 (1981) 393
31. Noor A., and Peters J. Tracking post-linit-paths with reduced basis technique. Comput. Methods Appl. Mech. Engrg. 28 (1981) 217
32. Okabe A., Boots B., Sugihara K., and Chiu S. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams (2000), Wiley, Chichester
33. Park H., and Cho D. Low dimensional modeling of flow reactors. Int. J. Heat Mass Transfer 39 (1996) 3311-3323
34. Park H., and Chung J. A sequential method of solving inverse natural convection problems. Inverse Prob. 18 (2002) 529-546
35. Park H., and Jang Y. Control of Burgers equation by means of mode reduction. Int. J. Engrg. Sci. 38 (2000) 785-805
36. Park H., and Lee J. Solution of an inverse heat transfer problem by means of empirical reduction of modes. Z. Angew. Math. Phys. 51 (2000) 17-38
37. Park H., and Lee W. An efficient method of solving the Navier-Stokes equations for flow control. Int. J. Numer. Meth. Engrg. 41 (1998) 1133-1151
38. Park H., and Lee W. A new numerical method for the boundary optimal control problems of the heat conduction equation. Int. J. Numer. Meth. Engrg. 53 (2002) 1593-1613
39. Peterson J. The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777
40. M. Rathinam, L. Petzold, A new look at proper orthogonal decomposition, in press.
41. M. Rathinam, L. Petzold, Dynamic iteration using reduced order models: a method for simulation of large scale modular systems, in press.
42. Ravindran S. Proper orthogonal decomposition in optimal control of fluids. Int. J. Numer. Meth. Fluids 34 (2000) 425-448
43. Ravindran S. Reduced-order adaptive controllers for fluid flows using POD. SIAM J. Sci. Comput. 15 (2000) 457-478
44. Rodríguez J., and Sirovich L. Low-dimensional dynamics for the complex Ginzburg-Landau equations. Physica D 43 (1990) 77-86
45. Sirovich L. Turbulence and the dynamics of coherent structures, I-III. Quart. J. Appl. Math. 45 (1987) 561-590
46. Smaoui N., and Armbruster D. Symmetry and the Karhunen-Loève analysis. SIAM J. Sci. Comput. 18 (1997) 1526-1532
47. Späth H. Cluster Analysis Algorithms for Data Reduction and Classification of Objects (1980), Ellis Horwood
48. Späth H. Cluster Dissection and Analysis, Theory, FORTRAN Programs, Examples (1985), Ellis Horwood
49. Sparks D. Algorithm AS 58: Euclidean cluster analysis. Appl. Stat. 22 (1973) 126-130
50. Volkwein S. Optimal control of a phase field model using the proper orthogonal decomposition. ZAMM 81 (2001) 83-97
51. S. Volkwein, Proper orthogonal decomposition and singular value decomposition, Spezialforschungsbereich F003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 153, Graz, 1999.