Interpolation among reduced-order matrices to obtain parameterized models for design, optimization and probabilistic analysis

International Journal for Numerical Methods in Fluids

Volumn 63, Issue 2, 2010, Pages 207-230

  Degroote, Joris ^a,b   Vierendeels, Jan ^a   Willcox, Karen ^a  

^a GHENT UNIVERSITY   (Belgium)

^b Massachusetts Institute of Technology   (United States)

Author keywords

Design; Interpolation; Kriging; Parameterized reduced order models; Riemannian manifold; Spline

Indexed keywords

KRIGING; PARAMETERIZED; REDUCED ORDER MODELS; RIEMANNIAN MANIFOLD;

COMPUTER SIMULATION; DESIGN; MONTE CARLO METHODS; OPTIMIZATION; PARAMETERIZATION; SPLINES;

INTERPOLATION;

EID: 77953583956     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.2089     Document Type: Article

References (51)

1. Barrault M, Maday Y, Nguyen N, Patera A. An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathéematique, Académie des Sciences, Paris 2004; I:339-667.
2. Grepl M, Maday Y, Nguyen N, Patera A. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Mathematical Modelling and Numerical Analysis 2007; 41(3):575-605.
3. Antoulas A, Sorensen D, Gugercin S. A survey of model reduction methods for large-scale systems. Contemporary Mathematics, Volume: Structured Matrices in Operator Theory, Numerical Analysis, Control, Signal and Image Processing, Chapter 280. American Mathematical Society: Providence, RI, 2001; 193-219.
4. Berkooz G, Holmes P, Lumley J. The proper orthogonal decomposition in the analysis of turbulent flows. Annual Review of Fluid Mechanics 1993; 25(1):539-575.
5. Sirovich L. Turbulence and the dynamics of coherent structures, part 1: coherent structures. Quarterly of Applied Mathematics 1987; 45(3):561-571.
6. Moore B. Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Transactions on Automatic Control 1981; AC-26(1):17-32.
7. Feldmann P, Freund R. Efficient linear circuit analysis by Padé-approximation via the Lanczos process. IEEE Transaction on Computer-Aided Design of Integrated Circuits and Systems 1995; 14(5):639-649.
8. Gallivan K, Grimme E, Van Dooren P. Padé approximation of large-scale dynamic systems with Lanczos methods. Proceedings of the 33rd IEEE Conference on Decision and Control, Lake Buena Vista, FL, December 1994.
9. Farhat C, Amsallem D. Reduced-order modeling, differential geometry and physics-based near-real-time predictions. Eighth World Congress on Computational Mechanics, 5th European Congress on Computational Methods in Applied Sciences and Engineering, Venice, Italy, 2008; 1.
10. Daniel L, Ong C, Low S, Lee K, White J. A multiparameter moment matching model reduction approach for generating geometrically parameterized interconnect performance models. IEEE Transaction on Computer-Aided Design of Integrated Circuits and Systems 2004; 23(5):678-693.
11. Hinze M, Volkwein S. Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In Dimension Reduction of Large-Scale Systems, Benner P, Mehrmann V, Sorensen D (eds). Lecture Notes in Computational and Applied Mathematics. Springer-Verlag: Berlin, Heidelberg, 2005; 261-306.
12. Kunisch K, Volkwein S. Control of Burgers' equation by reduced order approach using proper orthogonal decomposition. Journal of Optimization Theory and Applications 1999; 102:345-371.
13. Prud'homme C, Rovas D, Veroy K, Maday Y, Patera A, Turinici G. Reliable real-time solution of parameterized partial differential equations: reduced-basis output bound methods. Journal of Fluids Engineering 2002; 124:70-80.
14. Bui-Thanh T, Willcox K, Ghattas O. Model reduction for large-scale systems with high-dimensional parametric input space. SIAM Journal on Scientific Computing 2008; 30(6):3270-3288.
15. Ly H, Tran H. Modeling and control of physical processes using proper orthogonal decomposition. Journal of Mathematical and Computer Modeling 2001; 33:223-236.
16. Veroy K, Prudhomme C, Rovas D, Patera A. A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. The 16th AIAA Computational Fluid Dynamics Conference, Orlando, FL, U.S.A., 2003; 1-18. AIAA 2003-3847.
17. Grepl M, Patera A. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. Mathematical Modelling and Numerical Analysis 2005; 39(1):157-181.
18. Daniel L, White J. Automatic generation of geometrically parameterized reduced order models for integrated spiral RF inductors. Proceedings of the 2003 IEEE International Workshop on Behavioral Modeling and Simulation, San Jose, CA, U.S.A., 2003; 18-23.
19. Rewienski M, White J. A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micro machined devices. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 2003; 22(2):155-170.
20. Bond B, Daniel L. Parameterized model order reduction of nonlinear dynamical systems. Proceedings of the IEEE Conference on Computer-Aided Design, San Jose, CA, U.S.A., 2005; 1-8.
21. Bond B, Daniel L. A piecewise-linear moment-matching approach to parameterized model-order reduction for highly nonlinear systems. IEEE Transaction on Computer-Aided Design of Integrated Circuits and Systems 2007; 26(12):2116-2129.
22. Galletti B, Bruneau C, Zannetti L, Iollo A. Low-order modelling of laminar flow regimes past a confined square cylinder. Journal of Fluid Mechanics 2004; 503:161-170.
23. Allen M, Weickum G, Maute K. Application of reduced order models for the stochastic design optimization of dynamic systems. Tenth AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference,Albany,NY, U.S.A., 2004; 1-19. AIAA 2004-4614.
24. Kirsch U. Combined approximations-a general reanalysis approach for structural optimization. Structural and Multidisciplinary Optimization 2000; 20(2):97-106.
25. Keerti A, Nikolaidis E, Ghiocel D, Kirsch U. Combined approximations for efficient probabilistic analysis of structures. AIAA Journal 2004; 42(7):1321-1330.
26. Weickum G, Eldred M, Maute K. Multi-point extended reduced order modeling for design optimization and uncertainty analysis. Structures, Structural Dynamics and Materials Conference, AIAA/ASME/ASCE/AHS/ASC, Newport, RI, U.S.A., 2006.
27. Taylor J. Dynamics of large scale structures in turbulent shear layers. Ph.D. Thesis, Clarkson University, 2001.
28. Taylor J, Glauser M. Towards practical flow sensing and control via POD and LSE based low-dimensional tools. Journal of Fluids Engineering 2004; 126(3):337-345.
29. Lieu T, Lesoinne M. Parameter adaptation of reduced order models to three-dimensional flutter analysis. The 42nd AIAA Aerospace Sciences Meeting and Exhibit, vol. AIAA 2004-0888, Reno, NV, U.S.A., 2004.
30. Bjorck A, Golub G. Numerical methods for computing angles between linear subspaces. Mathematics of Computation 1973; 27(123):579-594.
31. Lieu T, Farhat C. Adaptation of POD-based aeroelastic ROMs for varying mach number and angle of attack: application to a complete F-16 configuration. AIAA Paper 2005-7666, 2005.
32. Lieu T, Farhat C, Lesoinne M. POD-based aeroelastic analysis of a complete F-16 configuration: ROM adaptation and demonstration. The 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, vol. AIAA 2005-2295, Austin, TX, U.S.A., 2005.
33. Lieu T, Farhat C, Lesoinne M. Reduced-order fluid/structure modeling of a complete aircraft configuration. Computer Methods in Applied Mechanics and Engineering 2006; 195:5730-5742.
34. Lieu T, Farhat C. Adaptation of aeroelastic reduced-order models and application to an F-16 configuration. AIAA Journal 2007; 45(6):1244-1257.
35. Amsallem D, Farhat C, Lieu T. Aeroelastic analysis of F-16 and F-18/A configurations using adapted CFD-based reduced order models. The 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, vol. AIAA 2007-2364, Honolulu, Hawaii, 2007; 1-20.
36. Carlson R, Fritsch F. Monotone piecewise bicubic interpolation. SIAM Journal on Numerical Analysis 1985; 22(2):386-400.
37. De Boor C. A Practical Guide to Splines. Springer: New York, NY, U.S.A., 2001.
38. Chui C, Wang RH. On smooth multivariate spline functions. Mathematics of Computation 1983; 41(163):131-142.
39. Spivak M. A Comprehensive Introduction to Differential Geometry (3rd edn), vol. 5. Publish or Perish: Houston, TX, U.S.A., 1999.
40. O'Neill B. Elementary Differential Geometry (Revised 2nd edn). Elsevier: Amsterdam, 2006.
41. Begelfor E, Werman M. Affine invariance revisited. Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York, NY, U.S.A., 2006.
42. Rahman I, Drori I, Stodden V, Donoho D, Schröder P. Multiscale representation for manifold-valued data. Multiscale Modeling and Simulation 2005; 4(4):1201-1232.
43. Davies P, Higham N. A Schur-Parlett algorithm for computing matrix functions. SIAM Journal on Matrix Analysis and Applications 2003; 25(2):464-485.
44. Higham N. The scaling and squaring method for the matrix exponential revisited. SIAM Journal on Matrix Analysis and Applications 2005; 26(4):1179-1193.
45. Eldred M, Giunta S, Collis S. Second-order corrections for surrogate-based optimization with model hierarchies. Proceedings of the 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference,Albany,NY, 2004.
46. Venter G, Haftka R, Starnes J. Construction of response surface approximations for design optimization. AIAA Journal 1998; 36(12):2242-2249.
47. Lophaven S, Nielsen H, Sündergaard J. Aspects of the Matlab Toolbox DACE. Informatics and Mathematical Modelling, DTU, Report imm-rep-2002-13 edn, 2002. Available from: http://www.imm.dtu.dk/hbn/publ/ TR0213.ps.
48. Sacks J, Welch W, Mitchell T, Wynn H. Design and analysis of computer experiments. Statistical Science 1989; 4(4):409-435.
49. Giunta A, Watson L. A comparison of approximation modeling techniques: polynomial versus interpolating models. Seventh AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, MO, U.S.A., 1998; 1-13. AIAA 98-4758.
50. Bashir O, Willcox K, Ghattas O, van Bloemen Waanders B, Hill J. Hessian-based model reduction for large-scale systems with initial condition inputs. International Journal for Numerical Methods in Engineering 2008; 73(6):844-868. DOI:10.1002/nme.2100.
51. Brooks A, Hughes T. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 1982; 32:199-259.