Fictitious domain with least-squares spectral element method to explore geometric uncertainties by non-intrusive polynomial chaos method

CMES - Computer Modeling in Engineering and Sciences

Volumn 22, Issue 1, 2007, Pages 41-63

  Parussini, L ^a   Pediroda, V ^a  

^a UNIVERSITY OF TRIESTE   (Italy)

Author keywords

Fictitious domain; Lagrange multipliers; Least Squares spectral element method; Non intrusive polynomial chaos

Indexed keywords

CHAOS THEORY; DIFFERENTIAL EQUATIONS; LAGRANGE MULTIPLIERS; LEAST SQUARES APPROXIMATIONS; NUMERICAL METHODS; POLYNOMIALS; UNCERTAINTY ANALYSIS;

DIFFERENTIAL PROBLEMS; FICTITIOUS DOMAIN; LEAST-SQUARES SPECTRAL ELEMENT METHOD; NON-INTRUSIVE POLYNOMIAL CHAOS;

COMPUTATIONAL GEOMETRY;

EID: 36448959779     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (43)

1. Abramowitz, M.; Stegun, I. A. (1972): Handbook of mathematical functions. Dover publications, New York.
2. Askey, A.; Wilson, J. A. (1985): Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., vol. 319.
3. Blith, M.; Pozrikidis, C. (2003): Heat conduction across irregular and fractal-like surfaces. International Journal of Heat and Mass Transfer, vol. 46, pp. 1329-1339.
4. Canuto, C.; Kozubek, T. (2007): A fictitious domain approach to the numerical solution of PDEs in stochastic domains. Numerische Mathematik. DOI 10.1007/s00211-007-0086-xISSN 0029-599X (Print) 0945-3245 (Online).
5. Gerritsma, M.; Maerschalck, B. D. (2006): The least-squares spectral element method. In VKI Lecture Series 2006-01: 34th CFD - Higher Order Discretization Methods (ISSN : 0377-8312), Von Karman Institute, Bruxelles.
6. Glowinski, R.; Pan, T.-W.; Hesla, T. I.; Joseph, D. (1999): A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow, vol. 25, pp. 755-794.
7. Glowinski, R.; Pan, T.-W.; Hesla, T. I.; Joseph, D. D.; Periaux, J. (2000): A distributed Lagrange multiplier/fictitious domain method for the simulation of flows around moving rigid bodies: Application to particulate flow. Comp. Meth. Appl. Mech. Eng., vol. 184, pp. 241-268.
8. Haslinger, J.; Maitre, J. F.; Tomas, L. (2001): Fictitious domains methods with distributed Lagrange multipliers, part i : application to the solution of elliptic state problems. Mathematical Models and Methods In Applied Sciences, vol. 11, no. 3, pp. 521-547.
9. Hosder, S.; Walters, R.; Perez, R. (2006): A Non-Intrusive Polynomial Chaos Method For Uncertainty Propagation in CFD Simulation. In 44th AIAA Aerospace Science Meeting and Exhibit, 912 January 2006, no. 2006-891, Reno, Nevada.
10. Joly, P.; Rhaouti, L. (1999): Domaines fictifs, elements finis h (div) et condition de Neumann: le probleme de la condition inf-sup. C.R. Acad. Sci. Paris, vol. 328, no. 12, pp. 1225-1230.
11. Karniadakis, G. E.; Sherwin, S. J. (1999): Spectral/hp Element Methods for CFD. Oxford University Press, Oxford.
12. Komatitsch, D.; Vilotte, J.-P. (1998): The spectral element method: an efficient tool to simulate the seismic response of 2d and 3d geological structures. Bulletin of the Seismological Society of America, vol. 88, pp. 368-392.
13. Komatitsch, D.; Vilotte, J.-P.; Vai, R.; Castillo-Covarrubias, J. M.; Sánchez-Sesma, F. J. (1999): The spectral element method for elastic wave equations - application to 2-d and 3-d seismic problems. International Journal for Numerical Methods in Engineering, vol. 45, no. 9, pp. 1139-1164.
14. Lin, G.; Su, C.; Karniadakis, G. (2006): Modeling Random Roughness in Supersonic Flow Past a Wedge. In 44th AIAA Aerospace Science Meeting and Exhibit, 9-12 January 2006, no. 2006-0124, Reno, Nevada.
15. Lin, G.; Wan, X.; Su, C.; Karniadakis, G. (2007): Stochastic Computational Fluid Mechanics. Computing in Science and Engineering, vol. 9, no. 2, pp. 21-29.
16. Lin, W. H. (1998): A least-squares spectral element method for sound propagation in acoustic ducts. The Journal of the Acoustical Society of America, vol. 104, no. 5, pp. 3111-3114.
17. Loeven, G.; Witteveen, J.; Bijl, H. (2006): Efficient uncertainty quantification using a two-step approach with Chaos Collocation. In ECCOMAS CFD.
18. Loeven, G.; Witteveen, J.; Bijl, H. (2007): Probabilistic Collocation: An Efficient NonIntrusive Approach For Arbitrarily Distributed Parametric Uncertainties. In 45th AIAA Aerospace Science Meeting and Exhibit, 8-11 January 2007, Reno, Nevada.
19. Makinen, R. A. E.; Rossi, T.; Toivanen, J. (2000): A moving mesh fictitious domain approach for shape optimization problems. Mathematical Modeling and Numerical Analysis, vol. 34, no. 1, pp. 31-45.
20. Mathelin, L.; Hussaini, Y. (2003): A Stochastic Collocation Algorithm for Uncertainty Analysis. Technical Report NAS 1.26212153, NASA CR-2003-212153, NASA Center: Langley Research Center, 2003.
21. Maury, B. (2001): A Fat Boundary Method for the Poisson Equation in a Domain with Holes. J. of Sci. Computing, vol. 16, no. 3, pp. 319-339.
22. Mitra, M.; Gopalakrishnan, S. (2006): Wavelet Based 2-D Spectral Finite Element Formulation for Wave Propagation Analysis in Isotropic Plates. CMES: Computer Modeling in Engineering & Sciences, vol. 15, no. 1, pp. 49-68.
23. Parussini, L. (2007): Fictitious Domain Approach for Spectral/hp Element Method. CMES: Computer Modeling in Engineering & Sciences, vol. 17, no. 2, pp. 95-114.
24. Pontaza, J. P. (2007): A spectral element least-squares formulation for incompressible NavierStokes flows using triangular nodal elements. Journal of Computational Physics, vol. 221, pp. 649-665.
25. Pontaza, J. P.; Reddy, J. N. (2003): Spectral/hp least-squares finite element formulation for Navier-Stokes equations. Journal of Computational Physics, vol. 190, pp. 523-549.
26. Pontaza, J. P.; Reddy, J. N. (2004): Space-time coupled spectral/hp least-squares finite element formulation for the incompressible Navier-Stokes equations. Journal of Computational Physics, vol. 197, pp. 418-459.
27. Proot, M. M. J.; Gerritsma, M. I. (2002): A Least-Squares Spectral Element formulation for the Stokes problem. Journal of Scientifing Computing, vol. 17, pp. 285-296.
28. Proot, M. M. J.; Gerritsma, M. I. (2005): Application of least-squares spectral element method using Chebyshev polynomials to solve the incompressible Navier-Stokes equations. Numerical Algorithms, vol. 38, pp. 155-172.
29. Ramiere, I.; Angot, P.; Belliard, M. (2005): Fictitious domain methods to solve convectiondiffusion problems with general boundary conditions. In 17th AIAA Computational Fluid Dynamics Conference, no. 2005-4709, Toronto, Canada.
30. Rossi, T.; Toivanen, J. (1999): Parallel Fictitious Domain Method for a Nonlinear Elliptic Neumann Boundary Value Problem. Numerical linear algebra with applications, vol. 6, no. 1, pp. 51-60.
31. Rotondi, A.; Pedroni, P.; Pievatolo, A. (2001): Probabilita Statistka e Simulazione. Springer.
32. Schoutens, W. (2000): Stochastic Processes and Orthogonal Polynomial. Springer, 5 edition.
33. Stenberg, R. (1995): On some techniques for approximating boundary conditions in the finite element method. Journal of Computational and Applied Mathematics, vol. 63, pp. 139-148.
34. Wan, X.; Karniadakis, G. (2006): Stochastic heat transfer in a grooved channel. J. Fluid Mech., vol. 565, pp. 255-278.
35. Wiener, N. (1938): The homogeneous chaos. Am. J. math., vol. 60, pp. 897-936.
36. Wiener, N. (1958): Nonlinear Problems in Random Theory. MIT Technology Press and John Wiley ans Sons, New York.
37. Wu, C.-Y.; Al-Khoury, R.; Kasbergen, C.; Liu, X.-Y.; Scarpas, A. (2007): Spectral Element Approach for Inverse Models of 3D Layered Pavement. CMES: Computer Modeling in Engineering & Sciences, vol. 17, no. 3, pp. 163-172.
38. Wu, C.-Y.; Liu, X.-Y.; Scarpas, A.; Ge, X.-R. (2006): Spectral Element Approach for Forward Models of 3D Layered Pavement. CMES: Computer Modeling in Engineering & Sciences, vol. 12, no. 2, pp. 149-157.
39. Xiu, D.; Karniadakis, G. E. (2002): Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Computer methods in applied mechanics and engineering, vol. 191, pp. 4927-4948.
40. Xiu, D.; Karniadakis, G. E. (2003): Modeling uncertainties in flow simulation via generalized polynomial chaos. Journal of Computational Physics, vol. 187, pp. 137-167.
41. Xiu, D.; Karniadakis, G. E. (2003): A new stochastic approach to transient heat conduction modeling with uncertainty. International Journal of Heat and Mass Transfer, vol. 46, pp. 4681-4693.
42. Xiu, D.; Tartakovsky, D. M. (2006): Numerical methods for differential equations in random domains. SIAM J. Sci. Comput., vol. 28, no. 3, pp. 1167-1185.
43. Zienkiewicz, O. C.; Taylor, R. L. (2000): The finite element method. Oxford, 5 edition.