Abstract
In present day, the technology reduction of product development costs and design cycle time are essential ingredients in a competitive industrial environment. E.g. in aeronautics the objectives set by the EU are to reduce aircraft development costs by resp. 20 and 50 % in the short and long term. Virtual prototyping and advanced design optimization, which largely depend on the predictive performance and the reliability of simulation software, are essential tools to reach this goal.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
N. Wiener. The Homogeneous Chaos. Am. J. Math., 60:897–936, 1938.
D. Xiu and G.E. Karniadakis. Modeling uncertainty in flow simulations via generalized polynomial chaos. JCP, 187:137–167, 2003.
D. Lucor, D. Xiu, C.-H. Su, and G.E. Karniadakis. Predictability and uncertainty in CFD. Int. J. Numer. Meth. Fluids, 43:483–505, 2003.
O. Le Maitre, O. Knio, H. Najm, and R. Ghanem. A stochastic projection method for fluid flow i. basic formulation. J. Comput. Phys., 173:481–511, 2001.
L. Mathelin, M. Hussaini, and T. Zang. Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algorithms, 38:209–236, 2005.
R.W. Walters and L. Huyse. Uncertainty analysis for fluid mechanics with applications. ICASE Rep. no. 2002-1, 2002.
Witteveen J.A.S. and Bijl H. Efficient Quantification Of The Effect Of Uncertainties In Advection-Diffusion Problems Using Polynomial Chaos. Numerical Heat Transfer, Part B, 53:437–465, 2008.
D. Xiu and G.E. Karniadakis. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput., 24:619–644, 2002.
K. Karhunen. Zur spektraltheorie stochastischer prozesse. Ann. Acad. Sci. Fennicae, 34:1–7, 1946.
M. Loève. Fonctions aleatoires du seconde ordre. Processus Stochastiques et Movement Brownien, 1948. Ed. P. Levy, Paris.
Xiu D. FastNumericalMethods for Stochastic Computations: A Review. Commun. Comput. Phys., 5:242–272, 2009.
C. Lacor and S. Smirnov. Uncertainty Propagation in the Solution of Compressible Navier-Stokes Equations using Polynomial Chaos Decomposition. RTO-MP-AVT-147 Computational Uncertainty in Military Vehicle Design, 2007. Athens.
C. Lacor and S. Smirnov. Non-Deterministic Compressible Navier-Stokes Simulations using Polynomial Chaos. Proc. ECCOMAS Conf., 2008. Venice.
C. Dinescu, S. Smirnov, Ch. Hirsch, and C. Lacor. Assessment of intrusive and non-intrusive non-deterministic cfd methodologies based on polynomial chaos expansions. Int. J. Engineering Systems Modelling and Simulation, 2:87–98, 2010.
G. Onorato, G.J.A. Loeven, G. Ghorbaniasl, H. Bijl, and C. Lacor. Comparison of intrusive and non-intrusive polynomial chaos methods for CFD applications in aeronautics. Proc. ECCOMAS CFD Conf., 2010. Lisbon, June 2010.
O. Le Maitre, M. Reagan, H. Najm, R. Ghanem, and O. Knio. A stochastic projection method for fluid flow ii. random process. J. Comput. Phys., 181:9–44, 2002.
F. Nobile, R. Tempone, and C.G. Webster. A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data. SIAM J. Numer. Analysis, 46: 2309–2345, 2009.
M. Berveiller, B. Sudret, and M. Lemaire. Stochastic finite element: a non-intrusive approach by regression. Rev. Europ enne M canique Num rique, 15:81–92, 2006.
S. Hosder, R.W. Walters, and R. Perez. A Non-Intrusive Polynomial Chaos Method For Uncertainty Propagation in CFD Simulations. 44th AIAA Aerospace Sciences Meeting and Exhibit, 2006.
S. Hosder, R.W. Walters, and M. Balch. Efficient uncertainty quantification applied to the aeroelastic analysis of a transonic wing. 46th AIAA Aerospace Sciences Meeting and Exhibit, 2008.
C. Hu and B.D. Youn. Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct Multidisc Optim, 43:419–442, 2011.
R. Ghanem. Hybrid stochastic finite elements and generalized monte carlo simulation. ASME J. Appl. Mech., 65:1004–1009, 1998.
Field R.V. Numerical methods to estimate the coefficients of the polynomial chaos expansion. In Proc. 15th ASCE Engineering mechanics Conference, 2002.
D. Ghiocel and R. Ghanem. Stochastic finite-element analysis of seismic soil-structure interaction. J. Eng. mech., 128:66–77, 2002.
S.A. Smolyak. Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov Math Dokl, 4:240–243, 1963.
Gerstner T. and Griebel M. Dimension-adaptive tensor-product quadrature. Computing, 71(7):65–87, 2003.
Ma X. and Zabaras N. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys., 228:3084–3113, 2009.
Rahman S. and Xu H. A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probabilistic Engineering Mechanics, 19:393-408, 2004.
Xu H. and Rahman S. A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. Int. J. Numer. Meth. Engng., 61:1992-2019, 2004.
Mathelin L., Hussaini M.Y., Zang T.A., and Bataille F. Uncertainty propagation for turbulent, compressible nozzle flow using stochastic methods,. AIAA J, 42:1669–1676, 2004.
Alekseev A. K., Navon I. M., and Zelentsov M. E. The estimation of functional uncertainty using polynomial chaos and adjoint equations. Int. J. Numer. Meth. Fluids, 67:328–341, 2011.
W.L. Oberkampf, T.G. Trucano, and Ch. Hirsch. Verification, Validation, and Predictive Capability in Computational Engineering and Physics. Paper presented at FOUNDATIONS 02, Foundations for Verification and Validation in the 21st Century Workshop, USA, 2002.
D. Xiu and D.M. Tartakovsky. Numerical Methods for Differential Equations in Random Domains. SIAM Journal on Scientific Computation, 28:1167–1185, 2006.
Parussini L. and Pediroda V. Investigation of Multi Geometric Uncertainties by Different Polynomial Chaos Methodologies Using a Fictitious Domain Solver. Computer Modeling in Engineering and Sciences, 23:29-52, 2008.
Parussini L. Fictitious Domain Approach Via Lagrange Multipliers with Least Squares Spectral Element Method. Journal of Scientific Computing, 37:316-335, 2008.
Perez R. and Walters R. An Implicit Polynomial Chaos Formulation for the Euler Equations. AIAA 2005–1406, 2005.
Perez R. Uncertainty Analysis of Computational Fluid Dynamics Via Polynomial Chaos. PhD Thesis, Virginia Polytechnic Institute and State University, 2008.
A. Doostan and H. Owhadi. A non-adapted sparse approximation of PDEs with stochastic inputs. Journal of Computational Physics, 230:3015–3034, 2011.
M. Raets. Application of Polynomial Chaos to CFD. Master Thesis, Research Group Fluid Mechanics and Thermodynamics, VUB, 2007.
Hirsch Ch. and Dinescu C. Uncertainty Quantification and Non-deterministic Methodologies for CFD based Design - the NODESIM-CFD Project. Proceedings 7th European Symposium on Aerothermodynamics, SP-692, 2011.
J. Dunham. CFD validation for propulsion system components. AGARD-AR-355, 1998.
B. Tartinville and Ch. Hirsch. Rotor37 in FLOMANIA - A European Initiative on Flow Physics Modelling. Notes in Numerical Fluid Mechanics and Multidisciplinary Design, Springer, 2006. Vol. 94, pp. 193–202.
G.J.A. Loeven, J.A.S. Witteveen, and H. Bijl. Probabilistic Collocation: An Efficient Non-Intrusive Apporach For Arbitrarily Distributed Parametric Uncertainties. AIAA paper 2007–317, 2007. 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference.
X. Wan and G.E. Karniadakis. An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations. JCP, 209:617–642, 2005.
O. Le Maitre, O.M. Knio, H.N. Najm, and R.G. Ghanem. Uncertainty propagation using wiener-haar expansions. J. Comput. Phys., 197:28–57, 2004.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Lacor, C., Dinescu, C., Hirsch, C., Smirnov, S. (2013). Implementation of Intrusive Polynomial Chaos in CFD Codes and Application to 3D Navier-Stokes. In: Bijl, H., Lucor, D., Mishra, S., Schwab, C. (eds) Uncertainty Quantification in Computational Fluid Dynamics. Lecture Notes in Computational Science and Engineering, vol 92. Springer, Cham. https://doi.org/10.1007/978-3-319-00885-1_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-00885-1_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00884-4
Online ISBN: 978-3-319-00885-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)