Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods

Abstract : In the last three decades, computer simulation tools have achieved wide spread use in the design and analysis of engineering devices. This has shortened the overall product design cycle and it has also provided bet- ter understanding of the operating behavior of the systems of interest. As a consequence numerical simulations have lead to a reduction of physical prototyping and to lower costs. In spite of this considerable success, it remains difficult to provide objec- tive confidence levels in quantitative information obtained from numerical predictions. The complexity arises from the amount of uncertainties related to the inputs of any computation attempting to represent a physical system. As a result, especially in the area of reliability and safety, physical testing remains the dominant mechanism of certification of new devices. Rigor- ous quantification of the errors and uncertainties1 introduced in numerical simulations is required to establish objectively their predictive capabilities.

[1]  Jan S. Hesthaven,et al.  Padé-Legendre Interpolants for Gibbs Reconstruction , 2006, J. Sci. Comput..

[2]  Roger G. Ghanem,et al.  On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data , 2006, J. Comput. Phys..

[3]  A. Stroud,et al.  Nodes and Weights of Quadrature Formulas , 1965 .

[4]  Richard D. Deveaux,et al.  Applied Smoothing Techniques for Data Analysis , 1999, Technometrics.

[5]  Jon C. Helton,et al.  Investigation of Evidence Theory for Engineering Applications , 2002 .

[6]  Matthew F. Barone,et al.  Measures of agreement between computation and experiment: Validation metrics , 2004, J. Comput. Phys..

[7]  William L. Oberkampf,et al.  Guide for the verification and validation of computational fluid dynamics simulations , 1998 .

[8]  WALTER GAUTSCHI Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules , 1994, TOMS.

[9]  M. D. Salas,et al.  Multiple steady states for characteristic initial value problems , 1986 .

[10]  Gianluca Iaccarino,et al.  Padé-Legendre approximants for uncertainty analysis with discontinuous response surfaces , 2009, J. Comput. Phys..

[11]  Jon C. Helton,et al.  Sampling-based methods for uncertainty and sensitivity analysis. , 2000 .

[12]  Henryk Wozniakowski,et al.  Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems , 1995, J. Complex..

[13]  Richard M. Wilson,et al.  A course in combinatorics , 1992 .

[14]  J. Rice Mathematical Statistics and Data Analysis , 1988 .

[15]  Daniel M. Tartakovsky,et al.  Stochastic analysis of transport in tubes with rough walls , 2006, J. Comput. Phys..

[16]  P. András,et al.  Alternative sampling methods for estimating multivariate normal probabilities , 2003 .

[17]  Jan S. Hesthaven,et al.  Uncertainty analysis for the steady-state flows in a dual throat nozzle , 2005 .

[18]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

[19]  H. Keller,et al.  Analysis of Numerical Methods , 1967 .

[20]  William H. Press,et al.  Numerical recipes in C , 2002 .

[21]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[22]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[23]  Robert W. Walters,et al.  Uncertainty analysis for fluid mechanics with applications , 2002 .

[24]  Michel Loève,et al.  Probability Theory I , 1977 .

[25]  Baskar Ganapathysubramanian,et al.  Sparse grid collocation schemes for stochastic natural convection problems , 2007, J. Comput. Phys..

[26]  Patrick Knupp,et al.  Verification of Computer Codes in Computational Science and Engineering , 2002 .

[27]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[28]  Erich Novak,et al.  High dimensional polynomial interpolation on sparse grids , 2000, Adv. Comput. Math..

[29]  K. Ritter,et al.  High dimensional integration of smooth functions over cubes , 1996 .

[30]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[31]  K. Ritter,et al.  Simple Cubature Formulas with High Polynomial Exactness , 1999 .

[32]  N. Wiener The Homogeneous Chaos , 1938 .

[33]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[34]  Anna M. Bonner,et al.  Acknowledgments , 2019, The Neurodiagnostic journal.