Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains

We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non-intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three-dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high-quality approximations for the first two statistical moments at modest computational effort.

[1]  I. Babuska,et al.  Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation , 2005 .

[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]  Michael B. Giles,et al.  Implicit time-accurate solutions on unstructured dynamic grids , 1997 .

[4]  Herbert H. H. Homeier,et al.  Effects of local field variations on the contrast of a field-ion microscope , 1983 .

[5]  Roberto F. S. Andrade,et al.  Fractal properties of equipotentials close to a rough conducting surface , 1999 .

[6]  Suzanne M. Shontz,et al.  A Mesh Warping Algorithm Based on Weighted Laplacian Smoothing , 2003, IMR.

[7]  Stefan Reh,et al.  Probabilistic finite element analysis using ANSYS , 2006 .

[8]  A. Nouy.  Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations , 2009 .

[9]  D. Xiu.  Efficient collocational approach for parametric uncertainty analysis , 2007 .

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

[11]  M. Eldred,et al.  Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos. , 2008 .

[12]  S. Rahman.  A polynomial dimensional decomposition for stochastic computing , 2008 .

[13]  Terje Haukaas,et al.  Shape sensitivities in the reliability analysis of nonlinear frame structures , 2006 .

[14]  A. Genz,et al.  Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight , 1996 .

[15]  Clarence O. E. Burg,et al.  Analytic study of 2D and 3D grid motion using modified Laplacian , 2006 .

[16]  Carl J. G. Evertsz,et al.  Harmonic measure around a linearly self-similar tree , 1992 .

[17]  Alejandro F. Frangi,et al.  Automatic construction of multiple-object three-dimensional statistical shape models: application to cardiac modeling , 2002, IEEE Transactions on Medical Imaging.

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

[19]  Alexandre Clément,et al.  Identification of random shapes from images through polynomial chaos expansion of random level set functions , 2009 .

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

[21]  Spencer J. Sherwin,et al.  Regular Article: Free-Surface Flow Simulation Using hp/Spectral Elements , 1999 .

[22]  R. Honda.  Stochastic BEM with spectral approach in elastostatic and elastodynamic problems with geometrical uncertainty , 2005 .

[23]  Mark G. Blyth,et al.  Heat conduction across irregular and fractal-like surfaces , 2003 .

[24]  S. Richardson,et al.  A model for the boundary condition of a porous material. Part 2 , 1971, Journal of Fluid Mechanics.

[25]  Alejandro F Frangi,et al.  Automatic construction of 3-D statistical deformation models of the brain using nonrigid registration , 2003, IEEE Transactions on Medical Imaging.

[26]  P. Nair.  On the theoretical foundations of stochastic reduced basis methods , 2001 .

[27]  T. Tezduyar,et al.  Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements , 2003 .

[28]  Marios M. Fyrillas,et al.  Conductive heat transport across rough surfaces and interfaces between two conforming media , 2001 .

[29]  Branislav Jaramaz,et al.  Nearly orthogonal two-dimensional grid generation with aspect ratio control , 2001 .

[30]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[31]  A. Nouy.  A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations , 2007 .

[32]  R D'Ippolito,et al.  Modelling of a vehicle windshield with realistic uncertainty , 2006 .

[33]  Reinhold Schneider,et al.  Sparse second moment analysis for elliptic problems in stochastic domains , 2008, Numerische Mathematik.

[34]  Daniel Rueckert,et al.  Hierarchical Statistical Shape Analysis and Prediction of Sub-cortical Brain Structures , 2007 .

[35]  George Stefanou,et al.  Stochastic finite element analysis of shells with combined random material and geometric properties , 2004 .

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

[37]  Andy J. Keane,et al.  Hybridization of stochastic reduced basis methods with polynomial chaos expansions , 2006 .

[38]  C. Chauviere,et al.  Efficient Computation of RCS From Scatterers of Uncertain Shapes , 2007, IEEE Transactions on Antennas and Propagation.

[39]  Long Chen,et al.  Stability and accuracy of adapted finite element methods for singularly perturbed problems , 2008, Numerische Mathematik.

[40]  Claudio Canuto,et al.  A fictitious domain approach to the numerical solution of PDEs in stochastic domains , 2007, Numerische Mathematik.

[41]  D. Xiu,et al.  Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos , 2002 .

[42]  Nicolas Moës,et al.  An extended stochastic finite element method for solving stochastic partial differential equations on random domains , 2008 .

[43]  Ilya M. Sobol,et al.  Theorems and examples on high dimensional model representation , 2003, Reliab. Eng. Syst. Saf..

[44]  Brian T. Helenbrook,et al.  Mesh deformation using the biharmonic operator , 2003 .

[45]  Geoffrey Ingram Taylor,et al.  A model for the boundary condition of a porous material. Part 1 , 1971, Journal of Fluid Mechanics.

[46]  A. Keane,et al.  Stochastic Reduced Basis Methods , 2002 .

[47]  Pol D. Spanos,et al.  A stochastic Galerkin expansion for nonlinear random vibration analysis , 1993 .

[48]  Jean-Pierre Coyette,et al.  Modal approaches for the stochastic finite element analysis of structures with material and geometric uncertainties , 2003 .

[49]  P. Janssen,et al.  Theory of U-statistics , 1994 .

[50]  George E. Karniadakis,et al.  The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications , 2008, J. Comput. Phys..

[51]  Daniel M. Tartakovsky,et al.  Numerical Methods for Differential Equations in Random Domains , 2006, SIAM J. Sci. Comput..

[52]  H. Rabitz,et al.  General foundations of high‐dimensional model representations , 1999 .

[53]  Víctor E. Garzón,et al.  Probabilistic aerothermal design of compressor airfoils , 2003 .

[54]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[55]  Ronald Cools,et al.  An encyclopaedia of cubature formulas , 2003, J. Complex..