Snapshots location for Reduced Order Models: An approach based on Proper Orthogonal Decomposition and Mesh Adaptivity Techniques

Solving Partial Differential Equations (PDE), is a key issue in science and engineering since they are widely used in many real problem modelling including fluid mechanics, acoustics, heat and mass transfer ...etc. Exact solutions for PDEs and Ordinary Differential Equations (ODE) can be obtained in a very few and simplified cases only, numerical approximations are used instead (i.e. Finite Volume Method (FEM), Finite Volume Method (FVM)...). However, and despite the availability of important supercomputing facilities, due to the huge number of degrees of freedom these methods have, they may still suffer from cost-effectiveness performance. They are mainly two contexts where one need to solve PDEs with a lower computational cost: Real-time context and Many-query context. Examples for the former are: parameter-estimation, control, flying simulator..., and for the latter: optimization, multi model/scale simulation... To reduce significantly simulation time (often on the expense of accuracy) Reduced Order Modelling (ROM) techniques are introduced. The main idea is to reduce the initial solving space dimension (as in finite elements method) to a subspace with a significantly reduced dimension and then solve for the projected solutions. Proper Orthogonal Decomposition (POD) is one of the more used ROM strategies. The presented work focuses on this technique, which has two challenging steps: (i) the snapshot location and (ii) the error estimate on the parameter space that drive the process to search new snapshot location. As a consequence of these two steps, POD applied to PDEs is considered as belonging to the well-known Greedy Algorithm family. This thesis brings a mesh adaptivity approach as the process to find the new parameter space locations. This process will be driven by a new error estimate based on Leave One Out Cross Validation (LOOCV) technique. We could say that this error estimate is universal, in the sense that it is not problem dependent. In addition, it is well known POD lack of accuracy when dealing with PDEs which solution contains shocks. Here, a new interpolation approach improves this, for shocked solutions. Finally, we present that the POD reduced basis is optimal just in average, and a new local basis is presented (Sorted Gram-Schmidt (SGS)) to be coupled with POD one. The criteria to decide which basis is better to be used for each new parameter value is defined as well. The whole proposed strategy efficiency is validated against a mathematical (exact) solutions of an incompressible, steady state flow equations, and on CFD solutions of an inviscid flow around a NACA0012 airfoil.

[1]  Juan J. Alonso,et al.  SU2: the Open-Source Software for Non-ideal Compressible Flows , 2017 .

[2]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[3]  Matthew F. Barone,et al.  On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment , 2010 .

[4]  Francis X. Giraldo,et al.  Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations , 2014, J. Comput. Phys..

[5]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[6]  Nguyen Ngoc Cuong,et al.  Certified Real-Time Solution of Parametrized Partial Differential Equations , 2005 .

[7]  Kai Schneider,et al.  Review of Some Fundamentals of Data Processing , 2007 .

[8]  Muruhan Rathinam,et al.  A New Look at Proper Orthogonal Decomposition , 2003, SIAM J. Numer. Anal..

[9]  Alessandro Alla,et al.  A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations , 2015 .

[10]  J. Mercer.  Functions of Positive and Negative Type, and their Connection with the Theory of Integral Equations , 1909 .

[11]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[12]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[13]  Jens Nørkær Sørensen,et al.  Evaluation of Proper Orthogonal Decomposition-Based Decomposition Techniques Applied to Parameter-Dependent Nonturbulent Flows , 1999, SIAM J. Sci. Comput..

[14]  Janet S. Peterson,et al.  The Reduced Basis Method for Incompressible Viscous Flow Calculations , 1989 .

[15]  Giulio Gori,et al.  Experimental assessment of the open- source SU2 CFD suite for ORC applications , 2017 .

[16]  Stefan Volkwein,et al.  Galerkin proper orthogonal decomposition methods for parabolic problems , 2001, Numerische Mathematik.

[17]  Juan Du,et al.  Non-linear Petrov-Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods , 2013, J. Comput. Phys..

[18]  Charbel Farhat,et al.  Nonlinear model order reduction based on local reduced‐order bases , 2012 .

[19]  Timothy J. Baker,et al.  Mesh adaptation strategies for problems in fluid dynamics , 1997 .

[20]  O. C. Zienkiewicz,et al.  Adaptive remeshing for compressible flow computations , 1987 .

[21]  Yingzheng Liu,et al.  The identification of coherent structures using proper orthogonal decomposition and dynamic mode decomposition , 2014 .

[22]  Karen Willcox,et al.  Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics , 2003 .

[23]  Alfio Quarteroni.  Reduced basis approximation for parametrized partial differential equations , 2014 .

[24]  L. Sirovich.  TURBULENCE AND THE DYNAMICS OF COHERENT STRUCTURES PART I : COHERENT STRUCTURES , 2016 .

[25]  P. Astrid,et al.  Reduction of process simulation models : a proper orthogonal decomposition approach , 2004 .

[26]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[27]  Stefan Volkwein,et al.  Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics , 2002, SIAM J. Numer. Anal..

[28]  D. Rovas,et al.  A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations , 2003 .

[29]  LAKHDAR REMAKI,et al.  3-D Mesh Adaptation on Multiple Weak Discontinuities and Boundary Layers , 2006, SIAM J. Sci. Comput..

[30]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[31]  Oubay Hassan,et al.  A High Order Finite Volume -HLLC Solver and Anisotropic Delaunay Mesh Adaptation , 2009 .

[32]  Peter Benner,et al.  Interpolatory Projection Methods for Parameterized Model Reduction , 2011, SIAM J. Sci. Comput..

[33]  Benjamin Peherstorfer,et al.  Localized Discrete Empirical Interpolation Method , 2014, SIAM J. Sci. Comput..

[34]  Abubakr Gafar Abdalla,et al.  Probability Theory , 2017, Encyclopedia of GIS.

[35]  R. Pinnau.  Model Reduction via Proper Orthogonal Decomposition , 2008 .

[36]  Jonathan A. Parker,et al.  Euler equations ∗ , 2007 .

[37]  Bernard Haasdonk,et al.  Adaptive Reduced Basis Methods for Nonlinear Convection–Diffusion Equations , 2011 .

[38]  A. Patera,et al.  A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations , 2005 .

[39]  J. Hahn,et al.  State-preserving nonlinear model reduction procedure , 2011 .

[40]  M. Grepl,et al.  Offline Error Bounds for the Reduced Basis Method , 2017 .

[41]  Angelo Iollo.  Remarks on the Approximation of the Euler Equations by a Low Order Model , 1997 .

[42]  S. Ravindran,et al.  A Reduced-Order Method for Simulation and Control of Fluid Flows , 1998 .

[43]  C. Farhat,et al.  Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations , 2011 .

[44]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[45]  Nirmal J. Nair,et al.  Transported snapshot model order reduction approach for parametric, steady‐state fluid flows containing parameter‐dependent shocks , 2017, International Journal for Numerical Methods in Engineering.

[46]  A. Patera,et al.  A Successive Constraint Linear Optimization Method for Lower Bounds of Parametric Coercivity and Inf-Sup Stability Constants , 2007 .

[47]  Gianluigi Rozza,et al.  Certified reduced basis approximation for parametrized partial differential equations and applications , 2011 .

[48]  Y. Maday,et al.  Reduced Basis Techniques for Stochastic Problems , 2010, 1004.0357.

[49]  M. Carbonaro,et al.  von Karman Institute for Fluid Dynamics , 2004 .

[50]  Marco Fossati,et al.  Local Reduced-Order Modeling and Iterative Sampling for Parametric Analyses of Aero-Icing Problems , 2015 .

[51]  Jean-Antoine Désidéri,et al.  Stability Properties of POD–Galerkin Approximations for the Compressible Navier–Stokes Equations , 2000 .

[52]  Eusebio Valero,et al.  A review on design of experiments and surrogate models in aircraft real-time and many-query aerodynamic analyses , 2018 .

[53]  Benjamin Stamm,et al.  Parameter multi‐domain ‘hp’ empirical interpolation , 2012 .

[54]  Karen Willcox,et al.  A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems , 2015, SIAM Rev..

[55]  D. Sorensen,et al.  A Survey of Model Reduction Methods for Large-Scale Systems , 2000 .

[56]  Peter Benner,et al.  Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective , 2014, Archives of Computational Methods in Engineering.

[57]  Peter Benner,et al.  Some a posteriori error bounds for reduced-order modelling of (non-)parametrized linear systems , 2017 .

[58]  Charbel Farhat,et al.  Nonlinear Model Reduction for CFD Problems Using Local Reduced Order Bases , 2012 .

[59]  Anthony T. Patera,et al.  An "hp" Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations , 2010, SIAM J. Sci. Comput..

[60]  K. Kunisch,et al.  Optimal snapshot location for computing POD basis functions , 2010 .

[61]  Bui-Thanh Tan,et al.  Model-Constrained Optimization Methods for Reduction of Parameterized Large-Scale Systems , 2007 .

[62]  B. Haasdonk,et al.  REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATIONS OF PARAMETRIZED LINEAR EVOLUTION EQUATIONS , 2008 .

[63]  Karen Willcox,et al.  Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space , 2008, SIAM J. Sci. Comput..

[64]  A. Patera,et al.  Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds , 2005 .

[65]  Saifon Chaturantabut.  Temporal localized nonlinear model reduction with a priori error estimate , 2017 .

[66]  R. Verfiirth.  A posteriori error estimation and adaptive mesh-refinement techniques , 2001 .

[67]  Ionel M. Navon,et al.  Non-linear Petrov-Galerkin methods for reduced order modelling of the Navier-Stokes equations using a mixed finite element pair , 2013 .

[68]  Michel Fortin.  Anisotropic mesh adaptation through hierarchical error estimators , 2001 .

[69]  M. Fortin,et al.  Anisotropic mesh adaptation: towards user‐independent, mesh‐independent and solver‐independent CFD. Part I: general principles , 2000 .

[70]  A. Quarteroni,et al.  Reduced Basis Methods for Partial Differential Equations: An Introduction , 2015 .

[71]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[72]  Bernard Haasdonk,et al.  A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space , 2011 .

[73]  Oliver Lass,et al.  Adaptive POD basis computation for parametrized nonlinear systems using optimal snapshot location , 2014, Comput. Optim. Appl..

[75]  Gianluigi Rozza,et al.  Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity , 2005 .

[76]  Laurent Cordier,et al.  Proper Orthogonal Decomposition: an overview , 2008 .

[77]  J. Hesthaven,et al.  Certified Reduced Basis Methods for Parametrized Partial Differential Equations , 2015 .

[78]  Luca Daniel,et al.  A Piecewise-Linear Moment-Matching Approach to Parameterized Model-Order Reduction for Highly Nonlinear Systems , 2007, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[79]  Etienne Balmes,et al.  PARAMETRIC FAMILIES OF REDUCED FINITE ELEMENT MODELS. THEORY AND APPLICATIONS , 1996 .

[80]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[81]  Jeanne A. Atwell,et al.  Proper Orthogonal Decomposition for Reduced Order Control of Partial Differential Equations , 2000 .

[82]  Nadine Aubry,et al.  The dynamics of coherent structures in the wall region of a turbulent boundary layer , 1988, Journal of Fluid Mechanics.

[83]  T. Simpson,et al.  Computationally Inexpensive Metamodel Assessment Strategies , 2002 .

[84]  Erwan Liberge,et al.  A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM-POD), for a 2D incompressible fluid flow , 2014, J. Comput. Appl. Math..