Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes

In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offline–online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model.

[1]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[2]  Niels Leergaard Pedersen,et al.  Discretizations in isogeometric analysis of Navier-Stokes flow , 2011 .

[3]  Alfio Quarteroni,et al.  A discontinuous Galerkin reduced basis element method for elliptic problems , 2016 .

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

[5]  Adrien Leygue,et al.  The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer , 2013 .

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

[7]  Numérisation de documents anciens mathématiques,et al.  Mathematical modelling and numerical analysis : Modélisation mathématique et analyse numérique. , 1985 .

[8]  Laura Iapichino.  Reduced Basis Methods for the Solution of Parametrized PDEs in Repetitive and Complex Networks with Application to CFD , 2012 .

[9]  F. Chinesta,et al.  A Short Review in Model Order Reduction Based on Proper Generalized Decomposition , 2018 .

[10]  Mihai Anitescu,et al.  Proper orthogonal decompositions in multifidelity uncertainty quantification of complex simulation models , 2014, Int. J. Comput. Math..

[11]  Gianluigi Rozza,et al.  Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows , 2014, J. Sci. Comput..

[12]  A. Fowler,et al.  Acid polishing of lead glass , 2011 .

[13]  A. Quarteroni,et al.  Numerical solution of parametrized Navier–Stokes equations by reduced basis methods , 2007 .

[14]  Thomas W. Sederberg,et al.  Free-form deformation of solid geometric models , 1986, SIGGRAPH.

[15]  A. Quarteroni,et al.  Shape optimization for viscous flows by reduced basis methods and free‐form deformation , 2012 .

[16]  Luca Heltai,et al.  Isogeometric analysis based reduced order modelling for incompressible viscous flows in parametrized shapes: applications to underwater shape design , 2015 .

[17]  David J. Benson,et al.  A multi-patch nonsingular isogeometric boundary element method using trimmed elements , 2015, Computational Mechanics.

[18]  Zhendong Luo,et al.  A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems , 2013, Math. Comput. Simul..

[19]  Nadine Aubry,et al.  On The Hidden Beauty of the Proper Orthogonal Decomposition , 1991 .

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

[21]  Thomas J. R. Hughes,et al.  Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology , 2012, Comput. Aided Des..

[22]  Ionel M. Navon,et al.  An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model , 2007 .

[23]  Gianluigi Rozza,et al.  A Reduced Basis Method for Evolution Schemes with Parameter-Dependent Explicit Operators , 2008 .

[24]  Patrick Laborde,et al.  A Reduced Basis Enrichment for the eXtended Finite Element Method , 2009 .

[25]  G. Rozza,et al.  Parametric free-form shape design with PDE models and reduced basis method , 2010 .

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

[27]  S. Ravindran.  A reduced-order approach for optimal control of fluids using proper orthogonal decomposition , 2000 .

[28]  S Niroomandi,et al.  Real‐time simulation of surgery by reduced‐order modeling and X‐FEM techniques , 2012, International journal for numerical methods in biomedical engineering.

[29]  Alfio Quarteroni,et al.  Isogeometric analysis and proper orthogonal decomposition for parabolic problems , 2017, Numerische Mathematik.

[30]  Gianluigi Rozza,et al.  Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations , 2015 .

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

[32]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[33]  Simona Perotto,et al.  Space–time adaptive hierarchical model reduction for parabolic equations , 2015, Advanced Modeling and Simulation in Engineering Sciences.

[34]  Thomas J. R. Hughes,et al.  Patient-Specific Vascular NURBS Modeling for Isogeometric Analysis of Blood Flow , 2007, IMR.

[35]  O. Anwar Bég,et al.  Modeling and simulation in engineering sciences , 2016 .

[36]  Gianluigi Rozza,et al.  Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization , 2016, J. Comput. Phys..

[37]  Zhendong Luo,et al.  Reduced-Order Modeling of the Upper Tropical Pacific Ocean Model using Proper Orthogonal Decomposition , 2006, Comput. Math. Appl..

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

[39]  Alfio Quarteroni,et al.  Spectral based Discontinuous Galerkin Reduced Basis Element method for parametrized Stokes problems , 2016, Comput. Math. Appl..

[40]  Gianluigi Rozza,et al.  Reduced basis methods for Stokes equations in domains with non-affine parameter dependence , 2009 .

[41]  L. Heltai,et al.  Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils , 2015 .

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

[43]  Bernard Haasdonk,et al.  Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation , 2012, SIAM J. Sci. Comput..

[44]  Simona Perotto,et al.  Hierarchical Model Reduction for Advection-Diffusion-Reaction Problems , 2008 .

[45]  G. Rozza,et al.  On the stability of the reduced basis method for Stokes equations in parametrized domains , 2007 .

[46]  Giancarlo Sangalli,et al.  IsoGeometric Analysis: Stable elements for the 2D Stokes equation , 2011 .

[47]  L. Heltai,et al.  Nonsingular isogeometric boundary element method for Stokes flows in 3D , 2014 .

[48]  Gianluigi Rozza,et al.  Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants , 2013, Numerische Mathematik.

[49]  Simona Perotto,et al.  Hierarchical Local Model Reduction for Elliptic Problems: A Domain Decomposition Approach , 2010, Multiscale Model. Simul..

[50]  B. Haasdonk,et al.  A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems , 2011 .

[51]  Gianluigi Rozza,et al.  A Reduced Basis Model with Parametric Coupling for Fluid-Structure Interaction Problems , 2012, SIAM J. Sci. Comput..

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

[53]  John A. Evans,et al.  ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS , 2013 .

[54]  Marina Rinaldi,et al.  Reduced Basis method for Isogeometric Analysis: application to structural problems , 2015 .

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

[56]  Henry J. Lamousin,et al.  NURBS-based free-form deformations , 1994, IEEE Computer Graphics and Applications.

[57]  Gianluigi Rozza,et al.  Reduced Order Methods for Modeling and Computational Reduction , 2013 .

[58]  Ionel Michael Navon,et al.  An optimizing finite difference scheme based on proper orthogonal decomposition for CVD equations , 2011 .

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

[60]  Thomas J. R. Hughes,et al.  Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations , 2013, J. Comput. Phys..

[61]  Stefan Turek,et al.  Isogeometric Analysis of the Navier-Stokes equations with Taylor-Hood B-spline elements , 2015, Appl. Math. Comput..