Reduced basis methods for Stokes equations in domains with non-affine parameter dependence

In this paper we deal with reduced basis techniques applied to Stokes equations. We consider domains with different shape, parametrized by affine and non-affine maps with respect to a reference domain. The proposed method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. An “empirical”, stable and inexpensive interpolation procedure has permitted to replace non-affine coefficient functions with an expansion which leads to a computational decomposition between the off-line (parameter independent) stage for reduced basis generation and the on-line (parameter dependent) approximation stage based on Galerkin projection, used to find a new solution for a new set of parameters by a combination of previously computed stored solutions. As in the affine case this computational decomposition leads us to preserve reduced basis properties: rapid and accurate convergence and computational economies. The applications and results are based on parametrized geometries describing domains with curved walls, for example a stenosed channel and a bypass configuration. This method is well suited to treat also problems in fixed domain with non-affine parameters dependence expressing varying physical coefficients.

[1]  T. A. Porsching,et al.  Estimation of the error in the reduced basis method solution of nonlinear equations , 1985 .

[2]  Anthony T. Patera,et al.  A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations , 2002, J. Sci. Comput..

[3]  S. Ravindran,et al.  A Reduced Basis Method for Control Problems Governed by PDEs , 1998 .

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

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

[6]  A. T. Patera,et al.  Reduced-basis output bounds for approximately parametrized elliptic coercive partial differential equations , 2004 .

[7]  Max Gunzburger,et al.  3 – Finite Element Spaces , 1989 .

[8]  G. Reddien,et al.  On the reduced basis method , 1995 .

[9]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[10]  Max Gunzburger,et al.  Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms , 1989 .

[11]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[12]  A. Patera,et al.  Reduced-Basis Approximation of the Viscosity-Parametrized Incompressible Navier-Stokes Equation: Rigorous A Posteriori Error Bounds , 2004 .

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

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

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

[16]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[17]  O. Pironneau,et al.  Applied Shape Optimization for Fluids , 2001 .

[18]  Anthony T. Patera,et al.  Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations , 2002 .

[19]  柴田 良弘.  書評/G.P.Galdi:An Introduction to the Mathematical Theory of the Navier-Stokes Equations Volume 1:Linearized Steady Problem, Volume 2:Nonlinear Steady Problem , 1999 .

[20]  Karl Kunisch,et al.  Control and estimation of distributed parameter systems , 1998 .

[21]  Gianluigi Rozza,et al.  Real time reduced basis techniques for arterial bypass geometries , 2005 .

[22]  Dennis A. Nagy,et al.  Modal representation of geometrically nonlinear behavior by the finite element method , 1979 .

[23]  Sidney Yip,et al.  Handbook of Materials Modeling , 2005 .

[24]  Gianluigi Rozza,et al.  Shape Design in Aorto-Coronaric Bypass Anastomoses Using Perturbation Theory , 2006, SIAM J. Numer. Anal..

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

[26]  Anthony T. Patera,et al.  A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations , 2002 .

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

[28]  Gianluigi Rozza,et al.  On optimization, control and shape design of an arterial bypass , 2005 .

[29]  D. Rovas,et al.  Reduced--Basis Output Bound Methods for Parametrized Partial Differential Equations , 2002 .

[30]  A. Quarteroni,et al.  OPTIMAL CONTROL AND SHAPE OPTIMIZATION OF AORTO-CORONARIC BYPASS ANASTOMOSES , 2003 .

[31]  S. Fischler.  Formes linéaires en polyzêtas et intégrales multiples , 2002 .

[32]  Werner C. Rheinboldt,et al.  On the Error Behavior of the Reduced Basis Technique for Nonlinear Finite Element Approximations , 1983 .

[33]  P. Stern,et al.  Automatic choice of global shape functions in structural analysis , 1978 .

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

[35]  P. Hood,et al.  A numerical solution of the Navier-Stokes equations using the finite element technique , 1973 .

[36]  Werner C. Rheinboldt,et al.  On the theory and error estimation of the reduced basis method for multi-parameter problems , 1993 .

[37]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .