Real-time reduced basis techniques for Navier-Stokes equations: Optimization of parametrized bypass configurations

The reduced basis method on parametrized domains is applied to approximate blood flow through an arterial bypass. The aim is to provide (a) a sensitivity analysis for relevant geometrical quantities of interest in bypass configurations and (b) rapid and reliable prediction of integral functional outputs ( such as fluid mechanics indexes). The goal of this investigation is (i) to achieve design indications for arterial surgery in the perspective of future development for prosthetic bypasses, (ii) to develop numerical methods for optimization and design in biomechanics, and (iii) to provide an input-output relationship led by models with lower complexity and computational costs than the complete solution of fluid dynamics equations by a classical finite element method.

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

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

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

[4]  Gianluigi Rozza,et al.  A Mathematical Approach in the Design of Arterial Bypass Using Unsteady Stokes Equations , 2006, J. Sci. Comput..

[5]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

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

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

[8]  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 .

[9]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

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

[11]  Gianluigi Rozza.  Shape design by optimal flow control and reduced basis techniques , 2005 .

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

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

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

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