Proper Generalized Decomposition method for incompressible Navier–Stokes equations with a spectral discretization

Abstract

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. Proper Generalized Decomposition (PGD) is a method which consists in looking for the solution to a problem in a separate form. This approach has been increasingly used over the last few years to solve mathematical problems. The originality of this work consists in the association of PGD with a spectral collocation method to solve transfer equations as well as Navier-Stokes equations. In the first stage, the PGD method and its association with spectral discretization is detailed. This approach was tested for several problems: the Poisson equation, the Darcy problem, Navier-Stokes equations (the Taylor Green problem and the lid-driven cavity). In the Navier-Stokes problems, the coupling between velocity and pressure was performed using a fractional step scheme and a P N -P NÀ2 discretization. For all problems considered, the results from PGD simulations were compared with those obtained by a standard solver and/or with the results found in the literature. The simulations performed showed that PGD is as accurate as standard solvers. PGD preserves the spectral behavior of the errors in velocity and pressure when the time step or the space step decreases. Moreover, for a given number of discretization nodes, PGD is faster than the standard solvers.

Generalized parametric solutions in Stokes flow

“…for all v and q ranging in some suitable spaces. Equations (24) are solved with a PGD strategy, exactly as described in the previous sections for the Stokes problem and equation (3). Typically, this operation is performed selectingũ PGD andp PGD in the same functional spaces as u PGD and p PGD .…”

“…NACA-4 example. Velocity (quiver fields) and pressure (color maps) solutions of the parametrized NACA-4 airfoils for four representative sets of parameter values{c,t,m,p}:[1]={0.53, 0.13, 0.10, 0.51};[2]={0.55, 0.17, 0.00, 0.50};[3]={0.71, 0.17, 0.07, 0.51};[4]={0.77, 0.23, 0.14, 0.38}.…”

Non-intrusive proper generalized decomposition involving space and parameters: application to the mechanical modeling of 3D woven fabrics

“…The in-plane-out-of-plane decomposition was then extended to many other physics: thermal models were considered in [18]; squeeze flows of Newtonian and non Newtonian fluids in laminates in [21]; flows in stratified porous media in [22] an nonlinear viscoplastic flows in plate domains in [23]. A full space decomposition was also efficiently applied for solving the Navier-Stokes equations in the lid-driven cavity problem in [24][25][26].…”

The method of dynamic mode decomposition in shallow water and a swirling flow problem

“…The use of this approach allows avoiding the use of the QR-decomposition when the matrix V N 1 0 is rank deficient [22]. (5) This step aims to compute S , and for the reader information, we will detail it in the following. Relations AV N 1…”