Cours Numerical methods for fluid mechanics  (5AF03 )

F. HECHT & S. ZALESKI 

UPMC / Master II  Fluid Mechanics / SPI 

Version  21 Oct..  2019  F. Hecht part. 

Université Pierre et Marie Curie

F. Hecht,  (Class Room 24-34 310)

F. HECHT (Cours)

            Monday          room: 24.34.310          8h30 – 12h30  is the , Planning

Modelling, Mathematical Analysis and Scientific Calculation in Household Waste Management Cours CIMPA, Kénitra 2019 G 03-13Juilet 2019 le fichier archive des presentations, exemples et le dossier .

Week 1) 23 Sep 2019 

Presentation of  Heat equation. 

Writing the weak form or the variational form

Presentation of the finite element and how tho build a finite element space.

Presentation of FreeFem++

First  solve the stationary Heat equation with all kind of boundary condition (B.C)

Dirichlet, Neumann,

Verified  the classical error of Finite element. 

The freefem++ example is in directory of w1/

To do

1) install freefem++ precompile from the web site http://www.freefem.org/ff++/ for mac or windows systems

and for linux  compile le version using  http://www.freefem.org/ff++/linux.php (with ./configure --enable_download

to have a full version of freefem++, the mpi version is not mandatory). 

2)  launch FreeFem++ on fish.edp example on you computer.

3)  try do add the computation of the potenteil field in file w1/Potentiel-flow-NACA12.edp

Week 2) 30 sept 2019

explication of the regularization to full Neuman Boundary condition.

The computation of irrotational flow with Joukowski condition.

see section 2. 2 of NS3book.pdf

the full solution is in w2/irrot-low-adap.edp

Week 3) 6 october 

Variational formulation  of Stokes Problem.

Approximation of the Stationnary  Stokes  problem 

Problem of compatibility  on discretization on Velocity and Pressure.

 see example w3/Stokes-v0.edp

Boundary  condition  problem of compatibility  ( zero output flux) 

a discrete level see w3/Stokes-bug.edp

a first way to do no linear problem (fixe point loop) w3/Navier-Stokes.edp

for more information of boundary condition for Stokes problem see  BC-STOKES.pdf

Week 4) 14 october

Newton algorithm to speed up the convergence

  page 153 of Cours-FFv4.pdf

Computation of derivative of operator. 

Example on Minimal surface problem. with fixe point algorithme 

is directory s4, they are example of steady state problem of Navier-Stokes , Rayleigh-Bernard flow

Home Work for next week, Implement the newton method of the minimal surface problem

after few iteration of fixed point method in s4/MinSurf.edp. And Change the func g , 

func g= sin(3*atan2(y,x));

in

func g= sin(10*atan2(y,x));

the err test : the err test if(err < 1e-7) break;

...

try to optimize the speed of resolution ..

Week 5) 21 october

Introduction of time scheme

Euler explicite (Warning the stability condition dt/h^2 < Cte for elliptique equation)

Euler implicite (alway stable)

The theta-schema (alway stable if theta > 0.5)

 BDF schema for higher order   see Backward differentiation formula - Wikipedia, the free encyclopedia