Pratique de l’optimisation numérique et du calcul scientifique

4AE04

1

Plan

1. Introduction

2. Problème modèle : Les équations de Burgers

1. Les équations

2. Les solutions exactes

3. Méthodes de Résolutions

3. Adaptation de maillage automatique

1. Les équations

2. La résolutions

3. Exemples

4. Projet sous matlab

2

Pourquoi des méthodes numériques

3

Pourquoi des méthodes numériques

4

Pourquoi des méthodes numériques

5

Pourquoi des méthodes numériques

6

Problèmatiques liés aux méthodes numériques

7

Maillage

Complexité...

8

(Jouhaud & Sagaut, J. Fluid Engng, 2008)

Maillage et calcul numérique

Principe : Raffiner le maillage dans les zones de forts gradients

Objectif : minimiser les coûts de calculMéthodes :

1. Raffinement à priori

2. Raffinement dynamique en cours de simulation

9

Problème modèle :Les équations de Burgers

Modèle 2D

10

An Adaptive Mesh-Moving and Local Refinement Method l 49

flows, boundary and shear layers in viscous flows, and reaction zones in combus-

tion processes. The numerical solution of these problems is usually difficult

because the nature, location, and duration of the structures are often not known

in advance. Thus, conventional numerical approaches that calculate solutions on

a prescribed (typically uniform) mesh often fail to adequately resolve the fine-

scale phenomena, have excessive computational costs, or produce incorrect

results. Adaptive procedures that evolve with the solution offer a robust, reliable,

and efficient alternative. Such techniques have been the subject of a great deal

of recent attention (cf. Babuska et al. [8,9] and Thompson [29]) and are generally

capable of introducing finer meshes in regions where greater resolution is needed

[l, 2, 3, 6, 14, 15, 22, 261, moving meshes in order to follow isolated dynamic

phenomena [l, 2, 5, 20, 22, 26, 281, or changing the order of methods in specific

regions of the problem domain [17, 211. The utility of such adaptive techniques

is greatly enhanced when they are capable of providing an estimate of the accuracy

of the computed solution. Local error estimates are often used as refinement

indicators and to produce solutions that satisfy either local or global accuracy

specifications [ 1,2,3,6,14,15,26]. Successful error estimates have been obtained

using h-refinement [6,14,15], where the difference between solutions on different

meshes is used to estimate the error, and p-refinement [l, 2, 3, 15, 211 where the

difference between methods of different orders are used to estimate the error.

We discuss an adaptive procedure that combines mesh movement and local

refinement for m-dimensional vector systems of partial differential equations having the form

ut + fb, Y, 4 u, u,, uy) = P’b, Y, 4 u)u,lx + P2k Y, t, uhl,, (Ha)

for t > 0, (x, y) E 0,

with initial conditions

u(x, Y, 0) = UO(& YL for (x, y) E D U dQ, (l.lb)

and appropriate well-posed boundary conditions on the boundary d R of a rectan-

gular region Q.

We suppose that a numerical method is available for calculating approximate

solutions and error indicators of (1.1) at each node of a moving mesh of

quadrilateral cells. Most numerical methods are applicable, and the error indi-

cator can either be an estimate of the local discretization error or another function

(e.g., an estimate of the solution gradient or curvature) that is large where

additional resolution is needed and small where less resolution is desired. Our

adaptive algorithm consists of three parts: (i) movement of a coarse base mesh,

(ii) local refinement of the base mesh in regions where resolution is inadequate,

and (iii) creation and regeneration of the base mesh when it becomes overly

distorted. Mesh motion can substantially reduce errors of hyperbolic problems

for a very modest computational cost. Mesh motion alone, however, cannot

produce a solution that will satisfy a prescribed error tolerance in all situations.

For this reason, we have combined mesh motion with local mesh refinement and recursively solve local problems in regions where error tolerances are not satisfied.

The local solution scheme successively reduces the domain size, and thus further

ACM Transactions on Mathematical Software, Vol. 16, No. 1, March 1990.

An Adaptive Mesh-Moving and Local Refinement Method l 49

flows, boundary and shear layers in viscous flows, and reaction zones in combus-

tion processes. The numerical solution of these problems is usually difficult

because the nature, location, and duration of the structures are often not known

in advance. Thus, conventional numerical approaches that calculate solutions on

a prescribed (typically uniform) mesh often fail to adequately resolve the fine-

scale phenomena, have excessive computational costs, or produce incorrect

results. Adaptive procedures that evolve with the solution offer a robust, reliable,

and efficient alternative. Such techniques have been the subject of a great deal

of recent attention (cf. Babuska et al. [8,9] and Thompson [29]) and are generally

capable of introducing finer meshes in regions where greater resolution is needed

[l, 2, 3, 6, 14, 15, 22, 261, moving meshes in order to follow isolated dynamic

phenomena [l, 2, 5, 20, 22, 26, 281, or changing the order of methods in specific

regions of the problem domain [17, 211. The utility of such adaptive techniques

is greatly enhanced when they are capable of providing an estimate of the accuracy

of the computed solution. Local error estimates are often used as refinement

indicators and to produce solutions that satisfy either local or global accuracy

specifications [ 1,2,3,6,14,15,26]. Successful error estimates have been obtained

using h-refinement [6,14,15], where the difference between solutions on different

meshes is used to estimate the error, and p-refinement [l, 2, 3, 15, 211 where the

difference between methods of different orders are used to estimate the error.

We discuss an adaptive procedure that combines mesh movement and local

refinement for m-dimensional vector systems of partial differential equations having the form

ut + fb, Y, 4 u, u,, uy) = P’b, Y, 4 u)u,lx + P2k Y, t, uhl,, (Ha)

for t > 0, (x, y) E 0,

with initial conditions

u(x, Y, 0) = UO(& YL for (x, y) E D U dQ, (l.lb)

and appropriate well-posed boundary conditions on the boundary d R of a rectan-

gular region Q.

We suppose that a numerical method is available for calculating approximate

solutions and error indicators of (1.1) at each node of a moving mesh of

quadrilateral cells. Most numerical methods are applicable, and the error indi-

cator can either be an estimate of the local discretization error or another function

(e.g., an estimate of the solution gradient or curvature) that is large where

additional resolution is needed and small where less resolution is desired. Our

adaptive algorithm consists of three parts: (i) movement of a coarse base mesh,

(ii) local refinement of the base mesh in regions where resolution is inadequate,

and (iii) creation and regeneration of the base mesh when it becomes overly

distorted. Mesh motion can substantially reduce errors of hyperbolic problems

for a very modest computational cost. Mesh motion alone, however, cannot

produce a solution that will satisfy a prescribed error tolerance in all situations.

For this reason, we have combined mesh motion with local mesh refinement and recursively solve local problems in regions where error tolerances are not satisfied.

The local solution scheme successively reduces the domain size, and thus further

ACM Transactions on Mathematical Software, Vol. 16, No. 1, March 1990.

Conditions initiales :

Modèle 1D

Stationnaire

Instationnaire

11

Solution exacte

Burgers stationnaire avec conditions de Robin

Solution :

12

avec

Solution

Comparaison Analytique / Numerique :

13

Burgers 1D instationnaire

Problème de déplacement de front

Solution

14

Stratégie de Raffinement

Topologie des mailles élémentaires

Méthode : Eléments finis, Volume finis, Spectral

Forme des mailles :

1D

2D : quadrilatère, triangle, ...

3D : Tétraèdre, Héxaèdres

Raffinement

à priori : statique au cours de la simulation

adaptatif

Critère de raffinement : Erreur, gradient, ...

15

Maillage

Complexité...

16

(Jouhaud & Sagaut, J. Fluid Engng, 2008)

Probème physique et maillage

Densité baryonique

17

Stratégie de raffinement dynamique

Structure arborescente : Hiérarchie Quad Trees

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1D 2D

Ondes de choc

Méthode : AMR

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Exemple : Jet

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Exemple : Rayleigh - Taylor

Méthode AMR : fort gradient de concentration.

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Exemple : MHD 2D

Nappes de courant : gradients intenses.

22

Exemple : dynamo solaire

composante radiale de la vitesse dans le soleil (3D)

23

Méthodes Numériques etéquations de Burgers

Shémas : u,t + a u,x = 0

Conditions initialesfor i=1:nx

x(i) = (i-1)*dx ; u(i) = sin(x(i)*2*pi) ; end

24

Notations :

w est la solution exacte

si u est le vecteur des valeurs u(i) alors up est le vecteur des u(i+1) um est le vecteur des u(i-1)

Schémas classiques

Lax-Friedrichs

25

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% marche en temps : Lax-Friedrichs avec cfl=0.9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w est la solution exacte% si u est le vecteur des valeurs u(i) alors % up est le vecteur des u(i+1) % um est le vecteur des u(i-1)% for n=1:nt%up = wshift2('1D',u,+1) ;um = wshift2('1D',u,-1) ;v = (up+um)/2 - (0.5*dt/dx)*a*(up-um) ;u = v ; % affichage de la solution approchee et exacteif mod(n,5) == 0 w = soltr(w,nx,dx,dt,a,n) ; plot(x,u,'o',x,w) ; axis([0 lg -1.1 1.1]) ; title('Lax-Friedrichs, cfl=0.9') pause(0) ;end%end

Lax Friedrichs

Evolution temporelle : CFL = 0.9

26

Schémas classiques

Lax Friedrichs

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nouveau schema: Lax-Friedrichs avec cfl=1.1 (explose)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dt = 1.1*cfl ;nt = 400 ;%% for n=1:nt%up = wshift2('1D',u,+1) ;um = wshift2('1D',u,-1) ;v = (up+um)/2 - (0.5*dt/dx)*a*(up-um) ;u = v ; if mod(n,5) == 0w = soltr(w,nx,dx,dt,a,n) ;plot(x,u,'o',x,w) ;axis([0 lg -1.1 1.1]) ;title('Lax-Friedrichs, cfl=1.1')pause(0) ;end%end

Lax Friedrichs

Evolution temporelle : CFL = 1.1

28

Lax Friedrichs

CFL = 0.1 (très dissipatif )

29

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nouveau schema: Lax-Friedrichs avec cfl=0.1 (tres dissipatif)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dt = 0.1*cfl ;nt = 1000 ;

Lax Friedrichs

Evolution temporelle : CFL = 0.1

30

Schémas classiques

Centré explicite

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nouveau schema: centre explicite avec cfl=0.5 (explose)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dt = 0.5*cfl ;nt = 400 ;

for n=1:nt%up = wshift2('1D',u,+1) ;um = wshift2('1D',u,-1) ;v = u - (0.5*dt/dx)*a*(up-um) ;u = v ; if mod(n,5) == 0w = soltr(w,nx,dx,dt,a,n) ;plot(x,u,'o',x,w) ;axis([0 lg -1.1 1.1]) ;title('centre explicite, cfl=0.5')pause(0) ;end%end

Centré Explicite

CFL = 0.5

32

Schémas classiques

Centré explicité : CFL =0.1

33

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nouveau schema: centre explicite avec cfl=0.1 (n'explose pas)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dt = 0.1*cfl ;nt = 1000 ;

Centré explicite

CFL = 0.1

34

Centré implicite

Centré impliciteCFL=0.9

35

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nouveau schema: centre implicite avec cfl=0.9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dt = 0.9*cfl ;nt = 400 ;%%% initialisation%x=zeros(1,nx) ;u=zeros(1,nx) ;v=zeros(1,nx) ;w=zeros(1,nx) ;for i=1:nx x(i) = (i-1)*dx ; u(i) = sin(x(i)*2*pi) ;endw = u ;mat=zeros(nx,nx) ;for i=2:nx-1 mat(i,i) = 1. ; mat(i,i+1) = (0.5*dt/dx)*a ; mat(i,i-1) = -(0.5*dt/dx)*a ;endmat(1,1) = 1. ;mat(1,2) = (0.5*dt/dx)*a ;mat(1,nx) = -(0.5*dt/dx)*a ;mat(nx,nx-1)= -(0.5*dt/dx)*a ;mat(nx,1) = (0.5*dt/dx)*a ;mat(nx,nx) = 1. ;smat = sparse(mat) ; plot(x,u) ;axis([0 lg -1.1 1.1]) ;title('centre implicite, cfl=0.9')

for n=1:nt%u0=u';v=smat\u0 ;u = v' ; if mod(n,5) == 0w = soltr(w,nx,dx,dt,a,n) ;plot(x,u,'o',x,w) ;axis([0 lg -1.1 1.1]) ;title('centre implicite, cfl=0.9')pause(0) ;end%F = getframe(fig); aviobj = addframe(aviobj,F);end

Centré implicite

CFL = 0.9

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Centré implicité

CFL = 2

37

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nouveau schema: centre implicite avec cfl=2.0%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dt = 2.0*cfl ;nt = 400 ;

Centré implicité

CFL = 2

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Lax Wendroff

Lax-Wendroff

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nouveau schema: Lax-Wendroff avec cfl=0.9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dt = 0.9*cfl ;nt = 800 ;%% for n=1:nt%up = wshift2('1D',u,+1) ;um = wshift2('1D',u,-1) ;v = u - (0.5*dt/dx)*a*(up-um) + ... (0.5*dt*dt/(dx*dx))*( a*a*(up-u) ... - a*a*(u-um) ) ;u = v ; if mod(n,5) == 0w = soltr(w,nx,dx,dt,a,n) ;plot(x,u,'o',x,w) ;axis([0 lg -1.1 1.1]) ;title('Lax-Wendroff, cfl=0.9')pause(0) ;end%end

Lax Wendroff

CFL = 0.9

40

Solution discontinue

Condition initiale créneau

41

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% donnee initiale = creneau% nouveau schema: Lax-Friedrichs avec cfl=0.9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dt = 0.9*cfl ;nt = 400 ;%% initialisation%x=zeros(1,nx) ;u=zeros(1,nx) ;v=zeros(1,nx) ;w=zeros(1,nx) ;up=zeros(1,nx) ;um=zeros(1,nx) ;for i=1:nx x(i) = (i-1)*dx ; if x(i) < lg/2 u(i) = -1. ; else u(i) = 1. ; endendw = u ; plot(x,u) ;axis([0 lg -1.1 1.1]) ;title('Lax-Friedrichs, cfl=0.9') pause ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% marche en temps : Lax-Friedrichs%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for n=1:nt%up = wshift2('1D',u,+1) ;um = wshift2('1D',u,-1) ;v = (up+um)/2 - (0.5*dt/dx)*a*(up-um) ;u = v ; if mod(n,5) == 0 for i=1:nx xi = (i-1)*dx-a*n*dt ; xint = floor(xi/lg) ; xi = xi - xint*lg ; if xi < lg/2 w(i) = -1. ; else w(i) = 1. ; end endplot(x,u,'o',x,w) ;axis([0 lg -1.1 1.1]) ;title('Lax-Friedrichs, cfl=0.9')pause(0) ;end%end pause ;

Lax Friedrichs

Créneau

42

Lax Wendroff

CréneauCFL = 0.9

43

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% donnee initiale = creneau% nouveau schema: Lax-Wendroff avec cfl=0.9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dt = 0.9*cfl ;nt = 400 ;%

for n=1:nt%up = wshift2('1D',u,+1) ;um = wshift2('1D',u,-1) ;v = u - (0.5*dt/dx)*a*(up-um) + ... (0.5*dt*dt/(dx*dx))*( a*a*(up-u) ... - a*a*(u-um) ) ;u = v ; if mod(n,5) == 0 for i=1:nx xi = (i-1)*dx-a*n*dt ; xint = floor(xi/lg) ; xi = xi - xint*lg ; if xi < lg/2 w(i) = -1. ; else w(i) = 1. ; end endplot(x,u,'o',x,w) ;axis([0 lg -1.4 1.4]) ;title('Lax-Wendroff, cfl=0.9')pause(0) ;end%end

Lax Wendroff

Créneau

44

Adaptation automatique

Méthode du maillage mobile

Points de grille mobiles

Dérivée temporelle de u sur cette grille

Fonction de monitoring

45

Advection de front d’onde

Adaptation du maillage

46

Exemples d’applications

transport linéaire en 2-d

 transport linéaire non diffusif

trafic routier

équations d'Euler 2-d

Euler avec friction et Darcy

 schémas VFRoe et VFRoe-ncv

chocs non classiques

schéma exact de Godunov

 approximation d'ordre élevé

schéma équilibre

dynamique des gaz sans pression

couplage de deux systèmes

MHD -> modèle de Thomas

47

Projet Numerique : 3x4h

48

UPMC M1 EE

UE : 4AE04

Pratique de l’optimisation numerique et du calcul

scientifique

Resolution d’EDP

et maillage adaptatif

1. Maillage Adaptatif

(a) Lire le code matlab et faire le lien avec l’article de Huang et al..

(b) Donner la signification du parametre ⌧ et etudier son influence sur la simulation.

(c) Donner la signification du parametre � et etudier son influence sur la simulation.

(d) Donner la signification du parametre p et etudier son influence sur la simulation.

(e) Tracer les evolutions temporelles des quantites H(t) et E(t) de l’article de Huanget al.. Comparer les resultats obtenus pour la simulation avec le meme jeu deparametres (fig. 6 p287). Commenter vos resultats.

(f) Comparer la methodes a maillage adaptatif avec les methodes a maillage fixe clas-sique : Euler, Lax-Wendro↵, Lax-Friedrichs en regime stationnaire et instation-naire.

(g) Que peut-on conclure sur l’interet des methodes a maillage adaptatif.

1

Projet Numerique : 3x4h

49

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

x

solu

tion

u

Burgers equation. Mass matrix M(t,y). MMPDE6

t = 0t = 0.2t = 0.6t = 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

xt

Burgers equation. Grid trajectories

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Lax−Wendroff

2

50