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, Pour obtenir un tel réseau, une matrice d'adjacence est créée (section A.5) puis les bords du réseau sont tendus et l'équation de la dynamique est appliquée (section A.3) avec dissipation de façon à ce que le réseau atteigne un

, Dans un premier temps, la phase aller du retournement temporel est effectuée (ligne 66 à 74) (section A.2). La source utilisée est définie aux lignes 15 à 17, les dissipations frictionnelle et visqueuse aux lignes 29 et 30. Les signaux (déplacements et vitesses) reçus au miroir à retournement temporel (défini à la ligne 26) sont alors stockés

, Nous présentons ici le script permettant de ne renvoyer qu'une fenêtre temporelle donnée (lignes 23) du signal transmis total. Les déplacements et vitesses reçus au miroir sont tronqués (lignes 102 et 103), retournés temporellement (lignes 106 et 107) et formatés pour être compris par la fonction wave.m (lignes 109 à 115). Puis l'équation de la dynamique est appliquée, Dans un second temps, la phase retour est effectuée (ligne 90 à 126)

, U_x_RT_store sous la forme d'une matrice tri-dimensionnelle, la première dimension étant le temps, la deuxième y, et la troisième x

A. A. Implémentation-de-l'expérience and . Numérique,

, Time Reversal Mirror

;. Trm=horzcat and . Meshgrid, * a); _ ref,sources _ mat _ p _ ref,time _ source,time _ wave ] = sources _ burst _ mat( source _ frequency, source _ amplitude,source _ nb _ cycles,sources _ positions, :n)',ones(n,1) * 66)

, % Applying dynamic equation

U. _-x,u-_-y,u-_-p-_-x,u-_-p-_-y,u-_-x-_-wave-_-store, U. _-y-_-wave-_-store, U. _-p-_-x-_-wave-_-store, U. _-p-_-y-_-wave-_-store, ~. et al., U _ x _ relaxed,U _ y _ relaxed,zeros(n), zeros(n), wall _ displacement,delta _ t _ wave,alpha _ wave,beta _ wave, cluster,K _ mat,m _ mat,live _ plot _ wave,X,Y,a,window _ center, window _ range,sources _ positions, sources _ mat _ ref,sources _ mat _ p _ ref

, % Storing the signals at the Time Reversal Mirror (TRM)

, TRM _ signal _ ref=zeros(nb _ steps _ wave,nb _ elem _ in _ TRM, vol.2

, TRM _ signal _ p _ ref=zeros(nb _ steps _ wave,nb _ elem _ in _ TRM, vol.2

, TRM _ signal _ ref(:,p,1)=U _ x _ wave _ store(:,TRM(p,1),TRM(p,2))?U _ x _ relaxed(TRM, p.2

, TRM _ signal _ ref(:,p,2)=U _ y _ wave _ store(:,TRM(p,1),TRM(p,2))?U _ y _ relaxed(TRM, p.2

, TRM _ signal _ p _ ref(:,p,1)=U _ p _ x _ wave _ store(:,TRM(p,1), p.2

, TRM _ signal _ p _ ref(:,p,2)=U _ p _ y _ wave _ store(:,TRM(p,1), p.2

, TR _ window _ start _ time=TR _ window

, TR _ window _ end _ time=TR _ window

, duration _ TR _ window=TR _ window _ end _ time?TR _ window _ start _ time

~. and T. R. ,

~. and T. R. ,

, % Windowing the signals at TRM (x and y components)

, TRM _ signal _ window=TRM _ signal _ ref(TR _ slice _ start _ index:TR _ slice _ end _ index

, TRM _ signal _ p _ window=TRM _ signal _ p _ ref(TR _ slice _ start _ index:TR _ slice _ end _ index

, % Time reversing the signals recorded at TRM

, TRM _ signal _ window _ flipped _ mat=flipud(TRM _ signal _ window)

, TRM _ signal _ p _ window _ flipped _ mat=flipud(TRM _ signal _ p _ window

, % Putting the TRM signal in cells

, TRM _ signal _ window _ flipped=cell(1,nb _ elem _ in _ TRM)

, TRM _ signal _ p _ window _ flipped=cell(1,nb _ elem _ in _ TRM)

, 112 for TRM _ element=1:nb _ elem _ in _ TRM

, TRM _ signal _ window _ flipped{TRM _ element}=squeeze(TRM _ signal _ window _ flipped _ mat(:,TRM _ element

, TRM _ signal _ p _ window _ flipped{TRM _ element}=squeeze(TRM _ signal _ p _ window _ flipped _ mat(:,TRM _ element

, time _ RT _ window _ record=?TR _ window _ end _ time:delta _ t _ RT:TR _ window _ end _ time

, time _ RT _ window _ source=0:delta _ t _ RT:duration _ TR _ window

, % Initial velocities set to zero before TR

U. _-p-_-x-_-init=zeros,

U. _-p-_-y-_-init=zeros,

U. _-x-_-rt, U. _-y-_-rt, U. _-p-_-x-_-rt, U. _-p-_-y-_-rt, U. _-x-_-rt-_-store et al., TRM,TRM _ signal _ window _ flipped,TRM _ signal _ p _ window _ flipped, time _ RT _ window _ record

, Elle peut aussi fournir l'état du réseau à chaque instant de la propagation sous la forme d'une matrice 3D (U_x_store, U_y_store, etc.) ainsi que l'évolution au cours du temps des déplacements et vitesses de certaines masses données (probe_U et probe_Up), A.2 Propagation d'une onde La fonction wave.m applique l'équation de la dynamique

, puis les accélérations déduites (ligne 94). L'équation de la dynamique est alors appliquée, grâce à la méthode d'Euler (lignes 105 à 113), pour en déduire les vitesses et positions au pas de temps suivant, Les longueurs des ressorts sont calculées (ligne 91 et section A.6)

U. _-x,u-_-y,u-_-p-_-x,u-_-p-_-y,u-_-x-_-store, U. _-y-_-store, U. _-p-_-x-_-store, U. _-p-_-y-_-store, . Up et al.,

, U _ x _ store=zeros(nb _ steps _ record

, U _ y _ store=zeros(nb _ steps _ record

, U _ p _ x _ store=zeros(nb _ steps _ record

, U _ p _ y _ store=zeros(nb _ steps _ record

, %% Wall displacements vector 25 walls _ displacements _ vector=linspace(?wall _ displacement,wall _ displacement

K. _-left,k-_-up,k-_-right,k-_-down,k-_-up-_-left and K. _-up-_-right,

, probe _ U=zeros(nb _ steps _ record,nb _ probes, vol.2

, probe _ Up=zeros(nb _ steps _ record,nb _ probes, vol.2

, %% Starting of the simulation 44 disp('Wave propagation

, Number of steps : ' num2str(nb _ steps _ record)

, accel _ y=accel _ y+contact _ friction _ y + viscous _ friction _

A. A. Implémentation-de-l'expérience-numérique-a, 4 Création d'un réseau tendu Le script suivant permet de créer un réseau bidimensionnel percolé tendu et équilibré. Le réseau percolant est défini entre les lignes 61 et 85. On créé d'abord une matrice d'adjacence (voir section A.5)

X. and Y. Meshgrid,

A. 4. Création-d'un-réseau and . Tendu,

K. _-mat-=-k-_-mat-_-create, (. K0, and K. Rand,

, cluster=spanning _ cluster(n,percolation); i=1:size(masses _ to _ add,1) 75 cluster(masses _ to _ add(i,1),masses _ to _ add, p.1

K. Mat, missing _ mass)=0

K. Mat, , p.0

K. _-left,k-_-up,k-_-right,k-_-down,k-_-up-_-left and K. _-up-_-right,

U. _-x,u-_-y, ~. , ~. , ~. , ~. et al., = pulling _ walls( zeros(n),zeros(n),zeros(n), zeros(n), wall _ displacement,wall _ speed ,delta _ t _ pulling _ walls,alpha _ pulling _ walls,beta _ pulling _ walls,cluster,K _ mat,m _ mat, live _ plot _ pulling _ walls, %% Pulling the walls 91

, %% Relaxation of the network %%

U. _-x-_-relaxed, U. _-y-_-relaxed, U. _-p-_-x-_-relaxed, U. _-p-_-y-_-relaxed, ~. et al., zeros(n,n ), wall _ displacement,duration _ relaxation,delta _ t _ relaxation,alpha _ relaxation,beta _ relaxation,cluster, K _ mat,m _ mat,live _ plot _ relaxation

A. A. Implémentation-de-l'expérience and . Numérique,

?. Ux,

?. Ux+a,

?. Uy, l _ right _ y=circshift(Uy

B. , RÉSOLUTION DES MODES PROPRES du système sont alors donnés par les vecteurs propres de D, et les fréquences propres par ses valeurs propres

B. La-figure, La première colonne présente pour chacun des modes présentés l'amplitude (vectorielle) de vibration en chaque site du réseau. La seconde colonne présente uniquement la composante selon l'axe x et la troisième colonne la composante selon l'axe y. On peut de plus tracer la fréquence des modes en fonction du numéro des modes (figure B, vol.4

, La densité présentée ici est à comparer à celle de la figure 3.23, en notant cependant deux différences dans les conditions du calcul