cavitynewtow.edp

FreeFem++可以生成高质量的有限元网格。可以用于流体力学

mesh Th=square(8,8);
fespace Xh(Th,P2);
fespace Mh(Th,P1);
fespace XXMh(Th,[P2,P2,P1]);
XXMh [u1,u2,p];
XXMh [v1,v2,q]; 
//Mh p,q;
macro div(u1,u2) (dx(u1)+dy(u2))//
macro grad(u1,u2) [dx(u1),dy(u2)]//
macro ugrad(u1,u2,v) (u1*dx(v)+u2*dy(v)) //
macro Ugrad(u1,u2,v1,v2) [ugrad(u1,u2,v1),ugrad(u1,u2,v2)]//

solve Stokes ([u1,u2,p],[v1,v2,q],solver=UMFPACK) =
    int2d(Th)( ( dx(u1)*dx(v1) + dy(u1)*dy(v1)
            +  dx(u2)*dx(v2) + dy(u2)*dy(v2) )
            + p*q*(0.000001) 
            - p*div(v1,v2)-q*div(u1,u2)
           )
  + on(3,u1=4*x*(1-x),u2=0) 
  + on(1,2,4,u1=0,u2=0);
 Xh uu1=u1,uu2=u2;  
plot(coef=0.2,cmm=" [u1,u2] et p  ",p,[uu1,uu2],wait=1);

Xh psi,phi;


solve streamlines(psi,phi) = 
      int2d(Th)( dx(psi)*dx(phi) + dy(psi)*dy(phi))
   +  int2d(Th)( -phi*(dy(u1)-dx(u2)))
   +  on(1,2,3,4,psi=0);

plot(psi,wait=1);
int i=0;
real  nu=1./100.;
real dt=0.1;
real alpha=1/dt;



/* NL 
 varf   vNS ([u1,u2,p],[v1,v2,q],solver=Crout,init=i) =
    int2d(Th)(
             alpha*( u1*v1 + u2*v2) 
            + nu * ( dx(u1)*dx(v1) + dy(u1)*dy(v1)
            +  dx(u2)*dx(v2) + dy(u2)*dy(v2) )
            + p*q*(0.000001) 
            + p*dx(v1)+ p*dy(v2)
            + dx(u1)*q+ dy(u2)*q
            + Ugrad(u1,u2,u1,u2)'*[v1,v2]
           )   
  + on(3,u1=1,u2=0) 
  + on(1,2,4,u1=0,u2=0) 
*/

XXMh [up1,up2,pp];
varf   vDNS ([u1,u2,p],[v1,v2,q]) =
    int2d(Th)(
            
            + nu * ( dx(u1)*dx(v1) + dy(u1)*dy(v1)
            +  dx(u2)*dx(v2) + dy(u2)*dy(v2) )
            + p*q*(0.000001) 
            + p*dx(v1)+ p*dy(v2)
            + dx(u1)*q+ dy(u2)*q
            + Ugrad(u1,u2,up1,up2)'*[v1,v2]
            + Ugrad(up1,up2,u1,u2)'*[v1,v2]
           )
  + on(1,2,3,4,u1=0,u2=0) 
;


varf   vNS ([u1,u2,p],[v1,v2,q]) =
    int2d(Th)(
          
            + nu * ( dx(up1)*dx(v1) + dy(up1)*dy(v1)
            +  dx(up2)*dx(v2) + dy(up2)*dy(v2) )
            + pp*q*(0.000001) 
            + pp*dx(v1)+ pp*dy(v2)
            + dx(up1)*q+ dy(up2)*q
            + Ugrad(up1,up2,up1,up2)'*[v1,v2]
           )
  + on(1,2,3,4,u1=0,u2=0) 
;

for(real re=100;re<25000;re *=2)
{ 

real lerr=0.04;

if(re>8000) lerr=0.01;
if(re>10000) lerr=0.005; 
for(int step=0;step<2;step++)
{
     Th=adaptmesh(Th,[u1,u2],p,err=lerr,nbvx=100000);
     //plot(Th,wait=0);
     [u1,u2,p]=[u1,u2,p];
     [up1,up2,pp]=[up1,up2,pp];
	
for (i=0;i<=20;i++)
 { nu =1./re;
   up1[]=u1[];
    real[int] b = vNS(0,XXMh);
   matrix Ans=vDNS(XXMh,XXMh);
   set(Ans,solver=UMFPACK);
   real[int] w = Ans^-1*b;
   u1[] -= w;
   cout << " iter = "<< i << "  " << w.l2 <<  " rey = " << re << endl;
    if(w.l2<1e-6) break; 
   // uu1=u1;uu2=u2;
    //plot(coef=0.2,cmm=" [u1,u2] et p  ",p,[uu1,uu2]);  
    
 } ;
}
 uu1=u1;uu2=u2;
 streamlines;
  plot(coef=0.2,cmm="rey="+re+" [u1,u2] et p  ",psi,[uu1,uu2],wait=0,nbiso=20,ps="cavity-"+re+".ps");  
}