Numerical Study of a Monolithic Fluid–Structure Formulation

Abstract

The conservation laws of continuum mechanic are naturally written in an Eulerian frame where the difference between a fluid and a solid is only in the expression of the stress tensors, usually with Newton’s hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. There are currently two favored approaches to Fluid Structured Interactions (FSI) both working with the equations for the solid in the initial domain; one uses an ALE formulation for the fluid and the other matches the fluid–structure interfaces using Lagrange multipliers and the immersed boundary method. By contrast the proposed formulation works in the frame of physically deformed solids and proposes a discretization where the structures have large displacements computed in the deformed domain together with the fluid in the same; in such a monolithic formulation velocities of solids and fluids are computed all at once in a single variational formulation by a semi-implicit in time and the finite element method. Besides the simplicity of the formulation the advantage is a single algorithm for a variety of problems including multi-fluids, free boundaries, and FSI. The idea is not new but the progress of mesh generators renders this approach feasible and even reasonably robust. In this article the method and its discretization are presented, stability is discussed showing in a loose fashion were are the difficulties and why one is able to show convergence of monolithic algorithms on fixed domains for fluids in compliant shell vessels restricted to small displacements. A numerical section discusses implementation issues and presents a few simple tests.

Appendix: The FreeFem++ Script

// FSI with same variable for fluid and structure

border a(t=10,3)  { x=0; y=t;};  //  left

border b(t=0,10) { x=t; y=3 ;};  //  bottom

border c(t=3,7)  { x=10; y=t ;}; //  right low

border d(t=10,1) { x=t; y=7; };  //  low beam

border e(t=7,8) { x=1; y=t;  };  // left beam

border f(t=1,10) { x=t; y=8; };  // top beam

border g(t=8,10) { x=10; y=t;};  // right up

border hh(t=10,0) { x=t; y=10 ;}; //  top

border ee(t=7,8) { x=10; y=t;  };  // left beam

int m=1;

mesh th = buildmesh( a(m*30)+b(m*20)+c(m*16)+d(m*30)+e(m*5)

              +f(m*30)

+g(m*5)+hh(m*20)+ee(m*5));

real h=0.3;

int fluid=th(1,4).region, beam=th(9,7.5).region;

fespace V2h(th,P2);

fespace Vh(th,P1);

fespace Wh(th,P1);

Vh p,ph;

V2h u=0,v=0,uh,vh,d1=0,d2=0, uold=0, vold=0,

               uu,vv, uuold=0,vvold=0;

real nu=0.1;

real E = 2.15;

real sigma = 0.29;

real mu = E/(2*(1+sigma));

real c1=2*mu/2;

real penal=1e-6;

//real lambda = E*sigma/((1+sigma)*(1-2*sigma));

real gravity = -0.2;

real rhof=0.5, rhos=1.;

macro div(u,v) ( dx(u)+dy(v) ) // EOM

macro DD(u,v)  [[2*dx(u),div(v,u)],[div(v,u),2*dy(v)]] // EOM

macro Grad(u,v)[[dx(u),dy(u)],[dx(v),dy(v)]] // EOM

Vh g11=1,g12=0,g21=0,g22=1, g11aux,g22aux,g12aux,g21aux,

    f11,f12,f21,f22;

macro G[[g11,g12],[g12,g22]]//EOM

int NN=100;

real T=300, dt=T/NN;

problem aa([u,v,p],[uh,vh,ph]) =

int2d(th,beam)( rhos*[u,v]’*[uh,vh]/dt - div(uh,vh)*p

- div(u,v)*ph+ penal*p*ph

+dt*c1*trace(DD(uh,vh)*(DD(u,v)-Grad(u,v)*Grad(d1,d2)’

- Grad(d1,d2)*Grad(u,v)’)))

  + int2d(th,beam) ( -rhos*gravity*vh +c1*trace(DD(uh,vh)

    *(DD(d1,d2)

- Grad(d1,d2)*Grad(d1,d2)’))

-  rhos*[uold,vold]’*[uh,vh]/dt)

 +  int2d(th,fluid)(rhof*[u,v]’*[uh,vh]/dt- div(uh,vh)*p

     -div(u,v)*ph

+ penal*p*ph + nu/2*trace(DD(uh,vh)’*DD(u,v)))

  -  int2d(th,fluid)(rhof*gravity*vh

+rhof*[convect([uuold,vvold],-dt,uuold),

convect([uuold,vvold],-dt,vvold)]’*[uh,vh]/dt)

 + on(1,3,7,8,9,u=0,v=0) + on(2,u=0,v=0) ;

// Computation time loop

for(int n=0;n<NN;n++){

aa;

solve bb([uu,vv],[uh,vh]) = int2d(th,fluid)(

trace(Grad(uu,vv)*Grad(uh,vh)’) )

+ int2d(th,beam)(10000*[uu,vv]’*[uh,vh])

- int2d(th,beam)(10000*[u,v]’*[uh,vh])

+ on(1,2,3,7,8,uu=0,vv=0) + on(4,5,6,9,uu=u,vv=v);

  real mintcc = checkmovemesh(th,[x,y])/5.;

  real mint = checkmovemesh(th,[x+uu*dt,y+vv*dt]);

  uh=d1;

  vh=d2;

  if (mint<mintcc) {

    th=adaptmesh(th,h,IsMetric=1) ;// plot(th);

    }

  else {

th = movemesh(th,[x+uu*dt,y+vv*dt]);

  d1=0; d1[]=uh[]+dt*u[];

  d2=0; d2[]=vh[]+dt*v[];

  uold=0;  uold[]=u[];

  vold=0;  vold[]=v[];

uuold=u;

vvold=v;

  f11=1+dt*dx(uold); f12= dt*dx(vold); f21=dt*dy(uold);

  f22=1+dt*dy(vold);

     g11aux=g11*f11+g12*f21;

 g22aux=g12*f12+g22*f22;

 g12aux=g11*f12+g12*f22 ;

 g21aux=g12*f11+g12*f21 ;

     g11=f11*g11aux+f21*g21aux;

 g22=f12*g12aux+f22*g22aux;

 g12=f11*g12aux+f21*g22aux ;

  }

if((n/10)*10==n) plot(th,[uold,vold]);

vh=det(Grad(d1,d2));

cout<<n*dt<<" <- time, det d -> " << vh[].max<<

" pmax= "<<ph[].max<<" area= "<<int2d(th,beam)(1.)<<endl;

}

}