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# Free boundary problems

The domain $$\Omega$$ is defined with:

 1// Parameters
2real L = 10; //length
3real hl = 2.1; //left height
4real hr = 0.35; //right height
5int n = 4;
6
7// Mesh
8border a(t=0, L){x=t; y=0;}; //bottom: Gamma_a
9border b(t=0, hr){x=L; y=t;}; //right: Gamma_b
10border f(t=L, 0){x=t; y=t*(hr-hl)/L+hl;}; //free surface: Gamma_f
11border d(t=hl, 0){x=0; y=t;}; // left: Gamma_d
12mesh Th = buildmesh(a(10*n) + b(6*n) + f(8*n) + d(3*n));
13plot(Th);


The free boundary problem is:

Find $$u$$ and $$\Omega$$ such that:

$\begin{split}\left\{ \begin{array}{rcll} -\Delta u &=& 0 & \mbox{ in }\Omega\\ u &=& y & \mbox{ on }\Gamma_b\\ \partial u \over \partial n &=& 0 & \mbox{ on }\Gamma_d\cup\Gamma_a\\ \partial u \over \partial n &=& {q \over K} n_x & \mbox{ on }\Gamma_f\\ u &=& y & \mbox{ on }\Gamma_f \end{array} \right.\end{split}$

We use a fixed point method;

$$\Omega^0 = \Omega$$

In two step, fist we solve the classical following problem:

$\begin{split}\left\{ \begin{array}{rcll} -\Delta u &=& 0 & \mbox{ in }\Omega^n\\ u &=& y & \mbox{ on }\Gamma^n_b\\ \partial u \over \partial n &=& 0 & \mbox{ on }\Gamma^n_d\cup\Gamma^n_a\\ u &=& y & \mbox{ on }\Gamma^n_f \end{array} \right.\end{split}$

The variational formulation is:

Find $$u$$ on $$V=H^1(\Omega^n)$$, such than $$u=y$$ on $$\Gamma^n_b$$ and $$\Gamma^n_f$$

$\int_{\Omega^n}\nabla u \nabla u' = 0,\ \forall u' \in V \mbox{ with } u' =0 \mbox{ on }\Gamma^n_b \cup \Gamma^n_f$

And secondly to construct a domain deformation $$\mathcal{F}(x,y)=[x,y-v(x,y)]$$ where $$v$$ is solution of the following problem:

$\begin{split}\left\{ \begin{array}{rcll} -\Delta v &=& 0 & \mbox{ in }\Omega^n\\ v &=& 0 & \mbox{ on }\Gamma^n_a\\ \partial v \over \partial n &=& 0 & \mbox{ on }\Gamma^n_b\cup\Gamma^n_d\\ \partial v \over \partial n &=& {\partial u \over \partial n} - {q\over K} n_x & \mbox{ on }\Gamma^n_f \end{array} \right.\end{split}$

The variational formulation is:

Find $$v$$ on $$V$$, such than $$v=0$$ on $$\Gamma^n_a$$:

$\int_{\Omega^n} \nabla v \nabla v' = \int_{\Gamma_f^n}({\partial u \over \partial n} - { q\over K} n_x )v',\ \quad \forall v' \in V \mbox{ with } v' =0 \mbox{ on }\Gamma^n_a$

Finally the new domain $$\Omega^{n+1} = \mathcal{F}(\Omega^n)$$

Tip

Free boundary

The FreeFEM implementation is:

 1// Parameters
2real L = 10; //length
3real hr = 2.1; //left height
4real hl = 0.35; //right height
5int n = 4;
6
7real q = 0.02; //incoming flow
8real K = 0.5; //permeability
9
10// Mesh
11border a(t=0, L){x=t; y=0;}; //bottom: Gamma_a
12border b(t=0, hr){x=L; y=t;}; //right: Gamma_b
13border f(t=L, 0){x=t; y=t*(hr-hl)/L+hl;}; //free surface: Gamma_f
14border d(t=hl, 0){x=0; y=t;}; // left: Gamma_d
15mesh Th = buildmesh(a(10*n) + b(6*n) + f(8*n) + d(3*n));
16plot(Th);
17
18// Fespace
19fespace Vh(Th, P1);
20Vh u, v;
21Vh uu, vv;
22
23// Problem
24problem Pu (u, uu, solver=CG)
25    = int2d(Th)(
26          dx(u)*dx(uu)
27        + dy(u)*dy(uu)
28    )
29    + on(b, f, u=y)
30    ;
31
32problem Pv (v, vv, solver=CG)
33    = int2d(Th)(
34          dx(v)*dx(vv)
35        + dy(v)*dy(vv)
36    )
37    + on(a, v=0)
38    + int1d(Th, f)(
39          vv*((q/K)*N.y - (dx(u)*N.x + dy(u)*N.y))
40    )
41    ;
42
43// Loop
44int j = 0;
45real errv = 1.;
47while (errv > 1e-6){
48    // Update
49    j++;
50
51    // Solve
52    Pu;
53    Pv;
54
55    // Plot
56    plot(Th, u, v);
57
58    // Error
59    errv = int1d(Th, f)(v*v);
60
61    // Movemesh
62    real coef = 1.;
63    real mintcc = checkmovemesh(Th, [x, y])/5.;
64    real mint = checkmovemesh(Th, [x, y-v*coef]);
65
66    if (mint < mintcc || j%10==0){ //mesh too bad => remeshing
68        mintcc = checkmovemesh(Th, [x, y])/5.;
69    }
70
71    while (1){
72        real mint = checkmovemesh(Th, [x, y-v*coef]);
73
74        if (mint > mintcc) break;
75
76        cout << "min |T| = " << mint << endl;
77        coef /= 1.5;
78    }
79
80    Th=movemesh(Th, [x, y-coef*v]);
81
82    // Display
83    cout << endl << j << " - errv = " << errv << endl;
84}
85
86// Plot
87plot(Th);
88plot(u, wait=true);

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