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An Example with Complex Numbers

In a microwave oven heat comes from molecular excitation by an electromagnetic field. For a plane monochromatic wave, amplitude is given by Helmholtz’s equation:

\[\beta v + \Delta v = 0.\]

We consider a rectangular oven where the wave is emitted by part of the upper wall. So the boundary of the domain is made up of a part \(\Gamma_1\) where \(v=0\) and of another part \(\Gamma_2=[c,d]\) where for instance \(\displaystyle v=\sin\left(\pi{y-c\over c-d}\right)\).

Within an object to be cooked, denoted by \(B\), the heat source is proportional to \(v^2\). At equilibrium, one has :

\[\begin{split}\begin{array}{rcl} -\Delta\theta &=& v^2 I_B\\ \theta_\Gamma &=& 0 \end{array}\end{split}\]

where \(I_B\) is \(1\) in the object and \(0\) elsewhere.

In the program below \(\beta = 1/(1-i/2)\) in the air and \(2/(1-i/2)\) in the object (\(i=\sqrt{-1}\)):

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// Parameters
int nn = 2;
real a = 20.;
real b = 20.;
real c = 15.;
real d = 8.;
real e = 2.;
real l = 12.;
real f = 2.;
real g = 2.;

// Mesh
border a0(t=0, 1){x=a*t; y=0; label=1;}
border a1(t=1, 2){x=a; y=b*(t-1); label=1;}
border a2(t=2, 3){ x=a*(3-t); y=b; label=1;}
border a3(t=3, 4){x=0; y=b-(b-c)*(t-3); label=1;}
border a4(t=4, 5){x=0; y=c-(c-d)*(t-4); label=2;}
border a5(t=5, 6){x=0; y=d*(6-t); label=1;}

border b0(t=0, 1){x=a-f+e*(t-1); y=g; label=3;}
border b1(t=1, 4){x=a-f; y=g+l*(t-1)/3; label=3;}
border b2(t=4, 5){x=a-f-e*(t-4); y=l+g; label=3;}
border b3(t=5, 8){x=a-e-f; y=l+g-l*(t-5)/3; label=3;}

mesh Th = buildmesh(a0(10*nn) + a1(10*nn) + a2(10*nn) + a3(10*nn) +a4(10*nn) + a5(10*nn)
   + b0(5*nn) + b1(10*nn) + b2(5*nn) + b3(10*nn));
real meat = Th(a-f-e/2, g+l/2).region;
real air= Th(0.01,0.01).region;
plot(Th, wait=1);

// Fespace
fespace Vh(Th, P1);
Vh R=(region-air)/(meat-air);
Vh<complex> v, w;
Vh vr, vi;

fespace Uh(Th, P1);
Uh u, uu, ff;

// Problem
solve muwave(v, w)
   = int2d(Th)(
        v*w*(1+R)
      - (dx(v)*dx(w) + dy(v)*dy(w))*(1 - 0.5i)
   )
   + on(1, v=0)
   + on(2, v=sin(pi*(y-c)/(c-d)))
   ;

vr = real(v);
vi = imag(v);

// Plot
plot(vr, wait=1, ps="rmuonde.ps", fill=true);
plot(vi, wait=1, ps="imuonde.ps", fill=true);

// Problem (temperature)
ff=1e5*(vr^2 + vi^2)*R;

solve temperature(u, uu)
   = int2d(Th)(
        dx(u)* dx(uu)+ dy(u)* dy(uu)
   )
   - int2d(Th)(
        ff*uu
   )
   + on(1, 2, u=0)
   ;

// Plot
plot(u, wait=1, ps="tempmuonde.ps", fill=true);

Results are shown on Fig. 37, Fig. 38 and Fig. 39.

RealMicroWave

Fig. 37 Real part

ImaginaryMicrowave

Fig. 38 Imaginary part

TemperatureMicrowave

Fig. 39 Temperature

Microwave

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