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Suppose we wish to discover the time-dependent behavior of our example Cartesian blob due to the application of a time-sinusoidal applied temperature.

The time-dependent heat equation is Div(K*Grad(Phi)) = Cp*dt(Phi)

If we assume that the boundary values and solutions can be represented as

Phi(x,y,t) = Cphi(x,y)*exp(i*omega*t)

Substituting in the heat equation and dividing out the exponential term, we are left with a complex equation

Div(K*Grad(Cphi)) - Complex(0,1)*Cphi = 0

The time-varying temperature Phi can be recovered from the complex Cphi simply by multiplying by the appropriate time exponential and taking the real part of the result.

The modified script becomes:

TITLE 'Heat flow around an Insulating blob'

VARIABLES

Phi = Complex(Phir,Phii)        { the complex temperature amplitude }

DEFINITIONS

K = 1                { default conductivity }

R = 0.5                { blob radius }

EQUATIONS

Phi: Div(-k*grad(phi)) - Complex(0,1)*Phi = 0

BOUNDARIES

REGION 1 'box'

START(-1,-1)

VALUE(Phi)=Complex(0,0)        LINE TO (1,-1)

NATURAL(Phi)=Complex(0,0)        LINE TO (1,1)

VALUE(Phi)=Complex(1,0)        LINE TO (-1,1)

NATURAL(Phi)=Complex(0,0)        LINE TO CLOSE

REGION 2 'blob'        { the embedded blob }

k = 0.01        { change K for prettier pictures }

START 'ring' (R,0)

ARC(CENTER=0,0) ANGLE=360 TO CLOSE

PLOTS

CONTOUR(Phir) CONTOUR(Phii)

VECTOR(-k*grad(Phir))

ELEVATION(Phi) FROM (0,-1) to (0,1)

ELEVATION(Normal(-k*grad(Phir))) ON 'ring'

END

 

Running this script produces the following results for the real and imaginary components:

 

 

 

The ELEVATION trace through the center shows: