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{ COMPLEX_SINUSOIDAL_HEAT.PDE

 

 This example demonstrates the use of COMPLEX variables and ARRAY definitions

 to compute the time-sinusoidal behavior of a rod in a box.

 

 The heat equation is

   div(k*grad(temp)) = cp*dt(temp)

 

 If we assume that the sources and solutions are in steady oscillation at a frequency

 omega, then we can write

   temp(x,y,t) = phi(x,y)*exp(i*omega*t) = phi(x,y)*(cos(omega*t) + i*sin(omega*t))

 

 Substituting this into the heat equation and dividing the exp(i*omega*t) out of the

 result leaves

   div(k*grad(phi)) - i*omega*cp*phi = 0

 

 The temperature temp(x,y,t) can be reconstructed at any time by expanding the above

 definition.

 

 In this example, we construct an array of sample times and the associated arrays

 of sine and cosine factors.  These arrays are then used to display a time history of

 temperature at various points in the domain.

}  

 

TITLE 'Time Sinusoidal Heat flow around an Insulating blob '      

 

VARIABLES  

  ! define the complex amplitude function phi and its real and imaginary components

   phi=complex(phir,phii)  

 

DEFINITIONS  

   k=1  

   ts = array (0 by pi/20 to 2*pi) ! an array of sample times

   fr = cos(ts)               ! sine and cosine arrays

   fi = sin(ts)  

  ! define a function for evaluating a time array of temp(px,py,t) at a point

   temp(px, py) = eval(phir,px,py)*fr + eval(phii,px,py)*fi  

 

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)   { Phi=0 in base line }  

    NATURAL(Phi)=complex(0,0) LINE TO (1,1)   { normal derivative =0 on right side }  

    VALUE(Phi)=complex(1,0)   LINE TO (-1,1) { Phi = 1 on top }  

    NATURAL(Phi)=complex(0,0) LINE TO CLOSE   { normal derivative =0 on left side }  

 

REGION 2 'rod'   { the embedded circular rod }  

     k=0.01  

    START 'ring' (1/2,0)  

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

 

PLOTS  

  CONTOUR(Phir)       ! plot the real part of phi

      REPORT(k) REPORT(INTEGRAL(Phir, 'rod'))  

  CONTOUR(Phii)       ! plot the imaginary part of phi

      REPORT(k) REPORT(INTEGRAL(Phii, 'rod'))  

 

  ! reconstruct the temperature distribution at a few selected times

  REPEAT tx=0 by pi/2 to 2*pi

      SURFACE(phir*cos(tx)+phii*sin(tx)) as "Phi at t="+$[4]tx  

  ENDREPEAT  

 

  ! plot the time history at a few selected positions

  ELEVATION(temp(0,0), temp(0,0.2), temp(0,0.4), temp(0,0.5)) vs ts as "Phi(t) at X=0, Y=(0, 0.2 ,0.4, 0.5)"  

 

  VECTOR(-k*grad(Phir))  

       

  ! plot a lineout of phir and phii through the domain

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

  ! plot the real component of flux on the surface of the rod

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

 

END