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{  CONTROL_TRANSIENT.PDE  

   This example shows the use of a GLOBAL VARIABLE in a control application.

   We wish to find the required power input to a heater, such that the resulting

   average temperature over the domain is a specified value.

   The temperature on the outer surface is prescribed, with a time-sinusoidal

   oscillation.  

 

   The input power is driven by a time-relaxation equation.  The coefficient

   of the right hand side is the reciprocal of the response time of the power.

   This problem is a modification of CONTROL_STEADY.PDE, showing the use of

   time-dependent GLOBAL equations.

}

 

TITLE "Time-dependent Control test"

 

VARIABLES

 temp     { The temperature field }

 

GLOBAL VARIABLES

 power     { a single value for input power }

 

DEFINITIONS

 setpoint = 700           { the desired average temperature }

 skintemp = 325+20*sin(t) { oscillating outer boundary temperature }

 responsetime = 0.1       { response time of the power input }

 k=1       { conductivity }

 heat=0   { the heat function for the temperature.

               it is non-zero only in the heater region }

{ the control function, average temperature }

 tcontrol = integral(temp)/integral(1)  

{  tcontrol = val(temp,0,0)   -- an alternative control method, which tracks the

                               temperature value at a specified point (unused here) }

 

{initial guess for temperature distribution }

 tinit1=min(1767-400*abs(x), 1767-400*abs(y))  

 

INITIAL VALUES

 temp=min(1500,tinit1)  

 power=137     { initial guess for power  }

 

EQUATIONS

 temp:   div(-k*grad(temp))-heat = 0   { diffusion of temperature field }

{ single equation defining power.  response time is 1/100  }

 power:  dt(power) = (setpoint - tcontrol)/responsetime  

 

BOUNDARIES

 

  REGION 'Insulation'

       k=0.1

       heat=0

      start(-4,-4)

          value(temp)=skintemp

      line to (4,-4) to (4,4) to (-4,4) to close

 

  REGION 'Heater'

       k=50

       heat = power

      start(-1,-1) line to (1,-1) to (1,1) to (-1,1) to close

 

TIME 0 to 20 by 1e-4

 

PLOTS

for cycle=10

contour(temp)

  report power

  report tcontrol as "Avg Temp"

 

HISTORIES

History(tcontrol-setpoint, skintemp-325) as "Skin Temperature and Error"

History(tcontrol-setpoint) as "Controlled temperature error"

History(power)

 

END