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{ BOUNDARY_CONSTRAINT.PDE      

   

 This problem demonstrates the use of boundary-integral CONSTRAINTS.

 

 A heat equation is solved subject to the constraint that the average temperature

 on the outer boundary must be 1.0.  

 

 Only natural (derivative) boundary conditions are applied, so the solution is  

 underdetermined subject to an arbitrary additive constant.

 

 The constraint provides the additional information necessary to make the  

 solution unique.

 

}  

 

title 'Boundary Constraint Test'  

 

variables  

    u  

 

equations  

    U: div(grad(u)) +x = 0;  

 

constraints  

    { force the average boundary value to 1 }  

    bintegral(u,"outer") = bintegral(1,"outer")  

 

boundaries  

    Region 1  

      start "outer" (-1,-1)  

      natural(u) = 0       line to (1,-1) to (1,1) to (-1,1) to close  

 

monitors  

  contour(u) report(bintegral(u,"outer"))  

plots  

  contour(u)   surface(u)  

  elevation(u) on "outer" report(bintegral(u,"outer")/bintegral(1,"outer")) as "Average"  

  summary  

    report("Constraint Validation:")  

    report(bintegral(u,"outer")/bintegral(1,"outer")) as "Average boundary value"  

 

end