[Getdp] Problem definition of static heat flow

[Kenny Erleben]

Hi All,

I have been reading the reference manual and looked through the examples,
but I just don't get it. How do I write a formulation for the PDE of
static heat flow? That is

  u_xx + u_yy = - q

Below I have attached my .msh and .pro files

Regards

Kenny

-------- heat.msh --------
$NOD
20
 1 0 3 0
 2 0 2 0
 3 0 1 0
 4 0 0 0
 5 1 3 0
 6 1 2 0
 7 1 1 0
 8 1 0 0
 9 2 3 0
10 2 2 0
11 2 1 0
12 2 0 0
13 3 3 0
14 3 2 0
15 3 1 0
16 3 0 0
17 4 3 0
18 4 2 0
19 4 1 0
20 4 0 0
$ENDNOD

$ELM
38

 1 1 1001 0 2 17 13
 2 1 1001 0 2 13  9
 3 1 1001 0 2  9  5
 4 1 1001 0 2  5  1

 5 1 1002 0 2  4  8
 6 1 1002 0 2  8 12
 7 1 1002 0 2 12 16
 8 1 1002 0 2 16 20

 9 1 1003 0 2  1  2
10 1 1003 0 2  2  3
11 1 1003 0 2  3  4

12 1 1004 0 2 20 19
13 1 1004 0 2 19 18
14 1 1004 0 2 18 17

15 2 1000 0 3  5  1  6
16 2 1000 0 3  2  6  1
17 2 1000 0 3  6  2  7
18 2 1000 0 3  3  7  2
19 2 1000 0 3  7  3  8
20 2 1000 0 3  4  8  3
21 2 1000 0 3  9  5 10
22 2 1000 0 3  6 10  5
23 2 1000 0 3 10  6 11
24 2 1000 0 3  7 11  6
25 2 1000 0 3 11  7 12
26 2 1000 0 3  8 12  7
27 2 1000 0 3 13  9 14
28 2 1000 0 3 10 14  9
29 2 1000 0 3 14 10 15
30 2 1000 0 3 11 15 10
31 2 1000 0 3 15 11 16
32 2 1000 0 3 12 16 11
33 2 1000 0 3 17 13 18
34 2 1000 0 3 14 18 13
35 2 1000 0 3 18 14 19
36 2 1000 0 3 15 19 14
37 2 1000 0 3 19 15 20
38 2 1000 0 3 16 20 15
$ENDELM

----- heat.pro -----

Group {
  Top     = Region[ 1001 ];
  Bottom  = Region[ 1002 ];
  Left    = Region[ 1003 ];
  Right   = Region[ 1004 ];

  Boundary          = Region[ {Top, Bottom, Left, Right} ];
  DirichletBoundary = Region[ {Top, Right} ] ;
  NeumannBoundary   = Region[ {Left, Bottom} ] ;
  Domain            = Region[ 1000 ];
}

Function {
  lambda1 [ Domain ] = 1.;
  lambda2 [ Domain ] = 1.;
   qtilde [ Domain ] = 1.;
        q [ NeumannBoundary ] = 1.;
        f [ DirichletBoundary ] = 0.;
        f [ Domain ] = 0.;
}

Constraint {
  { Name DirichletConstraint ;
    Case {
      { Region DirichletBoundary ; Value f[] ; }
    }
  }
  { Name NeumannConstraint ;
    Case {
      { Region NeumannBoundary ; Value q[] ; }
    }
  }
}

Jacobian {
  { Name JacobianVol ;
    Case {
      { Region All ; Jacobian Vol ; }
    }
  }
}

Integration {
  { Name I1 ;
    Case {
      { Type Gauss ;
        Case {
	  { GeoElement Triangle    ; NumberOfPoints  4 ; }
	}
      }
    }
  }
}

FunctionSpace {
  { Name Temperature ; Type Form0 ;
    BasisFunction {
      { Name N ; NameOfCoef uHat ; Function BF_Node ;
        Support Domain ; Entity NodesOf[ All ] ; }
    }
    Constraint {
      { NameOfCoef uHat ; EntityType NodesOf ; NameOfConstraint
DirichletConstraint ; }
    }
  }
}

Formulation {
  { Name StaticHeatFlow ; Type FemEquation ;
    Quantity {
      { Name u ; Type Local ; NameOfSpace Temperature ; }
    }
    Equation {
      Galerkin { [ - lambda1[] * Dof{d u} , {d u} ] ;
                 In Domain ; Jacobian JacobianVol ; Integration I1 ; }

/*    Galerkin { [ - mu[] * hc[] , {d phi} ] ;
                 In NeumannBoundary ; Jacobian JacobianVol ; Integration
I1 ; }
*/
    }
  }
}

Resolution {
  { Name StaticHeatSolution ;
    System {
      { Name StaticHeatEquations ; NameOfFormulation StaticHeatFlow ; }
    }
    Operation {
      Generate[StaticHeatEquations] ; Solve[StaticHeatEquations] ;
SaveSolution[StaticHeatEquations] ;
    }
  }
}

PostProcessing {
  { Name StaticHeatPP ; NameOfFormulation StaticHeatFlow ;
    Quantity {
      { Name u   ; Value { Local { [ { u } ] ; In Domain ; Jacobian
JacobianVol ; } } }
    }
  }
}

PostOperation {
  { Name magic ; NameOfPostProcessing StaticHeatPP;
    Operation {
      Print[ u, OnElementsOf Domain, File "u.pos"] ;
    }
  }
}