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ElectrostaticsFloating/floating.pro

@@ -20,12 +20,10 @@
    identical nodal value "v_electrode", a global (non-local) basis function
    "BF_electrode" is obtained as factor which is the sum of the shape functions
    of all the nodes in the electrode region. This basis function "BF_electrode"
-
    - is a continuous function, scalar in this case,
-   - is equal to 1 at the nodes of the electrode region, and to 0 at all other
-     nodes,
-   - decreases from 1 to 0 over the one element thick layer of outside finite
-     elements immediately in contact with the electrode region.
+   - is equal to 1 at the nodes of the electrode, and to 0 at all other nodes,
+   - decreases from 1 to 0 over the one-element-thick layer of elements sharing
+     at least one node with the electrode region.
 
    One such glabal basis function can be associated with each electrode in the
    system, so that the finite element expansion of the electric scalar potential
@@ -33,6 +31,9 @@
 
    v = Sum_k sn_k vn_k + Sum_electrode v_electrode BF_electrode
 
+   with the the sum_k running over all nodes except those of the electrode
+   regions.
+
    We show in this tutorial how GetDP takes advantage of global quantities and
    the associated global basis functions
    - to reduce the number of unknowns
@@ -76,6 +77,7 @@ DisplayGlobalBF = DefineNumber[0, Name "3Options/Display global basis functions"
                                Choices {0,1} ] ;
 OverwriteOutput = DefineNumber[1, Name "3Options/Overwrite output.txt file",
                                Choices {0,1} ] ;
+
 Function {
   eps0 = 8.854187818e-12;  // permittivity of empty space
   epsilon[Air] = eps0;
@@ -83,9 +85,9 @@ Function {
 }
 
 Constraint {
-  /* Dirichlet boundary condition on the local electric potential is no longer
-     used. The microstrip and the ground are now treated as electrodes, whose
-     voltage is imposed with the "SetGlobalPotential" constraint below. */
+  /* The Dirichlet boundary condition on the local electric potential is no
+     longer used. The microstrip and the ground are now treated as electrodes,
+     whose voltage is imposed with the "SetGlobalPotential" constraint below. */
   { Name Dirichlet_Ele; Type Assign;
     Case {
     }
@@ -94,8 +96,8 @@ Constraint {
   { Name SetGlobalPotential; Type Assign;
     Case {
       /* Define the imposed potential regionwise on the different parts of
-	 "Sur_Electrodes_Ele". No voltage imposed to the Microstrip electrode when
-	 the "Fixed charge" option is enabled (if MicrostripTypeBC != 0). */
+	 "Sur_Electrodes_Ele". No voltage is imposed to the Microstrip electrode
+	 when the "Fixed charge" option is enabled (if MicrostripTypeBC != 0). */
       { Region Ground; Value 0; }
       If(!MicrostripTypeBC)
 	{ Region Microstrip; Value MicrostripValueBC; }
@@ -135,14 +137,20 @@ FunctionSpace {
      "Integral" term, but where no integration needs to be performed. The
      "AssociatedWith" statement manifests the fact that the global potential of
      an electrode is the (electrostatic) energy dual of the electric charge
-     carried by that electrode. Indeed, in the weak formulation, when the
-     test-function v' is BF_electrode,
+     carried by that electrode. Indeed, let us consider the electrostatic weak
+     formulation derived in Tutorial 1: find v in Hgradv_Ele such that
+
+     (epsilon Grad v, Grad v')_Vol_Ele + (epsilon n.Grad v, v')_Bnd_Vol_Ele = 0
+
+     holds for all test functions v'. When the test-function v' is BF_electrode,
+     the boundary term reduces to
 
-     (epsilon n.Grad v, BF_electrode)_Bnd_Vol_Ele =
-       (epsilon n.Grad v, BF_electrode)_Sur_Electrodes_Ele =
-       (epsilon n.Grad v, 1)_Sur_Electrodes_Ele = Q_electrode,
+       (epsilon n.Grad v, BF_electrode)_Sur_Electrodes_Ele.
 
-     the charge carried by the electrodes.
+     Since BF_electrode == 1 on the electrode, the boundary term is actually
+     simply equal to the integral of (epsilon n.Grad v) on the electrode,
+     i.e. the flux of the displacement field, which is by definition the
+     charge Q_electrode carried by the electrodes.
 
      By checking the "Display global basis functions" checkbox and running the
      model, you can take a look on how the two "BF_electrode" basis functions in
@@ -199,13 +207,13 @@ Integration {
 }
 
 Formulation {
-  /* The formulation only contains minor changes compared to the first tutorial.
-     The global quantities are declared as "Global" in the "Quantity" section,
-     and a "GlobalTerm" is added that triggers the assembly of the additional
-     equation per electrode (the "pre-integrated" surface Neumann term) in the
-     system to compute the charge Q_electrode. Considering the equation
-     corresponding to the test function BF_electrode leads to the following
-     expression for the electrode charge:
+  /* The formulation only contains minor changes compared to formulation from
+     the first tutorial.  The global quantities are declared as "Global" in the
+     "Quantity" section, and a "GlobalTerm" is added that triggers the assembly
+     of the additional equation per electrode (the "pre-integrated" boundary
+     term) in the system to compute the charge Q_electrode, which
+     satisfies (just consider the equation corresponding to the test function
+     BF_electrode):
 
      Q_electrode = (-epsilon[] Grad v, Grad BF_electrode)_Vol_Ele */
   { Name Electrostatics_v; Type FemEquation;
@@ -263,7 +271,7 @@ PostProcessing {
         }
       }
       { Name energy; Value {
-          Integral { Type Global; // not per sub-region in Vol_Ele
+          Integral { Type Global; // global integral over Vol_Ele
             [ epsilon[] / 2. * SquNorm[{d v}] ];
             In Vol_Ele; Jacobian Vol; Integration Int;
           }