reworked Gelerkin {} description and moved tuto1 into a new subdirectory

Electrostatic/microstrip.geo → Electrostatic/Laplacian/microstrip.geo

 /* -------------------------------------------------------------------
-   File "mStrip.geo"
+   File "microstrip.geo"
 
    This file is the geometrical description used by GMSH to produce
-   the file "mStrip.msh".
+   the file "microstrip.msh".
    ------------------------------------------------------------------- */
 
 /* Definition of some parameters for geometrical dimensions, i.e.
@@ -43,11 +43,12 @@ Line(11) = {5,7};
 Line Loop(12) = {8,-2,-1,-7,9};   Plane Surface(13) = {12};
 Line Loop(14) = {10,-11,8,3,4,5}; Plane Surface(15) = {14};
 
-/* Definition of Physical entities (surfaces, lines). The Physical
-   entities tell GMSH the elements and their associated region numbers
-   to save in the file 'mStrip.msh'. For example, the Region 
-   111 is made of elements of surface 13, while the Region 121 is 
-   made of elements of lines 9, 10 and 11 */
+/* Definition of Physical entities (surfaces, lines). 
+   The definition of Physical entities (Surfaces and Lines) 
+   tells GMSH the elements and associated region numbers
+   that have to be saved in the mesh file 'microstrip.msh'. 
+   For example, Region 111 is made of the triangle elements of the geometric surface 13, 
+   whereas Region 121 is made of line elements of the geometric lines 9, 10 and 11. */
 
 Physical Surface (101) = {15} ;   /* Air */
 Physical Surface (111) = {13} ;   /* Diel1 */

Electrostatic/microstrip.pro → Electrostatic/Laplacian/microstrip.pro

@@ -33,7 +33,8 @@ Group {
   /* We now define abstract regions to be used below 
      in the definition of the scalar electric potential formulation:
 
-     Vol_Dielectric_Ele : dielectric volume regions where "Div ( epsr[] Grad v)" is solved
+     Vol_Dielectric_Ele : dielectric volume regions where 
+                          "Div ( epsr[] Grad v)" is solved
      Sur_Dir_Ele        : Dirichlet boundary condition (v imposed)
      Sur_Neu_Ele        : Neumann bondary condition ( epsr[] n.Grad v = 0 )
 
@@ -55,7 +56,8 @@ Function {
 }
 
 Constraint {
-  /* As for material laws, the Dirichlet boundary condition is defined piecewise. 
+  /* As for material laws, the Dirichlet boundary condition 
+     is defined piecewise. 
      The constraint "Dirichlet_Ele" is invoked in the FunctionSpace below */
 
   { Name Dirichlet_Ele; Type Assign;
@@ -72,14 +74,16 @@ Group{
 
      Contrary to Vol_xxx and Sur_xxx regions, 
      which contain only volume or surface regions, resp.,
-     domains of definition Dom_xxx regions may contain both volume and surface regions. 
+     domains of definition Dom_xxx regions may contain 
+     both volume and surface regions. 
      Hence the use of the prefixes Vol_, Sur_ and Dom_ to avoid confusions.*/
 
   Dom_Hgrad_v_Ele =  Region[ {Vol_Dielectric_Ele, Sur_Dir_Ele, Sur_Neu_Ele} ];
 }
 
 FunctionSpace {
-  /* The function space in which we shall pick the electric scalar potential "v" solution
+  /* The function space in which we shall pick 
+     the electric scalar potential "v" solution
      is definied by 
      - a domain of definition ("Dom_Hgrad_v_Ele")
      - a type ("Form0" means scalar field)
@@ -88,12 +92,15 @@ FunctionSpace {
 
      The FE representation of the unknown field "v" reads
      
-     v = Sum_k sn_k vn_k 
+     v = Sum_k vn_k sn_k 
 
-     where the "vn_k" are the nodal values and "sn_k" the basis functions.
-     Not all nodal values are unknowns of the FE problem, due to the "Constraint"
-     which assigns particular values to the nodes of the region "Sur_Dir_Ele". 
-     GetDP deals with that automatically on basis of the definition of the FunctionSpace.
+     where the "vn_k" are the nodal values (connectors)
+     and "sn_k" the basis functions.
+     Not all connectors are unknowns of the FE problem, 
+     due to the "Constraint", which assigns particular values 
+     to the nodes of the region "Sur_Dir_Ele". 
+     GetDP deals with that automatically 
+     on basis of the definition of the FunctionSpace.
   */
 
   { Name Hgrad_v_Ele; Type Form0;
@@ -130,37 +137,73 @@ Formulation {
   /* The syntax of the Formulation{} section is a hard nut to crack.
      So let's deal with it carefully.
 
-     The weak form of the electrostatic problem is particularly simple in this model,
-     as it has only one term:
+     The weak form of the electrostatic problem is particularly simple
+     in this model, as it has only one term:
 
-     (epsr[] Grad v, Grad vp)_Vol_Dielectric_Ele = 0 for all vp in S0(Vol_Dielectric_Ele).
+     (epsr[] Grad v, Grad vp)_Vol_Dielectric_Ele = 0 
+     for all vp in S0(Vol_Dielectric_Ele).
 
-     The corresponding Euler-Lagrange are:
+     The corresponding Euler-Lagrange equations are:
      * Div ( epsr[] Grad v) = 0  on Vol_Dielectric_Ele
      * epsr[] n.Grad v = 0       on Sur_Neu_Ele
 
-     The Galerkin{} statement is a symbolic representation of this weak formulation term. 
-     The first argument represents the density to be integrated over finite elements,
-     the result of which being the bilinear form, i.e., the element-based matrix 
-     that will be assembled in the linear system of FE equations. 
-     
-     What is a bit confusing is that the two comma-separated arguments of the bracket [] 
-     are not exactly interpreted the same way. 
-     The symbol "d" represents the exterior derivative and is a synonym of "Grad".
-     The expression "d v" stands then for the gradient of the v field, 
+     The Galerkin{} statement is a symbolic representation 
+     of this weak formulation term.
+     It has got 4 semicolon separated arguments: 
+     * the density [,] to be integrated,
+     * the domain of integration,
+     * the jacobian of the transformation reference element -> real element,
+     * the integration method to be used.
+     The symbol "d" represents the exterior derivative,
+     and it is a synonym of "Grad" when applied to a scalar function. 
+     The expression "d v" stands thus for the gradient of the v field, 
      i.e., for the electric field up to a minus sign.
-     The Dof{} operator, finally, returns the array of basis functions of its argument.
 
-     As the Galerkin method uses as trial (test) functions the basis functions 
-     of the unknown field, the density of the bilinear form should be
+     What is a bit confusing is that the two comma-separated terms 
+     of the bracket [,], in the first argument,
+     are not interpreted the same way. Let us unravel this in detail. 
+
+     As the Galerkin method uses as trial (test) functions 
+     the basis functions "sn_k" of the unknown field "v", 
+     the density should be something like this
+
+     [ epsr[] * {d v} , basis_functions_of {d v} ].
+
+     Since the second argument is devoted to the trial functions,
+     the operator "basis_functions_of" would always be there.
+     It can therefore be made implicit, and, 
+     according to the GetDP syntax, it is omitted.
+     So, one writes simply "{d v}".
+
+     The first argument, on the other hand, can contain
+     a much wider variety of expressions than the second one.  
+     In this problem, it contains
+
+     epsr[] * {d v} = Sum_k   vn_k  epsr[]*{d sn_k}
+
+     which is the eletric displacement vector, up to a sign.
+     Here, we have two valid syntaxes, with very different meanings.
 
-     [ epsr[] * Dof{d v} , Dof{d v} ]
+     The first argument can be expressed in terms of 
+     the FE expansion of "v" at the present system solution.
+     This is indicated by invoking the Dof{} operator:
 
-     to highlight the symmetry.
-     But the second argument of the density is always reserved for the trial functions
-     and the Dof{} operator should therefore always be there.
-     It can thus be made implicit and, according to the GetDP syntax, it must be omitted.
+     [ epsr[] * Dof{d v} , {d v} ].
+
+     Another option, which would not work here,
+     is to evaluate the first argument 
+     with the last available already computed solution. 
+     For this the Dof{} operator is omitted:
+
+     [ epsr[] * {d v} , {d v} ].
+
+     Both choices are commonly used in different contexts,
+     and we shall come back on this often in subsequent tutorials. 
+
+     To express the stiffness matrix of the electrostatic problem at hand,
+     we have to take the first option. 
      Hence the final expression of the density below. 
+
    */
   { Name Electrostatics_v; Type FemEquation;
     Quantity {
@@ -193,9 +236,9 @@ eps0 = 8.854187818e-12;  // permittivity of empty space
    which itself refers to the FunctionSpace, 
    a number of quantities that can be evaluated at the postprocessing stage.
    The three quantities defined here are :
-   - the scalar vector potential
-   - the electric fied
-   - the electric displacement
+   - the scalar vector potential,
+   - the electric field,
+   - the electric displacement.
    The second part consists in defining groups of post-processing operations,
    which can be invoked separately. 
    The first group is invoked by default when Gmsh is run interactively.
@@ -204,7 +247,8 @@ eps0 = 8.854187818e-12;  // permittivity of empty space
    - the region on which the evaluation is done,
    - the name of the output file.
    The generated post-processing files are automatically displayed by Gmsh
-   if the "Merge result automatically" option is enabled (which is the default). */
+   if the "Merge result automatically" option is enabled 
+   (which is the default). */
 
 PostProcessing {
   { Name EleSta_v; NameOfFormulation Electrostatics_v;
@@ -237,11 +281,11 @@ h = 5.e-3; // vertical position of the cut
 PostOperation {
   { Name Map; NameOfPostProcessing EleSta_v;
      Operation {
+       Print [ v, OnElementsOf Dom_Hgrad_v_Ele, File "mStrip_v.pos" ];
        Echo[ StrCat["l=PostProcessing.NbViews-1;", 
 		    "View[l].IntervalsType = 1;",
 		    "View[l].NbIso = 40;"],
 	     File "tmp.geo", LastTimeStepOnly] ;
-       Print [ v, OnElementsOf Dom_Hgrad_v_Ele, File "mStrip_v.pos" ];
        Print [ e, OnLine {{e,h,0}{14.e-3,h,0}}{60}, File "Cut_e.pos" ];
      }
   }