electrostatic tutorial

Electrostatic/microstrip.geo

+/* -------------------------------------------------------------------
+   File "mStrip.geo"
+
+   This file is the geometrical description used by GMSH to produce
+   the file "mStrip.msh".
+   ------------------------------------------------------------------- */
+
+/* Definition of some parameters for geometrical dimensions, i.e.
+   h (height of 'Diel1'), w (width of 'Line'), t (thickness of 'Line')
+   xBox (width of the air box) and yBox (height of the air box) */
+
+h = 1.e-3 ; w = 4.72e-3 ;  t = 0.035e-3 ;
+xBox = w/2. * 6. ;  yBox = h * 12. ;
+
+/* Definition of parameters for local mesh dimensions */
+
+s = 1. ;
+p0 = h / 10. * s ;
+pLine0 = w/2. / 10. * s ;  pLine1 = w/2. / 50. * s ;
+pxBox = xBox / 10. * s ;  pyBox = yBox / 8. * s ;
+
+/* Definition of gemetrical points */
+
+Point(1) = { 0   , 0, 0, p0} ;
+Point(2) = { xBox, 0, 0, pxBox} ;
+Point(3) = { xBox, h, 0, pxBox} ;
+Point(4) = { 0   , h, 0, pLine0} ;
+Point(5) = { w/2., h, 0, pLine1} ;
+Point(6) = { 0   , h+t, 0, pLine0} ;
+Point(7) = { w/2., h+t, 0, pLine1} ;
+Point(8) = { 0   , yBox, 0, pyBox} ;
+Point(9) = { xBox, yBox, 0, pyBox} ;
+
+/* Definition of gemetrical lines */
+
+Line(1) = {1,2};   Line(2) = {2,3};  Line(3) = {3,9};
+Line(4) = {9,8};   Line(5) = {8,6};  Line(7) = {4,1};
+Line(8) = {5,3};   Line(9) = {4,5};  Line(10) = {6,7};
+Line(11) = {5,7};
+
+/* Definition of geometrical surfaces */
+
+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 */
+
+Physical Surface (101) = {15} ;   /* Air */
+Physical Surface (111) = {13} ;   /* Diel1 */
+
+Physical Line (120) = {1} ;       /* Ground */
+Physical Line (121) = {9,10,11} ; /* Line */
+Physical Line (130) = {2,3,4} ;   /* SurfInf */

Electrostatic/microstrip.pro

+/* -------------------------------------------------------------------
+   File "microstrip.pro"
+
+   This file defines the problem dependent data structures for the
+   microstrip problem.
+   
+   To compute the solution: 
+       getdp microstrip -solve EleSta_v
+
+   To compute post-processing results:
+       getdp microstrip -pos Map
+       getdp microstrip -pos Cut
+   ------------------------------------------------------------------- */
+
+
+
+Group {
+  /* One starts by giving explicit meaningful names to 
+     the Physical regions defined in the "microstrip.msh" mesh file. 
+     There are 2 volume regions and 3 surface regions in this model. */ 
+
+  Air = Region[101]; 
+  Diel1 = Region[111];
+
+  Ground = Region[120]; 
+  Electrode = Region[121]; 
+  SurfInf = Region[130];
+
+  /* We now define abstract regions to be used in the definition
+     of the scalar electric potential formulation:
+
+     Vol_Dielectric_Ele : dielectric volume regions where "div epsr[] grad v = 0" is solved
+     Sur_Dir_Ele        : Dirichlet boundary condition (v imposed)
+     Sur_Neu_Ele        : Neumann bondary condition ( epsr[] n.grad v = 0 )
+
+     Vol_xxx groups contain only volume elements of the mesh (triangles here).
+     Sur_xxx groups contain only surface elements of the mesh (lines here).
+  */
+ 
+  Vol_Dielectric_Ele = Region[ {Air, Diel1} ];
+  Sur_Dir_Ele = Region[ {Ground, Electrode} ];
+  Sur_Neu_Ele = Region[ {SurfInf} ];
+}
+
+Function {
+
+  /* The relative permittivity is defined piecewise using the above defined Groups */
+  epsr[Air] = 1.;
+  epsr[Diel1] = 9.8;
+}
+
+Constraint {
+  /* As for material laws, the Dirichlet boundary conditions can be defined piecewise 
+     The constraint "Dirichlet_Ele" is invoked in the FunctionSpace below */
+
+  { Name Dirichlet_Ele; Type Assign;
+    Case {
+      { Region Ground; Value 0.; }
+      { Region Electrode; Value 1.e-3; }
+    }
+  }
+}
+
+
+/* The function space of the electric scalar potential v 
+   is definied by 
+   - a domain of definition, which is the Group "Dom_Hgrad_v_Ele",
+   - a type (Form0 means scalar field)
+   - a set of scalar basis functions (here nodal basis functions "BF_Node")
+   - constraints (here the Dirichlet boundary conditions) 
+
+   Contrary to the above defined groups, which contain either volure or surface elements,
+   Dom_xxx groups contain both volume and surface elements */
+
+Group{
+  Dom_Hgrad_v_Ele =  Region[ {Vol_Dielectric_Ele, Sur_Dir_Ele, Sur_Neu_Ele} ];
+}
+
+FunctionSpace {
+  { Name Hgrad_v_Ele; Type Form0;
+    BasisFunction {
+      { Name sn; NameOfCoef vn; Function BF_Node;
+        Support Dom_Hgrad_v_Ele; Entity NodesOf[ All ]; }
+    }
+    Constraint {
+      { NameOfCoef vn; EntityType NodesOf; 
+        NameOfConstraint Dirichlet_Ele; }
+    }
+  }
+}
+
+Jacobian {
+  { Name Vol ;
+    Case { 
+      { Region All ; Jacobian Vol ; }
+    }
+  }
+}
+
+Integration {
+  { Name GradGrad ;
+    Case { {Type Gauss ;
+            Case { { GeoElement Triangle    ; NumberOfPoints  4 ; }
+                   { GeoElement Quadrangle  ; NumberOfPoints  4 ; } }
+      }
+    }
+  }
+}
+
+Formulation {
+  { Name Electrostatics_v; Type FemEquation;
+    Quantity {
+      { Name v; Type Local; NameOfSpace Hgrad_v_Ele; }
+    }
+    Equation {
+      Galerkin { [ epsr[] * Dof{d v} , {d v} ]; 
+	In Vol_Dielectric_Ele; 
+	Jacobian Vol; Integration GradGrad; }
+    }
+  }
+}
+
+Resolution {
+  { Name EleSta_v;
+    System {
+      { Name Sys_Ele; NameOfFormulation Electrostatics_v; }
+    }
+    Operation { 
+      Generate[Sys_Ele]; Solve[Sys_Ele]; SaveSolution[Sys_Ele];
+    }
+  }
+}
+
+
+eps0 = 8.854187818e-12;  // permittivity of empty space
+
+
+PostProcessing {
+  { Name EleSta_v; NameOfFormulation Electrostatics_v;
+    Quantity {
+      { Name v; 
+        Value { 
+          Local { [ {v} ]; 
+	    In Dom_Hgrad_v_Ele; Jacobian Vol; } 
+        }
+      }
+      { Name e; 
+        Value { 
+          Local { [ -{d v} ]; 
+	    In Dom_Hgrad_v_Ele; Jacobian Vol; }
+        }
+      }
+      { Name d; 
+        Value { 
+          Local { [ -eps0*epsr[] * {d v} ]; 
+	    In Dom_Hgrad_v_Ele; Jacobian Vol; } 
+        } 
+      }
+    }
+  }
+}
+
+e = 1.e-7; // parameter to ensure the cut is inside the simulation domain
+
+PostOperation {
+  { Name Map; NameOfPostProcessing EleSta_v;
+     Operation {
+       Print [ v, OnElementsOf Dom_Hgrad_v_Ele, File "mStrip_v.pos" ];
+       Print [ e, OnElementsOf Dom_Hgrad_v_Ele, File "mStrip_e.pos" ];
+     }
+  }
+  { Name Cut; NameOfPostProcessing EleSta_v;
+     Operation {
+       Print [ e, OnLine {{e,e,0}{10.e-3,e,0}} {500}, File "Cut_e" ];
+     }
+  }
+}