From 2eb45f59c2685554c64698578b160956c04c7293 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Fran=C3=A7ois=20Henrotte?= <francois.henrotte@uclouvain.be>
Date: Fri, 20 Oct 2017 17:00:27 +0200
Subject: [PATCH] new tuto: electrostatics with global quantities
---
Electrostatics/microstrip.pro | 4 +-
ElectrostaticsFloating/floating.geo | 58 ++++++
ElectrostaticsFloating/floating.pro | 303 ++++++++++++++++++++++++++++
3 files changed, 362 insertions(+), 3 deletions(-)
create mode 100644 ElectrostaticsFloating/floating.geo
create mode 100644 ElectrostaticsFloating/floating.pro
diff --git a/Electrostatics/microstrip.pro b/Electrostatics/microstrip.pro
index 445b20e..5eb0caa 100644
--- a/Electrostatics/microstrip.pro
+++ b/Electrostatics/microstrip.pro
@@ -14,8 +14,6 @@
To compute the solution interactively from the Gmsh GUI:
File > Open > microstrip.pro
Run (button at the bottom of the left panel)
-
- The microstrip.pro file describes the finite element (FE) model.
------------------------------------------------------------------- */
Group {
@@ -90,7 +88,7 @@ FunctionSpace {
- a set of scalar basis functions ("BF_Node" means nodal basis functions)
- a constraint (here the Dirichlet boundary conditions)
- The FE representation of the unknown field "v" reads
+ The FE expansion of the unknown field "v" reads
v = Sum_k vn_k sn_k
diff --git a/ElectrostaticsFloating/floating.geo b/ElectrostaticsFloating/floating.geo
new file mode 100644
index 0000000..8b791bd
--- /dev/null
+++ b/ElectrostaticsFloating/floating.geo
@@ -0,0 +1,58 @@
+/* -------------------------------------------------------------------
+ File "microstrip.geo"
+
+ This file is the geometrical description used by GMSH to produce
+ the file "microstrip.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) = {1, 2, -8, -9, 7}; Plane Surface(13) = {12};
+Line Loop(14) = {10,-11,8,3,4,5}; Plane Surface(15) = {14};
+
+/* 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 ("Air", 101) = {15};
+Physical Surface ("Dielectric", 111) = {13};
+
+Physical Line ("Ground", 120) = {1} ;
+Physical Line ("Electrode", 121) = {9,10,11} ;
+Physical Line ("Surface infinity", 130) = {2,3,4} ;
diff --git a/ElectrostaticsFloating/floating.pro b/ElectrostaticsFloating/floating.pro
new file mode 100644
index 0000000..1b5ea46
--- /dev/null
+++ b/ElectrostaticsFloating/floating.pro
@@ -0,0 +1,303 @@
+/* -------------------------------------------------------------------
+ Tutorial 4 : floating potential of a microstrip electrode
+
+ Features:
+ - Global quantities and their special shape functions
+ - Duality
+ - More on ONELAB parameters (flags, model options, check boxes, menus, ...)
+
+ To compute the solution interactively from the Gmsh GUI:
+ File > Open > floating.pro
+ Run (button at the bottom of the left panel)
+
+ ------------------------------------------------------------------- */
+
+/*
+ A Thing GetDP is pretty good at is the management of global (non-local) basis functions.
+ Finite element expansions typically associate basis functions to individual nodes
+ or edges in the mesh. But consider the situation where a scalar field is set to be uniform
+ over a region of the problem (a floating potential electrode in an Electrostatics problem,
+ to fix the idea).
+ By factorizing the 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.
+ BF_electrode
+ - is a continuous function
+ - 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
+ 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 reads:
+
+ v = Sum_k sn_k vn_k + Sum_electrode v_electrode BF_electrode
+
+ We show in this tutorial how GetDP takes advantage of global quantities
+ and the associated global basis functions
+ - to reduce the number of unknowns
+ - to deal with floating potentials
+ - to compute efficiently the electrode charges "Q_electrode",
+ which are precisely the energy duals of the global "v_electrode" quantities
+ - to provide output quantities (charges, armature voltages, capacitances, ...)
+ that can be immediately used in a external circuit.
+ */
+
+
+Group {
+ /* Geometrical regions: */
+
+ Air = Region[101];
+ Diel1 = Region[111];
+
+ Ground = Region[120];
+ Microstrip = Region[121];
+ SurfInf = Region[130];
+
+ /* Abstract regions:
+ Vol_Dielectric_Ele : dielectric volume regions where
+ "Div ( epsr[] Grad v)" is solved
+ Sur_Neu_Ele : Neumann bondary condition ( epsr[] n.Grad v = 0 )
+ Electrodes_Ele : electrode regions
+
+
+ No prefix (Vol_ or Sur_) for the region "Electrodes_Ele",
+ which can contain both surface or volume regions.
+ There are two electrodes in this model: Ground and Microstrip
+ */
+
+ Vol_Dielectric_Ele = Region[ {Air, Diel1} ];
+ Sur_Neu_Ele = Region[ {SurfInf} ];
+ Electrodes_Ele = Region [ {Ground, Microstrip} ];
+}
+
+/* A number of ONELAB parameters are defined to define model parameters
+ or model options interactively. */
+MicrostripTypeBC =
+ DefineNumber[0, Name "1Microstrip excitation/Type",
+ Choices{ 0="Fixed voltage", 1="Fixed charge"} ] ;
+MicrostripValueBC =
+ DefineNumber[1e-3, Name "1Microstrip excitation/Value"] ;
+EpsilonRelDiel =
+ DefineNumber[9.8, Name "2Dielectric/Relative permittivity"] ;
+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 {
+ epsr[Air] = 1.;
+ epsr[Diel1] = EpsilonRelDiel;
+ eps0 = 8.854187818e-12; // permittivity of empty space
+}
+
+
+Constraint {
+ /* Dirichlet boundary condition is no longer used.
+ The microstrip and the ground are now treated as electrodes */
+ { Name Dirichlet_Ele; Type Assign;
+ Case {}
+ }
+
+ { Name SetGlobalPotential; Type Assign;
+ Case {
+ /* Define the imposed potential regionwise on the different parts of "Electrodes_Ele".
+ No voltage imposed to the Microstrip electrode
+ when the "Fixed charge" option is enabled ( MicrostripTypeBC = true ). */
+ { Region Ground; Value 0; }
+ If( !MicrostripTypeBC )
+ { Region Microstrip; Value MicrostripValueBC; }
+ EndIf
+ }
+ }
+ { Name SetArmatureCharge; Type Assign;
+ Case {
+ If( MicrostripTypeBC )
+ { Region Microstrip; Value MicrostripValueBC; }
+ EndIf
+ }
+ }
+}
+
+Group{
+ /* The domain of definition of lists all regions
+ on which the field "v" is defined.*/
+ Dom_Hgrad_v_Ele = Region[ {Vol_Dielectric_Ele,
+ Sur_Neu_Ele,
+ Electrodes_Ele } ];
+}
+
+FunctionSpace {
+ /* The magic in the treatment of global quantitities by GetDP is in the fact
+ that nearly all is done at the level of the FunctionSpace definition.
+
+ The finite element expansion is
+
+ v = Sum_k sn_k vn_k + Sum_electrode v_electrode BF_electrode
+
+ The sum_k runs over all nodes except those of the electrode regions.
+ "sf" stands for "BF_electrode"
+ "vf" stands for "v_electrode"
+ */
+
+ { Name Hgrad_v_Ele; Type Form0;
+ BasisFunction {
+ { Name sn; NameOfCoef vn; Function BF_Node;
+ Support Dom_Hgrad_v_Ele; Entity NodesOf[ All, Not Electrodes_Ele ]; }
+ { Name sf; NameOfCoef vf; Function BF_GroupOfNodes;
+ Support Dom_Hgrad_v_Ele; Entity GroupsOfNodesOf[ Electrodes_Ele ]; }
+ }
+ GlobalQuantity {
+ { Name GlobalPotential; Type AliasOf ; NameOfCoef vf; }
+ { Name ArmatureCharge ; Type AssociatedWith; NameOfCoef vf; }
+ }
+ Constraint {
+ { NameOfCoef vn; EntityType NodesOf;
+ NameOfConstraint Dirichlet_Ele; } // unused in this model
+ { NameOfCoef GlobalPotential; EntityType GroupsOfNodesOf;
+ NameOfConstraint SetGlobalPotential; }
+ { NameOfCoef ArmatureCharge; EntityType GroupsOfNodesOf;
+ NameOfConstraint SetArmatureCharge; }
+ }
+ // Subspace definition only needed to display "vf" in PostProcessing
+ SubSpace {
+ { Name vf; NameOfBasisFunction sf; }
+ }
+ }
+}
+
+Jacobian {
+ { Name Vol ;
+ Case {
+ { Region All ; Jacobian Vol ; }
+ }
+ }
+}
+
+Integration {
+ { Name Int ;
+ Case { {Type Gauss ;
+ Case { { GeoElement Triangle ; NumberOfPoints 4 ; }
+ { GeoElement Quadrangle ; NumberOfPoints 4 ; } }
+ }
+ }
+ }
+}
+
+Formulation {
+ /* Minor changes in the formulation.
+ Global quantities are declared in the "Quantity{}" section.
+ The Global term triggers the creation of the addition equation
+ in the system that computes the charge Q_electrode carried by each electrode
+
+ Q_electrode = (-epsr[] Grad v, Grad BF_electrode)_Vol_Dielectric_Ele
+ */
+ { Name Electrostatics_v; Type FemEquation;
+ Quantity {
+ { Name v; Type Local; NameOfSpace Hgrad_v_Ele; }
+ { Name U; Type Global; NameOfSpace Hgrad_v_Ele [GlobalPotential]; }
+ { Name Q; Type Global; NameOfSpace Hgrad_v_Ele [ArmatureCharge]; }
+ // next line only needed to display the global BF in PostProcessing
+ { Name vf; Type Local; NameOfSpace Hgrad_v_Ele [vf]; }
+ }
+ Equation {
+ Galerkin { [ epsr[] * Dof{d v} , {d v} ]; In Vol_Dielectric_Ele;
+ Jacobian Vol; Integration Int; }
+ GlobalTerm { [ -Dof{Q}/eps0 , {U} ]; In Electrodes_Ele; }
+ }
+ }
+}
+
+Resolution {
+ { Name EleSta_v;
+ System {
+ { Name Sys_Ele; NameOfFormulation Electrostatics_v; }
+ }
+ Operation {
+ Generate[Sys_Ele]; Solve[Sys_Ele]; SaveSolution[Sys_Ele];
+ If( OverwriteOutput )
+ DeleteFile[ "output.txt" ];
+ EndIf
+ }
+ }
+}
+
+PostProcessing {
+ { Name EleSta_v; NameOfFormulation Electrostatics_v;
+ Quantity {
+ { Name v; Value { Term { [ {v} ]; In Dom_Hgrad_v_Ele; Jacobian Vol; } } }
+ { Name e; Value { Term { [ -{d v} ]; In Dom_Hgrad_v_Ele; Jacobian Vol; } } }
+ { Name d; Value { Term { [ -eps0*epsr[] * {d v} ];
+ In Dom_Hgrad_v_Ele; Jacobian Vol; } } }
+ { Name Q; Value { Term { [ {Q} ]; In Electrodes_Ele; } } }
+ { Name U; Value { Term { [ {U} ]; In Electrodes_Ele; } } }
+ { Name C; Value { Term { [ {Q}/{U} ];
+ In Electrodes_Ele; } } }
+ { Name energy;
+ Value { Integral { Type Global;
+ [ eps0*epsr[] / 2. * SquNorm[{d v}] ];
+ In Vol_Dielectric_Ele; Jacobian Vol; Integration Int;
+ }
+ }
+ }
+ // next lines only needed to display global basis functions in PostProcessing
+ { Name BFGround; Value { Term { [ BF {vf} ]; In Dom_Hgrad_v_Ele;
+ SubRegion Ground; Jacobian Vol; } } }
+ { Name BFMicrostrip; Value { Term { [ BF {vf} ]; In Dom_Hgrad_v_Ele;
+ SubRegion Microstrip; Jacobian Vol; } } }
+
+ }
+ }
+}
+
+/* Various output results associated with the global quantities are generated.
+ They are both displayed in the graphical user interface, and stored in disk files.
+ In particular, all global quantities related results
+ are stored in the "output.txt" file.
+ There is a user option to display the global Basis functions of the two electrodes
+ in the system. Another user option allow the user to chose to not overwrite
+ the "output.txt" file when running a new simulation. */
+
+PostOperation {
+ { Name Map; NameOfPostProcessing EleSta_v;
+ Operation {
+ If( DisplayGlobalBF )
+ Print[ BFGround, OnElementsOf Dom_Hgrad_v_Ele, File "BFGround.pos" ];
+ Echo[ StrCat["l=PostProcessing.NbViews-1;",
+ "View[l].IntervalsType = 1;",
+ "View[l].NbIso = 40;",
+ "View[l].ShowElement = 1;"],
+ File "BFGround.opt", LastTimeStepOnly] ;
+
+ Print[ BFMicrostrip, OnElementsOf Dom_Hgrad_v_Ele, File "BFMicrostrip.pos" ];
+ Echo[ StrCat["l=PostProcessing.NbViews-1;",
+ "View[l].IntervalsType = 1;",
+ "View[l].NbIso = 40;",
+ "View[l].ShowElement = 1;"],
+ File "BFMicrostrip.opt", LastTimeStepOnly] ;
+ EndIf
+
+ Print[ v, OnElementsOf Dom_Hgrad_v_Ele, File "v.pos" ];
+ Echo[ StrCat["l=PostProcessing.NbViews-1;",
+ "View[l].IntervalsType = 3;",
+ "View[l].NbIso = 40;"],
+ File "mStrip_v.geo", LastTimeStepOnly] ;
+
+ Echo[ "Microstrip charge [C]:", Format Table, File > "output.txt"] ;
+ Print[ Q, OnRegion Microstrip, File > "output.txt", Color "AliceBlue",
+ Format Table, SendToServer "Output/Microstrip/Charge [C]" ];
+ Echo[ "Microstrip potential [V]:", Format Table, File > "output.txt"] ;
+ Print[ U, OnRegion Microstrip, File > "output.txt", Color "AliceBlue",
+ Format Table, SendToServer "Output/Microstrip/Potential [V]" ];
+ Echo[ "Ground charge [C]:", Format Table, File > "output.txt"] ;
+ Print[ Q, OnRegion Ground, File > "output.txt", Color "AliceBlue",
+ Format Table, SendToServer "Output/Ground/Charge [C]" ];
+ Echo[ "Microstrip capacitance [F]:", Format Table, File > "output.txt"] ;
+ Print[ C, OnRegion Microstrip, File > "output.txt", Color "AliceBlue",
+ Format Table, SendToServer "Output/Global/Capacitance [F]" ];
+ Echo[ "Electrostatic energy [J]:", Format Table, File > "output.txt"] ;
+ Print[ energy, OnRegion Vol_Dielectric_Ele, File > "output.txt", Color "AliceBlue",
+ Format Table, SendToServer "Output/Global/Energy [J]" ];
+ }
+ }
+}
+
+
--
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