Transcript Inductively coupled plasma - Pohang University of Science

Finite Element Method
(FEM)
BELA: Finite Element Electrostatic Solver
FEMM: Finite Element Method Magnetics
17.07.2015
Contents
 Introduction to BELA and FEMM packages
 Step 1. Drawing the problem geometry
 Step 2. Solve the problem
 Step 3. Results analysis
 Some more examples
 Numerical methods
17.07.2015
Plasma Application
Modeling Group
POSTECH
Contents
 Introduction to BELA and FEMM packages
 Step 1. Drawing the problem geometry
 Step 2. Solve the problem
 Step 3. Results analysis
 Some more examples
 Numerical methods
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Plasma Application
Modeling Group
POSTECH
BELA: Introduction
BELA and FEMM are freeware software packages for 2D
analysis of electrostatic and magnetostatic linear problems.
These packages were written by David Meeker.
The homepage is located at:
http://femm.foster-miller.net
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BELA: Triangle
Triangle is a 2D mesh generator and Delaunay Triangulator.
It was written by Jonathan Shewchuk.
Winner of the 2003 James Hardy Wilkinson
Prize in Numerical Software
The homepage: http://www-2.cs.cmu.edu/~quake/triangle.html
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BELA: Boundary Conditions
•Dirichlet, the value is explicitly defined on the boundary, e.g.
  100
•Neumann, the normal derivative is defined on the boundary, e.g.

0
n
•Mixed,

 c0  c1  0
n
• If no boundary conditions are defined, Neumann BC is used.
17.07.2015
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Modeling Group
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Contents
 Introduction to BELA and FEMM packages
 Step 1. Drawing the problem geometry
 Step 2. Solve the problem
 Step 3. Results analysis
 Some more examples
 Numerical methods
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Modeling Group
POSTECH
BELA: Geometry of the Problem
Problem Type: Planar or Axisymmetric
Length Units: mils, micrometers, millimeters, centimeters, inches,
and meters
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BELA: Geometry of the Problem
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BELA: Object Properties (1)
Boundary Properties:
•Fixed Voltage
•Mixed
•Surface Charge Density
•Periodic
•Antiperiodic
Materials Library
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BELA: Object Properties (2)
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BELA: Object Properties (3)
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Contents
 Introduction to BELA and FEMM packages
 Step 1. Drawing the problem geometry
 Step 2. Solve the problem
 Step 3. Results analysis
 Some more examples
 Numerical methods
17.07.2015
Plasma Application
Modeling Group
POSTECH
BELA: Mesh and Solver
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Modeling Group
POSTECH
Contents
 Introduction to BELA and FEMM packages
 Step 1. Drawing the problem geometry
 Step 2. Solve the problem
 Step 3. Results analysis
 Some more examples
 Numerical methods
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Modeling Group
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BELA: Results
Contour Plot
Density Plots:
• Voltage (V)
• Electric Field Intensity (E)
• Electric Flux Density (D)
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BELA: Results (1)
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BELA: Results (1)
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BELA: Results (2)
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BELA: Results (3)
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BELA: Results (4)
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BELA: Results
Line Plots:
• Potential along the contour
•
Magnitude of the flux density along the contour (|D|)
•
Component of flux normal to the contour (D.n)
•
Component of flux density tangential to the contour (D.t)
•
Magnitude of the field intensity along the contour (|E|)
•
Component field intensity normal to the contour (E.n)
• Component of field intensity tangential to the contour
(E.t)
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BELA: Results (5)
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BELA: Results (5)
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BELA: Results (6)
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BELA: Results (6)
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BELA: Results
Line Integrals:
• Voltage drop along the contour (E.t)
• Total electric flux passing through the contour (D.n). If the
contour is closed, the result is equal to the charge inside this
contour
•
Contour Length and/or Area
•
Force from stress tensor
•
Torque from stress tensor
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BELA: Results
Block Integrals:
• Storage Energy
•
Block cross-section area
•
Block Volume
•
Average E over the volume
•
Average D over the volume
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BELA: Results (7)
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Contents
 Introduction to BELA and FEMM packages
 Step 1. Drawing the problem geometry
 Step 2. Solve the problem
 Step 3. Results analysis
 Some more examples
 Numerical methods
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BELA: Example – One Quarter
top  100
Air
Silicon
bottom  0
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BELA: Example – One Quarter
Air
Silicon
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BELA: Example – One Quarter
9.500e+001 : >1.000e+002
9.000e+001 : 9.500e+001
8.500e+001 : 9.000e+001
8.000e+001 : 8.500e+001
7.500e+001 : 8.000e+001
7.000e+001 : 7.500e+001
6.500e+001 : 7.000e+001
6.000e+001 : 6.500e+001
5.500e+001 : 6.000e+001
5.000e+001 : 5.500e+001
4.500e+001 : 5.000e+001
4.000e+001 : 4.500e+001
3.500e+001 : 4.000e+001
3.000e+001 : 3.500e+001
2.500e+001 : 3.000e+001
2.000e+001 : 2.500e+001
1.500e+001 : 2.000e+001
1.000e+001 : 1.500e+001
5.000e+000 : 1.000e+001
<0.000e+000 : 5.000e+000
Density Plot: V, Volts
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BELA: Example – Circle
top  100
Air
Silicon
bottom  100
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BELA: Example – Circle
Air
Silicon
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BELA: Example – Circle
9.000e+001 : >1.000e+002
8.000e+001 : 9.000e+001
7.000e+001 : 8.000e+001
6.000e+001 : 7.000e+001
5.000e+001 : 6.000e+001
4.000e+001 : 5.000e+001
3.000e+001 : 4.000e+001
2.000e+001 : 3.000e+001
1.000e+001 : 2.000e+001
-1.421e-014 : 1.000e+001
-1.000e+001 : -1.421e-014
-2.000e+001 : -1.000e+001
-3.000e+001 : -2.000e+001
-4.000e+001 : -3.000e+001
-5.000e+001 : -4.000e+001
-6.000e+001 : -5.000e+001
-7.000e+001 : -6.000e+001
-8.000e+001 : -7.000e+001
-9.000e+001 : -8.000e+001
<-1.000e+002 : -9.000e+001
Density Plot: V, Volts
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Modeling Group
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Contents
 Introduction to BELA and FEMM packages
 Step 1. Drawing the problem geometry
 Step 2. Solve the problem
 Step 3. Results analysis
 Some more examples
 Numerical methods
17.07.2015
Plasma Application
Modeling Group
POSTECH
BELA: Numerical Methods
General description of the finite element method:
Step 1. The problem is discretized by dividing the total space
domain into simple subdomains, the elements. In 2D problems
the basic region is divided into triangles, parallelograms or
curved-sided triangles. For 3D problems the region is
discretized into tetrahedral or cubic elements.
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BELA: Numerical Methods
General description of the finite element method:
BELA (and FEMM) uses triangular elements with linear
approximation of the potential by the expression
  a  bx  cy
Potential along any triangle edge is the linear interpolate
between its two vertex values, so if two triangles share the
same vertices, the potential will be continuous across the
interelement boundary.
The linear algebra problem is formed by choosing the potential
on the basis of minimizing the total energy of the problem.
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BELA: Numerical Methods
General description of the finite element method:
BELA (and FEMM) uses the Cuthill-McKee method for
renumbering the nodes.
  a  bx  cy
Source file: cuthill.cpp
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BELA: Numerical Methods
General description of the finite element method:
Step 2. For each of the elements a suitable approximation to the
functions which describe the problem, has to be chosen. In
general the form of the trial function in the element is controlled
by function value at certain points of the element, the nodes.
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BELA: Numerical Methods
General description of the finite element method:
Step 3. Solving the system of equations is the final step in a
FEM. Once the system of equations is solved, the desired
parameters can be compute and display in for of the curves,
plots, etc. This stage is often referred to as postprocessing.
To solve the set of linear equations Symmetric Successive Over
Relaxation (SSOR) method is used.
Source file: spars.cpp
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