Finite Element Method (FEM)

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Finite Element Method (FEM)

• BELA Finite Element Electrostatic Solver
• FEMM Finite Element Method Magnetics

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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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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 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/trian
gle.html
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BELA Triangle
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BELA Boundary Conditions

• Dirichlet, the value is explicitly defined on the
  boundary, e.g.

• Neumann, the normal derivative is defined on the
  boundary, e.g.

• Mixed,

• If no boundary conditions are defined, Neumann
  BC is used.

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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 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

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BELA Mesh and Solver
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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 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 (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
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BELA Example One Quarter
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BELA Example One Quarter
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BELA Example Circle
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BELA Example Circle
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BELA Example Circle
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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 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 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. 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