# Finite Element Method (FEM) - PowerPoint PPT Presentation

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Finite Element Method (FEM). BELA: Finite Element Electrostatic Solver FEMM: Finite Element Method Magnetics. Plasma Application Modeling Group POSTECH. Contents. Introduction to BELA and FEMM packages Step 1. Drawing the problem geometry Step 2. Solve the problem

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Finite Element Method (FEM)

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## Finite Element Method (FEM)

BELA: Finite Element Electrostatic Solver

FEMM: Finite Element Method Magnetics

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

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

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

Plasma Application

Modeling Group

POSTECH

### 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

Plasma Application

Modeling Group

POSTECH

### 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.

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

Plasma Application

Modeling Group

POSTECH

### BELA: Geometry of the Problem

Problem Type: Planar or Axisymmetric

Length Units: mils, micrometers, millimeters, centimeters, inches,

and meters

Plasma Application

Modeling Group

POSTECH

### BELA: Geometry of the Problem

Plasma Application

Modeling Group

POSTECH

### BELA: Object Properties (1)

• Boundary Properties:

• Fixed Voltage

• Mixed

• Surface Charge Density

• Periodic

• Antiperiodic

• Materials Library

Plasma Application

Modeling Group

POSTECH

### BELA: Object Properties (2)

Plasma Application

Modeling Group

POSTECH

### BELA: Object Properties (3)

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

Plasma Application

Modeling Group

POSTECH

### BELA: Mesh and Solver

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

Plasma Application

Modeling Group

POSTECH

### BELA: Results

• Contour Plot

• Density Plots:

• Voltage (V)

• Electric Field Intensity (E)

• Electric Flux Density (D)

Plasma Application

Modeling Group

POSTECH

### BELA: Results (1)

Plasma Application

Modeling Group

POSTECH

### BELA: Results (1)

Plasma Application

Modeling Group

POSTECH

### BELA: Results (2)

Plasma Application

Modeling Group

POSTECH

### BELA: Results (3)

Plasma Application

Modeling Group

POSTECH

### BELA: Results (4)

Plasma Application

Modeling Group

POSTECH

### 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)

Plasma Application

Modeling Group

POSTECH

### BELA: Results (5)

Plasma Application

Modeling Group

POSTECH

### BELA: Results (5)

Plasma Application

Modeling Group

POSTECH

### BELA: Results (6)

Plasma Application

Modeling Group

POSTECH

### BELA: Results (6)

Plasma Application

Modeling Group

POSTECH

### 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

Plasma Application

Modeling Group

POSTECH

### BELA: Results

• Block Integrals:

• Storage Energy

• Block cross-section area

• Block Volume

• Average E over the volume

• Average D over the volume

Plasma Application

Modeling Group

POSTECH

### BELA: Results (7)

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

Plasma Application

Modeling Group

POSTECH

### BELA: Example – One Quarter

Plasma Application

Modeling Group

POSTECH

### BELA: Example – One Quarter

Plasma Application

Modeling Group

POSTECH

### BELA: Example – One Quarter

Plasma Application

Modeling Group

POSTECH

### BELA: Example – Circle

Plasma Application

Modeling Group

POSTECH

### BELA: Example – Circle

Plasma Application

Modeling Group

POSTECH

### BELA: Example – Circle

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

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.

Plasma Application

Modeling Group

POSTECH

### 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.

Plasma Application

Modeling Group

POSTECH

### 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

Plasma Application

Modeling Group

POSTECH

### 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.

Plasma Application

Modeling Group

POSTECH

### 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