1. Introduction to the Finite Element Method

2. History of Field Simulation
Maxwell’s Equations (1873) define and solve electromagnetic fields exactly and completely except...
THERE IS NO CLOSED FORM
SOLUTION FOR THEM!
Field Simulation techniques (1950’s) can be used for complicated geometries and complicated boundary conditions where “textbook equations” are not valid

3. Field Simulation Software
Design Tool for Engineers
Solves Electrical, Structural, Thermal, Fluid, and other Engineering Problems
Can be used for complicated geometries and complicated boundary conditions where “textbook equations” are not valid.

4. Field Simulation Software
The Advantages:
Evaluate and Understand Design
Optimize Design
Reduce Product Development Time
Reduce Prototyping Costs

5. Major Industries Automotive Power Transmission & Distribution
Consumer Electronics
Communications
Integrated Circuitry
Medical
Universities

6. Different Methods of Electromagnetic Analysis
Analytical Techniques
Numerical Techniques
Closed Form
Iterative
Integral Equations
Differential Equations
Boundary Elements
Finite Difference
Finite Elements
BEM
FEM
FDM
Scalar Potentials
Vector Potentials
Components of H-Field
2D Electrostatic
2D/3D Thermal
3D Electrostatic
2D Magnetostatic
2D Eddy
2D Transient
3D Magnetostatic
3D Eddy
3D Transient

7. Different Methods of Field Simulation
Many different methods of field simulation exist with various benefits and limitations such as:
Finite Element Method (FEM)
Boundary Element Method (BEM)
Finite Difference Method (FDM)

8. FEM What are Finite Elements?
Complicated shapes are broken up into simple pieces called finite elements
Equilateral triangles (or tetrahedron) work best for the 2nd order quadratic interpolation between nodes
GOOD
BAD

9. FEM Delaunay Tessellation
Delaunay tessellation is the method used to create triangles
Of all possible triangulations for an arbitrary set of points, Delaunay triangles maximize the sum of the minimum angles
Virtually eliminates long isosceles triangles

10. FEM Delaunay Tessellation
Typical model to be triangulated

11. FEM Delaunay Tessellation
Initial points are located on object vertices

12. FEM Delaunay Tessellation
What is best way to triangulate initial points?

13. FEM Delaunay Tessellation
Create Voronoi polygons by drawing perpendicular bisector between all points

14. FEM Delaunay Tessellation
Triangles formed by connecting all points that are neighbors

15. FEM Delaunay Tessellation
Object boundaries are preserved by adding additional points and triangles

16. FEM Delaunay Tessellation
Circle condition - no point may be inside any circle connecting corners of a Delaunay triangle

17. FEM Delaunay Tessellation
Circle condition proof - the center of the circle is at the vertex of a Voronoi polygon

18. FEM Maxwell’s Equations
Differential Form of Maxwell’s Equations

19. FEM Approximation Functions
The desired field in each element is approximated with a 2nd order quadratic polynomial
Az(x,y) = ao + a1x + a2y + a3x2 + a4xy + a5 y2
Field quantities are calculated for 6 points (3 corners and 3 midpoints) in 2D and 10 points in 3D

20. FEM Variational Principle
Poisson’s equation
is replaced with energy functional
This functional is minimized with respect to value of A at each node in every triangle

21. FEM Matrix Equation
Now, over all the triangles, the result is a large, sparse matrix equation
This can be solved using standard matrix solution techniques such as:
Sparse Gaussian Elimination (direct solver)
Incomplete Choleski Conjugate Gradient Method (ICCG iterative solver)

22. FEM Error Evaluation
Put the approximate solution back into Poisson’s equation
Since A is a quadratic function, R is a constant in each triangle.
The local error in each triangle is proportional to R.

23. FEM Percent Error Energy
Summation of local error in each triangle divided by total energy
Local errors can exceed Percent Error Energy

24. FEM Solver Residuals
Residuals specify how close a solution must come before moving on to the next iteration
Two types of residuals:
Linear - used only for ICCG iterative matrix solver
Nonlinear - used only for problems with nonlinear BH materials

25. FEM Adaptive Refinement Process
Triangles are automatically refined to reduce energy error
Solution continues until one of two stopping criteria is met:
the specified number of passes are completed OR
percent error energy AND delta energy are less than specified
Start Field Solution
Generate Initial Mesh
Compute Fields
Perform Error Analysis
Stop Field Solution
Has Stopping Criteria been met?
Refine Mesh
Yes No