Introduction to the Finite Element Method

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

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.

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

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

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

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)

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

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

FEM Delaunay Tessellation Typical model to be triangulated

FEM Delaunay Tessellation Initial points are located on object vertices

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

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

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

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

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

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

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

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

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

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)

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.

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

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

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