Finite Elements Method - FEM

FEM minimizes the energy stored in the field and instead of solving a second order partial differential equation it minimizes an expression containing the square of the first derivatives of potential.
The potential is evaluated at the mesh points of a mesh made of small quadrilateral elements, each with its own material properties. The computation is based on the assumption that the potential changes linearly inside a triangular finite element.
Fast solution methods get the potential by solving a set of linear equations of the order of the number of mesh points, half a million equations in less than a minute.
We determine the potential only in 2 dimensions.

Information about FEM

A number of papers was published, e.g. at CPO (Charged Particle Optics) conferences between 1990-2006
Computation of coefficients for the first order FEM was improved for all relevant problems
Fine mesh for computation can have very many points
Fast preconditioned conjugate gradient method is used to solve large sets of equations
Accuracy estimate of solution from 2 meshes (one is two times denser) can be used for lenses

The coarse mesh is made of horizontal and vertical lines
The quadrilaterals so made cannot be degenerate
Each of them can be filled with its own material (electrodes can be only a part of a line)
Magnetic materials are specified by magnetization curve and they can saturate
A number of lens excitations can be calculated one after another

The fine mesh is made by subdividing the coarse mesh by additional lines
The fine mesh is automatically computed in EOD from one or more gaps in which the mesh density is specified by the user
The rest of the fine mesh has graded mesh step that expands from the gap
Each of small quadrilaterals is for computations divided into four triangles, in each of them the potential is supposed to be a linear function of coordinates
From the condition that the energy stored is minimum we get a system of linear equations for potentials at the nodes of the fine mesh