Home FEM

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

FEM mesh – specification of geometry

  • 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

Fine mesh for computation of potential

  • 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
Graded fine mesh in part of the lens