Numerical methods for electrical field calculation (FDM) :

Finite Difference Method (FDM) :

• The finite method is a numerical method for solving partial differential equations.
• This method is used to solve linear and non-linear problem.
• An FDM method divides the solution domain into finite discrete points and repaces the partial differential equations with a set of difference equations. Thus the solutions obtained by FDM are not exact but approximate.
• In this method, the field space between the electrodes is covered with a uniform spaced mesh which may be square, rectangular, a combination of the two.
• This gives rise to a large number of nodes.
• The original Laplace differential equation is then transformed into difference equations for potential  at these nodes.
• To cover an irregular three dimensional field so that these nodes are laid upon the boundaries becomes extremely difficult.
• Further, to cover such fields by a proposed mesh, a unlimited number of  values of potential distribution are necessary, requiring a large memory and time for computation.
• The FDM is, therefore, found suitable only for two dimensional symmetrical field.
• This method is rarely applied for electrostatic fields.
• Apart from other numerical methods for solving partial differential equations, the Finite Difference Method (FDM) is universally applied to solve linear and even non-linear problems.
• Although the applicability of difference equations to solve the Laplace's equation was used earlier, it was not untill 1940s that FDMs have been widely used.
• The field problem for which the Laplace's or Poisson's equation applies isgiven within a (say x, y), plane, the area of which is limited by given boundary conditions, i.e. by contours on which some field quantities are known.
• Every potential f and its distribution within the area under consideration will be continuous. Therefore, an unlimited number of f (x, y) values will be necessary to describe the complete potential distribution.
• Since any numerical computation can provide only a limited amount of information, discretization of the area will be necessary to represent all the nodes for which the solution is needed.
• Such nodes are generally produced by net or grid laid down on the area as shown in FIGURE.

• The unknown potential  can be expressed by the surrounding potentials which are assumed to be known for the single difference equations.
• For every two-dimensional, most of the field region can be subdivided by a regular square net.
• Then, if the step size chosen for discretization is h, the following approximate equation becomes valid.

         

• In the above equation,  are the potentials at the immediate neighbourhood nodes with respect to the node p of interest (of which the potential  needs to be determined.)
• The potentials at the neighbourhood points are expected to be known a priori, either from given boundary conditions or from any previous computational results.
• The term F(p) arises if the field region is governed by the Poisson's equation, (i.e. the relation  holds good).
• Thus, any general field problem to be treated needs sub-division of the finite plane by  a predominantly regular grid, which is supplemented by irregular elements at the boundaries, if required.
• The whole grid will the contain n nodes, for which the potential  is to be calculated. Then, a system of n simultaneous equations would result. Here, it may be noted that simple problems with small number of unknowns can be treated by long hand computation using the concept of residuals and point relaxation.
• For more complex problems, machine computation is necessary and iteractive schemes are most efficient in combination with successive relaxation methods.