Numerical Methods of Calculation of Electric field

Numerical Methods of Calculation of Electric field

There are numerous to solve partial differential equation having Laplace and Poisson equation but there are many difficulties in solving these equations for 2D and 3D fields with complex boundary conditions or for insulating material with different permittivities and conductivities.

There are 3 types of numerical methods commonly employed :

1. Finite Difference Method (FDM)
2. Finite Element Method (FEM)
3. Charge Simulation Method (CSM)

1. Finite Difference Method (FDM)

• This method is universally applicable to linear and even non linear problems. This method is illustrated by 2D problems for which Laplace’s equation of Poisson’s equation applies :

  ∇² Φ = ∂²/dx² +
∂²/dy²  + ∂²/dz²

• The field problem is then given within an x-y plane, the area of this is limited by the boundary conditions on which some field quantities are known.
• Every potential and its distribution within the area under consideration will be continuous in nature. Hence an unlimited number of (x, y) values would be necessary to assign the potential distribution.
• Discretization of the area is necessary to exhibit nodes for which the solution may be found. Such nodes are produced by any net or grid laid down upon area.
• As any irregular net would lead to inadequate difference equations replacing the original partial differential equation and thus would be prohibitive for numerical computations the FDM is in general applied to regular nets or polygons only.
• Thus in FDM the partial derivatives of basic field equations have been replaced by their algebraic difference form resulting in a system of algebraic equation which have to be solved. Due to the approximation mode during this derivation the algebraic was linear of the first order.
• FDM method is time consuming and therefore is now seldom used.

2. Finite Element Method (FEM)