Numerical methods for electric field calculation (FEM) :

Finite Element Method (FEM) :

• Finite Element Method is widely used in the numerical solution of electric field problems and became very popular.
• The finite element analysis of any problem involves basically four steps :

1. Finite Element Discretization
2. Governing Equations
3. Assembling of All Elements
4. Solving the Resulting Equations

• The Finite Element Method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for partial differential equations.
• For one-dimensional (1-D) problems, the elements are nothing but line segments having only length and no shape. 
• For two-dimensional (2-D) problems, the elements used are triangles, rectangles and quadrilateral having straight or curved boundaries.
• For three-dimensional (3-D) problems, the shapes used are tetrahedron and parallelepiped having straight or curved surfaces. Divison of the domain into elements is called a mesh.
• Finite Element Method is widely used in the numerical solution of Electric Field Equation, and became very popular.
• In contrast to other numerical methods, FEM is a very general method and therefore is a versatile tool for solving wide range of Electric Field Equation.
• The finite element analysis of any problem involves basically four steps :

(a) Finite Element Discretization :

• To start with, the whole problem domain is ficticiously divided into small areas/ volumes called elements. (see FIGURE-A).
• The potential, which is unknown throughout the problem domain, is approcimated in each of these elements in terms of the potential at their vertices called nodes.
• As a result of this the potential function will be unknown only at the nodes. Normally, a certain class of polynomials, is used for the interpolation of the potential inside each element in terms of their nodal values. The coefficients of this interpolation function are then epressed in terms of the unknown nodal potentials.
• As a result of this, the interpolation can be directly carried out in terms of the nodal values. The associated algebric functions are called shape frictions.
• The elements derive their names through their shape, i.e., bar elements in one dimension (1D), triangular and quadrilateral elements in 2D, and tetrahedron and hexahedron elements for 3D problems.

(b) Governing Equations :

• The potential Ve within an element is first approximated and then interrelated to the potential distributions in various elements such that the potential is continuous across inter-element boundaries. The approximate solution for the whole region then

        
          becomes

• where N is the number of elements into which the solution region is divided. The most common form of approximation for the voltage V within an element is a polynomial approximation.

       

• For the triangular element, and for the quadrilateral element the equation becomes

       

             

• The potential Ve in general is not zero within the element e but it is zero outside the element in view of the fact that the quadrilateral elements are non-confirming elements (see FIGURE-A)
• Consider a typical triangular element shown in FIGURE-B. The potentials Ve1, Ve2 and Ve3 at nodes 1,2 and 3 are obtained from equation-2, as

     
          the coefficient a, b and c are determined from the above equation as

     

• Substituting this equation in equation-8, we get

   

          where,

   

          and A is the area of the element e, that is,

 

• The value of A is positive if the nodes are numbered counterclockwise (starting from any node) s shown by the arrow in FIGURE-B. It may be noted that equation-6 gives the potential at any point (x, y) within the element provided that the potentials at the vertices are known.
• These are called the element shape functions. They have the following properties :

• The energy per unit length associated with the element e is given by the following equation :

 

          where, T denotes the transpose of the matrix

• The matrix given above is normally called as element coefficient matrix. The matrix element of the coefficient matrix is considered as the coupling between nodes i and j.

             

(c) Assembling of All Elements :  

• Having considered a typical element, the next stage is to assemble all such elements in the solution region. The energy associated with all the elements will then be

  

         and, n is the number of nodes, N is number of elements and [C] is called the global coefficient               matrix which is the sum of the individual coefficient matrices.

(d) Solving the Resulting Equations :

• It can be shown that the Laplace's (and Poisson's) equation is satisfied when the total energy in the solution region is minimum.
• Thus, we require that the partial derivatives of W with respect to each nodal value of the potential is zero, i.e.

   

• In general,

   

         where, n is the number of nodes in the mesh. By writing the above equation-15 for all the nodes,             k=1, 2, .... n, we obtain a set of simultaneous equations from which the solution for V1, V2....Vn             can be found.

• This can be done either by using the Iteration Method or the Band Matrix Method.
• Now, for solving the nodal unknowns, one cannot resort directly to the governing partial differential equations, as a piece-wise approximation has been made to the unknown potential.
• Therefore, alternative approaches have to be sought. One such classical approach is the calculus of variation.
• This approach is based on the fact that potential will distribute in the domain such that the associated energy will reach extreme values.
• Based on this approach, Euler has showed that the potential function that satisfies the above criteria will be the solution of corresponding governing equation.
• In FEM, with the approximated potential function, extremization of the energy function is sought with respect to each of the unknown nodal potential.
• This potential leand to a set of linear algebric equations. In this matrix form, these equations form normally a symmetric sparse matrix, which is then solved for the nodal potentials.
• Within the individual elements the unknown potential function is approximated by the shape functions of lower order depending on the type of element.
• An approximate solution of the exact potential is then given in the form of an epression whose terms are the products of the shape function and the unknown nodal potentials.
• It can be shown that the solution of the differential equation describing the probelm corresponds to minimization of the field energy.
• This leads to a system of algebric equations the solution for which under the corresponding boundary conditions gives the required nodal potentials.
• Thus, this procedure results in a potential distribution in the form of disrete potential value at the nodal points of the FEM mesh.
• The related field strengths at the centres of all elements are then obtained from the potential gradient. The values of the field thus obtained are dependent on the distance between the centres of the elements and the electrode surface, and thus on the sizes of the elements. 

• Advantages :

1. Accurate representation of complex geometry
2. Easy representation of the total solution.
3. Domains consting of more than one materail can be easily analyzed.

• Disadvantages :

1. The FEM obtains only "approximate" solutions.
2. The FEM has "inherent" errors.