computational electromagnetics

Module 04

Computational Electromagnetics

Awab Sir (www.awabsir.com) 8976104646


• The evaluation of electric and magnetic fields in an electromagnetic system is of utmost importance.

• Depending on the nature of the electromagnetic system, Laplace or Poisson equation may be suitable to model the system for low frequency operating conditions.

• In high frequency applications we must solve the wave equation in either the time domain or the frequency domain to accurately predict the electric and magnetic fields.

• All these solutions are subject to boundary conditions. • Analytical solutions are available only for problems of

regular geometry with simple boundary conditions.



When the complexities of theoretical formulas make analytic solution intractable, we resort to non analytic methods, which include

• Graphical methods

• Experimental methods

• Analog methods

• Numerical methods



Graphical, experimental, and analog methods are applicable to solving relatively few problems.

Numerical methods have come into prominence

and become more attractive with the advent of fast digital computers. The three most commonly used simple numerical techniques in EM are

• Moment method • Finite difference method • Finite element method



EM Problems

Partial differential equations

Finite Difference

Method

Finite Element Method

Integral equations

Method of Moments



We now use numerical techniques to compute electric and magnetic fields

In principle, each method discretizes a continuous domain into finite number of sections and then requires a solution of a set of algebraic equations instead of differential or integral equations.



Consider the Laplace equation which is given as follows:

And a source free equation given as

Where, u is the electrostatic potential.


Why do we need to use numerical methods?

If we take an example of a parallel plate capacitor. When we neglect the fringing field we get the following equation.

Where, V is the closed form analytical solution (can see the effects of varying any quantity on the RHS by significant change in the LHS). For solving the equation 3 we do not need the use of numerical techniques. Awab Sir (www.awabsir.com) 8976104646

Why do we need to use numerical methods?

However if we now take into account the fringing field the solution for every point x & y such that equation 4 is satisfied is not manually possible

To find the field intensity at any point we need to use

numerical techniques.


Step 1: Divide the given problem domain into

sub-domains

It is a tough job to approximate the potential for the entire domain at a glance. Therefore any domain in which the field is to be calculated is divided into small elements. We use sub domain approximation instead of whole domain approximation.

Considering a one dimensional function


Step 2: Approximate the potential for each

element The approximate potential for an element can be

given as

OR Where, a, b and c are constants. Considering the first equation for each element, the

potential distribution will be approximated as a straight line for every element.


Step 3: Find the potential u for every element

in terms of end point potentials Assuming

The above equation can also be written as

We can write



in terms of end point potentials

We can write equation 1 and 2 in matrix form as follows

Rearranging the terms we get



in terms of end point potentials

Substituting equation 5 in equation 1 we get

Similarly we can find the electrostatic potential for each element.


Step 4: Find the energy for every element

The energy for a capacitor is given as follows:

The field is distributed such that the energy is minimized. The electric field intensity is related to the electrostatic potential as follows


Step 5: Find the total energy

The total energy is the summation of the

individual electrostatic energy of every element in the domain.


Step 6: Obtain the general solution

The field within the domain is distributed such that the energy is minimized. For minimum energy the differentiation of electrostatic energy with respect to the electrostatic potential is equated to zero for every element.

Solving this we get a matrix


Step 6: Obtain the general solution

The matrix [K] is a function of geometry and the material properties. The curly brackets denote column matrix. Equation 9 does not lead to a unique solution. For a unique solution we have to apply boundary conditions.


Step 7: Obtain unique solution

We need to apply boundary conditions to equation 9 to obtain a unique solution. For example we assume u1=1 V and u4=5 V. On applying boundary conditions the RHS becomes a non zero matrix and a unique solution can be obtained.


Steps for Finite Element Method

1. Divide the given problem domain into sub-domains

2. Approximate the potential for each element

3. Find the potential u for every element in terms of end point potentials

4. Find the energy for every element

5. Find the total energy

6. Obtain the general solution

7. Obtain unique solution


FINITE ELEMENT METHOD


FINITE ELEMENT METHOD

The finite element analysis of any problem involves basically four steps:

1. Discretizing the solution region into a finite number of sub regions or elements

2. Deriving governing equations for a typical element

3. Assembling of all elements in the solution region

4. Solving the system of equations obtained.


1. Finite Element Discretization



We divide the solution region into a number of finite elements as illustrated in the figure above, where the region is subdivided into four non overlapping elements (two triangular and two quadrilateral) and seven nodes. We seek an approximation for the potential Ve within an element e and then inter-relate the potential distributions in various elements such that the potential is continuous across inter-element boundaries. The approximate solution for the whole region is



Where, N is the number of triangular elements into which the solution region is divided. The most common form of approximation for Ve within an element is polynomial approximation, namely

for a triangular element and for a quadrilateral element.



The potential Ve in general is nonzero within element e but zero outside e. It is difficult to approximate the boundary of the solution region with quadrilateral elements; such elements are useful for problems whose boundaries are sufficiently regular. In view of this, we prefer to use triangular elements throughout our analysis in this section. Notice that our assumption of linear variation of potential within the triangular element as in eq. (2) is the same as assuming that the electric field is uniform within the element; that is,


2. Element Governing Equations



The potential Ve1, Ve2, and Ve3 at nodes 1, 2, and 3, respectively, are obtained using eq. (2); that is,



We can obtain the values of a, b and c. Substituting these values in equation 2 we get



And A is the area of the element e



The value of A is positive if the nodes are numbered counterclockwise. Note that eq. (5) gives the potential at any point (x, y) within the element provided that the potentials at the vertices are known. This is unlike the situation in finite difference analysis where the potential is known at the grid points only. Also note that α, are linear interpolation functions. They are called the element shape functions.



The shape functions α1 and α2 for example, are illustrated in the figure below



The energy per unit length can be given as

Where

and



The matrix C(e) is usually called the element coefficient matrix. The matrix element Cij

(e) of the coefficient matrix may be regarded as the coupling between nodes i and j.


3. Assembling of all Elements

Having considered a typical element, the next step is to assemble all such elements in the solution region. The energy associated with the assemblage of all elements in the mesh is

Where n is the number of nodes, N is the number of elements, and [C] is called the overall or global coefficient matrix, which

is the assemblage of individual element coefficient matrices.


3. Assembling of all Elements

The properties of matrix [C] are

1. It is symmetric (Cij = Cji) just as the element coefficient matrix.

2. Since Cij = 0 if no coupling exists between nodes i and j, it is evident that for a large number of elements [C] becomes sparse and banded.

3. It is singular.


4. Solving the Resulting Equations

From variational calculus, it is known that Laplace's (or 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 be zero; that is,

To find the solution we can use either the iteration method or the band matrix method.


Advantages

FEM has the following advantages over FDM and MoM

1. FEM can easily handle complex solution region.

2. The generality of FEM makes it possible to construct a general-purpose program for solving a wide range of problems.


Drawbacks

1. It is harder to understand and program than FDM and MOM.

2. It also requires preparing input data, a process that could be tedious.


FINITE DIFFERENCE METHOD

Boundary Conditions

A unique solution can be obtained only with a specified set of boundary conditions. There are basically three kinds of boundary conditions:

1. Dirichlet type of boundary

2. Neumann type of boundary

3. Mixed boundary conditions


Boundary Conditions

Dirichlet Boundary Condition

Consider a region s bounded by a curve l. If we want to determine the potential distribution V in region s such that the potential along l is V=g. Where, g is prespecified continuous potential function. Then the condition along the boundary l is known as Dirichlet Boundary condition.


Boundary Conditions

Neumann Boundary Condition

Neumann boundary condition is mathematically represented as

Where, the conditions along the boundary are such that the normal derivative of the potential function at the boundary is specified as a continuous function.


Boundary Conditions

Mixed Boundary Condition

There are problems having the Dirichlet condition and Neumann condition along l1 and l2 portions of l respectively. This is defined as mixed boundary condition.



A problem is uniquely defined by three things:

1. A partial differential equation such as Laplace's or Poisson's equations.

2. A solution region.

3. Boundary and/or initial conditions.



A finite difference solution to Poisson's or Laplace's equation, for example, proceeds in three steps:

1. Dividing the solution region into a grid of nodes.

2. Approximating the differential equation and boundary conditions by a set of linear algebraic equations (called difference equations) on grid points within the solution region.

3. Solving this set of algebraic equations.


Step 1

Suppose we intend to apply the finite difference method to determine the electric potential in a region, shown in the figure below. The solution region is divided into rectangular meshes with grid points or nodes as shown. A node on the boundary of the region where the potential is specified is called a fixed node (fixed by the problem) and interior points in the region are called free points (free in that the potential is unknown).


Step 1


Step 2

Our objective is to obtain the finite difference approximation to Poisson's equation and use this to determine the potentials at all the free points.


Step 3

To apply the following equation, to a given problem, one of the following two methods is commonly used.

)


Iteration Method

We start by setting initial values of the potentials at the free nodes equal to zero or to any reasonable guessed value. Keeping the potentials at the fixed nodes unchanged at all times, we apply eq. (1) to every free node in turn until the potentials at all free nodes are calculated. The potentials obtained at the end of this first iteration are not accurate but just approximate. To increase the accuracy of the potentials, we repeat the calculation at every free node using old values to determine new ones. The iterative or repeated modification of the potential at each free node is continued until a prescribed degree of accuracy is achieved or until the old and the new values at each node are satisfactorily close.


Band Matrix Method

Equation (1) applied to all free nodes results in a set of simultaneous equations of the form

Where: [A] is a sparse matrix (i.e., one having many

zero terms), [V] consists of the unknown potentials at the free

nodes, and [B] is another column matrix formed by the known

potentials at the fixed nodes.


Band Matrix Method

Matrix [A] is also banded in that its nonzero terms appear clustered near the main diagonal because only nearest neighboring nodes affect the potential at each node. The sparse, band matrix is easily inverted to determine [V]. Thus we obtain the potentials at the free nodes from matrix [V] as


The concept of FDM can be extended to Poisson's, Laplace's, or wave equations in other coordinate systems. The accuracy of the method depends on the fineness of the grid and the amount of time spent in refining the potentials. We can reduce computer time and increase the accuracy and convergence rate by the method of successive over relaxation, by making reasonable guesses at initial values, by taking advantage of symmetry if possible, by making the mesh size as small as possible, and by using more complex finite difference molecules. One limitation of the finite difference method is that interpolation of some kind must be used to determine solutions at points not on the grid. One obvious way to overcome this is to use a finer grid, but this would require a greater number of computations and a larger amount of computer storage.


METHOD OF MOMENTS


METHOD OF MOMENTS

Like the finite difference method, the moment method or the method of moments (MOM) has the advantage of being conceptually simple. While the finite difference method is used in solving differential equations, the moment method is commonly used in solving integral equations.

MoM uses integral method. The advantage of this method is that the order of the problem is reduced by one. For example a parallel plate capacitor is a 3-D domain. However, we will be working only on the surface of the capacitor plates, it becomes reduced to 2-D domain. Awab Sir (www.awabsir.com) 8976104646

Steps for MoM

The potential at any point on the plate is a function of the charge distribution


Step 1

The charge can be given as


Step 2

To determine the charge distribution, we divide the plate section into smaller rectangular elements. The charge in any section is concentrated at the centre of the section. The potential V at the centre of any section is given by


Step 3

Assuming there is uniform charge distribution on each subsection we get

This equation can be rearranged to obtain Where, [B]: column matrix defining the potentials [A]: square matrix In MoM, the potential at any point is the function of

potential distribution at all points, this was not done in FDM and FEM.


Finite Difference

Method (FDM)

Finite Element Method

(FEM)

Method of Moments

(MoM)

Basic

principle

Based on differentiation

i.e. the differential

equation is converted to a

difference equation.

1. Energy based (energy

minimization)

2. Weighted residual

(reducing the error)

Based on integral

method

Advantage 1. Simplest method

2. Taylor series based

1. Computationally easier

than MoM

2. Can be applied to

unisotropic media.

3. [K] matrix is sparse

1. More accurate

(errors effectively

tend to cancel each

other)

2. Ideally suited for

open boundary

conditions

3. Popular for

antennas

Disadvantage 1. Need to have uniform

rectangular sections

(not possible for real

life structures)

2. Outdated

1. Difficult as compared to

FDM

1. Mathematically

complex


computational electromagnetics

Education