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Module 04. Computational Electromagnetics. Computational Electromagnetics. The evaluation of electric and magnetic fields in an electromagnetic system is of utmost importance.
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Module 04 Computational Electromagnetics
Computational Electromagnetics • 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.
Computational Electromagnetics 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
Computational Electromagnetics 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
Computational Electromagnetics 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.
Computational Electromagnetics 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.
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
Step 3:Find the potential u for every element in terms of end point potentials We can write equation 1 and 2 in matrix form as follows Rearranging the terms we get
Step 3:Find the potential u for every element 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 • Divide the given problem domain into sub-domains • Approximate the potential for each element • Find the potential u for every element in terms of end point potentials • Find the energy for every element • Find the total energy • Obtain the general solution • Obtain unique solution
FINITE ELEMENT METHOD The finite element analysis of any problem involves basically four steps: • Discretizingthe solution region into a finite number of sub regions or elements • Deriving governing equations for a typical element • Assembling of all elements in the solution region • 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 Vewithin 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
1. Finite Element Discretization Where, N is the number of triangular elements into which the solution region is divided. The most common form of approximation for Vewithin an element is polynomial approximation, namely for a triangular element and for a quadrilateral element.
1. Finite Element Discretization The potential Vein 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,
2. Element Governing Equations We can obtain the values of a, b and c. Substituting these values in equation 2 we get
2. Element Governing Equations And A is the area of the element e
2. Element Governing Equations 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.
2. Element Governing Equations The shape functions α1 and α2 for example, are illustrated in the figure below
2. Element Governing Equations The energy per unit length can be given as Where and
2. Element Governing Equations 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 • FEM can easily handle complex solution region. • The generality of FEM makes it possible to construct a general-purpose program for solving a wide range of problems.
Drawbacks • It is harder to understand and program than FDM and MOM. • 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: • Dirichlettype of boundary • Neumann type of boundary • 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.
FINITE DIFFERENCE METHOD A problem is uniquely defined by three things: • A partial differential equation such as Laplace's or Poisson's equations. • A solution region. • Boundary and/or initial conditions.
FINITE DIFFERENCE METHOD A finite difference solution to Poisson's or Laplace's equation, for example, proceeds in three steps: • Dividing the solution region into a grid of nodes. • Approximating the differential equation and boundary conditions by a set of linear algebraic equations (called difference equations) on grid points within the solution region. • 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).
