Computational Electromagnetics High frequency rigorous methods IE DE MoM FDTD TLM field based current based GO/GTD PO/PTD TD FD VM FEM

Computational Electromagnetics Electromagnetic problems are mostly described by three methods: Differential Equations (DE)  Finite difference (FD, FDTD) Integral Equations (IE)  Method of Moments (MoM) Minimization of a functional (VM)  Finite Element (FEM) Theoretical effort less more Computational effort more less

Numerical Differentiation “FINITE DIFFERENCES”

Introduction to differentiation Conventional Calculus The operation of diff. of a function is a well-defined procedure The operations highly depend on the form of the function involved Many different types of rules are needed for different functions For some complex function it can be very difficult to find closed form solutions Numerical differentiation Is a technique for approximating the derivative of functions by employing only arithmetic operations (e.g., addition, subtraction, multiplication, and division) Commonly known as “finite differences”

Taylor Series Problem: For a smooth function f(x), Given: Values of f(xi) and its derivatives at xi Find out: Value of f(x) in terms of f(xi), f(xi), f(xi), …. x y f(x) f(xi) xi

Taylor’s Theorem If the function f and its n+1 derivatives are continuous on an interval containing xi and x, then the value of the function f at x is given by

Finite Difference Approximations of the First Derivative using the Taylor Series (forward difference) Assume we can expand a function f(x) into a Taylor Series about the point xi+1 x y f(x) f(xi) xi xi+1 f(xi+1) h h

Finite Difference Approximations of the First Derivative using the Taylor Series (forward difference) Assume we can expand a function f(x) into a Taylor Series about the point xi+1 Ignore all of these terms

Finite Difference Approximations of the First Derivative using the Taylor Series (forward difference) y f(x) h xi xi+1 x f(xi) f(xi+1)

Finite Difference Approximations of the First Derivative using the forward difference: What is the error? The first term we ignored is of power h1. This is defined as first order accurate. First forward difference

Finite Difference Approximations of the First Derivative using the Taylor Series (backward difference) Assume we can expand a function f(x) into a Taylor Series about the point xi-1 x y f(x) f(xi-1) xi-1 xi f(xi) h -h

Finite Difference Approximations of the First Derivative using the Taylor Series (backward difference) Ignore all of these terms First backward difference

Finite Difference Approximations of the First Derivative using the Taylor Series (backward difference) y f(x) h xi-1 xi x f(xi-1) f(xi)

Finite Difference Approximations of the Second Derivative using the Taylor Series (forward difference) y x f(x) f(xi) xi xi+1 f(xi+1) h xi+2 f(xi+2) (1) (2) (2)-2* (1)

Finite Difference Approximations of the Second Derivative using the Taylor Series (forward difference) y x f(x) f(xi) xi xi+1 f(xi+1) h xi+2 f(xi+2) Recursive formula for any order derivative

Higher Order Finite Difference Approximations

Centered Difference Approximation (1) (2) (1)-(2)

Finite Difference Approximations of the First Derivative using the Taylor Series (central difference) y f(x) h xi-1 xi xi+1 x f(xi-1) f(xi+1) f(xi)

Second Derivative Centered Difference Approximation (central difference) (1) (2) (1)+(2)

Using Taylor Series Expansions we found the following finite-differences equations FORWARD DIFFERENCE BACKWARD DIFFERENCE CENTRAL DIFFERENCE CENTRAL DIFFERENCE

Forward finite-difference formulas

Centered finite difference formulas

Finite Difference Approx. Partial Derivatives Problem: Given a function u(x,y) of two independent variables how do we determine the derivative numerically (or more precisely PARTIAL DERIVATIVES) of u(x,y)

Pretty much the same way STEP #1: Discretize (or sample) U(x,y) on a 2D grid of evenly spaced points in the x-y plane

2D GRID xi xi+1 xi-1 xi+2 yj yj+1 yj-1 yj-2 y axis u(xi,yj) u(xi+1,yj) x axis

ui,j ui+1,j ui-1,j ui,j-1 ui,j+1 SHORT HAND NOTATION i i+1 i-1 i+2 j y axis i i+1 i-1 i+2 j j+1 j-1 j-2 ui,j ui+1,j ui-1,j ui,j-1 ui,j+1 x axis

Partial First Derivatives Problem: FIND recall:

Partial First Derivatives Problem: FIND Dy These are central difference formulas Are these the only formulas we could use? Could we use forward or backward difference formulas? Dx

Partial First Derivatives: short hand notation Problem: FIND Dy Dx

Partial Second Derivatives Problem: FIND recall:

Partial Second Derivatives Problem: FIND Dy Dx

Partial Second Derivatives: short hand notation Problem: FIND Dy Dx

FINITE DIFFERENCE ELECTROSTATICS Electrostatics deals with voltages and charges that do no vary as a function of time. Poisson’s equation Laplace’s equation Where, F is the electrical potential (voltage), r is the charge density and e is the permittivity.

FINITE DIFFERENCE ELECTROSTATICS: Example Fo F1 F2 F3 Find F(x,y) inside the box due to the voltages applied to its boundary. Then find the electric field strength in the box.

Electrostatic Example using FD Problem: FIND Dy Dx

Electrostatic Example using FD Problem: FIND If Dx = Dy

Electrostatic Example using FD Problem: FIND Iterative solution technique: Discretize domain into a grid of points Set boundary values to the fixed boundary values Set all interior nodes to some initial value (guess at it!) Solve the FD equation at all interior nodes Go back to step #4 until the solution stops changing DONE

Electrostatic Example using FD MATLAB CODE EXAMPLE

FINITE DIFFERENCE Waveguide TM modes: Example Where for TM modes and If then kz becomes imaginary and the mode does not propagate.

FINITE DIFFERENCE Waveguide TM modes: Example Goal: Find all permissible values of kt and the corresponding mode shape (F(x,y)) for that mode.

Waveguide Example using FD Problem: FIND Dy Dx

Waveguide Example using FD Problem: FIND If Dx = Dy=h

Waveguide Example using FD let where N is the number of interior nodes (i.e. not on a boundary) If we now apply the FD equation at all interior nodes we can form a matrix equation Where I is the identity matrix

Waveguide Example using FD is an eigenvalue equation usually cast in the form The eigenvalues will provide the permissible values for the transverse wavenumber kt and the eigenvectors are the corresponding mode shapes (F(x,y))

Waveguide Example using FD h W

Waveguide Example using FD h W Node #1: Node #2: Node #3: Node #4: Solve using the Matlab “eig” function

Waveguide Example using FD is an eigenvalue equation usually cast in the form The eigenvalues will provide the permissible values for the transverse wavenumber kt and the eigenvectors are the corresponding mode shapes (F(x,y)) HOMEWORK: WRITE A MATLAB PROGRAM THAT CALCULATES THE TRANSVERSE WAVENUMBERS AND MODE SHAPES FOR A RECTANGULAR WAVEGUIDE FOR TM MODES

Finite Difference Time Domain Method (FDTD)

Some big advantages Broadband response with a single excitation. 3D models easily. Frequency dependent materials accommodated. Most parameters can be generated e.g. Scattered fields antenna patters RCS S-parameters etc…..

How does it work? Based on the 2 Maxwell curl equations in derivative form. These are linearized by central finite differencing. We only consider nearest neighbor interactions because all the fields are advanced temporally in discrete time steps over spatial cells. ie we sample in space & time embedding of an antenna in a FDTD space lattice (note that the whole volume is meshed!)

closed & open problems FDTD is especially suitable for computing transients for closed geometries. When an open geometry is required, such as when we are dealing with an antenna, we need a boundary condition to simulate infinity. In this case the FDTD requires an absorbing boundary condition (ABC) at the grid truncation. This means there is no reflection from the boundary where the mesh ends.

FDTD: The Basic Algorithm Maxwell’s Equations in the TIME Domain:

Equate Vector Components: Six E and H-Field Equations

2-D Equations: Assume that all fields are uniform in y direction (i. e 2-D Equations: Assume that all fields are uniform in y direction (i.e. d/dy = 0) 2D - TE 2D - TM

1-D Equations: Assume that all fields are uniform in y and x directions (i.e. d/dy =d/dx= 0) 1D - TE 1D - TM

Discretize Objects in Space using Cartesian Grid 2D Discretization 3D Discretization z x 1D Discretization

Define Locations of Field Components: FDTD Cell called Yee Cell Finite-Difference Space is divided into small cells One Cell: (dx)(dy)(dz) E and H components are distributed in space around the Yee cell (note: field components are not collocated) FDTD: Yee, K. S.: Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Transactions on Antennas Propagation, Vol. AP-14, pp. 302-307, 1966.

Replace Continuous Derivatives with Differences Derivatives in time and space are approximated as DIFFERENCES

Solution then evolves by time-marching difference equations Time is Discretized One Time Step: dt E and H fields are distributed in time This is called a “leap-frog” scheme.

1-D FDTD Assuming that field values can only vary in the z-direction (i.e. all spatial derivatives in x and z direction are zero), Maxwell’s Equations reduce to: z Hy Ex e(z), m(z)

1-D FDTD – Staggered Grid in Space Interleaving of the Ex and Hy field components in space and time in the 1-D FDTD formulation Time plane

Replace all continuous derivatives with finite differences 1-D FDTD Replace all continuous derivatives with finite differences

Substitution of difference equations in above yields: 1-D FDTD Substitution of difference equations in above yields:

After some simple algebra: 1-D FDTD After some simple algebra:

1-D FDTD Algorithm – Flow Chart Start Compute 1-D Faraday’s FDTD equation: For all nodes n inside the simulation region: Compute 1-D Ampère-Maxwell’s FDTD equation: For all nodes n inside the simulation region: Electric current density excitation: For all excitation nodes n: Boundary condition: For all PEC boundary nodes n: No Yes Stop

FDTD Solution of 1-D Maxwell’s Equations (EXAMPLE) FDTD equations Initial conditions initial-boundary-value problem For all values of i Boundary condition for a perfectly electrically conducting (PEC) material

FDTD Solution of the First Two 1-D Scalar Maxwell’s Equations Excitation pulse: RC2(t) – Time Domain Amplitude RC2(t) Excitation pulse: RC2(f) – Frequency Domain Magntiude |RC2(f)|

FDTD Solution of 1-D Maxwell’s Equations

FDTD Solution of 1-D Maxwell’s Equations

SOME OPEN QUESTIONS?? How do we determine what Dt and Dz should be? How do we implement real sources? How do we simulate open boundaries? How accurate is the solution?

Potential Source of Error: Numerical Dispersion In implementing any numerical technique, it is essential to understand the origin and nature of errors that are introduced as a result of the computational approach. In the FDTD errors arise from the discretization of the computational space and finite difference approximations to Maxwell’s equations. One particularly important source of error is call numerical dispersion. To see this consider the continuous wave equation in ID: Which has a solution of the form we have looked at before:

If we substitute this expression into the wave equation, we get: From this we see that the wave number is linearly proportional to the frequency. This is more widely known as the phase velocity, when expressed as np = w/k np=c To determine how the FDTD effects this relationship, we repeat the same procedure using the difference equations.

Let’s now look at the discrete wave equation Represent the second order difference equation (for both time and space) in terms of: Doing the same for the time derivative, we get:

Now defining the solution from in difference form: Substituting this into the difference form of the wave equation we get: Factor out:

Factor out:

We get: Group time and space terms and divide both sides by 2 Use Euler’s Identity: (FDTD) (continuous)

This is a nonlinear equation that represents the relationship between the wave number, and the frequency, w. It is a function of the time step Dt, spatial step Dx and frequency w This means that as a simulated wave propagate through the solution space it undergoes phase errors because the numerical wave either slows down or accelerates relative to the actual wave propagation in physical space. To examine the implication we consider 3 special cases: Case 1 Consider Δt  and Δx  0 In this case, use the first two terms of the Taylor series expansion of terms  Same as in real space

Case 2 Use the relation cΔt = Δx This is often referred to as the magic time step! Plugging in we get:  Implies that No dispersion Unfortunately the relation cΔt = Δx is unstable!

Case 3 In this case we consider the general case where there is no assumed relation between Δt and Δx. Therefore we need to solve for a general expression that can tell us the effect of numerical dispersion for a given relationship. To do this, we solve in terms of the wave number. Using this equation we can determine the numerical wave number for any relation between Δt and Δx for a given sampling rate. Substitute in the appropriate parameters and determine the effects of discretizing the computational space.

Where we have used k= w/c, use the sampling note of x Where we have used k= w/c, use the sampling note of Now define numerical phase velocity as

Over a distance of 10λ, which is in this case in no computational cells The numerical wave would only propagate 98.73 cells  phase error of 45.72º If we repeat this analysis for Now the phase error at wλ is 11.19º The conclusion to draw from this is that the larger the solution space, the greater the phase error is going to be. (MATLAB DEMO)

Stability As stated before, the FDTD method approximates M.E.’s as a set of completed difference equations. As such this method is useful only when the solution of the difference equations are convergent and stable. Convergence means that as time stepping continues, the solution of the difference equations asymptomatically approach the solution given by Maxwell’s equations. On the other hand, stability is basically stated as a set of condition under which the error generated does not grow in an unbounded fashion.

In order for the FDTD equations to remain stable the following relationship (called the Courant stability criteria must be met) This is the stability condition for 1D case For higher dimensional cases we have the form ,where D=1, 2, 3 depending on the number of dimensions used (MATLAB DEMO)

So far we have considered a sample 1D FDTD formulation, that physically represents the propagation of a plane wave. More often than not, EM BVP’s cannot be accurately represented as a 1D problem. 2D FDTD Formulation In the formulation of the two-dimensional FDTD, one can assume either TE ( electric field is to the plane of incidence, or TM; magnetic field is to the plane of incidence). This results from the fact that in 2D neither the fields nor the object contain any variations in the z-direction,consequently at

So in this context, Maxwell’s Equations can be reduced to:

For TM polarization we have:

Applying the central difference expression, for TE mode, we get: For the TM case, we have:

Numerical Dispersion 2D case Using same procedure as for the 1D case we obtain:

Numerical Dispersion 2D case Plot of the normalised phase velocity vs. travelling wave angle.

Writing a 2D FDTD Program Defining physical constants Define program constants magnitude of electric field : total # of time steps : # of nodes in the x, y direction If using a pulse source, define Ez and Hx, or we can implement a hard source. Start time marching nΔt, time value n = 1 : Nt Increment the spatial index First apply the absorbing boundary conditions Apply the ABC along the edges of the computational region

(MATLAB DEMO)

Yee’s 3D Mesh

Transition to 3D The 1D formulation involved 2 degrees of freedom (one field quantity E or H and one time variable). In 3D we have 4 degrees of freedom (3 in space and 1 in time). The following notation is somewhat standard where the increment is the lattice space increments and i,j,k are integers. Any field may be represented as

The Yee Cell seen in a more Macroscopic view – Faraday's and Ampere’s law Fall 2007 mp@es.aau.dk

So far we have considered the 1D and 2D formulations. General 3D Formation So far we have considered the 1D and 2D formulations. Now we will do the 3D formulation. Recall that Maxwell’s curl equations are: (i, j, k) (i, j, k)

For simplicity, consider a linear and isotropic medium. In Cartesian coordinates, the curl equations reduce to: i j k /x /y /z Ex Ey Ez

Also, for Ampere’s Law: 3D Discretization Given a function f(x, y, z, t) we write it in discrete form as: f(x, y, z, t) = f(iΔx, jΔy, kΔz, nΔt) = In general, we let: Δx = Δy = Δz = d

3D formulation These six coupled equations form the basis for the FDTD algorithm. 3D Discretization Given a function f(x, y, z, t) we write it in discrete form as: f(x, y, z, t) = f(iΔx, jΔy, kΔz, nΔt) = In general, we let: Δx = Δy = Δz = d

FDTD Discretization of 3D Curl Equations Where we have used time interpolation

We can simplify these expressions if we set: And we can define the material constants:

Substituting in we get:

ABC’s In general EM analysis of scattering structures often requires the solution of “open region” problems. As a result of limited computational resources, it becomes necessary to truncate the computational domain in such a way as to make it appear infinite. This is achieved by enclosing the structure in a suitable output boundary that absorbs all outward traveling waves ABC

In this lecture we will describe some of the more common ABC’s, such as: Bayliss-Turkel annihilation operators Enqquist-Majda one way operators Mur ABC Liao’s extrapolation method And the PML, perfectly matched layer The PML method which was introduced in 1994 by Berenger, represents one of the most significant advances in FDTD development, since it conception I 1966, by Kane Yee. The PML produces back reflection ~ over a very broad range of incident PMC Region

Early ABC’s When Yee first introduced the FDTD method, he used PEC boundary conditions. This technique is not very useful in a general sense. It wasn’t until the 70’s when several alternative ABC’s were introduced. However, these early ABC’s suffered from large back reflections, which limited the efficacy of the FDTD method.

Extrapolation from Interior Node (Taylor 1969) If we consider the electric field located on a 2D boundary Since and are determined in the FDTD calculations, and are therefore known values, we can solve this equation for the exterior node. This equation is only effective on normally incident waves and degrades rapidly when the incident wave is off-normal. y, (j) Wave x, (i)

Extending this method to 2D: This approach applies to a wide range of incident field angles, but due to the nature of the averaging process often gives rise to significant non-physical back reflections. j=3 j=2 j=1 i-1 i i+1

Dissapative Medium In this technique a lossy medium is used to surround the FDTD computational region. The idea is that as that waves propagate into this medium, they are dissipated before they can undergo back reflection

The problem is that there is often an input impedance between the FDTD region and the long media. The back reflection results from the ratio of electrical-conductivity and magnetic-conductivity parameters, respectively. Quadratic Linear

Here the are assumed to be free space. Therefore the input impedance is: which is not necessarily equal to Zo. To overcome this the conductivity values can be implemented in a non-uniform fashion. That is the can have a linear or quadratic profile. This tends to require relatively thick medium layers - computational requirements!