ELEG 840

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ELEG 840

• Lecture 11
• Professor Dennis W. Prather

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• Last Time
• Introduced the scattered field formulation.
• Introduced total-scattered field formulation.
• 1D FDTD method to introduce the source
• Today
• Discuss the absorbing boundary conditions.

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• a) Validate 2D FDTD with dielectric cylinder.
• b) Characterize 3 types of ABCs
• Determine the of back reflection vs.
• 1)
• Do with FDTD
• Analytic Methods
• 2) - Mur
• - Lias
• - PMCs ----- PMCs to 2 other methods
• - TAFLOVES
• - Averaging

Plot Ez
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• ABCs
• 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
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• In this lecture we will describe some of the more
  common ABCs, such as
• Bayliss-Turkel annihilation operators
• Enqquist-Majda one way operators
• Mar ABC
• Liaos 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
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• Early ABCs
• When Yee first introduced the FDTD method, he
  used PEC boundary conditions.
• This technique is not very useful in a general
  sense.
• It wasnt until the 70s when several alternative
  ABCs were introduced.
• However, these early ABCs suffered from large
  back reflections, which limited the efficacy of
  the FDTD method.

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• 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)
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• Averaging of Outgoing Waves (Totlove 1975)
• For this method, we consider the wave equation in
  1D
• This can be factored with the following
  expression
• We can look at both differential operators and
  see that the first one represents/corresponds to
  a traveling wave in the forward direction and the
  second a traveling wave in the backward
  direction.
• Therefore, at the last node of the computational
  space, N, we must satisfy the outgoing wave
  condition, which is represented by one of the
  differential operators.

Differential operators
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Radiator conditions positive going wave
operator
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In difference form, the forward going operator
reduces to By interpolating the half
steps and half space as
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Doing the same for the term and
substituting back in we get If we use the
magic time step then we have following
radiation condition. For the backward going
wave we can derive a similar relationship
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Extending this method to 2D In this case, we
have assumed that 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.
1/3
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• Radiation Boundary Conditions
• In this approach we apply the radiation condition
  to the field values in the boundary of the
  computational region.
• However, this assumes that the field values on
  the boundary are approximately far-field values.
• This assumption requires the extension of the
  computational space many wavelengths beyond the
  location of the scattering object.
• 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

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• 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
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• Here the are assumed to be free space.
• Therefore the input impedance is
• which is not necessarily equal to R0?.
• 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!

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• Bayliss-Turkel Annihilation Operators
• One of the first successful classes of ABCs,
  provided less than back reflection over
  a wide range of incident angles, was based
  analytically derived operators whose sole purpose
  was to annihilate the reflected field.
• The idea of the annihilation operator is to
  construct an operator that can be used to
  estimate the values of the field along the
  truncated boundary. The problem arises from the
  fact that the central difference requires nodal
  values immediately adjacent to the node that is
  being updated!
• Along the boundary there is no field immediately
  adjacent.
• So the annihilation operator used a weighed sum
  of partial derivatives to estimate the field at
  the boundary.
• The derivatives consist of
• 1. A spatial derivative in the direction of the
  outgoing wave.
• 2. A spatial derivative in the direction
  transverse to the outgoing wave.
• 3. A time partial derivative.
• By doing this the field values along the
  truncated boundary are determined, in the
  direction of outgoing wave propagation.

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• Spherical Coordinates
• In this case consider the spherical wave equation
  in 3D
• The radiating solution to this equation can be
  expressed in an expansion of the form
• At very large distances only the leading
  terms/term dominate the expansion
• We now introduce the operator
• which is known as the Sommerfeld radiation
  condition

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Note that this operator is true for a plane,
however, the field distribution is not
necessarily a plane wave. As a result, ERROR is
introduced!! To see how the error can be reduced
consider the application of the radiation
condition to the expansion. Canceling the
first two terms we get
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• This produces error to the order
• given this we can rewrite the radiation operator
  as
• Radial expansion
• What Bayliss and Turkel did was to devise a
  modification to the radiation condition that
  allowed it to delay more quickly.
• The modification was proposed as

Sommerfeld Radiation Condition
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• If we now apply this operator to the expansion,
  we get
• this is referred to as the Bayliss-Turkel
  Operator of Order 1.
• To improve it further, they proposed a higher
  operator
• If we apply this operator

2nd order operator B12 B1
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• In principle this process can be repeated
  infinitely according to the following operator
• In general the most popular order is the 2nd
  order, which is a good trade between accuracy and
  complexity of implementation.

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• Engquist-Majda One-Way Wave Operators
• This technique also consists of a partial
  differential equation that permits propagation in
  only one direction.
• In this method consider a 2D wave equation in
  Cartesian coordinates
• We can define a partial differential operator.

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We can then express the wave equation
as where Engquist and Majda showed that
when applied to the appropriate boundary L- and
L act as exact ABCs for outgoing waves!
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• Unfortunately, their direct application in
  numerical is not possible due to the square root
  operator, which is non-linear in time and space.
• So we approximate these operators using
  expansions.
• One approximation is a One-Term Taylor Expansion,
  if we set
• then
• In this case the operator becomes
• which is simply the Sommerfeld Radiation
  Condition!
• If we take a two-term expansion we get

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• In this case the operator is capable of handling
  layer of Ks of incidence, because the first
  approximation implied the DtgtgtDy, which means
  that the field is at an angle very close to the
  normal tot he truncated boundary.
• The resulting operator is
• By performing LU0 we can obtain a corresponding
  PDE that can be implemented as a second order
  accurate ABC
• Analogous ABCs can be derived for the remaining
  boundaries.
• To be useful in the FDTD one must use
  finite-difference scheme, which was introduced by
  Mur.

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• Mur ABC
• As we discussed the 1st and 2nd order
  approximation to the Engquist-Majda one-way wave
  operators can be extendedto 31) and applied to a
  rectangular region.
• The truncated boundaries consist of six sides
  located in the planes x0, a, y0, b, z0, c.

c
c
b
a
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• So the 1st and 2nd order operators have the
  following form

 
 
 
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