EEE 431 Computational Methods in Electrodynamics

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EEE 431Computational Methods in Electrodynamics

• Lecture 7
• By
• Dr. Rasime Uyguroglu
• Rasime.uyguroglu_at_emu.edu.tr

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)
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FDTD References

• 1M.N.O. Sadiku, Numerical Techniques in
  Electromagnetics, CRC Press, 2001, pp. 159-192
• 2 A. Taflove, Computational Elertodynamics the
  Finite-Difference
• Time-Domain Method, Artech, 1995
• 3 A. Taflove, S.C. Hagness, same as above, 2nd
  ed., Artech, 2000
• 4 K. Kunz and R. Luebbers, Finite-Difference
  Time-Domain Method
• for Electromagnetics, CRC Press, 1993
• 5Kane S. Yee, Numerical solution of initial
  boundary value
• problems involving Maxwells equations in
  isotropic media, IEEE Trans. Antennas Propagat.,
  vol. AP-14, No. 3, pp. 302-307, May,1966.

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References

• Introduction to the Finite Time Domain (FDTD)
  Technique, S.Connor

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• The FDTD method, proposed by Yee, 1966, is
  another numerical method, used widely for the
  solution of EM problems. It is used to solve
  open-region scattering, radiation, diffusion,
  microwave circuit modeling, biomedical etc.
  problems.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• One of the most important concerns of the FDTD
  method is the requirement of the artificial mesh
  truncation (boundary) conditions. These
  conditions are used to truncate the solution
  domain and they are known as absorbing boundary
  conditions (ABCs), as they theoretically absorb
  fields. Imperfect ABCs create reflections and the
  accuracy of the FDTD method depends on the
  accuracy of the ABCs.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• The following advantages make the FDTD method
  popular
• Its a direct solution of Maxwells equations, no
  integral equations are required and no matrix
  inversions are necessary.
• Its implementation is easy and it is conceptually
  simple.
• It can be applied to the three-dimensional,
  arbitrary geometries.
• It can be applied to materials with any
  conductivity.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• The FDTD method has also the following
  disadvantages
• When the FDTD method is applied, the object and
  its surroundings must be defined.
• Since computational meshes are rectangular in
  shape it is difficult to apply the method to the
  curved scaterers.
• It has low order of accuracy and stability unless
  fine mesh is used

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• In this method the coupled Maxwells curl
  equations in the differential form are
  discretized, approximating the derivatives with
  centered difference approximations in both time
  and space domains.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• The six scalar components of electric and
  magnetic fields are obtained in a time-stepped
  manner.
• The space domain includes the object and it is
  terminated by Absorbing Boundary Conditions
  (ABCs).

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Basic Finite-Difference Time-Domain Algorithm
• Differential Forms of Maxwells Equations

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• In linear, isotropic and homogeneous materials
  and are related to and with the following
  constitutive relations

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Also is related to as,
• in a conducting medium.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Substituting the constitutive relations into the
  Maxwells equations, we can write six scalar
  equations in the Cartesian coordinate system.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Electric Field Intensity

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Magnetic field intensity

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Yee Algorithm
• Yee algorithm solves for both electric and
  magnetic fields in time and space using the
  coupled Maxwells curl equations.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• the Yee algorithm centers its and field
  components in three-dimensional space so that
  every component is surrounded by four
  circulating components, and every component is
  surrounded by four circulating components.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Unit cell of the Yee space lattice.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• The computational domain is divided into a number
  of rectangular unit cells
• According to Yee algorithm and field
  components are separated by in time.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)
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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Finite Differences ( Discretization )
• A space point in a uniform rectangular lattice is
  denoted as

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Here , and are the lattice space
  increments in the x, y, and z coordinate
  directions respectively and i, j, and k are
  integers. If any scalar function of space and
  time evaluated at a discrete point in the grid
  and at a discrete point in time is denoted by u,
  then

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Using central, finite difference approximation in
  space, i.e w.r.t.x

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Using central, finite difference approximation in
  time

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Ex

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Ey

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Ez

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Hx

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Hy

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Hz