Warnick Homeworks
Homework #1
Due Sep. 2
1. 2.1
2. 2.2
3. 2.3
4. 2.4
5. If you have not already done this, write the code from the ECEn 462 FDTD lab.
Homework #2
Due Sep. 7
1. 3.1
2. 3.2
3. 3.3
Homework #3
Due Sep. 14
3.4
Homework #4
Due Sep. 16
3.5
Homework #5
Due Sep. 21
1. 3.6
2. 3.7
Homework #6
Due Sep. 23
3.8
Homework #7
Due Sep. 30
1. 3.9
2. 3.10
Homework #7 helps
1. Here are the basic steps required to add the near-to-far transformation in post-processing
to the 2D FDTD code:
- Step through the FDTD algorithm until the transient part of the solution propagates
away from the scatterer.
- At this point in time, inside the loop over the time index, begin storing values of Hx,
Hy, and Ez at grid samples on the Huygens contour around the scatterer. This could be
done using eight arrays (top, borrom, right, left sides of the contour for each of the
tangential field components). Continue this for exactly one time cycle of the incident
field (1/f/dt time steps). After one time cycle of the fields on the contour have been
stored, end the FDTD loop and go to post processing.
- In the post-processing of the code, use the FFT command to transform the time samples
of the fields. The third argument of the matlab fft command allows you to select which
dimension of an array to transform. Pull out the second sample of each FFT and scale by
2/N. You now have one complex phasor at each point on the contour for Hx, Hy, and Ez.
- Loop over 360 or so scattering angles. For each scattering angle (inside the loop),
compute the far-field radiation integral over the Huygens contour using a Riemann sum
to approximate the integral. In the radiation integral, the angle phi is the scattering angle,
and the point x',y' represents the grid points on the Huygens contour that you are
summing over. The value of this integral will be the scattering amplitude at the current
scattering angle. Convert the scattering amplitude to scattering width by taking the
magnitude squared and multiplying by the appropriate constant. You now have the
bistatic scattering width at various angles around the scatterer.
- Compare the scattering width to the analytical solution using Bessel functions derived
in the notes.
2. Signs for radiation integral:
Top: +Ez sin + eta Hx
Bottom: -Ez sin - eta Hx
Right: +Ez cos - eta Hy
Left: -Ez cos + eta Hy
3. Common errors:
In the radiation integral, the x' and y' points used to compute the phase term must be the
same as the physical points corresponding to the samples of the fields Hx, Hy, and Ez.
In matlab, do not use the apostrophe (') to change column vectors to row vectors or vice
versa, because it will conjugate the elements (it is the Hermitian transpose operation).
Use period apostrophe (.') instead.
4. For best accuracy, put the Huygens contour for the radiation integral three or four cells
away from the PEC cylinder. Let the FDTD simulation run long enough that the
transients propagate away from the scatterer but not so long that the reflection from the
ABC makes it back to the contour. For additional accuracy, if the contour is on the Ez
points, average the magnetic field values in space and time to get values at the same
space and time points as Ez.
Homework #8
Due Oct. 7
1. 3.11
2. 3.12
Homework #9
Due Oct. 12
1. 4.1
2. 4.2
HW #9 helps: http://mathworld.wolfram.com/Legendre-GaussQuadrature.html
Homework #10
Due Oct. 14
1. 4.4
2. 4.5
Homework #11
Due Oct. 21
1. 5.1
2. 5.2
3. 5.3
Homework #12
Due Oct. 26
1. 5.4
2. 5.6
Homework #13
Due Nov. 2
1. For the volume MOM with a cylinder of radius ¸=2 and ²r = 2, replace the matlab slash
operator with the Jacobi iteration. Does it converge? Next, try the Gauss-Seidel iteration. (see
notes)
2. Raise the relative permittivity of the cylinder to 10. Repeat the Jacobi and Gauss-Seidel
iterative solutions.
Homework #14
Due Nov. 4
1. 6.2
2. 6.3
Homework #15
Due Nov. 9
Derive the Helmholtz equation from the functional
Homework #16
Due Nov. 16
1. Assemble the FEM T matrix by hand for a 2D mesh with two or three triangles (leave the area
A as a variable but otherwise give numerical values for the matrix elements).
2. Use the matlab command [p,e,t] = poimesh(’rect’,N); to create a finite element
mesh for the rectangular waveguide in HW4. The geometry function rect.m can be downloaded
from the course webpage. Write a code to assemble the finite element stiffness matrix and
compute the cutoff frequencies for the waveguide. Turn your code and a semilogy plot of the
cutoff frequencies of the first twenty modes from FEM overlaid with the exact solution.
http://ece563web.groups.et.byu.net/hw/rect.m
Homework #17
Due Nov. 24
Implement the FEM-BEM method and use it to compute the bistatic scattering width of a
circular PEC cylinder and a dielectric cylinder of relative permittivity 2, both of radius one half
wavelength.
Compare to the exact solution or previous results. Hint: debug the code first for the PEC cylinder
and plot the near fields using
a = [];
a([interior nodes,boundary nodes]) = x(1:N);
pdesurf(p,t,real(a))
Homework #17 helps
http://ece563web.groups.et.byu.net/hw/fembem.html
Homework #18
Due Dec. 2
10.3
Homework #19
Due Dec. 7
1. Please complete the online course evaluation for this class for extra credit equal to one
homework problem.
2. Apply the regularized sampling method to the forward scattering data given on the course
website. Can you identify the objects?
http://ece563web.groups.et.byu.net/hw/forward_data.html
Homework #20
Due Dec. 9
(saved in folder)