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 matlabfft 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.

  1. Signs for radiation integral:

Top: +Ez sin + eta Hx

Bottom: -Ez sin - eta Hx

Right: +Ezcos - eta Hy

Left: -Ezcos + eta Hy

  1. 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.

  1. 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:

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 iterativesolutions.

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.mcan 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 semilogyplot of the cutoff frequencies of the first twenty modes from FEM overlaid with the exact solution.

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

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?

Homework #20

Due Dec. 9

(saved in folder)