Finding Reflection factor for the PML technique

[Vijuma]

reflection factor 2d fdtd pml

URGENT!!!!
Can anyone send the formula to calculate the reflection factor for 2D PML technique?
Write the formula for both theoretical calculation as well as for the FDTD technique.

 

[iyami]

finding the factor

Write the formula for both theoretical calculation as well as for the FDTD technique.

copied from some homework exercise? I hope so, otherwise it sounds pretty rude.

How about letting us know some more details. Do you need it for all angles?, all frequencies?,
dispersive materials?, different gradings?, non-uniform grids, UPML/CPML/NPML?,
different stable time steps?.....

 

[Vijuma]

This is not a homework exercise.. I am solving Berenger paper (1994)- A perfectly matched layer for the absorption of electromagnetic waves...
I would like to know how to compute the reflection factors using theoretical as well as FDTD technique.. To be precise, I want to know how to compute Table 1 values given in that paper...

For the FDTD technique it is mentioned in the paper that reflection factor is computed by means of fourier transformation of the reflected pulse.. If any one has already computed this , pl, help me.

Thanks.

 

[iyami]

I think the Morgan&Claypool book by Berenger:

Perfectly Matched Layer (PML) for Computational Electromagnetics

has all the details (look at pages 72-85).

If you have a pulse P and its reflection R, then FT(R,omega)/FT(P,omega) is the
reflection factor.

The book has a more general treatment.

 

[Vijuma]

Thanks for your suggestion.. I am going through the suggested book.. I am able to calculate theoretical value, but I still have problems in finding the reflection factor using FDTD technique (I feel the book gives a very general approach).

 

[Vijuma]

I am still unable to compute the reflection factor for a plane wave striking a plane boundary using FDTD. If any one has solved Berenger paper (1994), pl. help me in this regard.. I am trying to compute values of Tables given in the paper and plot the results..

Thanks.

 

[iyami]

How about describing how you tried it. I thought

If you have a pulse P and its reflection R, then FT(R,omega)/FT(P,omega) is the
reflection factor.

was a complete answer already. Here P and R are the time signals you measured.

 

[Vijuma]

Hi iyami,
Thanks a lot for ur reply.

Can u pl. elaborate on the statement u have mentioned... I am attaching Berenger paper alongwith this message.. Can u pl. explain me according to the details given in page 193 of the paper. I am interested in retrieving the values of table 1. Hope u help me with this.

 

[Vijuma]

Iyami,
I am not able to proceed further on anything without
completing this part. Can u pl. explain the calculation.. Pl. reply at
the earliest.

Thanks in advance.

 

[iyami]

I am not able to proceed further on anything without
completing this part. Can u pl. explain the calculation.. Pl. reply at
the earliest.

I do not have time to explain everything from scratch. How about you tell
me what you don't understand and I see what I can do. I don't have lots of
time, usually 5 minutes here and there.

 

[Vijuma]

I am sorry to bother u so much. It is quite true that each person in this field (in any field for that matter) is quite busy and gets very less time for all other stuff. As I see that u are in this field, I thought u could solve my problem faster..

To find the reflection factor using fourier transformation of the reflected pulse, we are given dx, dt, and other usual constants like speed of light, epsilon, mu0 , theta, omega, frequency. we are also given R(0), number of cells.
My question is: in the formula which u have mentioned: FT(R, w)/FT(P, w),
what is the value of R? is it R(0)?? How do u get P? Can u pl. elaborate on the formula which u have mentioned?
For constant conductivity, with theta =45, Reflection factor is found as 4.953. (this is just one value of reflection factor found, just to mention one result which is in the paper).

 

[iyami]

To find the reflection factor using fourier transformation of the reflected pulse, we are given dx, dt, and other usual constants like speed of light, epsilon, mu0 , theta, omega, frequency. we are also given R(0), number of cells.
My question is: in the formula which u have mentioned: FT(R, w)/FT(P, w),
what is the value of R? is it R(0)?? How do u get P? Can u pl. elaborate on the formula which u have mentioned?

If you want to measure the reflection you have to do the following

1) run a FDTD simulation where you can measure the incoming pulse P and the reflected pulse R.
(often you use short pulses or pulses generated some distance from the reflecting region.

2) Both P and R are time signals. To get the frequency content at various frequencies you compute
either FFT or DFT of both P and R. (Above I used FT(R,w) to denote the value of the Fourier transform of R at the frequency w.)
.
Then the reflection factor is simply the quotient as I wrote above.

So far the easy part.

The more difficult part is how to get P and R. I am not sure what Berenger did since I don't have
access to the reference he cites. I assume the term Huyghens surface may be a TF/SF like method to introduce a planewave.

He used a Gaussian pulse. He measured the incoming pulse at a fixed location during the initial
part of the simulation. Then he made sure that the model was big enough that unwanted reflection
had no chance to reach the measurement point when he measured the reflection at the same point
(timesteps 100-200).

These measurements were the time signals P and R I mentioned above. The total measured signal
should roughly have the following shape

_/\____/\_____*******

the first peak is P the next R and eventually you will have lots of noise.

 

[Vijuma]

Thanks a lot for the explanation.