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Entry  Sun Jun 22 23:44:56 2014, Sam Moore, Optics, , Going through Heinert's 'TR noise of cylindrical test masses' paper 
    Reply  Mon Jun 23 11:40:17 2014, Sam Moore, Optics, , Going through Heinert's 'TR noise of cylindrical test masses' paper 
    Reply  Mon Jun 23 16:10:17 2014, Matt A., Optics, , Going through Heinert's 'TR noise of cylindrical test masses' paper SamEq1.pngSamEq2.pngSamEq3.pngSamEq4.png
Message ID: 85     Entry time: Mon Jun 23 16:10:17 2014     In reply to: 83
Author: Matt A. 
Type: Optics 
Category:  
Subject: Going through Heinert's 'TR noise of cylindrical test masses' paper 

Quote:

At this point, my are goals are to 1) convert the time-dependent heat equation into stationary, complex form, 2) use the Levin approach to calculate the TR noise given this stationary PDE, and 3) verify the results in COMSOL

 I have looked at the Heinert paper and converted the time-dependent partial differential Heat equation to a stationary, complex one.  I did this by assuming a temperature distribution of the form T(\vec{r},t) = T_0 ( \vec{r} ) e^{i \omega t }, where  \omega is the frequency of oscillation of the input heat field: q( \vec{r} , t) = q( \vec{r} ) e^{i \omega t}.  Hence, I am assuming the steady-state solution of the temperature field.  From this ansatz, the PDE becomes

C_p T(\vec{r}) + \frac{ i \kappa}{ \omega }  \nabla^2 T ( \vec{r} ) = q ( \vec{ r} ).  

This complex conversion already seems to have been done in the paper, since the stationary solution of the form

T(r, z) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{ \infty} T_{n m} J_0 (k_n r) \cos ( \ell_m z) is assumed.

k_n and \ell_m are obtained from the adiabatic boundary conditions.  Moreover, if I plug this form for T(r, z) into the stationary PDE, I get Heinert's eq. 12. ( although I'm not quite sure how to take the second derivative of a zeroth-order bessel function).

 

I've started to try out the cylindrical geometry in COMSOL, using the structural mechanics and heat transfer modules.  At this point, I'm not quite sure how to specify the PDE that I'm interested in, nor am I sure about how to specify the boundary conditions.

 

 

 

Hi Sam,

I know it seems a bit distracting, but it's sometimes nice to write up your elog entries in a more readable form. I've re-edited your comments to make them more easily read.

I used http://www.sciweavers.org/free-online-latex-equation-editor to convert your latex to png files for easy reading.

 

At this point, my are goals are to:

 
 1) convert the time-dependent heat equation into stationary, complex form, 
 2) use the Levin approach to calculate the TR noise given this stationary PDE, and 
 3) verify the results in COMSOL
 
I have looked at the Heinert paper and converted the time-dependent partial differential Heat equation to a stationary, complex one.  
I did this by assuming a temperature distribution of the form:
 
SamEq1.png
 
where ω is the frequency of oscillation of the input heat field: 
 
SamEq2.png
 
Hence, I am assuming the steady-state solution of the temperature field.  
 
From this ansatz, the PDE becomes:
 
SamEq3.png
 
This complex conversion already seems to have been done in the paper, since the stationary solution of the form:
 
SamEq4.png
 
is assumed. kn and ℓm are obtained from the adiabatic boundary conditions.
 Moreover, if I plug this form for T(r, z) into the stationary PDE, I get Heinert's eq. 12.
( although I'm not quite sure how to take the second derivative of a zeroth-order bessel function).
 
I've started to try out the cylindrical geometry in COMSOL, using the structural mechanics and heat transfer modules.  
At this point, I'm not quite sure how to specify the PDE that I'm interested in, nor am I sure about how to specify the boundary conditions.
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