ID 
Date 
Author 
Type 
Category 
Subject 
71

Thu Jul 25 13:21:46 2013 
Deep Chatterjee  Optics  General  TR results for different dimensions 
Quote: 
In this post I simulate the procedure of calculating the TR noise for finite cavities as proposed by Heinert and check for a
match.
The technique of performing all necessary calculations in COMSOL and exporting the results was applied to the TR codes.
It was seen that the codes gives similar output as the technique of extraction of Fourier coefficients in place of time averaging
as has been done in the codes of Koji Arai. One can see the output as the present code runs to be similar to the previous ones
found in the SVN.
However, the results in the present case were off by a constant factor close to 100. This maybe due to some 'm'  'cm' or similar difference between
analytic calculations and COMSOL values of parameters. Although, it has not been found yet, the correction is hopeful to be
found soon.
The codes give results similar to the analytic result for other values of the mirror radius and beam radii (apart from the constant
factor I have mentioned above). One may have a look at the trend of the graphs between analytic and simulated values in the plots
attached. These plots are for the case when the mirror radius = 25m while the beam radius = 9 cm i.e. the original radii were 0.25m
and 9cm respectively i.e. the ratio has been changed by a order of 2.
As mentioned before the reason for the constant factor difference will be looked into.

The discrepancy related to the difference between the analytic and COMSOL results has been partially addressed. Attached is another
plot showing the comparison. The ratio this time between the COMSOL results and the analytic results is between 0.7  0.8. This difference
will be looked into. It is, however, observed that the difference is not a constant factor  it has to do with the model file. 
Attachment 1: Jul_25.pdf


72

Thu Jul 25 15:54:58 2013 
Deep Chatterjee  Optics  General  TR results for different dimensions 
Quote: 
Quote: 
In this post I simulate the procedure of calculating the TR noise for finite cavities as proposed by Heinert and check for a
match.
The technique of performing all necessary calculations in COMSOL and exporting the results was applied to the TR codes.
It was seen that the codes gives similar output as the technique of extraction of Fourier coefficients in place of time averaging
as has been done in the codes of Koji Arai. One can see the output as the present code runs to be similar to the previous ones
found in the SVN.
However, the results in the present case were off by a constant factor close to 100. This maybe due to some 'm'  'cm' or similar difference between
analytic calculations and COMSOL values of parameters. Although, it has not been found yet, the correction is hopeful to be
found soon.
The codes give results similar to the analytic result for other values of the mirror radius and beam radii (apart from the constant
factor I have mentioned above). One may have a look at the trend of the graphs between analytic and simulated values in the plots
attached. These plots are for the case when the mirror radius = 25m while the beam radius = 9 cm i.e. the original radii were 0.25m
and 9cm respectively i.e. the ratio has been changed by a order of 2.
As mentioned before the reason for the constant factor difference will be looked into.

The discrepancy related to the difference between the analytic and COMSOL results has been partially addressed. Attached is another
plot showing the comparison. The ratio this time between the COMSOL results and the analytic results is between 0.7  0.8. This difference
will be looked into. It is, however, observed that the difference is not a constant factor  it has to do with the model file.

The issue related to the difference between the analytic and simulated values has been resolved. The codes seems to give reasonable match
between the analytic and simulated case. There is, however, a difference between the formulas being used from the previous cases. Note that
the 1/2 in front of Eq.(15) of Heinert is a because the time average has already been considered. However, in the present codes, the volume
integral of grad_T is evaluated in COMSOL and exported as a function of time. It is then averaged in MATLAB. Thus the factor of 1/2 is to be
omitted in this case(see Liu and Thorne Eq.(5). The presence of this extra factor of 1/2 was giving error in the last upoaded plots. From the
relative difference plot, one can see the maximum difference between COMSOL and analytic results go upto 7% but for most of the graph
it is close to 1% which is a fair result. 
Attachment 1: Jul_25_1.pdf


Attachment 2: Jul_25_2.pdf


Attachment 3: relative_plot.pdf


73

Mon Jul 29 22:42:57 2013 
Deep Chatterjee  Optics  General  Avoiding transient solutions in the Computation of TE/TR noise  An error was being encountered in the computation of the TR noise lately. It was observed that while running the simulations in the case of the materials which have a lower value of the thermal diffusivity (silicon / sapphire at room temperature), the simulated result were slightly off from the analytic result. On the other hand, if the simulation was run with a material of higher diffusivity(same materials at lower temperature), the match would be better. The reason being the transient solution not dying off significantly during the period of the simulation. Since a time average was being taken of the quantity integral{grad_T ^2}, the transient contributed to the integral. To get the correct value, the fourier coefficient of the time signal of integral{grad_T ^2}, was extracted at twice the frequency of the pressure oscillation. The reason being that the signal was squared. Extracting the response at this frequency after the integration is logical since the integration is over space while the response we extract is over time.
The same procedure was also applied to the TE noise calculation. However, this time we obtained similar result as the case where this procedure was not applied but a simple time averaging was performed. The tail of the plot, at high frequency, is still seen to deviate from the analytic result of Liu and Thorne as was the case previously. A plot is attached showing the spectrum for Fused silica at 290K. The conclusion being that the transients do not affect the TE noise calculation  the plot stayed the same even after filtering them out. This is probably because unlike the TR case which has a heat source present along the cylinder axis, the TE noise calculation involves applying pressure only on the face of the cylinder, and the transient do not contribute much to the volume integral.

Attachment 1: Jul29_2.pdf


74

Wed Jul 31 15:39:11 2013 
Deep Chatterjee  Optics  General  First try with paramter optimization for TE and TR noise profiles  After the simulations have been found to match to a fair extent with the analytic results by Heinert and Liu and Thorne, the attempt is check out the parameters for which the TE and TR noise are
close to each other. This was done with the analytic results. The frequency range 10  1000 Hz was looked into. In between this range, the quantity that was minimized was the absolite value
of the logarithm of the ratio between the TR and TE noise. The fminsearch function was used to minimize the mentioned quantity. The parameters that were changed were  conductivity, thermo
optic coefficient and coefficient of linear expansion. The reason for choosing these three were 
> TR noise is independent of coefficient of linear expansion
> TE noise is independent of thermo optic coefficient
> The power law dependence on conductivity is different for the TE and TR cases as can be seen from the analytic expressions
once the code returned the optimized parameters, these values were plugged in and the results were plotted.
**Note that the minimization was done for frequencies between 10 to 1000 Hz 
Attachment 1: Jul31_param_opt.pdf


75

Thu Aug 8 17:17:19 2013 
Deep Chatterjee  Optics  General  Something like cancellation  For the material parameters of Sapphire at 300K, the TE and TR Noise profiles, though not very close, lie close to within an order of magnitude. Sapphire has a positive coefficient of linear expansion. We just inverted the sign of this quantity
and the ran the codes that puts heat and pressure simultaneously to the test mass. The total noise looks to be lower than the TR noise which is greater.
Mirror radius 0.25[m]
Mirror height 0.46[m]
Beam Radius 0.09[m]
If we have physical parameters which make the two Noise sources come closer to each other and then flip the sign of alpha, we may be able to see some noise reduction to a greater extent. 
Attachment 1: suspected_cance_AUg8l.pdf


77

Wed Nov 13 23:55:44 2013 
Chris Couste  Optics  Analysis  Optical Mount Vibrational Analysis  Project: Vibrational Analysis of Optical Mounts
Goal: Use COMSOL to run finite element analysis on simplified models of different types of optical mounts available to us, in order to find which shape/style/material reacts the least to external sound pollution. Once the few best candidates have been identified, develop test to experimentally determine optimal mount configuration and material.
Part 1: FEM
Process: Simultaneously build models in SolidWorks and design analysis in COMSOL
Solidworks: base/shaft models based on measurements of actual optical mounts, optic holder models matching mass and moment of inertia from downloaded models from their manufacturers. ~Progress: base/shaft done, optic holders almost done.
COMSOL: Design analysis that first does a general eigenfrequency analysis to find general vicinity of modes, then does a full modal frequency analysis. ~Progress: finished, ready for models to be imported. 
78

Mon Nov 18 15:56:59 2013 
Chris Couste  Optics  Analysis  Representative Models  The simpler models of the optical mounts are finished, they will be run through the comsol analysis software soon.
see pictures below:

Attachment 1: Custom_Closed.JPG


Attachment 2: custom_open.JPG


Attachment 3: Stock_Closed.JPG


Attachment 4: Stock_Open.JPG


79

Fri Nov 22 21:04:53 2013 
Chris Couste  Optics  Analysis  analysis  The analysis is making nice eigenmode and stress mode models, but the displacement experiment needs work. Should be fixed by monday. 
80

Fri Jan 24 17:26:38 2014 
Chris Cousté  Optics  Analysis  Mount Analysis Functional!  The comsol eigenmode analysis is complete, and the only thing left to do on this part of the project is to run the analysis on a range of different configurations of optical mounts as well as a range of materials. This compilation will be posted on this elog in the next few weeks, due to the fairly long runtime of the analysis software. 
81

Sun Feb 16 21:10:56 2014 
Chris Cousté  Optics  Analysis  Optical Mount data compilation 1: Aluminum  The time is finally here! this is the highest displacement of each mount in its lowest few eigenfrequencies, using 6061 Aluminum as a material. pictures will be added in a future log, because I'm going to make them into one file. Other materials will also be tested to see if there is variance in these findings, but only relevant data will be posted.
tl;dr findings: thickopen is the best combination because it has less displacement in eigenmodes and its eigenmodes are all very high/spread apart in frequency.
long:
(key) Eigenfrequency (Hz), highest displacement (mm), direction
Thin base, open mount:
315, 8.07, leaning back (laser will angle up)
316, 7.94, leaning to the side (laser won't change direction if the optic is flat, i.e. point of contact moves along surface of optic)
924, 10.50, twist to the side (laser will angle to the side)
1949, 8.89, twist about optic, slight lean forward (point of contact will move slightly, laser may angle down)
2144, 10.47, another leaning forward mode
thin base, open mount:
322, 13.29, leaning forward and translating to the side
328, 13.36, same as above
994, 17.66, twisting to the side (laser will angle to the side)
1943, 13.86, intense leaning forward
2100, 14.27, twist about optic
thick base, closed mount:
1329, 13.23, leaning back
1347, 13.32, leaning back and to the side
3577, 15.46, twisting to the side
thick base, open mount:
1326, 9.69, leaning to the side
1335, 9.77, leaning back
3504, 13.88, twisting to the side

All of these frequencies were checked for all four models, and the max displacements in those cases ranged from 100 picometers to 10 femtometers, so it's pretty reasonable to base decisions off of the displacements at the eigenmodes.
Based on this, it seems that for both thick and thin, open is the best, considering it displaces significantly less than closed (which is reasonable because there is less mass at the top to be thrown around). The choice between thick and thin depends on what frequencies we believe the mounts will be subjected to. If we are probably not going to have much sound at frequencies above 1000Hz, then thick is the better option (which is nice because it is the stock stalk instead of the custom made one). However, if there will be plenty of high noise, the thick is still probably the better option because it has fewer eigenmodes in the same range of frequencies.
here's a sample of two pictures of the relevant eigenmodes of the winner.

Attachment 1: aluminum2.png


Attachment 2: aluminum3.png


83

Sun Jun 22 23:44:56 2014 
Sam Moore  Optics   Going through Heinert's 'TR noise of cylindrical test masses' paper  At this point, my are goals are to 1) convert the timedependent 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 timedependent 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 steadystate 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 zerothorder 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.

84

Mon Jun 23 11:40:17 2014 
Sam Moore  Optics   Going through Heinert's 'TR noise of cylindrical test masses' paper 
Quote: 
At this point, my are goals are to 1) convert the timedependent 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 timedependent 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 steadystate 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 zerothorder 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.

I carried out the calculations for Heinert's simple model in greater detail and was able to obtain equation 12 (with the exception of a plus sign instead of a minus sign in the denominator; I guess the Fourier transform method gives a minus sign. In the end, that doesn't matter, since the modulus square yields the same result in the denominator.) I obtained the same spectral density S_z (\omega), with the exception of a missing factor of R^2. I don't know where this R^2 went, or why it is that the spectral density is associated with the extrinsic variable z (instead of r, say). It's supposed to be a "homogeneous [noise] readout along the z direction." 
85

Mon Jun 23 16:10:17 2014 
Matt A.  Optics   Going through Heinert's 'TR noise of cylindrical test masses' paper 
Quote: 
At this point, my are goals are to 1) convert the timedependent 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 timedependent 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 steadystate 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 zerothorder 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 reedited your comments to make them more easily read.
I used http://www.sciweavers.org/freeonlinelatexequationeditor to convert your latex to png files for easy reading.
At this point, my are goals are to:
1) convert the timedependent 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 timedependent partial differential Heat equation to a stationary, complex one.
I did this by assuming a temperature distribution of the form:
where ω is the frequency of oscillation of the input heat field:
Hence, I am assuming the steadystate solution of the temperature field.
From this ansatz, the PDE becomes:
This complex conversion already seems to have been done in the paper, since the stationary solution of the form:
is assumed. k_{n} and ℓ_{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 zerothorder 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.

86

Tue Jun 24 14:35:42 2014 
Sam Moore  Optics  General  Trying to Verify the Heinert Model  
Attachment 1: 06_23_14.pdf


87

Tue Jun 24 17:05:24 2014 
Sam Moore  Optics  General  Trying to Verify the Heinert Model  It does appear that the simplified model is only relevant for the simulations. To quote Heinert: "An efficient computation is only possible for the simple model, as the advanced model would require an element of size more than 10^{6 }." I have run Koji's code that replicates Heinert's figure 3. I have attached the resulting temperature distribution and noise amplitude curve. In the noise amplitude curve, the red line is the analytical result, while the dots are from COMSOL.
The next step is to convert this code to an efficient complex timeindependent solution. As stated before, my main concern here is whether COMSOL actually solves the right equation in the stationary case. 
Attachment 1: temperature.png


Attachment 2: noiseAmplitude_agreement.png


88

Sat Jun 28 21:59:11 2014 
Sam Moore  Optics  General  Difficulty with the COMSOL stationary module; Test Cases  Here, I describe some test cases to see if COMSOL's solutions are agreeing with some simple analytical solutions. Right now, I have two plots showing COMSOL's solution and my analytical solution on separate plots. I will be plotting there difference to see if they really match up.

Attachment 1: 6_27_14.pdf


89

Sun Jun 29 15:37:18 2014 
Sam Moore  Optics  General  Difficulty with the COMSOL stationary module; Test Cases 
Quote: 
Here, I describe some test cases to see if COMSOL's solutions are agreeing with some simple analytical solutions. Right now, I have two plots showing COMSOL's solution and my analytical solution on separate plots. I will be plotting there difference to see if they really match up.

The following document shows the relative difference between these two plots. 
Attachment 1: 6_29_14.pdf


90

Sun Jun 29 20:25:44 2014 
Koji  Optics  General  Difficulty with the COMSOL stationary module; Test Cases  What about this example? The result is easier to understand intuitively.
Consider a bar with the length of L.
Let's say there is no body heat applied, but the temperature of the bar at x=L is kept at T=0
and at x=0 is kept at T=T0 Exp[I w t].
The equation for the bar is
...(1)
Consider the solution with the form of T(x, t) = T(x) T0 Exp(I w t), where T(x) is the position dependent transfer function.
T(x) is a complex function.
Eq.1 is modified with T(x) as
With the boundary condition of
This can be analytically solved in the following form
where alpha is defined by
So kappa/Cp is the characteristic (angular) frequency of the system.
Here is the example plot for L=1 and alpha = 1 (red), 10 (yellow green), 100 (turquoise), 1000 (blue)
If the oscillation is slow enough, the temperature decay length is longer than the bar length and thus the temperature is linear to the position.
If the oscillation is fast, the decay length is significantly shorter than the bar length and the temperature dependence on the position is exponential.
Now what we need is to solve this in COMSOL 
91

Sat Jul 5 13:04:32 2014 
Sam Moore  Optics   Heinert Model TR Noise Verification  Agreement with Heinert's paper for cylindrical TR noise has now been achieved. Using the stationary state assumption to calculate the temperature profile, the computation time was reduced compared to the previous timedependent approach. Here are the plots showing the agreement. I have shown the plots for a 1D axisymmetric model, in addition to a full 3D model in COMSOL. Both give the same result.
What went wrong? In the 1D axisymmetric case, it turns out that COMSOL has the incorrect cylindrical coordinate Laplacian for the coefficient form PDE interface. I corrected for this by expanding the Laplacian with the product rule, giving an extra kappa/r ''convection coefficient'' in the PDE interface. Meanwhile, the 3D case worked from the beginning, since its Laplacian was in cartesian coordinates.
The next goal is to use a stationary state method to calculate TE noise for a test case.

Attachment 1: threeD_cylinderTRnoise.png


Attachment 2: oneDcylinderTRnoise_copy.png


92

Mon Jul 7 19:47:00 2014 
Koji  Optics   Heinert Model TR Noise Verification  How close are these FEA calculations with the analytical values?
Can you plot residual too? (Put analytical values, 1D, abs(1D  analytical), 3D, and abs(3D  analytical) all together.) 
93

Thu Jul 10 16:51:14 2014 
Sam Moore  Optics  General  Duan and Heinert Comparison  (See Plots in attached document)
My plan has been to replicate Duan's numerical thermoconductive (TE + TR) phase noise plot presented in his paper (section V). I am trying to match Duan's analytical expression with Heinert's analytical expression. This requires some rescaling of Heinert's TR displacement noise. (I also needed to divide Heinert's expression by 4 pi to match the Fourier Transform convention. ) Duan's analytical expression for the phase noise is obtained by evaluating the triple integral given in equation 13 of the Duan paper "General Treatment of Thermal Noise in Optical Fibers".
It turns out that an additional factor of 2 multiplies the phase noise because Duan's Fourier Transform only takes into account positive frequencies; there are also negative frequencies that occur in equal amplitude.
This integral was evaluated in Mathematica due to numerical noise in MATLAB's calculation. The calculation in Mathematica was very slow, so the upper limits on the integral were truncated. The following plots in the attached document show the resulting noise profile agreements for two different upper limits.
If the residual for the highest upper limit is considered acceptable for a match between the two plots, then I will use Heinert's plot as a reference when using the COMSOL steadystate method for Duan's numerical case (Heinert's plot runs much faster). 
Attachment 1: 7_10_14.pdf


94

Fri Jul 11 10:58:18 2014 
not Koji  Optics   Heinert Model TR Noise Verification 
Quote: 
How close are these FEA calculations with the analytical values?
Can you plot residual too? (Put analytical values, 1D, abs(1D  analytical), 3D, and abs(3D  analytical) all together.)

Here are the plots with their fractional residuals: abs(S_COMSOL  S_analytical)/S_analytical 
Attachment 1: heinert_analyticalTest_residual_threeD.eps


Attachment 2: heinert_analyticTest_residual_oneD.eps


95

Mon Jul 14 19:09:14 2014 
Sam Moore  Optics  General  Using Heinert's Solution for Duan's Parameters  I have plotted Heinert's analytical solution for TR noise using Duan's parameters. Since TO and TE noise can be found by simply rescaling TR noise, these have been included in the plot as well. The solid curve represents the analytical solution, while the tick marks represent COMSOL's solution. I have used COMSOL for both a 1D axisymmetric and a 3D model. Since Duan's cylinder has a radius of 125 microns, but a length of 1 m, the meshing was difficult for the 3D model. I ended up shortening the length of the cylinder, converting to the actual length when finally calculating the thermal noise. 
Attachment 1: oneD_duanParams_residualepsconvertedto.pdf


Attachment 2: threeD_duanParams_residualepsconvertedto.pdf


96

Mon Jul 14 19:14:31 2014 
Sam Moore  Optics  General  Duan and Heinert Comparison 
Quote: 
(See Plots in attached document)
My plan has been to replicate Duan's numerical thermoconductive (TE + TR) phase noise plot presented in his paper (section V). I am trying to match Duan's analytical expression with Heinert's analytical expression. This requires some rescaling of Heinert's TR displacement noise. (I also needed to divide Heinert's expression by 4 pi to match the Fourier Transform convention. ) Duan's analytical expression for the phase noise is obtained by evaluating the triple integral given in equation 13 of the Duan paper "General Treatment of Thermal Noise in Optical Fibers".
It turns out that an additional factor of 2 multiplies the phase noise because Duan's Fourier Transform only takes into account positive frequencies; there are also negative frequencies that occur in equal amplitude.
This integral was evaluated in Mathematica due to numerical noise in MATLAB's calculation. The calculation in Mathematica was very slow, so the upper limits on the integral were truncated. The following plots in the attached document show the resulting noise profile agreements for two different upper limits.
If the residual for the highest upper limit is considered acceptable for a match between the two plots, then I will use Heinert's plot as a reference when using the COMSOL steadystate method for Duan's numerical case (Heinert's plot runs much faster).

I have now plotted the DuanHeinert comparison for the case of an infinite upper bound. It turns out that the curves differ by a maximum of 10 percent for low frequencies. Such a discrepancy has been attributed to lack of experimental investigation into this regime (according to Duan). For our purposes, such a discrepancy is acceptable. We will therefore use the Heinert curve for subsequent calculations due to its faster computation time.

Attachment 1: duan_heinert_comparisonInfiniteepsconvertedto.pdf


97

Thu Jul 31 20:55:38 2014 
Sam Moore  Optics  General  Finding the Right Meshing for the TIR cavity  In this document, I try to identify I good mesh by comparing the numerical solution from that mesh with my analytical model. Since there are problems with carrying out the analytical calculation, it is still not entirely clear which mesh should be used.

Attachment 1: 7_30_14.pdf


98

Sat Aug 2 00:22:34 2014 
Sam Moore  Optics  General  Finding the Right Meshing for the TIR cavity 
Quote: 
In this document, I try to identify I good mesh by comparing the numerical solution from that mesh with my analytical model. Since there are problems with carrying out the analytical calculation, it is still not entirely clear which mesh should be used.

I have refined the analytical calculation procedure, as outlined in this new document. The procedure indicates that the discrepancy between the analytical and numerical solutions are more likely attributed to meshing inaccuracies. 
Attachment 1: 8_1_14.pdf


124

Thu Jan 18 21:13:59 2018 
aaron  Optics  PonderSqueeze  modifications to Gautam's 40m finesse model  I made a copy of Gautam's 40m model to add the unstable filter cavity for the ponderomotive squeezing project. I wanted to make a more explanatory record of the changes I've made because I think some of them might be necessary for other scripts using gautam's original model, but I have not implemented them in that file (also just for my own paper trail).
Changes:
 swapped the role of nXBS and nYBS; before I think it was sending the reflected beam at the main BS to the X arm, and the transmitted beam to the Y arm
 I kept nPOY, but I am a bit confused by itthis is the beam returning from the xarm that is transmitted in to the BS, but it isn't followed out of the BS; rather it seems this pick off beam is detected inside the BS substrate, which is odd. Anyway we aren't using it for now.
 Changed some labels to distinguish between the beamsplitter already in the IFO (IBS) and the BS used in the quantum phase compensator (QBS)
 Added a quantum phase compensator cavity, consisting of QP1 and QP2, as well as a mirror (QP0) to direct light transmitted from the QPC through QBS back to QBS

100

Sat Sep 5 15:04:43 2015 
Dennis Coyne  Mechanics  Configuration  summary of FEA modal model to State Space model  At the 2014 commissioning workshop, I presented a summary of my efforts in converting finite element modal models into state space models:
https://dcc.ligo.org/LIGOG1400099
I also provided the attached written summary for a report on the workshop that Aidan Brooks and Gabriele Vajente are preparing. 
Attachment 1: FEM2SS_summary.docx

105

Tue Jun 28 15:50:45 2016 
Joy Westland  Mechanics  Analysis  A Simple Model of the Modal Analysis of a Cantilever Circular/Cylinder Cross Section in ANSYS tutorial  Here is a tutorial to implement a simple Modal Analysis of a Cantilever Cylinder
1. Open the ANSYS workbench
2. Drop and drag the "Modal" analysis system into the project schematic
3. Right click on "Engineering Data" and edit the material. There are predetermined properties found in the "Engineering Data Sources" (Right click on the description or press the books in the top right hand corner). For this tutorial, fused silica was created with these parameters:
 Young's Modulus E: 7.2E10 Nm^2 (Isotropic Elasticity)
 Density p: 2200 kgm^3 (Density)
 Specific Heat c: 770 J/kgK (Specific Heat)
 Isotropic Thermal Conductivity k: 1.38 W/mk (Isotropic Thermal Conductivity)
 Coefficient of Thermal Expansion: 3.9E7 (Isotropic Secant Coefficient of Thermal Expansion)
 Reference Temperature: 21 C (Isotropic Secant Coefficient of Thermal Expansion)
 Poisson's Ratio: 0.17 (Isotropic Elasticity)
In order to put in the parameters, just drag and drop from the toolbox. Also make sure the filter is turned off (that's the filter symbol next to the engineering data sources/books in the top right hand corner) this will allow a user to few all the parameters.
4. Return to the project, and open/edit the geometry. The cantilever that was used for this tutorial is a cylinder with a diameter of 2 mm with a length of 200 mm, shown in image 1.
 How to Create a geometry:
 Open up the geometry
 Decide which plane you want to start drawing in: XY Plane, ZX Plane, YZ Plane
 Click on "new sketch" in the 4th row from the top after "Sketch1" it's blue sketch
 Whatever plane you decide to choose, press the "Look at Face/Plane/Sketch" icon in the top right hand corner (The person looking at a face)
 This will allow the view to go to the view where you want to sketch on
 Click on "Sketching" which is next to the "Modeling" right above the "Details View"
 Here will be Several Options
 Draw: This is where you can decide what shape you want to use
 Modify: This is the extras such as fillets, chamfers and corners
 Dimensions: This is where you can select what type of dimensions you will be using
 Constraints: This is where you can constrain lines or connect lines
 When you want to connect lines together, click on each end and make them "Coincident" this way the shape will connect when you change dimensions
 Settings: This is where you can show a grid or snap grid
 Sketch Color:
 Teal: Underdefined: The sketch does not have enough dimensions to constrain and tell the program what exactly the shape is
 Blue: Defined: The sketch is fully defined and does not need any more dimensions
 Red: Overdefined: The sketch has repeating dimensions that causes errors
 For this tutorial, draw a circle. I made mine coincident to the middle by pressing the vertex in the middle and one of the axis and then repeat the process with the second axis to center the circle
 Dimensions: The diameter is 2 mm, under dimensions to display the number, click "Display" and then "Value" instead of "name"
 To change the dimensions, the "Details View" in the bottom left hand corner will say D1 and from there you can change the diameter
 From there in order to create the length of the cylinder, press "Extrude" at the top (4th row) towards the right. From there it will ask what geometry, click on your sketch and click "Apply". Make sure the "Operation" is "Add Material" to Extrude.
 Then choose a "Depth" in this case it's 200 mm. Then click "Generate" the lightning bolt next to the "New Sketch" symbol.
 Your cylinder should look like image 1.
5. Return to the project and open the Model tab. With that open, go to the subtab of "Geometry", named "Solid". Under Solid go to "Assignment" under "Material" to assign Fused Silica or whatever material you decide to choose.
6. Go to the "Mesh" tab. In this case the "Element Size" under the "Sizing" tab is 0.04 and the "Relevance Center" as "Fine". Right click on the mesh and press "Generate mesh".
7. Under the Modal tab, go to the Analysis Settings and change the "Max Modes to Find" for ANSYS to calculate. In this tutorial, the amount of modes that were used was 17.
8. Right Click the Modal and press on "Fixed Support". This will make the bar a cantilever bar, once the geometry is set to fix one face of the bar. Press on one of the faces and click on "apply" on the Geometry.
9. Right click on the solution and press "Solve". Let ANSYS run the modes through.
10. On the Solution tab, the "Tabular Data" is listed but the Total Deformation has not been listed. In order to do so, select all the frequency of the modes in the column and right click and press "Create Mode Shape Results".
11. Once loaded, the total deformation will have a lightning bolt next to each entry, right click and press "Solve" or "Evaluate All Results". This will make all the entries have a green check mark.
12. This will give the user the ability to animate each entry. This is down by clicking the play (sideways triangle button). ANSYS will run through the simulation for that mode that is selected, shown in image 2.
13. Using Mathematica (or another computational system) input the analytical solution for a Cantilever bar fixed to one end.
 w_{n}=(k_{n}L)^{2}Sqrt(EI/mL^{4})
 Where:
 w_{n} is the frequency measured in radians/second
 E = Young's Modulus
 I = moment of cross section
 I_{rectangle }= ba^{3}/12 (b and a are the sides)
 I_{circle} = (Pi/64)(d^{4}) (d is the diameter)
 m = mass per unit length
 L = length of the bar
 k_{n} relates to the amount of nodes
 k_{1}L = 1.875, k_{2}L = 4.694, k_{3}L = 7.855, k_{4}L = 10.996, k_{5}L = 14.137
 n greater than 5: k_{n}L = (2n 1)(Pi/2)
 Convert w_{n} to f measured in Hz
14. Compare only to the bar actually bending, not twisting or contracting. There are modes that are the same due to the symmetry of the bar. In image 3 the underlined frequencies compare to the analytical calculations (Mathematica) and the computational calculations (ANSYS). 
Attachment 1: Simple_Cantilever_200_mm_Long_Geometry.PNG


Attachment 2: Simple_Cantilever_200_mm_Long.PNG


Attachment 3: Comparison_between_ANSYS_and_Calculations_Fused_Silica_200_mm_L.PNG


106

Tue Jun 28 16:54:23 2016 
Nikhil Mathur  Mechanics  Analysis  Ansys 14.5 Indroductory Tutorial: Modal Frequency Convergence  This tutorial will go through how to show frequency convergence for a cylindrical cantilever using Ansys 14.5 and Mathematica. 
Attachment 1: ConvergenceTutorial.pdf.gz

107

Wed Jun 29 14:14:44 2016 
Joy Westland  Mechanics  Analysis  ANSYS Tutorials with Basic Meshing  Here are a series of tutorials for basic meshing principles from ANSYS Meshing Basics:
 ANSYS Meshing Fine Mesh Basic Tutorial 1: https://www.youtube.com/watch?v=sZIX3CJkWBE
 ANSYS Meshing Method Basic Tutorial 2: https://www.youtube.com/watch?v=ATSnvFUbEk4
 ANSYS Meshing Refinement Mesh Basic Tutorial 3: https://www.youtube.com/watch?v=QVfJiydCu2g
 ANSYS Meshing Sizing Mesh Basic Tutorial 4: https://www.youtube.com/watch?v=ODJ11YDC6ac
 ANSYS Meshing Spheres of Influence Basic Tutorial 5: https://www.youtube.com/watch?v=AQlGccK_X0
 Body of Influnce 6: https://www.youtube.com/watch?v=3E1p1w32jt0
 Section Plane 7: https://www.youtube.com/watch?v=hhZOJSESJEU
 Mapped and Match Control 8: https://www.youtube.com/watch?v=3pogywR8Vgw
 MultiZone and Inflation 9: https://www.youtube.com/watch?v=vxVBvANHuB0
 Sizing Soft/Hard 10: https://www.youtube.com/watch?v=D8UzdC5Gmhs

108

Fri Jul 29 14:33:41 2016 
Joy Westland  Mechanics  General  A Tutorial in Importing SolidWorks Files and Applying a Gaussian Force in ANSYS for a LIGO Test Mass  Here is a tutorial in importing SolidWorks into ANSYS and the steps needed to apply a Gaussian Force to the LIGO test mass that’s imported.
Using SolidWorks:
 Download the SolidWorks Zip Folders from the DCC
 https://dcc.ligo.org/login/index.shtml?entityID=https%3A%2F%2Fdcc.ligo.org%2Fshibbolethsp&return=https%3A%2F%2Fdcc.ligo.org%2FShibboleth.sso%2FLogin%3FSAMLDS%3D1%26target%3Dss%253Amem%253A15b6c314d87e3fa8b3768d89cb6b9836fe39c754 (LIGOD1000760v4)
 Open/export the zip folders into a different file
 For only the test mass and ears, open the file called: D0902456 ITM OPTIC WITH EARS ASSEMBLY. This will represent the bottom mass.
 An assembly can be imported into ANSYS but it’s easier to convert an assembly into a part before importing it into ANSYS. Also this will allow you to perform a split line.
 To do this, resave the assembly but before pressing save, change the file to PRT extension instead of Assembly.
 Open the PRT file of the test mass, make a new sketch on the face of the test mass.
 Draw a circle (Or whatever shape) on the test mass. In this case a 0.1 m circle was made in the middle of the test mass.
 From there go to “Insert” and then press “Curve” and then go to “Split Line”. Split line allows a user to project the sketch onto the part. This makes it possible to press the sketch separately from the part. This is a useful tool because in ANSYS, once split line has been used on a sketch, a user can press that shape separately on the face of the object. The importance of this tool allows a user to click on the circle separately and apply a force in that particular area.
 In “Split Line” press the “sketch” you want to project onto the “face” of the object.
Importing SolidWorks into ANSYS:
 Once the part has been made, choose the “static structural analysis”.
 In the geometry: Import the test mass that was converted into a part
 Once imported, press the geometry again and “edit”
 In the edit module, “generate” the part and once the part has loaded notice the circle can be pressed such that it is a different part of the test mass surface.
 Exit geometry module
Applying Gaussian Force:
 Enter the “model” module to edit the setup of the analysis
 Assign the material that you want for the test mass under “Geometry” à “Solid” under the “Material” and “Assignment”
 Under the “Coordinate Systems”, make a new coordinate system
 “Coordinate System:
 Click on the face, in this case the circle made on the test mass for the “geometry” and “apply”
 Under “Definition” à “Type” à Change from Cartesian to Cylindrical. This allows the Gaussian Force to be distributed correctly
 Under “Analysis Settings” Turn on “Large Deflection”
 Add “gravity” under “Static Structural”
 Treat the test mass is if it were the bottom mass so make sure gravity is pointing the correct way, such that the wires on the ears would extend upward.
 Insert “Fix” to the top surfaces of the ears
 Insert “Pressure”
 Press on the circle made to apply for the pressure
 Under “Definition” change the “Magnitude” to “Function” (the arrow at the end of the Magnitude entry bar)
 Once function has been activated: A Gaussian force of: 1/((3.141592654*0.0156^2)*2.718281828^((x/0.0156)^2)) was used in SI units. Note that ANSYS does not use symbols. Once that’s entered into the “Magnitude”, under “Function” the “Coordinate System” will appear. Change that to the coordinate system that was made in step 4 of the cylindrical system.
 Under “Graph Controls” Make sure that the XAxis is changed from Time to X.
 Pick a range for the graph and the number of segments that you want to look at.
 “Solve” the system and right click on “Solution” and evaluate the different results you need.

Attachment 1: Bottom_Test_Mass_With_Ears.PNG


Attachment 2: Applying_Parameters.PNG


Attachment 3: Applying_Gaussian_Force.PNG


115

Thu Nov 2 17:23:56 2017 
Aaron  Mechanics  PonderSqueeze  Modelling suspension noise  aLIGO Suspensions Toy Model
On Wednesday I started making my own model of the aLIGO suspensions, with the top of the silica fibers attached to ears that are fixed rather attached to an additional suspension stage (so this will be a one stage suspension).
I grabbed the aLIGO ear design from the DCC: LIGOD080751v4
I am almost done with the model, should have it working tomorrow and will add it to the experimental gravity github in an appropriate place. 
116

Fri Nov 3 15:03:10 2017 
Aaron  Mechanics  PonderSqueeze  Modelling suspension noise  Model Geometry
Test Mass
I found the dimension of the test mass flat in the drawings of the mock test mass design here: LIGOD080687.
Fibers
I modelled the fibers with the profile described in LIGOD080751, fig 3.7.
Ears
I grabbed the dvalues from LIGOT1000545, but since the dvalue is defined as the distance from the center of mass (of the penultimate mass (PUM) or the E/ITM) to the bend point (BP) of the fiber (I believe the point on the fiber with maximal flexure in the fundamental mode), I did not go through the effort of figuring out where the bend point is but rather grabbed the hornCM distance from LIGOT1100407
I wanted to get the real aLIGO parameters for the first version of this model, and have parametrized the model in such a way that I can define all of the parameters that need to change (surface area of the earTM bond, length of the fibers, thickness and profile of fibers, dvalue, etc) and scale them with mass in some way for future iterations on this design.
I need to pare down the number of parameters, because I started by fully defining the ears and now am importing a 3D model of the ears and planning to scale these with mass.
Materials
For the material of the entire test mass and suspension, I used the fused silica that is specified as [solid,NIST SRM 739  Type I]. I wasn't sure the difference between the types of silica, but this one said SRM so I thought it might have been defined on my distribution of COMSOL by a LIGO person. A quick google search showed me that person may have been rana?? https://labcit.ligo.caltech.edu/~rana/research/etm.html
Physics
I'm using a solid mechanics model.
Fixed Boundary Constraint
I fixed the position of the bonding surfaces for the PUM ears, so it is as if they are contacted to a completely fixed PUM (the PUM is not included in the model, but the upper ears are included, so the constraint is on the ears not the fiber. See drawing).
Gravity
I added gravity to all parts of the model. Apparently, it is not trivial to use gravity in a frequency domain study in COMSOL, as described in this presentation here. Fortunately, the presentation in the link is interested in the transfer function for a mass on a string also, so I follow the simulation steps they describe below.
Boundary Load
I add a boundary load that will vary sinusoidally for the frequency domain study.
Mesh
I have not yet messed with the meshing for these models. Obviously the points with more flexture and smaller parts (like at the horns of the ears, the tapering parts of the fibers, etc) will require a finer mesh.
Study
I need to incorporate the advice on how to build this study described in the link above. The following might also be useful, though I haven't looked through them yet:
https://www.comsol.com/model/dynamicsofdoublependulum14021
https://www.comsol.se/forum/thread/4843/pendulumresponse?last=20100427T01:48:26Z 
117

Wed Nov 15 14:05:12 2017 
Aaron  Mechanics  PonderSqueeze  Modelling suspension noise  Model Geometry
I pared down the number of parameters in the model to only the necessary ones. These are the ones that remain:
TM_radius: Radius of the test mass
TM_width: Width of the test mass
TM_flats: length of TM flats
ear_length: length of the ear
horn_spacing: length of the ear
horn_gap: gap between the top of the horn and the TM on the near side
d_val: distance from the CM to the bend point
horn_BP: distance from the horn to the bend point
ear_height_tot_nominal: nominal total height of the ear and horn for the unscaled (aLIGO) design (this name made more sense in a previous version of the model)
fiber_stock_length: length of the fiber's stock
fiber_neck_length: length of the fiber's neck
fiber_thick_length: length of the thick section of the fiber
fiber_main_length: length of the main section of fiber (the thinnest part)
fiber_taper_length: length of the tapering section of fiber
fiber_stock_radius: radius of the fiber stock
fiber_thick_radius: radius of the thick section of fiber
fiber_main_radius: radius of the main section of fiber
F_load: force of the boundary load used for the excitation in the frequency domain study
ear_scale_height: scale the height of the ear
ear_scale_length: scale the length of the ear
ear_scale_width: scale the width of the ear
Materials
For the material of the entire test mass and suspension, I used the fused silica that is specified as [solid,NIST SRM 739  Type I]. I wasn't sure the difference between the types of silica, but this one said SRM so I thought it might have been defined on my distribution of COMSOL by a LIGO person. A quick google search showed me that person may have been rana?? https://labcit.ligo.caltech.edu/~rana/research/etm.html
Physics
Rana suggests that for the purpose of this study, it is not necessary to actually have COMSOL handle gravity as a restoring force... I'm not sure if I understand why this is yet? It seems that if we are interested in the relative strain energy in different parts of the wire compared to other parts of the system, it is important that the wire be under tension. If we have no gravity, the wire is effectively not under tension.
Mesh
I have not yet messed with the meshing for these models. Obviously the points with more flexture and smaller parts (like at the horns of the ears, the tapering parts of the fibers, etc) will require a finer mesh.
Study
I need to incorporate the advice on how to build this study described in the link above. The following might also be useful, though I haven't looked through them yet:
https://www.comsol.com/model/dynamicsofdoublependulum14021
https://www.comsol.se/forum/thread/4843/pendulumresponse?last=20100427T01:48:26Z 
118

Mon Dec 4 16:27:13 2017 
aaron  Mechanics  PonderSqueeze   Meshing Surface Layers
Defining New Selections
I don't know why I wasn't seeing this problem with previous models (perhaps because I wasn't importing any CAD or STEP files), but my latest attempts at meshing and selecting specific domains of my model were being thwarted by inconsistent domain definitions. I was previously always manually selecting domains, which is confusing because all domains just get assigned a number when they are created. Worse, sometimes the numbers assigned to domains change when the model rebuilds, especially if there has been a significant change in the model geometry. This results in later steps selecting the wrong set of domains (or boundaries, etc).
To fix this problem, I created new selections (sets of domains, boundaries, or etc that receive their own label and can be selected as a group later in the model). The new selections include:
Domain selections that separate surface and bulk layer for all domains in the fibers (fiber stock, neck, thick section, taper, and main section, in order from the horn to the center of the fiber)
Boundary selections for all domain selections described above
This is probably also a necessary step for getting reliable results when interfacing with MATLAB, and might explain some weird problems I was running in to a while ago and just made haphazard fixes for.
Mesh Steps
I use the following mesh steps to get what seems like a pretty reliable meshing:
 Mesh the upper tapers (bulk and surface separately) with a free tetrahedral mesh
 I mostly use the defaults for an extremely coarse mesh, but the only parameter that seems to make a large difference is the minimum mesh size. I set this to the skin_depth for the surface layer, and (fiber_main_radiusskin_depth) for the bulk. fiber_main_radiusskin_depth should be the radius of the bulk domain, and the skin_depth characterizes the smallest length scale in the surface layers. I have some limited ability to tweak the minimum mesh sizes when the surface layer is comparable in size to the fiber_main_radius (so the surface layer is comparable to the entire fiber radius), but it seems to be best to keep the mesh this fine when the surface layer becomes small.
 Mesh the main fiber (bulk and surface separately) with a swept mesh
 I create a distribution with a fixed number of elements, at ceil(fiber_main_length/fiber_stock_radius/2). This is somewhat arbitrarythe stock has the largest radial length scale, so I figured I'd divide up the main fiber into units that tall. A better thing would be to know how high a mode we are interested in studying, and break up the fiber into enough pieces to observe that mode. Seems fine for now, might want this distribution to be a bit coarser though.
 Mesh the lower tapers (bulk and surface separately) with a free tetrahedral mesh
 Use the same size settings as on the upper tapers
 Mesh the thick sections (bulk and surface separately) with a swept mesh
 The distribution uses a fixed number of ceil(fiber_thick_length/fiber_stock_radius/2) elements. Again perhaps this can be coarser; it also doesn't attempt to make a finer mesh at the thermoelastic cancellation region.
 Mesh the necks (bulk and surface separately) with a free tetrahedral mesh.
 I use an extremely coarse mesh with the minimum mesh element size set to fiber_thick_radiusskin_depth for the bulk; for the surface it is an extra coarse mesh instead, because the extremely coarse mesh gave low quality mesh elements. I'm not sure why this is, I can't see much difference and the problem only seems to happen to one of the 8 neck sections (why not in all sections?). The problem only arises when the skin depth is less than 20um, for the other parameters at the nominal aLIGO values.
 Mesh the stocks (bulk and surface separately) with a swept mesh
 Distribution is again ceil(fiber_stock_length/fiber_stock_radius/2), resulting in 2 vertical divisions of the stock.
 Mesh the horns with a free tetrahedral mesh
 Size is an extremely coarse mesh, where the minimum element size is set to the skin_depth (because the horn sees the boundary of the stock surface, so its smallest adjacent element can have a scale down to the skin_depth), and maximum growth rate is increased to 9 (any higher results in low quality mesh elements; I set it high because most of the horn does not need to be meshed as finely as the part directly in contact with the fiber stock).
Note on the skin depth: I defined the thickness of the surface layer by the parameter skin_depth. Since we are interested in how the energy in the surface layer changes with the radius of the fiber, and expect that the surface depth doesn't change much with fiber radius (a very thick fiber will have a few micron surface layer; so will a fiber half that diameter). I managed to get the mesh working with a 15um surface layer, but was having trouble getting a 10um layer. I wasn't completely sure how small a surface layer would be necessary, but for a test mass 1/100 the size of the aLIGO masses, the main part of the fiber would have a radius of 20um, so if we want to study the surface layer for masses down to that size I figure the skin depth should be at most around 1015um. It would be better to be motivated by the actual scale of the physics going on at the surfacehow deep to micro cracks and other lossy imperfections go?
Study
I'm running the frequency study on a small number (2) of frequencies in each decade from 60Hz to 10kHz (so there should be 58 total frequencies in the study). I started it at 4:25pm on my local machine, and it hasn't gotten very far in 30 minutes, so I may abort the study and try to make the mesh coarserespecially the distribution settings, which are easy to change. 
119

Mon Dec 4 17:42:53 2017 
gautam  Mechanics  PonderSqueeze  FEA on optimus  We could run the simulations on the 32 core machine in the 40m lab (optimus)? I think Mariia was running some of her studies on optimus, and even though we had some problems with the licensing initially, I think she resolved these and has detailed the procedure in her elogs...
Quote: 
Study
I'm running the frequency study on a small number (2) of frequencies in each decade from 60Hz to 10kHz (so there should be 58 total frequencies in the study). I started it at 4:25pm on my local machine, and it hasn't gotten very far in 30 minutes, so I may abort the study and try to make the mesh coarserespecially the distribution settings, which are easy to change.


120

Mon Dec 4 19:49:32 2017 
gautam  Mechanics  PonderSqueeze  FEA on optimus  It's a good idea, I'll check out her elogs and get it started tonight.
I found that I had the relative tolerance set too low (0.001, while the iterations were not converging further than 0.02 or so); I changed it to 0.1, which let me run over a few modes relatively quickly once I reduced the number of sections on the main part of the fiber to 10. This is not sufficientat 600Hz the fiber looks completely jagged because the mesh is not fine enough, so using optimus will be necessary.
Quote: 
We could run the simulations on the 32 core machine in the 40m lab (optimus)? I think Mariia was running some of her studies on optimus, and even though we had some problems with the licensing initially, I think she resolved these and has detailed the procedure in her elogs...


121

Tue Dec 5 10:50:54 2017 
aaron  Mechanics  PonderSqueeze  FEA on optimus  I had some trouble running this on optimus.
Optimus has COMSOL 5.1 installed, but I made these files in 5.3. I downloaded the comsol 5.3 dvd.iso file last night, but on install I'm now getting the error "No locks available." I wasn't sure if this is a file permissions issue (sometimes the file has been 'locked' when it is saved and I open it in the GUI), an svn issue (most of the information I find online is from people running into some problem with lockd or their NFS), or a licensing issue (I don't think this is likely, COMSOL on my laptop shows more licenses available).
Has anyone seen this? Do I need to do something else with /cvs after install? 
122

Tue Dec 5 19:50:47 2017 
aaron  Mechanics  PonderSqueeze  FEA on optimus  Gautam advised me against trying to install version 5.3, lest it break version 5.1I had already gone through the install, but looking at the install manual it says it shouldn't affect previous installs except that the default behavior when double clicking .mph files will always choose the latest version of COMSOL. Since we mostly run on the command line we should be fine. That said I haven't tested files with COMSOL 5.1.
Zach pointed me to sandbox1.ligo.caltech.edu at the lunch meeting, as well as these notes from Rana about running comsol on a remote server. I couldn't run the file myself on sandbox1 because I don't yet have a home directory there, but I asked Larry if he could set one up for me so I can use the clusters. Gautam helped me run them on sandbox1 by having me scp the file onto the shared 40m drive, then he moved it in to his home directory on sandbox1 and ran it. A version of the model asking for an analysis at 60Hz and 600Hz ran in 1020 min, and looked good (I was able to scp the output to my laptop and open it in the gui, though I have a script not quite to the point where Matlab can do this for me, it's not ready yet and anyway I wanted to directly see that the output was normal). I modified the study to take 100 steps between 60Hz and 5kHz, and Gautam has now started that run, which if it scales linearly (hopefully better than linear, since it won't have to remesh and etc) will take about 816 hours.
Thank you Gautam for the extended mattermost session helping me run these! 
123

Tue Dec 12 11:50:12 2017 
aaron  Mechanics  PonderSqueeze  FEA on optimus  Simulation results
First run
Gautam ran the COMSOL model on sandbox1 since we were trying to run it before I had a home directory there to run from my login. Since that first run, Larry set me up on sandbox1 so I was able to run a few more times with some tweaked model parameters.
Here are the results from the first run, which uses the nominal aLIGO test mass parameters:
I don't think I was expecting such a regular pattern here. Are the modes really that closely and regularly spaced? This might also be indicating some problem with my meshing, I could imagine a few scenarios. To go from this chart to something that looks like the suspension thermal noise plot, I'll have to scale this by the frequencydependent loss angle and a strain sensitivity TF probably? I need to remind myself for another few minutes on this, maybe chat about it.
Relevant properties of the file that might need to be tweaked:
 F_load
 The applied force may have been resulting in large displacements of the test mass (and thus large curvature in the fibers). Since the model contains geometric nonlinearities (I think it has to, no?), large displacements may not be able to be linearly extrapolated back to the small displacements we expect. Since I am running a frequency domain study, COMSOL expects a load force, which means I am specifying a force rather than a displacement. In future runs, I wanted to reduce the displacement by scaling down the force... I'll discuss my first attempt at this later.
 skin_depth
 Gabriele mentioned that when he models surface losses for wafers he does not explicitly mesh a boundary layer, but rather asks COMSOL for a boundary integral (so just the 2D integral on the outer surface of the wafer), and then presumably makes some assumption about the thickness of this layer to go from J/m to J if need be. In contrast, my suspension thermal noise model defines the 'surface' of the fiber as the 15um outer layer, and I explicitly mesh that surface volume and do a 3D (volume) integral over the surface domain.
 Using a 2D integral would significantly simplify the model and probably run faster and result in fewer bugs when the scale of the geometry changes the necessary fineness of the mesh. However, I am a bit uncertain about whether this is what we want. At least in my understanding, we are interested in asking about what happens when the surface layer depth becomes comparable to the radius of the fiber. Since the 2D integral would always represent an infinitesimal volume, and the geometry of an extended (deep) surface layer would differ significantly from that of an infinitesimal shell, so I am concerned that the strain energy density would vary significantly with the radial postition in the fiber. Nonetheless, even the mesh I am using now has only a few (24) mesh elements along the radial direction, so I'm not sure I'm doing much better even with the additional computation time. Therefore, in future runs I'd like to try Gabriele's suggestion.
 Frequencies
 I was using 50Hz spacing on the frequencies from 60Hz to 5kHz. I haven't been very systematic about this, but I'm getting some convergence issues going to lower frequenciesmaybe by taking the suggestion for eliminating the skin_depth I can make a finer mesh and go to lower frequencies. This frequency sweep is too coarse to find welldefined resonances, but they are suggested in the plot below.
More recent runs
I wanted to automate this loop over mass parameters in a matlab script, so I set up the Livelink handshake so Matlab would send the model to sandbox1 for solving, then MATLAB on my machine would work with the solved model to extract results. I realized later during running that this might not be optimal, since it will require me to keep the connection to the remote COMSOL server during the entire run, which is A. annoying and B. risky because these might take many hours to run and the connection can easily be severed, wrecking the job. I don't really have a workaround for this, so my current plan is to continue logging on to the remote server and running individual COMSOL files. As I'll discuss, this is probably necessary for now anyway since I ran in to some problems running these models, and might want to tweak the models on my local GUI before running them over a large (frequency) parameter space on sandbox1.
aLIGO parameters, but decrease F_load
Based on the plots generated from the first run, I estimated that the TM displacement was comparable to the TM thickness; just as a first pass I figured 'small displacement' could be easily defined as 'displacements smaller than the thickness of the fiber' or maybe 'smaller than the thickness of the surface layer'. If the displacement is directly proportional to the force, this meant I wanted to scale the previous F_load by 1e5, which I did. I wanted to just see what kinds of displacements I would get, so I asked to run the model only at 300Hz. After more than 45 min of running, the server threw an error that it couldn't find my STEP file for the horns, which were in a local directory but not in the remote (server's) directory.
I thought this was an odd error for two reasons. First, I had originally not even had the STEP in my local directory (it only needed to load once and then could be used many times for a given model in the GUI), and I was getting an error within a couple minutes of starting the job that it could not find the file. Adding the file to my local directory seemed to solve this problem, as the model was running for much longer. The error I got after it finally crashed was not the same error as before, but was still an error in loading the file, which makes me a bit confused about where it is actually looking for this file or when exactly the loading process happens. To solve this, I have copied the STEP to the remote directory and will run again with the import pointing to that remote file, which I suspect will solve the problem.
aLIGO with the mass scaled by 0.1, also decrease F_load by 1e5*0.1
Since the first run seemed to mostly work up to the outstanding questions mentioned above, I decided to also run the model in batch mode directly from the sandbox1 command line, just as Gautam had. That is, I ssh on to sandbox1 and run the command
$ /usr/local/comsol53/multiphysics/bin/glnxa64/comsol batch inputfile [inputname].mph outputfile [outputname].mph
For this run, I scaled the mass of the model by 0.1, so we now have a 4kg TM. Most lengths of the model then scale by 0.1^(1/3), but those related to the radius of the fibers (and the bend point of the fibers) scale by 0.1^(1/2) because we want to maintain a constaint stress in the fiber. The main length of the fiber remained the same so the modes would be in about the same place. I document these more thoroughly in the matlab script, which I should upload to the git. I scale the force by 1e5 as above (same back of the envelope calculation), with an additional factor of 0.1 to account for the lower mass.
The model ran in to some convergence issues after a few frequencies. I could solve this by changing the relative tolerance, but I think I will most likely instead pursue Gabriele's suggestion above and try to refine the mesh to improve the convergence. The solver nodes of the COMSOL models are still a bit mysterious to meI don't really know much about when different methods or measures of convergence are appropriate. Probably playing with these could both improve the accuracy and efficiency of the model... but it seems like a hefty undertaking. Might be worht the long term payoff though.
The first few frequencies did converge, but as I was extracting the resuls ancha, the COMSOL license server, went down so I'll have to extract them later.
NOTE TO SELF: WHAT'S WITH THE INLINE IMAGE QUALITY? DON't USE JPG !!!! 
125

Mon Jan 22 21:13:25 2018 
aaron  Mechanics  PonderSqueeze  tips from Shoaib  I talked with Shoaib about some changes I could make to the FEA model to improve convergence and reduce memory usage. Summary:
 use a hex mesh rather than tetrahedral
 Use more structured meshes. In particular, I can make an angled swept the mesh in the tapered portions rather than using a free mesh in these regions, defining the mesh only on one boundary
 Use a nonconformal mesh, so adjacent domains do not need to have matching meshes. This could allow transitions to coarser meshes as the fiber becomes thicker or contacts the horn
 try using curved elements so the curvature isn't just approximated by the number of regular mesh elements
 Might try scaling the model uniformly by some factor (100, 1000), which could avoid machine precision issues
I'll get these implemented this week and see if the computation goes through. 
126

Mon Jan 29 23:02:13 2018 
aaron  Mechanics  PonderSqueeze  tips from Shoaib  I started implemented some of these changes:
 Started the mesh with a boundary free quad mesh on the interface between the upper tapers and the main part of the fiber. I used the following size setting
 Maximum element size is fiber_taper_length, which I felt was a good characteristic maximum because it wouldn't make sense to have radial elements larger than the length of the taper itself. This setting does not limit the mesh (elements are much smaller than the maximum)
 Minimum element size: skin_depth/10, where I use the thickness of the surface layer as the characteristic length, and allow a mesh with elements substantially smaller than this characteristic length.
 Maximum element growth rate I set at 8, which from some trial and error isn't strongly affecting the mesh as long as it isn't brought much lower than this (so the mesh is allowed to get coarse relatively quickly, though other settings limit this)
 Curvature factor is 0.7, and actually does affect the mesh strongly; for now value seems to give a 'pretty' (symmetric to the eye) mesh.
 The resolution of narrow regions is set to 5, where this value again strongly affects the mesh and this value was chosen because it looks 'pretty'
 Note that in contrast to the previous way I had defined the mesh, I now am defining the surface and bulk of the mesh in the same step, which seems to help these two regions match up. Shoaib had mentioned using a nonconformal mesh, which I may end up implementing in a later update.
 Instead of doing a free mesh in each noncylindrical region, I continued with a swept mesh along the length of the fiber (still sweeping in sections, though I'm not sure this is necessary)
 I also slightly increased the number of mesh elements along the length of the fibers (ie, used more elements in each section of the mesh sweeping).
 In this model, I still have a free tet mesh on the horn and test mass, since those seemed a bit more involved to move over to a hex mesh (there is no free hex mesh in COMSOL that I could see right away, and I'm not sure that the horn can be meshed with a swept mesh due to its irregularity. I'll look in to this further, but I had no immediate solution for making a hex mesh in the horn) and anyway they are much coarser than the fibers so I don't expect them to cause much trouble.
These changes were fine with a 10um mesh, but ended up with much too fine a mesh (~80000 boundary mesh elements in the initial mesh between the taper and main fiber) due to meshing the surface and boundary in a single stage in the first step. I separate them out to get a coarser mesh in the bulk, and also make the resolution a bit coarser. 
129

Sun Aug 26 16:42:54 2018 
rana  Mechanics  Analysis  Test Mass Thermal Noise: Consistency Checks  For the Voyager test masses, we have been considering a barrel coating to increase the IR emissivity to increase the radiative cooling power. We also seek to estimate the added Brownian thermal noise that arises from this.
Dmitry Kopstov (from MSU) made a baseline model for this which we have been modifying. The latest is in the CryogenicLIGO git repo in the FEA directory as ETM_NASAbarrel.mph.
Comparison with Analytic Estimates:
Convergence Tests:

131

Thu Feb 14 12:38:51 2019 
Ching Pin  Mechanics   comsol modelling  So I did a simple comsol model of laser heating of a silicon disk, with only radiation, to see the temperature variation at steady state, which could be the limiting factor for high Q at 123 K, due to the thermalelastic effect.
The model just uses a simple 2 inch disc, at 0.028 cm thick, with the flats not incorporated in yet.
I had to search for silicon thermal conductivity and heat capacity at low temperatures, settling with k= 800 W/(m K) and C_p= 300 j/(kg K) from refering to papers. Will check LIGO documents for more accurate versions.
I put an arbitary boundary condition of constant temperature of 123 K on a spot .2 mm in radius, to simulate a beam.
Other arbitary values include 77 K for ambient and a surface emissivity of 0.5.
The laser is off center, because that it where the laser will enter the current setup.
We can see that the power required is .02 W, which seems reasonable.
The model is consistent with the analytic model I made with the laser beam at the centre of the disc. See last two figures.
I'm still trying to get the time dependence to work, as it is just giving me nonsense right now.
Some thoughts: beam radius affects the temperature variation quite significantly, with a fat beam (1 mm radius) having half the temperature variation as a beam of .2 mm radius
I think the halo is just a trick of the eye, but I could be wrong.
Things to do:
Find the time scale of the system, as we want to modulate the laser to adjust the temperature, which will then be run though the mode ringer to measure Q to find the zero crossing
Change the heat source to be an actual laser
Add in the solid mechanics part
Add in the sapphire lens underneath 
Attachment 1: Screenshot_from_20190214_124711.png


Attachment 2: Screenshot_from_20190214_150740.png


Attachment 3: Screenshot_from_20190214_154141.png


Attachment 4: graph.pdf


132

Fri Feb 15 21:05:31 2019 
Ching Pin  Mechanics   comsol modelling 
So I got the time dependence to work, but I'm not sure what went wrong in the first time anyways. I'll trying to get a sense of how long it takes for the temerature to semiequlibrate, and coming to grips with comsol as a whole. There seems to be some inaccuracies when the timing increases, so I'm having to figure out how to increase accuracy without drastically increasing compute time. On the bright side, I switched the model to heating via deposited beam, for a more accurate model.
Attached is comsol's handling of a deposited beam modulated sinusoidally with a frequency of 0.1 Hz and screwing up badly. Y axis is the average surface tempurature across the whole disc.

Attachment 1: Screenshot_from_20190215_214001.png


133

Tue Feb 19 19:52:53 2019 
Ching Pin  Mechanics   comsol modelling  The time step response to heating via laser (22.5 mW) is given in the attached picture, for 2 starting temperatures, 122.5 K and 122.8 K. We see that it takes fairly long to equlibrate, with a time constant of about 500 s, and is consistent across both temps. The y axis is average temperature across the surface of the disc, and the x axis is time. I believe that the heat distribution profile would be very similar with time, simply because of how much faster conduction is compared to radiation 
Attachment 1: Screenshot_from_20190219_195210.png


134

Fri Mar 1 19:33:40 2019 
Ching Pin  Mechanics   comsol modelling  I've changed the heating to be from two heat sources, to better model the situation with a heater and a laser. The heater deposits 22 mW, with the laser deposting .5 mW. The overall temperature distribution is smaller then before, as expected, but doesn't really change much. The heater is simulated with a deposited beam with a gassian beam profile with a standard deviation (s.d.) of 8 mm. The laser to the size has a .3 mm s.d. for contrast. I learned that while the deposited beam power doesn't care for emisitivity, it cares about the area the beam is incident with, so for example, if you increase the s.d. too much, you get less power deposited then what you enter.
I've also got modes to appear using solid mechanics, and I'm trying to see if there is a good way to get the frequency dependence with temperature nicely simulated. Changing the parameters of the simulation does give me my different frequencies, but I trying to find a way to do that over the time evolution of the model. I also got to check if the frequency shift is in line with what we are measuring. 
Attachment 1: Screenshot_from_20190301_193532.png


Attachment 2: Screenshot_from_20190301_193602.png


Attachment 3: Screenshot_from_20190301_194625.png


135

Mon Mar 4 17:22:07 2019 
Ching Pin  Mechanics   comsol modelling  I've updated the material properties to vary with temperature, mainly in the range of 90140 K. Using the parametric sweep function to vary the input power of the heater, we get the eigenfreqencies' dependence on temperature to show up. The fractional dependence of 1.3e5 /K around 123 K matches with what Aaron calculated in this elog entry, which is always a good sign that nothing is going horribly wrong. I've also added the flats to the silicon disc, for better accuracy. See the screenshots below showing the frequency shift with temperature. 
Attachment 1: Screenshot_from_20190304_191328.png


Attachment 2: Screenshot_from_20190304_191441.png


Attachment 3: Screenshot_from_20190304_191537.png


Attachment 4: Screenshot_from_20190304_191556.png


136

Wed Mar 6 09:51:18 2019 
Ching Pin  Mechanics   comsol modelling  So I tried adding the sapphire lens to the comsol model, and I am having teething issues. I can't seem to get the solver to converge, but I'm working on it. 
Attachment 1: Screenshot_from_20190306_093124.png


