Motivation: Before we can attempt to modify the mirror designs to reduce the thermal noise due to Brownian motion in them, we must verify that our model works and that its results match with the analytically calculated ones.
Methods: I had previously constructed a model of a cylindrical mirror in COMSOL: http://nodus.ligo.caltech.edu:8080/COMSOL/3. I updated the values used in the COMSOL model to match those from Levin's paper so that they would match the values I had already computed analytically. These values are listed below:
Kb=1.38065*10^23 (Boltzmann's constant)
T=300 (Temperature)
Sigma=0.16 (Poison's Ratio)
E0=71.8*10^9 (Young's Modulus)
r0=0.0156 (Gaussian Beam Size)
Phi=10^7 (Loss Angle)
RMirror=0.17 (Mirror Radius)
H=0.2 (Mirror Height)
Using the simplest of the boundary conditions I had attempted to implement, fixing the mirror face opposite the applied force, I ran a stationary solver on the model. After the solver had completed, I ran a volume integration of the strain energy in the mirror and obtained U_{max}=1.52887*10^10 J. I also ran a surface integral of the force applied to the surface to confirm that the total force, F_{0}, was the 1N that the COMSOL model had applied to the mirror's face, which it was.
Levin's equations 10 and 12 were combined to give S_{x}(f)=[(K_{b}*T) / (Pi*f)] * [U_{max} / F_{0}^2] * Phi
With the applied force of 1N and the value of U_{max}=1.52887*10^10 J, S_{x}(100 Hz)=2.0157*10^40 m^2 / Hz which agrees to within a factor of 4 with the results of the calculation based on Liu and Thorne's paper which gave a value of 7.80081*10^40 m^2 / Hz. A loglog plot of Sqrt[Sx[f]] with f ranging from 0.1 to 5000 Hz was plotted and is displayed below.
Future work:
The obvious next step to take in the project is to attempt to get the better boundary conditions to work, in particular the "gravitational" body load suggested by Liu and Thorne. We noted while working today that a much simpler case where we applied a force of 2N to one surface of a cylinder and an opposing force of 2N in the opposite direction did not converge. It may be worth working with this case and attempting to get it to converge in order to inform how we can make the more complicated case converge. If we can get that case to converge and it agrees with the analytic results, then we will be ready to start varying the relative sizes of the two mirror faces and determining the effect on thermal noise due to Brownian motion.
Levin, Y. (1998). Internal thermal noise in the LIGO test masses: A direct approach. Physical Review D, 57(2), 659.
Liu, Y. T., & Thorne, K. S. (2000). Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses. Physical Review D, 62(12), 122002.
