Motivation:
Reduction of Brownian thermal noise in future gravitational wave detectors is of significant interest. It has been suggested that changing the shape of the mirrors used may reduce the Brownian thermal noise. Before we can study how alterations in mirror shape effect Brownian thermal noise, we must be able to generate a finite element model which can compute the Brownian thermal noise in current mirror substrate so that the model may be tested against other calculations of that value using the fluctuationdissipation theorem.
Methods:
I began by constructing a model of a mirror design in COMSOL with the following parameters:
radius (R): 172.5 mm
height: 102 mm
Gaussian beam size (rBeam): 4 cm
TotalForce: 1 N (this is a placeholder force)
loge: log base 10 of e or about 0.4343
PressureCenter: TotalForce*(1/pi)*(rBeam^2)*(1e^(R^2/rBeam^2))^1
A cylinder was constructed using the radius and height above with its flat faces pointing in the z direction, and was given the properties of fused silica which COMSOL had built in.
Our first nondefault boundary condition was applying a force representing the laser beam onto one of the flat faces of the cylinder, surface 3, in the model. The force applied to this face was a Gaussian function PressureCenter*e^(sys2.r^2/rBeam^2) which was specified in COMSOL as a boundary load.
In order to keep the cylinder fixed in space, representing its being connected to wires supporting it and our ignoring violin modes, we need to apply another boundary condition. The simplest possibility would be to force surface 1 in the model, the face of the cylinder opposite the applied force, from moving, but page 5 of Liu and Thorne's paper demonstrates that we should instead apply a force of equal magnitude and opposite direction to that applied to surface 3, but distributed over the volume of the cylinder. To this end, we integrated the Gaussian force applied over the face of the mirror and determined that the total force applied was TotalForce=pi*PressureCenter(1e^(R^2/rBeam^2))rBeam^2/loge. It is more convenient to specify the total force than the pressure at the center of the mirror's face, so solving for PressureCenter we obtain PressureCenter=TotalForce*(loge/pi)*(rBeam^2)*(1e^(R^2/rBeam^2))^1. This opposing force was entered into COMSOL as a body load using the value TotalForce applied over the volume of the cylinder.
COMSOL was instructed to solve for a steady state given the above configuration and returned an error message that “The relative residual (19) is greater than the relative tolerance.” I increased the number of elements in the mesh and the analysis returned lower relative residuals (9.1) using a “finer” mesh. The computer being used did not have enough memory to use a finer mesh structure than that, but the lower relative residual indicates that using a finer mesh may solve the convergence problem.
Ideas for future work:
The simplest possibility is to perform the simulation again using a finer mesh on a computer with more memory and see if we can obtain a solution.
The function used to assign forces at the discrete points in the mesh is continuous, but the points are discrete. Using a discrete Gaussian function to determine the force at each point on the face may be worth trying.
Alternatively, the process could be handled by a Matlab script setup to first run a COMSOL simulation to determine the integrated force on the face of the cylinder, then normalize the central force based on that data such that the desired force is applied to the cylinder face. The script would then solve for the steady state solution.
We could also consider replacing the TotalForce applied over the cylinder's volume by a different boundary condition. Replacing it by a weak spring force on the back face of the mirror has been suggested. I think this is less likely to give good results than the above suggestions, but it may still be worth testing and seeing how its solution compares to values obtained from the fluctuationdissipation theorem.
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.
