Liu and Thorne, in their paper have analytically solved for the TE noise spectra considering the test mass to be a half infinite space.
An attempt was made to get similar results using a COMSOL model. In this post, the rough profile of the PSD plot is given and is compared
to Liu and Thorne's result.
Steps followed by Liu and Thorne:
> Oscillating pressure, scaled according to beam profile, is applied one of the surfaces.
> The stress balance equation is solved borrowing algorithm from Landau Lifshitz.
> The stress and the temperature perturbation is related using an equation given in Chapter 1 of LL
> The W_diss is calculated from the gradient of the temperature.
> PSD is calculated using FDT due to Levin
> The resulting plot looks like the following - ***Only the frequency dependence is plotted here leaving out the prefactors which will
appear just as an additive constant in the figure. The aim was to see how the plot of the simulation looks like.
Steps followed in the COMSOL model:
> 2D axisymmetric model was used since the applied was applied radially symmetrically.
> The dimensions of the test mass(cylinder) was defined larger as compared to the beam spot radius.
> The physics added was Solid Stress and Heat Transfer in Solids.
> Thermal expansion was enabled in the Solid Stress physics
> Pressure work was enabled in Heat Transfer physics which takes into consideration heat due to stresses i.e. adds this heat source to the heat eqn
> Heat insulation wsa applied to all surfaces.
> All surfaces apart from the one where pressure was applied was held fixed to mimic infinity
> A custom mesh was put.
> The model was exported as '.m'
> It was called for frequency range 10^(0 : 0.25 : 4)
> Following Liu and Thorne, the temperature gradient was integrated over the volume using mphint2() function in Matlab
> The time average was taken over the second half of the cycle to avoid transient which are assumed to die off after a couple of initial cycles.
> All the prefactors were ignored while plotting
> The plot shown below is *integral{grad_T}^2} / omega^2 *which is proportional to PSD
> We just wanted to see if at all we got anything looking similar to the analytic result of Liu and Thorne (shown above)
The above plot is when the radius is 100 times the beam spot and height of the cylinder is 200 times the beam spot
Other values of radius and height were also tried. The one below shows similar plot as above but for radius 10 times
the beam spot and height 20 times the same
While the one below is for radius = 1000 * beam spot and height = 2000 * beam spot
Here, it is seen that the straight line with a negative slope is seen to some extent.
However, the prefactors, when put in, will show the actual PSD.
Any comments or suggestions related to the model construction in COMSOL or otherwise is welcome. |