Spacer in BR noise

== COMSOL vs result from Kessler etal 2012==

     The analytical result from kessler2012, assume the force acts on whole surface of the spacer (with bore hole), I check this with COMSOL by comparing the result, similar to what I did in PSL:1075. The result agrees well within 2%. This verifies that COMSOL model is correct

==thermal noise level vs annulus thickness==

Typically, the contact surface between the spacer and the mirror is only a thin annulus, see psl:1199  . And the noise level is dependent on the actual area of contact. So I run the simulation to see the dependent of the stored energy (U) vs the annulus thickness. The annulus thickness is about 2 mm +/- 0.2 mm. The displacement noise is proportional to sqrt(U).

fig1: The stored energy as calculated by COMSOL, fitted with cubic polynomial.

The error from the contact area, the simulation result are small ~3% and 2%. These are smaller than the uncertainty of loss in bulk fused silica (can be from 10^-6 to 10^-7). The effect is still small in the total noise.