FE model implementation of seismically driven GG noise in subterranean gravitational wave detectors David Rabeling, Eric Hennes, and Jo van den Brand

Gravitational gradient noise Figure by: M. Lorenzini

2nd generation detectors

3rd generation detectors Picture adapted from “Pushing towards the ET sensitivity using conventional technology”, by Stefan Hild, Simon Chelkowski, and Andreas Freise: First generation Second generation Third generation

Previous work on GG noise First and second generation detectors: Saulson made first predictions and set upper limits to the expected GG noise levels in first generation detectors. Beccaria et. al. created a more accurate estimate of GG noise for VIRGO. Thorne and Hughes published a full analytic analysis of GG noise and human interaction with the detector. Third generation detectors: Cella presented various studies on subterranean gravitational wave detectors and accompanying noise reduction due to placing the detector test masses at various depths and in different types of cavities. Now we’re working on a FE model to verify more complex cavity models and soil compositions.

Important aspects for site selection Soil composition and dynamics Reflections between different soil layers Variations in E with pressure Seismic activity (use data from different sites)

FEM example for investigating GG noise FEM input: - 200m clay: E =80MPa  =3000 kg/m m granite: E =20GPa  = 3000 kg/m 3 - cavity: depth 260m Ø50m FEM Output: -Elements volumes -Node coordinates: x i, y i, z i -Node displacement: u i (t), v i (t), w i (t)

FE models: Plane waves Harmonic pressure wave: = 200m, Model parameters: 100 elements L=2000m A = 100m 2  = 2000kg/m 3 E = 80MPa = 0 f =1Hz Pulse shear wave: =141m,

FE models: Rayleigh waves Model parameters:  = 2000kg/m 3 E = 80MPa = 0 f =1Hz Harmonic Rayleigh wave: =123m, 0300depth

FEM data analysis Compute a(t) in the measuring point O by summing the contributions da(t) due to all mass displacements in the FEM results

Model verification: harmonic wave in straight line segment Compare FEM data analysis to analytical solution For a harmonic wave and dM=mdx (where m is the mass per unit length). The acceleration in the x plane at depth z is given by and verify: Dependence on depth z Dependence on  k For a pressure wave For a shear wave

Model verification: harmonic wave in straight line segment Local acceleration ampl. at depth z for a pressure (top) and shear (bottom) wave Remember that this comparison is in 1D, driven with an arbitrary excitation and does not bear any physical insight into the acceleration seen in other models. It’s just model verification!

Model verification: harmonic wave in straight line segment Looking at the wavelength dependence at a fixed point z. At a fixed depth z we can vary  For short wavelengths * average out at depth z. So with increasing depth comes a predefined wavelength averaging. * these are short compared to the depth z

Current models: 2D models illustrating surface (Rayleigh), pressure-, and shear waves Important aspects: - Make sure no reflections appear at the edges of the model. - Measure 2D velocity and displacement components and compare to Matlab and Maple models

Upcoming models and planned analysis Compare wave propagation through layered soils and soil impedance with known literature and analytic models. Reproduce and verify current gravity gradient noise estimations of VIRGO and LIGO type interferometers using FE analysis. Incorporate the subterranean cavity geometry in the analysis.

Conclusion FEM analysis is a useful tool for modeling GG noise. FEM and analysis of simple systems have been verified. Future plans: - Reproducing existing VIRGO GG noise predictions. - Incorporate more complex FEM models, including subterranean GG noise.