FE model implementation of seismically driven GG noise in subterranean

gravitational wave detectors

David Rabeling, Eric Hennes,

and Jo van den Brand

davidr@NIKHEF.nl

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: https://workarea.et-gw.eu/et/WG3Topology/document_dir/ET_sensitivity_v2_to_ET_workarea.pdf

First generationSecond generationThird 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 cavitymodels 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/m3

- 1300m granite: E =20GPa = 3000 kg/m3

- cavity: depth 260m Ø50m

FEM Output:-Elements volumes-Node coordinates: xi, yi, zi

-Node displacement: ui(t), vi(t), wi(t)

FE models: Plane waves

Harmonic pressure wave: = 200m, Model parameters: 100 elements L=2000m A = 100m2

= 2000kg/m3

E = 80MPa = 0 f =1Hz

Pulse shear wave: =141m,

/Ec

2/Ec

FE models: Rayleigh waves

Model parameters: = 2000kg/m3

E = 80MPa = 0 f =1Hz

Harmonic Rayleigh wave: =123m, /61.0 Ec

0 300depth

FEM data analysis

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)()(),()(3

tw

tv

tu

dtdwithtdr

dMGtd aggra 2/ sm

)2(33

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5

zyxzyzx

yzzyxyx

xzxyzyx

r

dMGda

Compute a(t) in the measuring point O by summing the contributionsda(t) due to all mass displacements in the FEM results

Model verification: harmonic wave in straight line segment

Compare FEM data analysis to analytical solutionFor a harmonic wave and dM=mdx (where m is the mass per unitlength). The acceleration in the x plane at depth z is given by

and verify: • Dependence on depth z• Dependence on k

2/

2/2/522

220 )cos(

)2()(

L

L

x dxkxtzx

zxmuGta

2/

2/2/522

0 )cos(3

)(L

L

x dxkxtzx

xzmuGta

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 outat 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.