pith. sign in

arxiv: 2603.15157 · v4 · submitted 2026-03-16 · ⚛️ physics.app-ph · astro-ph.IM· physics.comp-ph

An efficient finite element formulation for Newtonian noise analysis

Pieter Reumers , Xhorxha Kuci , Stijn Fran\c{c}ois , Geert Degrande This is my paper

Pith reviewed 2026-05-15 10:25 UTC · model grok-4.3

classification ⚛️ physics.app-ph astro-ph.IMphysics.comp-ph
keywords Newtonian noisefinite element methodgravitational wave detectorsseismic wavesdensity fluctuationscoupling matricesunderground observatoriesEinstein Telescope
0
0 comments X

The pith

Finite element coupling matrices reduce Newtonian noise integrals to geometry-only precomputations for any seismic wave field.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a finite element method that calculates Newtonian noise, the gravitational pull on detector mirrors caused by density changes from passing seismic waves. It evaluates the total effect along with separate bulk and surface contributions by converting the required volume and surface integrals into coupling matrices fixed by the mesh geometry and material properties alone. Once built, these matrices let new wave fields be processed through simple multiplications instead of repeating the full integration. This efficiency matters for underground gravitational wave observatories that must assess noise from many possible vibration sources in realistic soil conditions.

Core claim

A finite element formulation is presented which evaluates the total Newtonian noise, as well as the bulk and surface contributions, from a seismic wave field defined on a finite element mesh using Gaussian quadrature. Linear and quadratic tetrahedral and brick finite elements are supported. The approach computes the total, bulk, and surface contributions, and expresses the corresponding volume and surface integrals in terms of finite element coupling matrices that depend only on geometry and material properties. This allows efficient evaluation of the Newtonian noise for different seismic wave fields without recomputing the integrals.

What carries the argument

Finite element coupling matrices obtained via Gaussian quadrature on tetrahedral and brick elements, which isolate the geometric and material factors in the Newtonian noise volume and surface integrals so they can be reused across wave fields.

If this is right

  • Newtonian noise can be computed separately for bulk and surface contributions from any seismic displacement field on the mesh.
  • New wave fields require only matrix-vector multiplications after the initial coupling matrices are formed.
  • The method applies directly to both full-space plane-wave cases and half-space Rayleigh-wave cases under the long-wavelength limit.
  • The same matrices support rapid evaluation when material properties or mesh geometry change only slightly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation of bulk and surface terms could guide choices of cavity depth and shape to reduce overall Newtonian noise.
  • The framework can be extended to heterogeneous soil models by simply updating the material properties inside the existing matrices.
  • Stochastic ensembles of seismic sources could be processed quickly once the matrices exist, aiding statistical noise budgeting.

Load-bearing premise

Verification assumes seismic wavelengths are much larger than the cavity radius so wave scattering can be ignored and models the soil as homogeneous elastic material.

What would settle it

A full-wave numerical simulation of seismic propagation that includes scattering at the cavity boundary for wavelengths comparable to the cavity radius would show clear disagreement with the finite element Newtonian noise values if the long-wavelength assumption fails.

Figures

Figures reproduced from arXiv: 2603.15157 by Geert Degrande, Pieter Reumers, Stijn Fran\c{c}ois, Xhorxha Kuci.

Figure 1
Figure 1. Figure 1: Point mass m at a position x0 and volume Ω(x, t) with density ρ(x, t). wave field u(x, t) are continuously differentiable. Referring to equation (A.1), the scalar function g(x, t) in the present case is equal to −Gρ(x, t)/ ∥x − x0∥, and the total derivative of the gravitational potential Φ(x0, t) with respect to time is equal to: Dϕ(x0, t) Dt = −G Z Ω(x,t) " ∂ ∂t  ρ(x, t) ∥x − x0∥  + ∇ ·  ρ(x, t)v(x, t)… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Finite element mesh of a sphere with center at [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Real part of the total Newtonian noise [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Finite element mesh of a sphere with center at [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Real part of the total Newtonian noise [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Homogeneous halfspace with a test mass at a position [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the displacements uˆx(x, ω) and uˆz(x, ω) of a harmonic Rayleigh wave with unit amplitude and frequency of 5 Hz, propagating in the direction ek = {1, 0, 0} T in a homogeneous linear elastic halfspace with the same properties as introduced for the full space. The wave field is shown on a finite element mesh that will subsequently be used to compute the Newtonian noise. Both components exhibit exponen… view at source ↗
Figure 8
Figure 8. Figure 8: (a) Finite element mesh of a halfspace with radius [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Imaginary part of the total Newtonian noise [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Real part of the total Newtonian noise [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
read the original abstract

The Einstein Telescope is a third-generation underground gravitational wave observatory designed to achieve unprecedented sensitivity down to 3 Hz. Waves propagating in the soil due to anthropogenic or natural vibration sources generate density fluctuations which cause gravitational attraction, resulting in motion of the mirrors of the laser interferometer known as Newtonian noise. The latter is computed by integrating density fluctuations due to seismic wave fields over the soil domain surrounding the test mass. A finite element formulation is presented which evaluates the total Newtonian noise, as well as the bulk and surface contributions, from a seismic wave field defined on a finite element mesh using Gaussian quadrature. Linear and quadratic tetrahedral and brick finite elements are supported. The approach computes the total, bulk, and surface contributions, and expresses the corresponding volume and surface integrals in terms of finite element coupling matrices that depend only on geometry and material properties. This allows efficient evaluation of the Newtonian noise for different seismic wave fields without recomputing the integrals. The formulation is verified for plane P- and S-waves propagating in an elastic homogeneous full space with a mirror suspended in a spherical cavity, assuming the wavelength is much larger than the cavity radius, so that wave scattering can be ignored. Similar agreement is reported for the Newtonian noise on a test mass above the free surface of a homogeneous elastic halfspace in which a Rayleigh wave propagates. The methodology has been implemented in the ANNA Newtonian Noise analysis toolbox in MATLAB and is compatible with GNU Octave; a Python version is also available. The proposed finite element framework provides a physically consistent and computationally efficient approach for computing gravitational-seismic coupling in heterogeneous media.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a finite element formulation for computing Newtonian noise (NN) from seismic wave fields in gravitational-wave detectors. It evaluates the total, bulk, and surface contributions to NN via Gaussian quadrature on linear/quadratic tetrahedral and brick elements, reducing the volume and surface integrals to coupling matrices that depend only on geometry and material properties. This enables efficient reuse for arbitrary wave fields without recomputing the integrals. Verification is shown by direct comparison to analytical solutions for plane P- and S-waves in a homogeneous full space with a spherical cavity and for a Rayleigh wave on a homogeneous half-space, under the long-wavelength approximation that neglects scattering. The method is implemented in the ANNA toolbox for MATLAB/Octave with a Python version available.

Significance. If the formulation is correct, it provides a computationally efficient and physically consistent framework for NN analysis that is particularly valuable for heterogeneous media in underground observatories such as the Einstein Telescope. Reducing the integrals to precomputable geometry- and material-dependent matrices allows rapid assessment of NN for diverse seismic scenarios, which is essential for achieving the targeted sensitivity below 10 Hz. The open toolboxes support reproducibility and practical adoption.

major comments (2)
  1. [Verification] Verification section: All reported comparisons use homogeneous elastic media (plane P/S waves and Rayleigh wave). The central claim explicitly includes applicability to heterogeneous media via spatially varying density and moduli inside elements, yet no test compares the assembled coupling matrices or quadrature result against direct numerical integration of the Newtonian kernel over a non-uniform medium. This leaves the correctness of the matrix construction under material variation unconfirmed and is load-bearing for the heterogeneous claim.
  2. [Assumptions] Long-wavelength assumption (abstract and verification): The formulation assumes wavelength much larger than cavity radius to ignore scattering. No quantitative error estimate or validity range is provided for realistic seismic frequencies where this may not hold, which affects the reliability of the reported agreement with analytical solutions.
minor comments (1)
  1. The abstract mentions implementation in the ANNA toolbox and compatibility with GNU Octave; adding a brief note on key functions, input formats, or repository access would improve usability for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of our finite element formulation for Newtonian noise analysis. We address each major comment point by point below, proposing revisions where appropriate to strengthen the manuscript.

read point-by-point responses

  1. Referee: Verification section: All reported comparisons use homogeneous elastic media (plane P/S waves and Rayleigh wave). The central claim explicitly includes applicability to heterogeneous media via spatially varying density and moduli inside elements, yet no test compares the assembled coupling matrices or quadrature result against direct numerical integration of the Newtonian kernel over a non-uniform medium. This leaves the correctness of the matrix construction under material variation unconfirmed and is load-bearing for the heterogeneous claim.

    Authors: We agree that an explicit verification for heterogeneous media is needed to fully substantiate the claim. The formulation supports spatially varying density and elastic moduli through element-wise Gaussian quadrature, but the original manuscript did not include a direct comparison against numerical integration for a non-uniform case. In the revised version, we will add a new verification example using a heterogeneous medium (e.g., with piecewise constant or linearly varying properties across elements). The finite-element result via the precomputed coupling matrices will be compared to direct numerical integration of the Newtonian kernel over the same mesh, confirming the matrix construction under material variation. revision: yes

  2. Referee: Long-wavelength assumption (abstract and verification): The formulation assumes wavelength much larger than cavity radius to ignore scattering. No quantitative error estimate or validity range is provided for realistic seismic frequencies where this may not hold, which affects the reliability of the reported agreement with analytical solutions.

    Authors: We acknowledge that a quantitative discussion of the long-wavelength approximation's validity would improve the manuscript. This approximation, standard in the Newtonian noise literature, neglects scattering when the seismic wavelength greatly exceeds the cavity radius. In the revision, we will add a brief analysis in the verification section, referencing elastic scattering theory to provide error estimates and a validity range (in terms of wavelength-to-radius ratio) applicable to frequencies below 10 Hz relevant for the Einstein Telescope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard FE quadrature reformulation

full rationale

The paper reformulates the Newtonian noise volume and surface integrals using standard finite-element shape functions and Gaussian quadrature, yielding coupling matrices that depend only on geometry and material properties. This is a direct discretization of the gravitational integral with no fitted parameters, no self-definitional loops, and no load-bearing self-citations. Verification against analytical solutions in homogeneous media is independent of the matrix construction itself. The derivation chain is self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The formulation rests on standard finite-element discretization and quadrature rules for linear elastic wave propagation; no new physical constants, fitted parameters, or invented entities are introduced.

axioms (2)
  • domain assumption Seismic wave propagation obeys linear elasticity in the modeled domain
    Invoked for the homogeneous full-space and half-space verification cases.
  • domain assumption Wavelength much larger than cavity radius so scattering can be neglected
    Explicitly stated as the condition under which analytical comparisons are performed.

pith-pipeline@v0.9.0 · 5597 in / 1316 out tokens · 40792 ms · 2026-05-15T10:25:49.148692+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    September 2020

    ET Steering Committee Editorial Team.Design Report Update 2020 for the Einstein Telescope. September 2020

    work page 2020

  2. [2]

    Amann et al

    F. Amann et al. Site-selection criteria for the Einstein Telescope.Review of Scientific Instruments, 91:094504, 2020. 27

    work page 2020

  3. [3]

    Driggers, J

    J.C. Driggers, J. Harms, and R.X. Adhikari. Subtraction of Newtonian noise using optimized sensor arrays.Physical Review D, 86:102001, 2012

    work page 2012

  4. [4]

    J. Harms. Terrestrial gravity fluctuations.Living Reviews in Relativity, 22:6, 2019

    work page 2019

  5. [5]

    Kausel and J.M

    E. Kausel and J.M. Roësset. Stiffness matrices for layered soils.Bulletin of the Seis- mological Society of America, 71(6):1743–1761, 1981

    work page 1981

  6. [6]

    Schevenels, S

    M. Schevenels, S. François, and G. Degrande. EDT: An ElastoDynam- ics Toolbox for MATLAB.Computers & Geosciences, 35(8):1752–1754, 2009. http://dx.doi.org/10.1016/j.cageo.2008.10.012

    work page doi:10.1016/j.cageo.2008.10.012 2009

  7. [7]

    Lombaert, G

    G. Lombaert, G. Degrande, J. Kogut, and S. François. The experimental validation of a numerical model for the prediction of railway induced vibrations.Journal of Sound and Vibration, 297(3-5):512–535, 2006

    work page 2006

  8. [8]

    Lombaert and G

    G. Lombaert and G. Degrande. Ground-borne vibration due to static and dynamic axle loads of InterCity and high speed trains.Journal of Sound and Vibration, 319(3- 5):1036–1066, 2009

    work page 2009

  9. [9]

    Lombaert, S

    G. Lombaert, S. François, and G. Degrande. TRAFFIC Matlab toolbox for traffic in- duced vibrations. Report BWM-2012-10, Department of Civil Engineering, KU Leuven, November 2012. User’s Guide Traffic 5.2

    work page 2012

  10. [10]

    Sheng, C.J.C

    X. Sheng, C.J.C. Jones, and D.J. Thompson. A theoretical model for ground vibration from trains generated by vertical track irregularities.Journal of Sound and Vibration, 272(3):937–965, 2004

    work page 2004

  11. [11]

    Ntotsios, D.J

    E. Ntotsios, D.J. Thompson, and M.F.M. Hussein. The effect of track load correlation on ground-borne vibration from railways.Journal of Sound and Vibration, 402:142–163, 2017

    work page 2017

  12. [12]

    Ntotsios, D.J

    E. Ntotsios, D.J. Thompson, and M.F.M. Hussein. A comparison of ground vibration due to ballasted and slab tracks.Transportation Geotechnics, 21(100256), 2019

    work page 2019

  13. [13]

    P. Jean. A 3D FEM/BEM code for ground-structure interaction: implementation strategy including the multi-traction problem.Engineering Analysis with Boundary Elements, 59:52–61, 2015

    work page 2015

  14. [14]

    Yang and H.H

    Y.B. Yang and H.H. Hung. Soil vibrations caused by underground moving trains. Journal of Geotechnical and Geoenvironmental Engineering, Proceedings of the ASCE, 134(11):1633–1644, 2008

    work page 2008

  15. [15]

    Yang and H.H

    Y.B. Yang and H.H. Hung.Wave propagation for train-induced vibrations. World Scientific, Taipei, 2009. 28

    work page 2009

  16. [16]

    François, M

    S. François, M. Schevenels, G. Lombaert, and G. Degrande. A 2.5D displacement based PML for elastodynamic wave propagation.International Journal for Numerical Methods in Engineering, 90(7):819–837, 2012

    work page 2012

  17. [17]

    Andersen

    L. Andersen. Influence of dynamic soil-structure interaction on building response due to ground vibration. InProceedings of the 8th European Conference on Numerical Methods in Geotechnical Engineering, Delft, The Netherlands, June 2014

    work page 2014

  18. [18]

    François, M

    S. François, M. Schevenels, G. Lombaert, P. Galvín, and G. Degrande. A 2.5D cou- pled FE-BE methodology for the dynamic interaction between longitudinally invariant structures and a layered halfspace.Computer Methods in Applied Mechanics and En- gineering, 199(23-24):1536–1548, 2010

    work page 2010

  19. [19]

    Papadopoulos, S

    M. Papadopoulos, S. François, G. Degrande, and G. Lombaert. The influence of uncer- tain local subsoil conditions on the response of buildings to ground vibration.Journal of Sound and Vibration, 418:200–220, 2018

    work page 2018

  20. [20]

    Mazzieri, M

    I. Mazzieri, M. Stupazzini, R. Guidotti, and C. Smerzini. SPEED: SPectral Elements in Elastodynamics with Discontinuous Galerkin: a non-conforming approach for 3D multi-scale problems.International Journal for Numerical Methods in Engineering, 95(12):991–1010, 2013

    work page 2013

  21. [21]

    Newtonian-noise characterization at Terziet in Limburg - the Euregio Meuse–Rhine candidate site for Einstein Telescope.Classical and Quantum Gravity, 39(2):025009, 2022

    M.Bader, S.Koley, J.vandenBrand, X.Campman, H.J.Bulten, F.Linde, andB.Vink. Newtonian-noise characterization at Terziet in Limburg - the Euregio Meuse–Rhine candidate site for Einstein Telescope.Classical and Quantum Gravity, 39(2):025009, 2022

    work page 2022

  22. [22]

    Schillings, S

    P. Schillings, S. Yao, J. Erdmann, and A. Rietbrock. A numerical framework for Newtonian-noise estimation at the Einstein Telescope: 2-D simulations beyond the plane-wave approximation.arXiv, (2603.15424), 2026

    work page arXiv 2026

  23. [23]

    François, M

    S. François, M. Schevenels, D. Dooms, M. Jansen, J. Wambacq, G. Lombaert, G. De- grande, and G. De Roeck. Stabil: an educational Matlab toolbox for static and dynamic structural analysis.Computer Applications in Engineering Education, 29(5):1–18, 2021

    work page 2021

  24. [24]

    Quarteroni, F

    A. Quarteroni, F. Saleri, and P. Gervasio.Scientific computing with MATLAB and Octave, volume 2. Springer, 2006

    work page 2006

  25. [25]

    Python for scientific computing.Computing in science & engineer- ing, 9(3):10–20, 2007

    Travis E Oliphant. Python for scientific computing.Computing in science & engineer- ing, 9(3):10–20, 2007

    work page 2007

  26. [26]

    Array programming with numpy.Nature, 585(7825):357–362, 2020

    Charles R Harris, K Jarrod Millman, Stéfan J Van Der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J Smith, et al. Array programming with numpy.Nature, 585(7825):357–362, 2020. 29

    work page 2020

  27. [27]

    Geuzaine and J.-F

    C. Geuzaine and J.-F. Remacle. Gmsh: a three-dimensional finite element mesh genera- tor with built-in pre- and post-processing facilities.International Journal for Numerical Methods in Engineering, 79(11):1309–1331, 2009

    work page 2009

  28. [28]

    Rayleigh

    J.W.S. Rayleigh. On waves propagated along the plane surface of an elastic solid. Proceedings of the London Mathematical Society, 17:4–11, 1887

    work page

  29. [29]

    Papadopoulos.ME185 Introduction to Continuum Mechanics

    P. Papadopoulos.ME185 Introduction to Continuum Mechanics. Department of Me- chanical Engineering, University of California, Berkeley, USA, September 2020. 30

    work page 2020