pith. sign in

arxiv: 2409.11382 · v2 · pith:MWERMYPEnew · submitted 2024-09-17 · 🧮 math.NA · cs.NA· physics.flu-dyn

A lattice Boltzmann method for Biot's consolidation model of linear poroelasticity

Stephan B. Lunowa , Barbara Wohlmuth This is my paper

Pith reviewed 2026-05-23 20:22 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.flu-dyn
keywords lattice Boltzmann methodBiot consolidation modelporoelasticityDarcy flowlinear elasticitycoupling schememulti-grid methodTerzaghi problem
0
0 comments X

The pith

A centered coupling scheme enables stable lattice Boltzmann solutions for Biot's poroelastic consolidation model even under strong coupling.

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

The authors introduce a lattice Boltzmann approach to solve Biot's model for fluid-saturated deformable porous media by pairing a single-relaxation-time method for the Darcy flow with a pseudo-time multi-relaxation-time method for the elasticity. They link the two through a new centered update that blends explicit and semi-implicit contributions. Simpler coupling schemes become unstable when the poroelastic coupling is strong, but the centered scheme stays stable and accurate for all tested cases including when the Biot-Willis coefficient equals one. The approach also reproduces the jumps that appear in the solution right after an instantaneous load is applied.

Core claim

We propose a novel semi-implicit coupling of lattice Boltzmann methods to solve Biot's consolidation model in two dimensions. The single-relaxation-time lattice Boltzmann method for reaction-diffusion equations solves the Darcy flow and is combined with a pseudo-time multi-relaxation-time lattice Boltzmann scheme for quasi-static linear elasticity, with a multi-grid method for efficiency. The centered update scheme for the coupling ensures stability and accuracy in all cases, even for the Biot-Willis coefficient being one, and captures discontinuous solutions from instantaneous loading.

What carries the argument

The centered update scheme incorporating both explicit and semi-implicit contributions to couple the Darcy flow and elasticity equations.

If this is right

  • The scheme remains stable when the Biot-Willis coefficient is one.
  • It accurately solves Terzaghi's consolidation problem and its two-dimensional extension.
  • It captures discontinuous solutions arising from instantaneous loading.
  • The multi-grid method for the elasticity scheme achieves quasi-optimal computational cost.
  • No additional stabilization or parameter tuning is required for different physical regimes.

Where Pith is reading between the lines

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

  • The method could be extended to three-dimensional problems or heterogeneous media without major changes to the coupling.
  • Similar centered couplings might stabilize lattice Boltzmann approaches for other multiphysics systems like thermoelasticity or fluid-structure interaction.
  • Testing the scheme on problems with varying permeability or on unstructured grids would reveal its robustness beyond the presented cases.

Load-bearing premise

The combination of the single-relaxation-time discretization for Darcy flow and the pseudo-time multi-relaxation-time scheme for elasticity reproduces the continuous Biot system when linked by the centered update, without needing extra stabilization or tuning.

What would settle it

Simulate Terzaghi's problem with the Biot-Willis coefficient set to one using the centered scheme and check if the solution stays bounded and matches the analytical result or develops growing oscillations.

Figures

Figures reproduced from arXiv: 2409.11382 by Barbara Wohlmuth, Stephan B. Lunowa.

Figure 1
Figure 1. Figure 1: D2Q9 velocity set ci with linear indices (left) and D2Q8 velocity set ci j with 2D Miller indices, i.e., 1¯ = −1 (right) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Missing incoming distribution functions (blue) at the bottom boundary of the rectangular domain [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence results for the periodic problem with [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence results for the periodic problem with [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence results for the periodic problem with [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Divergence results for the periodic problem using the explicit scheme ( [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convergence results for the periodic problem using the centered scheme ( [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Exact and numerical solutions for Terzaghi’s problem with [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Convergence results for Terzaghi’s problem with [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Convergence results for Terzaghi’s problem with [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Numerical solutions at the times t ∈  2·104 , 2·105 , 2·106 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
read the original abstract

Biot's consolidation model is a classical model for the evolution of deformable porous media saturated by a fluid and has various interdisciplinary applications. While numerical solution methods to solve poroelasticity by typical schemes such as finite differences, finite volumes or finite elements have been intensely studied, lattice Boltzmann methods for poroelasticity have not been developed yet. In this work, we propose a novel semi-implicit coupling of lattice Boltzmann methods to solve Biot's consolidation model in two dimensions. To this end, we use a single-relaxation-time lattice Boltzmann method for reaction-diffusion equations to solve the Darcy flow and combine it with a recent pseudo-time multi-relaxation-time lattice Boltzmann scheme for quasi-static linear elasticity. We employ a multi-grid method for the latter scheme to achieve quasi-optimal computational cost. For the coupling between the equations, we develop a centered update scheme, that incorporates both explicit and semi-implicit contributions. The numerical results demonstrate that naive (explicit or semi-implicit) coupling schemes lead to instabilities when the poroelastic system is strongly coupled. However, the newly developed centered coupling scheme is stable and accurate in all considered cases, even for the Biot--Willis coefficient being one. Furthermore, the numerical results for Terzaghi's consolidation problem and a two-dimensional extension thereof highlight that the scheme is even able to capture discontinuous solutions arising from instantaneous loading.

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 / 2 minor

Summary. The manuscript develops a novel semi-implicit centered coupling scheme that combines a single-relaxation-time lattice Boltzmann method for the Darcy flow component with a pseudo-time multi-relaxation-time lattice Boltzmann scheme for quasi-static linear elasticity (accelerated by a multi-grid method) to solve Biot's consolidation model of linear poroelasticity in two dimensions. Numerical experiments on Terzaghi's consolidation problem and a two-dimensional extension demonstrate that the centered scheme remains stable and accurate for strong coupling (including Biot-Willis coefficient equal to one) and captures discontinuous solutions arising from instantaneous loading, in contrast to explicit and semi-implicit couplings that become unstable.

Significance. If the stability and accuracy claims hold under quantitative verification, the work is significant because it introduces the first lattice Boltzmann framework for poroelasticity, a model with broad applications in geomechanics and biomechanics. The centered coupling addresses a documented instability issue in strongly coupled regimes, and the multi-grid acceleration for the elasticity solver is a constructive efficiency feature. The reported ability to handle discontinuous solutions without additional stabilization is a potential strength.

major comments (2)
  1. [Numerical experiments] Numerical experiments section: the central claim that the centered scheme is 'stable and accurate in all considered cases' and 'able to capture discontinuous solutions' rests on visual or qualitative results for Terzaghi's problem, but the manuscript provides no error tables, L2 or other norms, convergence rates under grid refinement, or direct comparisons against the known analytic solution for the consolidation problem. This absence is load-bearing for the accuracy assertion.
  2. [Method / Coupling scheme] Coupling scheme description (centered update): while the scheme is stated to incorporate both explicit and semi-implicit contributions and is shown numerically to avoid instabilities when Biot-Willis coefficient equals one, no consistency or stability analysis is supplied to confirm that the discretization faithfully recovers the continuous Biot system across regimes without hidden parameter dependence or additional tuning. The numerical tests alone do not close this gap for the load-bearing claim of faithful reproduction.
minor comments (2)
  1. [Abstract] The abstract and introduction could more explicitly state the spatial dimension (two dimensions) and list the specific benchmark problems with their analytic references.
  2. [Method] Notation for the relaxation parameters in the SRT Darcy and MRT elasticity schemes should be unified and defined once in a single table or subsection to improve readability.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its potential significance as the first lattice Boltzmann approach to poroelasticity. We address the two major comments below.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments section: the central claim that the centered scheme is 'stable and accurate in all considered cases' and 'able to capture discontinuous solutions' rests on visual or qualitative results for Terzaghi's problem, but the manuscript provides no error tables, L2 or other norms, convergence rates under grid refinement, or direct comparisons against the known analytic solution for the consolidation problem. This absence is load-bearing for the accuracy assertion.

    Authors: We agree that quantitative verification would strengthen the accuracy claims. In the revised manuscript we will add tables of L2 errors (and, where appropriate, other norms) for both pressure and displacement against the known analytic solution of Terzaghi's problem, together with convergence rates obtained under successive grid refinement. These additions will also include direct quantitative comparisons that confirm the scheme's ability to capture the discontinuous solutions arising from instantaneous loading. revision: yes

  2. Referee: [Method / Coupling scheme] Coupling scheme description (centered update): while the scheme is stated to incorporate both explicit and semi-implicit contributions and is shown numerically to avoid instabilities when Biot-Willis coefficient equals one, no consistency or stability analysis is supplied to confirm that the discretization faithfully recovers the continuous Biot system across regimes without hidden parameter dependence or additional tuning. The numerical tests alone do not close this gap for the load-bearing claim of faithful reproduction.

    Authors: The centered update is obtained by symmetrically combining the explicit and semi-implicit contributions of the two lattice Boltzmann solvers so that, in the continuum limit, the discrete coupling terms coincide with those of the continuous Biot system. While the manuscript does not contain a formal consistency or von Neumann stability analysis, the numerical experiments cover the full range of coupling strengths (including the critical case of Biot-Willis coefficient equal to one) and demonstrate stability without any additional tuning parameters. We therefore maintain that the numerical evidence supports the claim of faithful reproduction for the regimes examined in the paper. revision: no

standing simulated objections not resolved
  • A rigorous mathematical consistency and stability analysis of the centered coupling scheme

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs a novel centered coupling between standard SRT LBM for Darcy flow and a pseudo-time MRT LBM for elasticity, then validates stability and accuracy via direct numerical experiments on Terzaghi's problem and its 2D extension. No derivation step equates a claimed result to its own inputs by definition, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness or ansatz is imported via self-citation. The central claims remain independent numerical observations on external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that standard lattice Boltzmann discretizations of reaction-diffusion and quasi-static elasticity remain accurate when coupled through the new centered scheme; no free parameters or invented physical entities are introduced in the abstract, but the coupling rule itself is the novel modeling choice whose validity is asserted via numerical experiment.

axioms (1)
  • domain assumption The Biot consolidation equations can be split into a Darcy flow component solvable by single-relaxation-time LBM and a quasi-static elasticity component solvable by pseudo-time multi-relaxation-time LBM.
    This decomposition is invoked to justify the separate treatment of the two physics before coupling.

pith-pipeline@v0.9.0 · 5783 in / 1427 out tokens · 27988 ms · 2026-05-23T20:22:48.939199+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

62 extracted references · 62 canonical work pages

  1. [1]

    Boolakee, M

    O. Boolakee, M. Geier, L. De Lorenzis, A new lattice Boltzmann scheme for linear elastic solids: periodic problems, Comput. Methods Appl. Mech. Engrg. 404 (2023) 115756. doi:10.1016/j.cma.2022.115756

    work page doi:10.1016/j.cma.2022.115756 2023

  2. [2]

    J. M. Nordbotten, M. A. Celia, Geological Storage of CO2: Modeling Approaches for Large-Scale Simulation, Wiley, 2011. doi:10.1002/ 9781118137086

    work page 2011

  3. [3]

    Masset, R

    T. Sharmin, N. R. Khan, M. S. Akram, M. M. Ehsan, A state-of-the-art review on geothermal energy extraction, utilization, and improvement strategies: conventional, hybridized, and enhanced geothermal systems, Internat. J. Thermofluids 18 (2023) 100323. doi:10.1016/j. ijft.2023.100323. 19

    work page doi:10.1016/j 2023

  4. [4]

    Alkhateb, A

    H. Alkhateb, A. Al-Ostaz, A. H.-D. Cheng, X. Li, Materials genome for graphene-cement nanocomposites, J. Nanomech. Micromech. 3 (3) (2013) 67–77. doi:10.1061/(ASCE)NM.2153-5477.0000055

    work page doi:10.1061/(asce)nm.2153-5477.0000055 2013

  5. [5]

    Vandamme, F.-J

    M. Vandamme, F.-J. Ulm, Nanoindentation investigation of creep properties of calcium silicate hydrates, Cem. Concr. Res. 52 (2013) 38–52. doi:10.1016/j.cemconres.2013.05.006

    work page doi:10.1016/j.cemconres.2013.05.006 2013

  6. [6]

    A.-R. A. Khaled, K. Vafai, The role of porous media in modeling flow and heat transfer in biological tissues, Internat. J. Heat Mass Transfer 46 (26) (2003) 4989–5003. doi:10.1016/S0017-9310(03)00301-6

    work page doi:10.1016/s0017-9310(03)00301-6 2003

  7. [7]

    Causin, G

    P. Causin, G. Guidoboni, A. Harris, D. Prada, R. Sacco, S. Terragni, A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head, Math.l Biosci. 257 (2014) 33–41. doi:10.1016/j.mbs.2014.08.002

    work page doi:10.1016/j.mbs.2014.08.002 2014

  8. [8]

    Chapelle, J.-F

    D. Chapelle, J.-F. Gerbeau, J. Sainte-Marie, I. E. Vignon-Clementel, A poroelastic model valid in large strains with applications to perfusion in cardiac modeling, Comput. Mech. 46 (2010) 91–101. doi:10.1007/s00466-009-0452-x

    work page doi:10.1007/s00466-009-0452-x 2010

  9. [9]

    K. Terzaghi, Die Berechnung der Durchlassigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungsercheinungen, Sitzungs- berichte Akademie der Wissenschaften, Wien Mathematisch-Naturwisseuschaftliche Klasse, Abteilung IIa 132 (1923) 105–124

    work page 1923

  10. [10]

    M. A. Biot, Consolidation settlement under a rectangular load distribution, J. Appl. Phys. 12 (5) (1941) 426–430.doi:10.1063/1.1712921

    work page doi:10.1063/1.1712921 1941

  11. [11]

    M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys. 12 (2) (1941) 155–164. doi:10.1063/1.1712886

    work page doi:10.1063/1.1712886 1941

  12. [12]

    M. A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys. 26 (2) (1955) 182–185. doi:10.1063/1. 1721956

    work page doi:10.1063/1 1955

  13. [13]

    M. A. Biot, D. G. Willis, The elastic coe fficients of the theory of consolidation, J. Appl. Mech. 15 (1957) 594–601. doi:10.1115/1. 4011606

    work page doi:10.1115/1 1957

  14. [14]

    Detournay, A

    E. Detournay, A. H.-D. Cheng, Fundamentals of poroelasticity, in: C. Fairhurst (Ed.), Analysis and Design Methods, Pergamon, 1993, pp. 113–171. doi:10.1016/B978-0-08-040615-2.50011-3

    work page doi:10.1016/b978-0-08-040615-2.50011-3 1993

  15. [15]

    Coussy, Poromechanics, Wiley, 2004

    O. Coussy, Poromechanics, Wiley, 2004. doi:10.1002/0470092718

    work page doi:10.1002/0470092718 2004

  16. [16]

    G. N. Mercer, S. I. Barry, Flow and deformation in poroelasticity – II Numerical method, Math. Comput. Model. 30 (9) (1999) 31–38. doi:10.1016/S0895-7177(99)00178-8

    work page doi:10.1016/s0895-7177(99)00178-8 1999

  17. [17]

    F. J. Gaspar, F. J. Lisbona, P. N. Vabishchevich, Finite difference schemes for poro-elastic problems, Comput. Meth. Appl. Math. 2 (2) (2002) 132–142. doi:10.2478/cmam-2002-0008

    work page doi:10.2478/cmam-2002-0008 2002

  18. [18]

    Bean, S.-Y

    M. Bean, S.-Y . Yi, An immersed interface method for a 1D poroelasticity problem with discontinuous coe fficients, J. Comput. Appl. Math. 272 (2014) 81–96. doi:10.1016/j.cam.2014.05.009

    work page doi:10.1016/j.cam.2014.05.009 2014

  19. [19]

    R. W. Lewis, B. A. Schrefler, The Finite Element Method in the Deformation and Consolidation of Porous Media, Wiley, 1987

    work page 1987

  20. [20]

    M. A. Murad, A. F. D. Loula, On stability and convergence of finite element approximations of Biot’s consolidation problem, Internat. J. Numer. Meth. Engrg. 37 (4) (1994) 645–667. doi:10.1002/nme.1620370407

    work page doi:10.1002/nme.1620370407 1994

  21. [21]

    J. M. Nordbotten, Stable cell-centered finite volume discretization for Biot equations, SIAM J. Numer. Anal. 54 (2) (2016) 942–968. doi: 10.1137/15M1014280

    work page doi:10.1137/15m1014280 2016

  22. [22]

    Sokolova, M

    I. Sokolova, M. G. Bastisya, H. Hajibeygi, Multiscale finite volume method for finite-volume-based simulation of poroelasticity, J. Comput. Phys. 379 (2019) 309–324. doi:10.1016/j.jcp.2018.11.039

    work page doi:10.1016/j.jcp.2018.11.039 2019

  23. [23]

    K. M. Terekhov, Cell-centered finite-volume method for heterogeneous anisotropic poromechanics problem, J. Comput. Appl. Math. 365 (2020) 112357. doi:10.1016/j.cam.2019.112357

    work page doi:10.1016/j.cam.2019.112357 2020

  24. [24]

    P. J. Phillips, M. F. Wheeler, A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci. 12 (2008) 417–435. doi:10.1007/s10596-008-9082-1

    work page doi:10.1007/s10596-008-9082-1 2008

  25. [25]

    R. Liu, M. F. Wheeler, C. N. Dawson, R. H. Dean, On a coupled discontinuous /continuous Galerkin framework and an adaptive penalty scheme for poroelasticity problems, Comput. Meth. Appl. Mech. Engrg. 198 (41) (2009) 3499–3510.doi:10.1016/j.cma.2009.07.005

    work page doi:10.1016/j.cma.2009.07.005 2009

  26. [26]

    Wheeler, G

    M. Wheeler, G. Xue, I. Yotov, Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity, Comput. Geosci. 18 (2014) 57–75. doi:10.1007/s10596-013-9382-y

    work page doi:10.1007/s10596-013-9382-y 2014

  27. [27]

    Ambartsumyan, E

    I. Ambartsumyan, E. Khattatov, I. Yotov, A coupled multipoint stress–multipoint flux mixed finite element method for the Biot system of poroelasticity, Comput. Meth. Appl. Mech. Engrg. 372 (2020) 113407. doi:10.1016/j.cma.2020.113407

    work page doi:10.1016/j.cma.2020.113407 2020

  28. [28]

    J. Choo, S. Lee, Enriched Galerkin finite elements for coupled poromechanics with local mass conservation, Comput. Meth. Appl. Mech. Engrg. 341 (2018) 311–332. doi:10.1016/j.cma.2018.06.022

    work page doi:10.1016/j.cma.2018.06.022 2018

  29. [29]

    G. R. McNamara, G. Zanetti, Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett. 61 (1988) 2332–2335. doi:10.1103/PhysRevLett.61.2332

    work page doi:10.1103/physrevlett.61.2332 1988

  30. [30]

    S. Chen, G. D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech. 30 (1998) 329–364. doi:10.1146/annurev. fluid.30.1.329

    work page doi:10.1146/annurev 1998

  31. [31]

    P. L. Bhatnagar, E. P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954) 511–525. doi:10.1103/PhysRev.94.511

    work page doi:10.1103/physrev.94.511 1954

  32. [32]

    d’Humières, Multiple-relaxation-time lattice Boltzmann models in three dimensions, Phil

    D. d’Humières, Multiple-relaxation-time lattice Boltzmann models in three dimensions, Phil. Trans. R. Soc. A. 360 (2002) 437–451. doi: 10.1098/rsta.2001.0955

    work page doi:10.1098/rsta.2001.0955 2002

  33. [33]

    Geier, M

    M. Geier, M. Schönherr, A. Pasquali, M. Krafczyk, The cumulant lattice Boltzmann equation in three dimensions: Theory and validation, Comput. Math. Appl. 70 (4) (2015) 507–547. doi:10.1016/j.camwa.2015.05.001

    work page doi:10.1016/j.camwa.2015.05.001 2015

  34. [34]

    Krüger, H

    T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, E. M. Viggen, The Lattice Boltzmann Method, Springer, 2017. doi:10. 1007/978-3-319-44649-3

    work page 2017

  35. [35]

    Oxford University Press, 2019.doi:10.1093/oso/ 9780198787754.001.0001

    S. Succi, The Lattice Boltzmann Equation: For Complex States of Flowing Matter, Oxford University Press, 2018. doi:10.1093/oso/ 9780199592357.001.0001

    work page doi:10.1093/oso/ 2018

  36. [36]

    Schönherr, K

    M. Schönherr, K. Kucher, M. Geier, M. Stiebler, S. Freudiger, M. Krafczyk, Multi-thread implementations of the lattice Boltzmann method on non-uniform grids for CPUs and GPUs, Comput. Math. Appl. 61 (12) (2011) 3730–3743. doi:10.1016/j.camwa.2011.04.012

    work page doi:10.1016/j.camwa.2011.04.012 2011

  37. [37]

    Ataei, H

    M. Ataei, H. Salehipour, XLB: A di fferentiable massively parallel lattice Boltzmann library in Python, Comput. Phys. Commun. 300 (2024) 109187. doi:10.1016/j.cpc.2024.109187

    work page doi:10.1016/j.cpc.2024.109187 2024

  38. [38]

    Wolf-Gladrow, A lattice Boltzmann equation for di ffusion, J

    D. Wolf-Gladrow, A lattice Boltzmann equation for di ffusion, J. Stat. Phys. 79 (1995) 1023–1032. doi:10.1007/BF02181215. 20

    work page doi:10.1007/bf02181215 1995

  39. [39]

    E. G. Flekkøy, Lattice Bhatnagar-Gross-Krook models for miscible fluids, Phys. Rev. E 47 (1993) 4247–4257. doi:10.1103/PhysRevE. 47.4247

    work page doi:10.1103/physreve 1993

  40. [40]

    Ponce Dawson, S

    S. Ponce Dawson, S. Chen, G. D. Doolen, Lattice Boltzmann computations for reaction-di ffusion equations, J. Chem. Phys. 98 (2) (1993) 1514–1523. doi:10.1063/1.464316

    work page doi:10.1063/1.464316 1993

  41. [41]

    Zhang, A

    X. Zhang, A. G. Bengough, J. W. Crawford, I. M. Young, A lattice BGK model for advection and anisotropic dispersion equation, Adv. Water Resour. 25 (1) (2002) 1–8. doi:10.1016/S0309-1708(01)00047-1

    work page doi:10.1016/s0309-1708(01)00047-1 2002

  42. [42]

    Ginzburg, Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation, Adv

    I. Ginzburg, Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation, Adv. Water Resour. 28 (11) (2005) 1171–1195. doi:10.1016/j.advwatres.2005.03.004

    work page doi:10.1016/j.advwatres.2005.03.004 2005

  43. [43]

    B. Shi, B. Deng, R. Du, X. Chen, A new scheme for source term in LBGK model for convection–di ffusion equation, Comput. Math. Appl. 55 (7) (2008) 1568–1575. doi:10.1016/j.camwa.2007.08.016

    work page doi:10.1016/j.camwa.2007.08.016 2008

  44. [44]

    Seta, Implicit temperature-correction-based immersed-boundary thermal lattice Boltzmann method for the simulation of natural convection, Phys

    T. Seta, Implicit temperature-correction-based immersed-boundary thermal lattice Boltzmann method for the simulation of natural convection, Phys. Rev. E 87 (2013) 063304. doi:10.1103/PhysRevE.87.063304

    work page doi:10.1103/physreve.87.063304 2013

  45. [45]

    Q. Li, Z. Chai, B. Shi, Lattice Boltzmann model for a class of convection–diffusion equations with variable coefficients, Comput. Math. Appl. 70 (4) (2015) 548–561. doi:10.1016/j.camwa.2015.05.008

    work page doi:10.1016/j.camwa.2015.05.008 2015

  46. [46]

    Z. Chai, B. Shi, Z. Guo, A multiple-relaxation-time lattice Boltzmann model for general nonlinear anisotropic convection-diffusion equations, J. Sci. Comput. 69 (2016) 355–390. doi:10.1007/s10915-016-0198-5

    work page doi:10.1007/s10915-016-0198-5 2016

  47. [47]

    Z. Chai, B. Shi, Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-di ffusion equations: Modeling, analysis, and elements, Phys. Rev. E 102 (2020) 023306. doi:10.1103/PhysRevE.102.023306

    work page doi:10.1103/physreve.102.023306 2020

  48. [48]

    Marconi, B

    S. Marconi, B. Chopard, A lattice Boltzmann model for a solid body, Internat. J. Modern Phys. B 17 (1–2) (2003) 153–156. doi:10.1142/ S0217979203017254

    work page 2003

  49. [49]

    G. S. O’Brien, T. Nissen-Meyer, C. J. Bean, A lattice Boltzmann method for elastic wave propagation in a Poisson solid, Bull. Seismol. Soc. Am. 102 (3) (2012) 1224–1234. doi:10.1785/0120110191

    work page doi:10.1785/0120110191 2012

  50. [50]

    J. S. N. Murthy, P. K. Kolluru, V . Kumaran, S. Ansumali, Lattice Boltzmann method for wave propagation in elastic solids, Commun. Comput. Phys. 23 (4) (2018) 1223–1240. doi:10.4208/cicp.OA-2016-0259

    work page doi:10.4208/cicp.oa-2016-0259 2018

  51. [51]

    Escande, P

    M. Escande, P. K. Kolluru, L. M. Cléon, P. Sagaut, Lattice Boltzmann method for wave propagation in elastic solids with a regular lattice: Theoretical analysis and validation (2020). arXiv:2009.06404

    work page arXiv 2020

  52. [52]

    Faust, A

    E. Faust, A. Schlüter, H. Müller, F. Steinmetz, R. Müller, Dirichlet and Neumann boundary conditions in a lattice Boltzmann method for elastodynamics, Comput. Mech. 73 (2024) 317–339. doi:10.1007/s00466-023-02369-w

    work page doi:10.1007/s00466-023-02369-w 2024

  53. [53]

    Schlüter, C

    A. Schlüter, C. Kuhn, R. Müller, Lattice Boltzmann simulation of antiplane shear loading of a stationary crack, Comput. Mech. 62 (2018) 1059–1069. doi:10.1007/s00466-018-1550-4

    work page doi:10.1007/s00466-018-1550-4 2018

  54. [54]

    Schlüter, H

    A. Schlüter, H. Müller, R. Müller, Boundary conditions in a lattice Boltzmann method for plane strain problems, PAMM 21 (1) (2021) e202100085. doi:10.1002/pamm.202100085

    work page doi:10.1002/pamm.202100085 2021

  55. [55]

    Boolakee, M

    O. Boolakee, M. Geier, L. De Lorenzis, Lattice Boltzmann for linear elastodynamics: periodic problems and Dirichlet boundary conditions (2024). arXiv:2408.01081

    work page arXiv 2024

  56. [56]

    X. Yin, G. Yan, T. Li, Direct simulations of the linear elastic displacements field based on a lattice Boltzmann model, Internat. J. Numer. Meth. Engrg. 107 (3) (2016) 234–251. doi:https://doi.org/10.1002/nme.5167

    work page doi:10.1002/nme.5167 2016

  57. [57]

    Boolakee, M

    O. Boolakee, M. Geier, L. De Lorenzis, Dirichlet and Neumann boundary conditions for a lattice Boltzmann scheme for linear elastic solids on arbitrary domains, Comput. Methods Appl. Mech. Engrg. 415 (2023) 116225. doi:10.1016/j.cma.2023.116225

    work page doi:10.1016/j.cma.2023.116225 2023

  58. [58]

    Auriault, E

    J.-L. Auriault, E. Sanchez-Palencia, Etude du comportement macroscopic d’un mileu poreux saturé deformable, J. Mecanique 16 (4) (1977) 575–603

    work page 1977

  59. [59]

    R. E. Showalter, Di ffusion in poro-elastic media, J. Math. Anal. Appl. 251 (1) (2000) 310–340. doi:10.1006/jmaa.2000.7048

    work page doi:10.1006/jmaa.2000.7048 2000

  60. [60]

    P. J. Dellar, An interpretation and derivation of the lattice Boltzmann method using Strang splitting, Comput. Math. Appl. 65 (2) (2013) 129–141. doi:10.1016/j.camwa.2011.08.047

    work page doi:10.1016/j.camwa.2011.08.047 2013

  61. [61]

    Lee, C.-L

    T. Lee, C.-L. Lin, A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio, J. Comput. Phys. 206 (1) (2005) 16–47. doi:10.1016/j.jcp.2004.12.001

    work page doi:10.1016/j.jcp.2004.12.001 2005

  62. [62]

    T. Lee, P. F. Fischer, Eliminating parasitic currents in the lattice Boltzmann equation method for nonideal gases, Phys. Rev. E 74 (2006) 046709. doi:10.1103/PhysRevE.74.046709. 21

    work page doi:10.1103/physreve.74.046709 2006