June  2019, 14(2): 205-264. doi: 10.3934/nhm.2019010

Wave propagation in fractal trees. Mathematical and numerical issues

1. 

POEMS (UMR 7231 CNRS-INRIA-ENSTA), ENSTA ParisTech, 828 Boulevard des Maréchaux, Palaiseau, F-91120, France

2. 

Technische Universität Darmstadt, Fachgebiet Mathematik, AG Numerik und Wissenschaftliches Rechnen, Dolivostraße 15, Darmstadt, D-64293, Germany

Received  December 2016 Revised  October 2018 Published  April 2019

We propose and analyze a mathematical model for wave propagation in infinite trees with self-similar structure at infinity. This emphasis is put on the construction and approximation of transparent boundary conditions. The performance of the constructed boundary conditions is then illustrated by numerical experiments.

Citation: Patrick Joly, Maryna Kachanovska, Adrien Semin. Wave propagation in fractal trees. Mathematical and numerical issues. Networks & Heterogeneous Media, 2019, 14 (2) : 205-264. doi: 10.3934/nhm.2019010
References:
[1]

Y. AchdouF. CamilliA. Cutrì and N. Tchou, Hamilton-Jacobi equations constrained on networks, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 413-445. doi: 10.1007/s00030-012-0158-1. Google Scholar

[2]

Y. AchdouC. Sabot and N. Tchou, Diffusion and propagation problems in some ramified domains with a fractal boundary, ESAIM: Mathematical Modelling and Numerical Analysis, 40 (2006), 623-652. doi: 10.1051/m2an:2006027. Google Scholar

[3]

Y. AchdouC. Sabot and N. Tchou, Transparent boundary conditions for a class of boundary value problems in some ramified domains with a fractal boundary, C. R. Math. Acad. Sci. Paris, 342 (2006), 605-610. doi: 10.1016/j.crma.2006.02.024. Google Scholar

[4]

Y. AchdouC. Sabot and N. Tchou, Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary, J. Comput. Phys., 220 (2007), 712-739. doi: 10.1016/j.jcp.2006.05.033. Google Scholar

[5]

Y. Achdou and N. Tchou, Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary, C. R. Math. Acad. Sci. Paris, 342 (2006), 611-616. doi: 10.1016/j.crma.2006.02.025. Google Scholar

[6]

Y. Achdou and N. Tchou, Neumann conditions on fractal boundaries, Asymptot. Anal., 53 (2007), 61-82. Google Scholar

[7]

Y. Achdou and N. Tchou, Boundary value problems in ramified domains with fractal boundaries, In Domain decomposition methods in science and engineering XVII, volume 60 of Lect. Notes Comput. Sci. Eng., pages 419-426. Springer, Berlin, 2008. doi: 10.1007/978-3-540-75199-1_53. Google Scholar

[8]

F. Ali Mehmeti and S. Nicaise, Nonlinear interaction problems, Nonlinear Anal., 20 (1993), 27-61. doi: 10.1016/0362-546X(93)90183-S. Google Scholar

[9]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, volume 96 of Monographs in Mathematics, Birkhäuser/Springer Basel AG, Basel, second edition, 2011. doi: 10.1007/978-3-0348-0087-7. Google Scholar

[10]

L. BanjaiC. Lubich and F.-J. Sayas, Stable numerical coupling of exterior and interior problems for the wave equation, Numer. Math., 129 (2015), 611-646. doi: 10.1007/s00211-014-0650-0. Google Scholar

[11]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. Google Scholar

[12]

J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976. Google Scholar

[13]

P. CazeauxC. Grandmont and Y. Maday, Homogenization of a model for the propagation of sound in the lungs, Multiscale Model. Simul., 13 (2015), 43-71. doi: 10.1137/130916576. Google Scholar

[14]

S. M. Cioabă and M. Ram Murty, A First Course in Graph Theory and Combinatorics, volume 55 of Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi, 2009. Google Scholar

[15]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651. doi: 10.1090/S0025-5718-1977-0436612-4. Google Scholar

[16]

A. Georgakopoulos, S. Haeseler, M. Keller, D. Lenz and R. K. Wojciechowski, Graphs of finite measure, J. Math. Pures Appl. (9), 103 (2015), 1093-1131. doi: 10.1016/j.matpur.2014.10.006. Google Scholar

[17]

F. Gesztesy and E. Tsekanovskii, On matrix-valued Herglotz functions, Math. Nachr., 218 (2000), 61-138. doi: 10.1002/1522-2616(200010)218:1<61::AID-MANA61>3.0.CO;2-D. Google Scholar

[18]

P. Joly and A. Semin, Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots, In ESAIM: Proceedings, volume 25, pages 44-67. EDP Sciences, 2008. doi: 10.1051/proc:082504. Google Scholar

[19]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966. Google Scholar

[20]

J. L. Kelley, General Topology, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27. Google Scholar

[21]

P. Kuchment, Graph models for waves in thin structures, Waves in Random Media, 12 (2002), R1-R24. doi: 10.1088/0959-7174/12/4/201. Google Scholar

[22]

B. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco, Calif., 1982. Google Scholar

[23]

B. MauryD. Salort and C. Vannier, Trace theorems for trees, application to the human lungs, Network and Heteregeneous Media, 4 (2009), 469-500. doi: 10.3934/nhm.2009.4.469. Google Scholar

[24]

K. Naimark and M. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees, Proc. London Math. Soc. (3), 80 (2000), 690-724. doi: 10.1112/S0024611500012272. Google Scholar

[25]

K. Naimark and M. Solomyak, Geometry of Sobolev spaces on regular trees and the Hardy inequalities, Russ. J. Math. Phys., 8 (2001), 322-335. Google Scholar

[26]

S. Nicaise, Elliptic operators on elementary ramified spaces, Integral Equations Operator Theory, 11 (1988), 230-257. doi: 10.1007/BF01272120. Google Scholar

[27]

H. PasterkampS. S. Kraman and G. R. Wodicka, Respiratory sounds: advances beyond the stethoscope, American journal of respiratory and critical care medicine, 156 (1997), 974-987. doi: 10.1164/ajrccm.156.3.9701115. Google Scholar

[28]

N. Pozin, S. Montesantos, I. Katz, M. Pichelin, I. Vignon-Clementel and C. Grandmont, A tree-parenchyma coupled model for lung ventilation simulation, Int. J. Numer. Methods Biomed. Eng., 33 (2017), e2873, 30pp. doi: 10.1002/cnm.2873. Google Scholar

[29] M. Redd and B. Simon, Methods of Modern Mathematical Physics, Academic Press, New York-London, 1978.
[30]

J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum, Arch. Ration. Mech. Anal., 160 (2001), 271-308. doi: 10.1007/s002050100164. Google Scholar

[31]

J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strips. â…¡. Convergence of the Ginzburg-Landau functional, Arch. Ration. Mech. Anal., 160 (2001), 309-324. doi: 10.1007/s002050100165. Google Scholar

[32]

D. RueterH.-P. HauberD. DroemanP. Zabel and S. Uhlig, Low-frequency ultrasound permeates the human thorax and lung: A novel approach to non-invasive monitoring, Ultraschall in der Medizin-European Journal of Ultrasound, 31 (2010), 53-62. Google Scholar

[33]

A. Semin, Propagation d'ondes dans des jonctions de fentes minces, PhD thesis, Université de Paris-Sud 11, 2010.Google Scholar

[34]

M. Solomyak, Laplace and Schrödinger operators on regular metric trees: The discrete spectrum case, In Function Spaces, Differential Operators and Nonlinear Analysis (Teistungen, 2001), pages 161-181. Birkhäuser, Basel, 2003. Google Scholar

[35]

M. Solomyak, On approximation of functions from Sobolev spaces on metric graphs, J. Approx. Theory, 121 (2003), 199-219. doi: 10.1016/S0021-9045(03)00033-9. Google Scholar

[36]

M. Solomyak, On the spectrum of the laplacian on regular metric trees, Waves in Random Media, 14 (2004), S155-S171. doi: 10.1088/0959-7174/14/1/017. Google Scholar

[37]

O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer, New York, 2008. Finite and boundary elements, Translated from the 2003 German original. doi: 10.1007/978-0-387-68805-3. Google Scholar

show all references

References:
[1]

Y. AchdouF. CamilliA. Cutrì and N. Tchou, Hamilton-Jacobi equations constrained on networks, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 413-445. doi: 10.1007/s00030-012-0158-1. Google Scholar

[2]

Y. AchdouC. Sabot and N. Tchou, Diffusion and propagation problems in some ramified domains with a fractal boundary, ESAIM: Mathematical Modelling and Numerical Analysis, 40 (2006), 623-652. doi: 10.1051/m2an:2006027. Google Scholar

[3]

Y. AchdouC. Sabot and N. Tchou, Transparent boundary conditions for a class of boundary value problems in some ramified domains with a fractal boundary, C. R. Math. Acad. Sci. Paris, 342 (2006), 605-610. doi: 10.1016/j.crma.2006.02.024. Google Scholar

[4]

Y. AchdouC. Sabot and N. Tchou, Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary, J. Comput. Phys., 220 (2007), 712-739. doi: 10.1016/j.jcp.2006.05.033. Google Scholar

[5]

Y. Achdou and N. Tchou, Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary, C. R. Math. Acad. Sci. Paris, 342 (2006), 611-616. doi: 10.1016/j.crma.2006.02.025. Google Scholar

[6]

Y. Achdou and N. Tchou, Neumann conditions on fractal boundaries, Asymptot. Anal., 53 (2007), 61-82. Google Scholar

[7]

Y. Achdou and N. Tchou, Boundary value problems in ramified domains with fractal boundaries, In Domain decomposition methods in science and engineering XVII, volume 60 of Lect. Notes Comput. Sci. Eng., pages 419-426. Springer, Berlin, 2008. doi: 10.1007/978-3-540-75199-1_53. Google Scholar

[8]

F. Ali Mehmeti and S. Nicaise, Nonlinear interaction problems, Nonlinear Anal., 20 (1993), 27-61. doi: 10.1016/0362-546X(93)90183-S. Google Scholar

[9]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, volume 96 of Monographs in Mathematics, Birkhäuser/Springer Basel AG, Basel, second edition, 2011. doi: 10.1007/978-3-0348-0087-7. Google Scholar

[10]

L. BanjaiC. Lubich and F.-J. Sayas, Stable numerical coupling of exterior and interior problems for the wave equation, Numer. Math., 129 (2015), 611-646. doi: 10.1007/s00211-014-0650-0. Google Scholar

[11]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. Google Scholar

[12]

J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976. Google Scholar

[13]

P. CazeauxC. Grandmont and Y. Maday, Homogenization of a model for the propagation of sound in the lungs, Multiscale Model. Simul., 13 (2015), 43-71. doi: 10.1137/130916576. Google Scholar

[14]

S. M. Cioabă and M. Ram Murty, A First Course in Graph Theory and Combinatorics, volume 55 of Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi, 2009. Google Scholar

[15]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651. doi: 10.1090/S0025-5718-1977-0436612-4. Google Scholar

[16]

A. Georgakopoulos, S. Haeseler, M. Keller, D. Lenz and R. K. Wojciechowski, Graphs of finite measure, J. Math. Pures Appl. (9), 103 (2015), 1093-1131. doi: 10.1016/j.matpur.2014.10.006. Google Scholar

[17]

F. Gesztesy and E. Tsekanovskii, On matrix-valued Herglotz functions, Math. Nachr., 218 (2000), 61-138. doi: 10.1002/1522-2616(200010)218:1<61::AID-MANA61>3.0.CO;2-D. Google Scholar

[18]

P. Joly and A. Semin, Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots, In ESAIM: Proceedings, volume 25, pages 44-67. EDP Sciences, 2008. doi: 10.1051/proc:082504. Google Scholar

[19]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966. Google Scholar

[20]

J. L. Kelley, General Topology, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27. Google Scholar

[21]

P. Kuchment, Graph models for waves in thin structures, Waves in Random Media, 12 (2002), R1-R24. doi: 10.1088/0959-7174/12/4/201. Google Scholar

[22]

B. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco, Calif., 1982. Google Scholar

[23]

B. MauryD. Salort and C. Vannier, Trace theorems for trees, application to the human lungs, Network and Heteregeneous Media, 4 (2009), 469-500. doi: 10.3934/nhm.2009.4.469. Google Scholar

[24]

K. Naimark and M. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees, Proc. London Math. Soc. (3), 80 (2000), 690-724. doi: 10.1112/S0024611500012272. Google Scholar

[25]

K. Naimark and M. Solomyak, Geometry of Sobolev spaces on regular trees and the Hardy inequalities, Russ. J. Math. Phys., 8 (2001), 322-335. Google Scholar

[26]

S. Nicaise, Elliptic operators on elementary ramified spaces, Integral Equations Operator Theory, 11 (1988), 230-257. doi: 10.1007/BF01272120. Google Scholar

[27]

H. PasterkampS. S. Kraman and G. R. Wodicka, Respiratory sounds: advances beyond the stethoscope, American journal of respiratory and critical care medicine, 156 (1997), 974-987. doi: 10.1164/ajrccm.156.3.9701115. Google Scholar

[28]

N. Pozin, S. Montesantos, I. Katz, M. Pichelin, I. Vignon-Clementel and C. Grandmont, A tree-parenchyma coupled model for lung ventilation simulation, Int. J. Numer. Methods Biomed. Eng., 33 (2017), e2873, 30pp. doi: 10.1002/cnm.2873. Google Scholar

[29] M. Redd and B. Simon, Methods of Modern Mathematical Physics, Academic Press, New York-London, 1978.
[30]

J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum, Arch. Ration. Mech. Anal., 160 (2001), 271-308. doi: 10.1007/s002050100164. Google Scholar

[31]

J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strips. â…¡. Convergence of the Ginzburg-Landau functional, Arch. Ration. Mech. Anal., 160 (2001), 309-324. doi: 10.1007/s002050100165. Google Scholar

[32]

D. RueterH.-P. HauberD. DroemanP. Zabel and S. Uhlig, Low-frequency ultrasound permeates the human thorax and lung: A novel approach to non-invasive monitoring, Ultraschall in der Medizin-European Journal of Ultrasound, 31 (2010), 53-62. Google Scholar

[33]

A. Semin, Propagation d'ondes dans des jonctions de fentes minces, PhD thesis, Université de Paris-Sud 11, 2010.Google Scholar

[34]

M. Solomyak, Laplace and Schrödinger operators on regular metric trees: The discrete spectrum case, In Function Spaces, Differential Operators and Nonlinear Analysis (Teistungen, 2001), pages 161-181. Birkhäuser, Basel, 2003. Google Scholar

[35]

M. Solomyak, On approximation of functions from Sobolev spaces on metric graphs, J. Approx. Theory, 121 (2003), 199-219. doi: 10.1016/S0021-9045(03)00033-9. Google Scholar

[36]

M. Solomyak, On the spectrum of the laplacian on regular metric trees, Waves in Random Media, 14 (2004), S155-S171. doi: 10.1088/0959-7174/14/1/017. Google Scholar

[37]

O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer, New York, 2008. Finite and boundary elements, Translated from the 2003 German original. doi: 10.1007/978-0-387-68805-3. Google Scholar

Figure 1.  Left: the limit tree $\mathbb{G} $. Right: the thick tree ${{\mathbb{G}}^{\delta }} $
Figure 2.  General tree. We numbered here the edges. We plotted in red the subtree $ \mathcal{T}_{2, 4} $ and in blue the truncated tree $ \mathcal{T}^1 $
Figure 3.  "1D tree" corresponding to the case $\alpha $ = 0.5
Figure 4.  Example of p-adic tree for p = 2. Left: iterative construction. Right: weight repartition
Figure 5.  Inductive construction of the mesh $\Gamma_n$
Figure 6.  A summary of the results of sections 3.1-3.3
Figure 7.  Polar mesh of the quarter plane
Figure 8.  Plots of $|\mathbf{\Lambda}_\mathfrak{d}(\omega)|$ (left) and $|\mathbf{\Lambda}_\mathfrak{n}(\omega)|$ (right), for $|\omega| < 2\pi$, $\alpha = \mu = 0.6$
Figure 9.  Plots of $\Im\left(\omega^{-1}\mathbf{\Lambda}_\mathfrak{d}(\omega)\right)$ (left) and $\Im\left(\omega^{-1}\mathbf{\Lambda}_\mathfrak{n}(\omega)\right)$ (right), for $|\omega| < 2\pi$, $\alpha = \mu = 0.6$. Remark that $\omega^{-1}\mathbf{\Lambda}_\mathfrak{d}(\omega)$ has a pole in $\omega = 0$, unlike $\omega^{-1}\mathbf{\Lambda}_\mathfrak{n}(\omega)$
Figure 10.  Plots of $ \left|\boldsymbol{\Lambda}_D(\omega)\right| $ (left) for $ \alpha = 0.6 $, $ \mu = 0.2 $ and of $ \left|\boldsymbol{\Lambda}_N(\omega)\right| $ (right) for $ \alpha = 0.6 $, $ \mu = 2 $
Figure 11.  Left row: the dependence of $u(M, t)$ on time for the exact (red solid line) and the truncated tree on 7 generations (blue dashed line). Top: Dirichlet condition. Middle: the first order DtN condition. Bottom: the second order DtN condition.
Right row: the dependence of $u(M, t)$ on time for the exact (red solid line) and the truncated tree on 9 generations (blue dashed line). Top: Dirichlet condition. Middle: the first order DtN condition. Bottom: the second order DtN condition
Figure 12.  ${{\text{L}}^{2}}$-error between exact and approximate solutions, with respect to the number of generations and the order of the approximate boundary condition
Table 1.  L2-error between the exact and approximate solutions, with respect to the number of generations and the order of the approximate boundary condition
Number of generations $n+1$ Dirichlet conditionFirst order conditionSecond order conditionGain with first orderGain with second order
$5$ $0.429$ $0.320$ $1.23\times10^{-1}$1.343.05
$6$ $0.370$ $0.205$ $5.01\times10^{-2}$1.807.35
$7$ $0.217$ $0.075$ $1.37\times10^{-2}$2.8915.83
$8$ $0.083$ $0.018$ $2.72\times10^{-3}$4.5330.5
$9$ $0.023$ $0.0031$ $3.84\times10^{-4}$7.4759.9
Number of generations $n+1$ Dirichlet conditionFirst order conditionSecond order conditionGain with first orderGain with second order
$5$ $0.429$ $0.320$ $1.23\times10^{-1}$1.343.05
$6$ $0.370$ $0.205$ $5.01\times10^{-2}$1.807.35
$7$ $0.217$ $0.075$ $1.37\times10^{-2}$2.8915.83
$8$ $0.083$ $0.018$ $2.72\times10^{-3}$4.5330.5
$9$ $0.023$ $0.0031$ $3.84\times10^{-4}$7.4759.9
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