Abstract
In this work we propose high-order transparent boundary conditions for the weighted wave equation on a fractal tree, with an application to the modeling of sound propagation in a human lung. This article follows the recent work (Joly et al. in Netw Heterog Media 14(2):205–264, 2019), dedicated to the mathematical analysis of the corresponding problem and the construction of low-order absorbing boundary conditions. The method proposed in this article consists in constructing the exact (transparent) boundary conditions for the semi-discretized problem, in the spirit of the convolution quadrature method developed by Ch. Lubich. We analyze the stability and convergence of the method, and propose an efficient algorithm for its implementation. The exposition is concluded with numerical experiments.
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Acknowledgements
We are deeply grateful to Adrien Semin (TU Darmstadt, Germany) for providing his code Netwaves.
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Appendices
Proof of Theorem 2.4
It remains to prove the upper bound on \(\varvec{\Lambda }_{{\mathfrak {a}}}(\omega )\). Without loss of generality, we will show it for \(\varvec{\Lambda }_{{\mathfrak {n}}}(\omega )\). First, \(\varvec{\Lambda }_{{\mathfrak {n}}}(\omega )\) can be defined via the solution of the frequency-domain problem:
where \(\lambda \in H _\mu ^1({\mathcal {T}})\) solves the boundary-value problem:
Let us define \(\Vert v\Vert _{\omega }:=\int \limits _{{\mathcal {T}}}\mu \left( |\partial _s v|^2+|\omega v|^2\right) \). We proceed as follows:
-
first prove the bound \(|\varvec{\Lambda }_{{\mathfrak {n}}}(\omega )|^2\) by the energy of the solution (notice that \(\lambda (M^*)=1\)):
$$\begin{aligned} \left| \varvec{\Lambda }_{{\mathfrak {n}}}(\omega )\right| ^2 \le |\omega |^2+C_0(1+{\text {Im}}{\omega })\Vert \lambda \Vert _{\omega }^2, \quad C_0>0. \end{aligned}$$(103) -
next show that the energy of the solution is bounded by \(\frac{1}{2}|\varvec{\Lambda }_{{\mathfrak {n}}}(\omega )|^2\), with \(C_0\) as above:
$$\begin{aligned} \begin{aligned} C_0(1+{\text {Im}}{\omega })\Vert \lambda \Vert _{\omega }^2&\le \frac{1}{2}|\varvec{\Lambda }_{{\mathfrak {n}}}(\omega )|^2+C_1\max (1, ({\text {Im}}{\omega })^{-2})|\omega |^2, \quad C_1>0. \end{aligned} \end{aligned}$$(104) -
combine (103) and (104) to obtain the desired bound:
$$\begin{aligned} \left| \varvec{\Lambda }_{{\mathfrak {n}}}(\omega )\right| ^2 \le C\max (1, ({\text {Im}}{\omega })^{-2})|\omega |^2. \end{aligned}$$
Proof of the bound (103) Let \(v_0(s)=\chi (s)\partial _s \lambda \), where \(\chi \in C^1({\mathcal {T}}; {\mathbb {R}})\), \({\text {supp}}\chi (s)\subseteq \varSigma _{0,0}\), \(\chi (M^*)=1\) and \(\chi (M_{0,0})=0\). The weak formulation (102) implies that \(\lambda \) satisfies
Testing the above with \(v_0(s)\), we obtain the following identity on the edge \(\varSigma _{0,0}\), parametrized by \(s\in [0,1]\) (recall that we work with the reference tree, and thus the length of \(\varSigma _{0,0}\) is 1):
Let \(\omega =\omega _r+i\omega _i\), \(\omega _r\in {\mathbb {R}}\) and \(\omega _i>0\). Let us consider the real part of the above:
where in the last identity we used \(\chi (0)=1\) and \(\chi (1)=0\). Combining (105), (106), we deduce
Similarly,
where we used \(\chi (0)=1\), \(\chi (1)=0\) and \(\lambda (0)=1\). Applying to the last integral the Young inequality we obtain the following bound, with \(c_2, c_3>0\),
Inserting (108) into (107) we prove (103).
Proof of the bound (104) Testing the Helmholtz equation corresponding to (102) with \(\omega \lambda (s)\) and integrating by parts we obtain the following identity (recall that \(\lambda (M^*)=1\)):
Taking the imaginary part of the above results in
Multiplying both sides of the above by \(-C_0(1+\omega _i)\omega _i^{-1}\), with \(C_0\) is as in (103), we obtain
It suffices to notice that the left hand side in the above equality is bounded:
where we used the Young inequality. In the above we bound further \(C_0(\omega _i^{-1}+1)\le 2\max (1, \omega _i^{-1})\). Inserting the bound into (109) gives
i.e. (104). Combining (103) and (104) proves the statement of the theorem.
Proof of Lemma 4.1
We first show (86). By definition, \(\tan \omega =i\frac{1-z}{1+z}\), with \(z=\mathrm {e}^{2i\omega }, \, \omega =\omega _r+i\omega _i\). Then,
hence the bound (86). Let us show (87). After straightforward computations,
where the last bound follows by noticing that, for \(\omega _i>0\),
Proof of Proposition 4.3
To prove Proposition 4.3, we need the following auxiliary result.
Lemma C.1
Let \(0<\rho <1\), \(\varepsilon >0\), and \(\lambda _{s,n}^{\varDelta t, \varepsilon }, \, n=0, \ldots , N_t\) be given by (93), with \(N\ge N_t+1\), where \(\max \limits _{k}\left| \varvec{\Lambda }^{s,\varepsilon }\left( \omega _k\right) -\varvec{\Lambda }^s\left( \omega _k\right) \right| <\varepsilon \). Then
Proof
For all \(n=0, \ldots , N_t\),
An upper bound for \(S_1\) follows from the triangle inequality and the assumption of the proposition: \(S_1\le \rho ^{-n}\varepsilon \le \rho ^{-N_t}\varepsilon \) (because \(\rho <1\)).
As for \(S_2\), it suffices to replace \(\varvec{\Lambda }^{s}\left( \omega _k\right) \) in the above sum by \(\sum \nolimits _{\ell =0}^{\infty }\lambda _{s,\ell }^{\varDelta t}\rho ^{\ell }\mathrm {e}^{i\frac{2\pi \ell k}{N}}\), cf. (92), and use the aliasing argument. In particular,
Since \(N^{-1}\sum \nolimits _{k=0}^{N-1}\mathrm {e}^{i\frac{2\pi k(\ell -n)}{N}}=1\) when \(\ell -n\) is a multiple of N and vanishes otherwise, and \(n\le N_t\le N-1\), the above can be rewritten as follows:
The above sum is then bounded:
The result follows by bounding in the above \(n\varDelta t\) by \(N_t\varDelta t\) and combining bounds for \(S_1\) and \(S_2\) into (113). \(\square \)
The bound of Lemma C.1 allows us to quantify the choice of \(\rho \), N in (93).
Proof of Proposition 4.3
The desired bound follows by applying the result of Lemma C.1. In particular, \(C_N(\rho )\) can be estimated by providing an adequate estimate on \(1-\rho ^N=1-\varepsilon ^{\frac{N}{N+N_t-1}}\). Because \(N_t\ge 1\), the function \(N\mapsto 1-\varepsilon ^{\frac{N}{N+N_t-1}}=1-\varepsilon \varepsilon ^{\frac{1-N_t}{N+N_t-1}}\) grows in N. Since, additionally, \(N\ge N_t+1\), we have
Plugging in this bound into (112) yields \(C_N(\rho )\le C(1+(N+N_t)\varDelta t)\) and
from which the desired bound is obtained immediately. \(\square \)
Proof of Lemma 4.2
Let us show (a), which basically follows from Section 5.2.1 in [3]. The frequencies \(\omega _k\) defined in (74), namely,
lie on the circle centered at \(c_{\rho , \varDelta t}\) of radius \(R_{\rho , \varDelta t}\) (this follows from the fact that \(z\mapsto \frac{1-z}{1+z}\) is a homography), with
i.e. \(\omega _k=c_{\rho ,\varDelta t}+R_{\rho ,\varDelta t}\mathrm {e}^{i\psi _k}\), for some \(\psi _k\in [0, 2\pi )\). Hence
and, as \(\rho <1\), \({\text {Im}}\omega _k>\frac{1-\rho }{\varDelta t}\). For \(\rho \) defined in (97),
where the last bound follows from (111). Therefore, as \(\varDelta t<1\), and \(\varepsilon <\frac{1}{2}\),
To show (b), we use the same property (114), which results in
Using (116), and then \(\varepsilon <\frac{1}{2}\), we deduce the following inequality, for some \(C, C'>0\),
\(\square \)
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Joly, P., Kachanovska, M. Transparent boundary conditions for wave propagation in fractal trees: convolution quadrature approach. Numer. Math. 146, 281–334 (2020). https://doi.org/10.1007/s00211-020-01145-9
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DOI: https://doi.org/10.1007/s00211-020-01145-9