Abstract
We consider the scattering of elastic waves by highly oscillating anisotropic periodic media with bounded support. Applying the two-scale homogenization, we first obtain a constant coefficient second-order partial differential elliptic equation that describes the wave propagation of the effective or overall wave field. We further pursue a higher-order homogenization with the help of complimentary boundary correctors and provide a detailed analysis on the rate of higher-order convergence. Finally we provide preliminary numerical examples to demonstrate the higher-order homogenization.
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Acknowledgements
The work was initiated when the authors participated the annual program on “Mathematics and Optics” (2017–2018) at the Institute for Mathematics and its Applications (IMA) at the University of Minnesota. Y.-H. Lin would like to thank the support from IMA for his stay at the University of Minnesota. S. Meng was partially supported by the Air Force Office of Scientific Research under award FA9550-18-1-0131.
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Appendix
Appendix
In the end of this paper, we offer basic materials in analysing the elastic scattering in periodic media.
1.1 5.1 The Dirichlet to Neumann Map
Let \(\mathbf{u}\) satisfy the Navier’s equation in the exterior domain
and \(\mathbf{u}\) has a decomposition that satisfies the Kupradze radiation condition. Let \(B_{R}\) be a sufficiently large ball such that \(\varOmega \subset B_{R}\). In the case that \(\varOmega \subset \mathbb{R} ^{3}\), we introduce the polar coordinates \(r\), \(\theta \), \(\phi \) and the unit vectors \(\widehat{r}\), \(\widehat{\theta }\), \(\widehat{\phi }\). The \(\theta \) coordinate corresponds to the angle from the \(z\)-axis, \(\theta \in [0, \pi ]\), and the \(\phi \) coordinate corresponds to the angle in the \((x, y)\)-plane, \(\phi \in [0, 2\pi ]\). Let \(Y_{nm}\) be the spherical harmonic
Now we let \(U_{nm}\) and \(V_{nm}\) be the vector spherical harmonics defined by
where \(\lambda _{n} = n(n+1)\). The vectors \(Y_{nm} \widehat{r}\), \(U_{nm}\), \(V_{nm}\) form an orthonormal basis for \(L^{2}(S)\) where \(S\) denotes the unit sphere. Then \(\mathbf{u}\) on \(\partial B_{R}\) has the following series expansion
where \((\cdot ,\cdot )\) denotes the \(L^{2}(S)\) inner product. One can correspondingly express \(T_{\boldsymbol{\nu }} \mathbf{u}\) on \(\partial B_{R}\) as (see [13])
The coefficients \(a_{n}\), \(b_{n}\), \(c_{n}\), \(d_{n}\) are given by
where
Now for any functions \(\mathbf{w}\) and \(\mathbf{u}\) that satisfy the Kupradze radiation condition (4), one can directly obtain from (91) and (92) that
We remark that when \(\varOmega \subset \mathbb{R}^{2}\), the above equality can be derived in a similar way [4].
Let \(B_{R}\subset \mathbb{R}^{d}\) be a ball of radius \(R>0\), then the Dirichlet to Neumann (DN) map was given by [4].
Definition 1
For any \(\mathbf{g}\in (H^{1/2}(\partial B_{R}) )^{d}\), the DN map
where \(\mathbf{u}\in (H^{1}_{\mathit{loc}}(\mathbb{R}^{d}\setminus \overline{B}_{R}) )^{d}\) is a solution of the Navier’s equation \(\Delta ^{*}\mathbf{u}+\omega ^{2}\mathbf{u}=0\) in \(\mathbb{R}^{d} \setminus \overline{B_{R}}\) and \(\mathbf{u}\) satisfies the Kupradze radiation condition (4) at infinity.
Notice that the DN map \(\varLambda \) is a bounded operator, so that it helps to reduce the scattering problem in unbounded domain to a bounded domain, and we refer readers to [4, Sect. 2] for detailed discussions.
1.2 5.2 Derivation of the Homogenized Equation
Consider the simplest linear elliptic system of the homogenization theory. The periodic homogenization theory was studied by [10, 14] and we refer readers to these references for the comprehensive study. We are concerned with the divergence form second order elliptic operators with rapidly oscillating periodic coefficients,
We assume the coefficients \(\mathbf{A}(y)= (a_{\mathit{ijk}\ell }(y) )\) with \(1\leq i,j,k, \ell \leq d\) for the dimension \(d\geq 2\) is real, bounded and measurable such that \(\mathbf{A}\) satisfies
for all symmetric matrix \((\varepsilon _{ij})_{1\leq i,j\leq d}\), and
for some constant \(\mu >0\).
Given \(\mathbf{F}\in (H^{-1}(\varOmega ) )^{d}\), let \(\mathbf{u} ^{\epsilon }\in (H_{0}^{1}(\varOmega ) )^{d}\) be a solution of
where \(\varOmega \) is a bounded Lipschitz domain in \(\mathbb{R}^{d}\). By the Lax–Milgram theorem, we have
where the constant \(C\) independent of \(\epsilon \). Note that \(\mathbf{u}^{\epsilon }\in (H_{0}^{1}(\varOmega ) )^{d}\) is a weak solution of (95) if for all \(\boldsymbol{\varphi }\in (H _{0}^{1}(\varOmega ) )^{d}\), we have
Next, we want to derive the homogenized equation by using the following asymptotic analysis. We consider \(\mathbf{u}^{\epsilon }\) to be the perturbation of \(\mathbf{u}^{{\scriptscriptstyle (0)}}\) with respect to \(\epsilon \)-parameter. Moreover, by observing the elliptic operator \(\mathcal{L}_{\epsilon }\), we introduce the famous two-scale homogenization method in the homogenization theory: Let us regard \(x=x\), and \(y=\frac{x}{\epsilon }\) as two independent parameters. Let
be the asymptotic expansion of \(u_{\epsilon }\), where
In addition,
which means under our two-scaled method, the operator \(\nabla =\nabla _{x}+\frac{1}{\epsilon }\nabla _{y}\). Therefore, (95) will become
We point out that the derivation of the homogenized equation did not need to take care of the boundary condition of certain equations. Expand (96) and compare it with the same \(\epsilon ^{N}\)-orders (for \(N=0,-1,-2\)), so we get
Recall that for the periodic elliptic equation
then we have
by using the divergence theorem. For \(O(\frac{1}{\epsilon ^{2}})\) term, this equation is solvable because the right hand side is zero. In further, we multiply \(\mathbf{u}^{{\scriptscriptstyle (0)}}(x,y)\) on both sides and integrate by parts, which will imply
which gives us the information that
and we know that \(\mathbf{u}_{0}\) is independent of \(y\).
Now, for the second term \(O(\frac{1}{\epsilon })\), the second term on the right hand side should be zero since \(\nabla _{y}\mathbf{u}^{{\scriptscriptstyle (0)}}(x)=0\). Solve the equation
formally. Note that since \(\mathbf{A}(y)\) is \(Y\)-periodic, then the equation is solvable for \(\mathbf{u}^{{\scriptscriptstyle (1)}}\) if
By using the separation of variables, we put the ansatz
with \(\mathbf{u}^{{\scriptscriptstyle (1)}}=(u^{{\scriptscriptstyle (1)}}_{\alpha })_{1\leq \alpha \leq d}\) such that
Moreover, the corrector \(\chi _{\alpha j \beta }\) is \(Y\)-periodic and solves the cell problem
and plug \(\mathbf{u}^{{\scriptscriptstyle (1)}}\) to the \(O(\frac{1}{\epsilon })\) equation (97) to obtain
Finally plug \(\mathbf{u}^{{\scriptscriptstyle (1)}}(x,y)=\boldsymbol{\chi }(y)\nabla _{x}\mathbf{u}^{{\scriptscriptstyle (0)}}\) into the \(O(1)\) equation and examine the solvability condition for \(\mathbf{u}^{{\scriptscriptstyle (2)}}(x,y)\), we have
where the first term vanishes by the periodicity of \(\mathbf{A}\) and \(\boldsymbol{\chi }\). Thus, we can obtain that \(\mathbf{u}^{{\scriptscriptstyle (0)}} \in ( H_{0}^{1}(\varOmega ) )^{d}\) is a solution of
where
where \(\overline{\mathbf{A}}\) is the (constant) homogenized operator and we call (98) to be the homogenized equation. In addition, \(\overline{\mathbf{A}}=(\overline{a}_{\mathit{ijk}\ell })_{1\leq i,j,k, \ell \leq d}\) and
For the rigorous derivation of the homogenized equation, we need to use a famous result, which is called the Div-Curl lemma. We skip the rigorous analysis here and refer readers to the lecture note [26] for more details.
Note that \(\overline{\mathcal{L}}:=-\nabla \cdot (\overline{ \mathbf{A}}\nabla )\) is the homogenized second order elliptic operator with respect to \(\mathbf{A}\) and we want to prove \(\overline{ \mathcal{L}}\) is an elliptic operator with constant coefficients.
Theorem 5
The homogenized operator \(\overline{\mathcal{L}}\) satisfies that
1. \(\overline{\mathcal{L}}\)is an elliptic operator, which means
for some constant\(\mu _{1}>0\).
2. The effective coefficient \(\overline{a}_{\mathit{ijk}\ell }\) is major and minor symmetric provided \(a_{\alpha \beta \gamma \delta }\) is major and minor symmetric.
Proof
It is easy to see that \(|\overline{a}_{\mathit{ijk}\ell }|\leq C\) by using (99) and the ellipticity of \(A(y)\), for some constant \(C>0\). It remains to show \(\overline{a}_{\mathit{ijk}\ell } \varepsilon _{ij}\varepsilon _{k\ell }\geq \mu _{1}\sum_{i,j=1}^{d}| \varepsilon _{ij}|^{2}\) for some constant \(\mu _{1}>0\). We can rewrite (99) as
where \(\delta _{s\alpha }\) is the standard Kronecker delta (i.e., \(\delta _{s\alpha }=1\) if \(s=\alpha \), and \(\delta _{s\alpha }=0\) otherwise). Hence, for \(\varepsilon =(\varepsilon _{ij})\in \mathbb{R} ^{d\times d}\), we have
If \(\overline{a}_{\mathit{ijk}\ell }\varepsilon _{ij}\varepsilon _{k\ell }=0\) for some \(\varepsilon =(\varepsilon _{ij})\in \mathbb{R}^{d\times d}\), then \(y_{i}\varepsilon _{i\beta }+\chi _{\beta ij}\) must be a constant. Recall that \(\chi _{\beta ij}(y)\) is \(Y\)-periodic, so this implies that \(\varepsilon =0\). This means that there exists \(\mu _{1}>0\) such that (100) holds. □
1.3 5.3 Tools and Estimates
In the last part, for the completeness of this paper, we provide some elliptic estimate where we have utilized in previous sections. The following theorem was proved in [6, Theorem 5.7] for the scalar case. It will hold for the vector case. For completeness, we provide the theorem and its proof as follows.
Theorem 6
Trace Theorem
Let\(\mathbf{A}=(a_{\mathit{ijk}\ell })_{1\leq i,j,k,\ell \leq d}\)be a four tensor satisfying the ellipticity condition (100) and\(\varOmega \subset \mathbb{R}^{d}\)be a bounded domain with a\(C^{\infty }\)-smooth boundary, for\(d\geq 2\). The (conormal) mapping\(\mathit{Tr}:\mathbf{u}\to \frac{\partial \mathbf{u}}{ \partial \boldsymbol{\nu }_{\mathbf{A}}}:=(\mathbf{A}\nabla \mathbf{u})\cdot \boldsymbol{\nu }\)defined in\(C^{\infty }(\overline{ \varOmega })\)can be continuously extended to a linearly continuous mapping (still denote by\(\mathit{Tr}\)) from\(H^{1}(\varOmega ,\mathbf{A})\)to\(H^{-1/2}( \partial \varOmega )\), where\(H^{1}(\varOmega ,\mathbf{A})\)is the space equipped with the graph norm
Proof
Let \(\boldsymbol{\varphi }\in (C^{\infty }(\overline{\varOmega }) )^{d}\) be a test function and \(\mathbf{u}\in C^{\infty }(\overline{ \varOmega };\mathbb{R}^{d})\). The integration by parts formula gives
By the standard density arguments, the above equation holds for \(\boldsymbol{\varphi }\in (H^{1}(\varOmega ) )^{d}\) so that
for any \(\boldsymbol{\varphi }\in (H^{1}(\varOmega ) )^{d}\), \(\mathbf{u}\in (C^{\infty }(\overline{\varOmega }) )^{d}\), where constant \(C>0\) is a constant independent of \(\boldsymbol{\varphi }\) and \(\mathbf{u}\). Let \(\mathbf{g}\in (H^{1/2}(\partial D) )^{d}\), by using the trace theorem, then there exists a function \(\boldsymbol{\varphi }\in (H^{1}(\varOmega ) )^{d}\) such that \(\gamma _{\partial \varOmega }\boldsymbol{\varphi }=\mathbf{f}\), where \(\gamma _{\partial \varOmega }\) stands for the trace operator. Continuing the inequality (101) and the trace theorem,
for any \(\mathbf{f} \in (H^{1/2}(\partial \mathbf{)} )^{d}\), \(\mathbf{u}\in (C^{\infty }(\overline{\varOmega }) )^{d}\).
Hence, the mapping
defines a continuous linear operator and from the duality argument,
Therefore, the linear mapping \(\mathit{Tr}: \mathbf{u} \to (\mathbf{A}\nabla u) \cdot \boldsymbol{\nu }\) defined on \((C^{\infty }(\overline{ \varOmega }) )^{d}\) is continuous under the norm \(H^{1}(\varOmega , \mathbf{A})\). Thus, the assertion follows from the density arguments. □
Let \(\mathbf{C}=(C_{\mathit{ijk}\ell })\) be an anisotropic elastic four tensor and \(\mathbf{C}_{0}\) be a constant isotropic elastic tensor defined by (2), which satisfy all the conditions given in Sect. 1. Next, we provide the stability estimate for the following transmission problem. The scalar case was demonstrated in [6, Sect. 5] and here we generalize the result to a system version.
Theorem 7
Let\(\varOmega \subset \mathbb{R}^{d}\)be a bounded\(C^{\infty }\)-smooth domain. Given\(\mathbf{f}\in (H^{1/2}(\partial \varOmega ) )^{d}\)and\(\mathbf{g}\in (H^{-1/2}(\partial \varOmega ) )^{d}\). Let\(\mathbf{u}\in (H^{1}(\varOmega ) )^{d}\)and\(\mathbf{v}\in (H^{1}_{\mathit{loc}}(\mathbb{R}^{d}\setminus \overline{\varOmega }) )^{d}\)be the solutions of the following transmission problem
where\(T_{\boldsymbol{\nu }}\)is the boundary traction operator given by (3), \(\omega \in \mathbb{R}\)is not an eigenvalue of the transmission problem (102) and\(v\)satisfies the Kupradze radiation condition (4). Then for any ball\(B_{R}\)with\(\varOmega \subset B_{R}\), there exists a constant\(C_{R}>0\)such that
Proof
Firstly, by using similar arguments in [4, Sect. 2] and [6, Sect. 5], the elastic scattering problem (102) is equivalent to the following transmission problem: Let \(\mathbf{u}\in (H^{1}(\varOmega ) )^{d}\) and \(\mathbf{v}\in (H^{1}(B_{R}\setminus \overline{ \varOmega }) )^{d}\) be the solutions of
where \(\varLambda \) is the DN map defined by (93) on \(\partial B_{R}\). Furthermore, by using [4, Lemma 2.8], the DN map \(\varLambda \) is a bounded operator and \(\varLambda \) can decomposed into \(\varLambda =\varLambda _{1}+ \varLambda _{2}\), where \(-\varLambda _{1}\) is a positive operator and \(\varLambda _{2}\) is a compact operator from \((H^{1/2}(\partial B_{R}) )^{d}\) to \((H^{-1/2}(\partial B_{R}) )^{d}\).
Next, let \(\mathbf{v}_{\mathbf{f}}\in (H^{1}(B_{R}\setminus \overline{ \varOmega }) )^{d}\) be the unique solution of the Navier’s equation in the exterior domain
By straight forward calculation, it is not hard to see that the variational formula of (104) can be written as follows: Find a function \(w\in H^{1}(B_{R})\) such that
for any test function \(\boldsymbol{\phi }\in (H^{1}(B_{R}) )^{d}\), where \(\mathbf{C} _{0}\) is a constant elastic tensor defined by (2). By using the integration by parts, one can easily see that \(\mathbf{u}=\mathbf{w}|_{\varOmega }\) and \(\mathbf{v}=\mathbf{w}|_{B_{R}\setminus \overline{\varOmega }}-\mathbf{v} _{\mathbf{f}}\) satisfy (104).
Now, let us consider two bilinear forms
and
Then we can rewrite the problem (105) as finding a function \(\mathbf{w}\in (H^{1}(B_{R}) )^{d}\) such that
Since \(-\varLambda _{1}\) is a positive operator, one can conclude that \(b_{1}(\cdot ,\cdot )\) is strictly coercive. Therefore, from the Lax–Milgram theorem, one can see that the operator \(A: (H^{1}(B _{R}) )^{d} \to (H^{1}(B_{R}) )^{d}\) defined by \(b_{1}( \mathbf{w},\boldsymbol{\phi })=(A\mathbf{w},\boldsymbol{\phi })_{H ^{1}(B_{R})}\) is invertible and has a bounded inverse. On the other hand, since \(\varLambda _{2}\) is a compact operator from \((H^{1/2}( \partial B_{R}) )^{d}\to (H^{-1/2}(\partial B_{R}) )^{d}\) and \((H^{1}(B_{R}) )^{d} \to (L^{2}(B_{R}) )^{d}\) is a compact embedding, then it is not hard to see that the operator \(B: (H^{1}(B_{R}) )^{d} \to (H^{1}(B_{R}) )^{d}\) defined by \(b_{2}(\mathbf{w},\boldsymbol{\phi })=(B\mathbf{w}, \boldsymbol{\phi })_{H^{1}(B_{R})}\) is compact. Hence, by using [6, Theorem 5.16], one can derive that the existence of the transmission problem (104) from the uniqueness of (104) and the stability estimate (103) holds automatically. □
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Lin, YH., Meng, S. Leading and Second Order Homogenization of an Elastic Scattering Problem for Highly Oscillating Anisotropic Medium. J Elast 137, 177–217 (2019). https://doi.org/10.1007/s10659-019-09725-z
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DOI: https://doi.org/10.1007/s10659-019-09725-z