An adaptive finite element DtN method for the elastic wave scattering problem

Abstract

Consider the scattering of an incident wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the scattering problem is formulated as a boundary value problem of the elastic wave equation in a bounded domain. By developing a new duality argument, an a posteriori error estimate is derived for the discrete problem by using the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed to solve the elastic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Appendix A. Transparent boundary conditions

In this section, we show the transparent boundary conditions for the scalar potential functions $\phi ^s, \psi ^s$ and the displacement of the scattered field ${\varvec{u}}^s$ on $\partial B_R$.

In the exterior domain ${\mathbb {R}}^2{\setminus }{{\overline{B}}}_R$, the solutions of the Helmholtz equations (2.5) have the Fourier series expansions in the polar coordinates:

$$\begin{aligned} \phi ^s(r,\theta )=\sum _{n\in {\mathbb {Z}}}\frac{H^{(1)}_n(\kappa _1 r)}{H_{n}^{(1)} (\kappa _1 R)} \phi ^s_n(R)e^{\mathrm{i}n\theta },\quad \psi ^s(r,\theta )=\sum _{n\in {\mathbb {Z}}}\frac{H^{(1)}_n(\kappa _2 r)}{H_{n}^{(1)}(\kappa _2 R)}\psi ^s_n(R)e^{\mathrm{i}n\theta }, \end{aligned}$$

(A.1)

where $H_n^{(1)}$ is the Hankel function of the first kind with order n. Taking the normal derivative of (A.1), we obtain the transparent boundary condition for the scalar potentials $\phi ^s, \psi ^s$ on $\partial B_R$:

$$\begin{aligned} \partial _r\phi ^s= & {} {\mathscr {T}}_1\phi ^s:=\sum \limits _{n\in {\mathbb {Z}}} \frac{\kappa _1 H_n^{(1)'}(\kappa _1 R)}{H_{n}^{(1)}(\kappa _1 R)} \phi ^s_n(R)e^{\mathrm{i}n\theta },\nonumber \\ \partial _r \psi ^s= & {} {\mathscr {T}}_2 \psi ^s:=\sum \limits _{n\in {\mathbb {Z}}} \frac{\kappa _2 H_n^{(1)'}(\kappa _2 R)}{H_{n}^{(1)}(\kappa _2 R)} \psi ^s_n(R)e^{\mathrm{i}n\theta }. \end{aligned}$$

(A.2)

The polar coordinates $(r, \theta )$ are related to the Cartesian coordinates ${\varvec{x}}=(x, y)$ by $x=r\cos \theta , y=r\sin \theta$ with the local orthonormal basis $\{{\varvec{e}}_r, \varvec{e}_\theta \}$, where ${\varvec{e}}_r=(\cos \theta , \sin \theta )^\top , {\varvec{e}}_\theta =(-\sin \theta , \cos \theta )^\top$.

Define a boundary operator for the displacement of the scattered wave

$$\begin{aligned} {\mathscr {B}} {\varvec{u}}^s =\mu \partial _r {\varvec{u}}^s +(\lambda +\mu )(\nabla \cdot {\varvec{u}}^s) \varvec{e}_r\quad \text {on}\,\partial B_R. \end{aligned}$$

Based on the Helmholtz decomposition (2.5) and the transparent boundary condition (A.2), it is shown in [38] that the scattered field ${\varvec{u}}$ satisfies the transparent boundary condition

$$\begin{aligned} {\mathscr {B}}{\varvec{u}}^s =({\mathscr {T}}{\varvec{u}}^s)(R, \theta ):=\sum _{n\in {\mathbb {Z}}}M_n{\varvec{u}}^s_n(R) e^{\mathrm{i}n\theta }\quad \text {on}\,\partial B_R, \end{aligned}$$

(A.3)

where

$$\begin{aligned} {\varvec{u}}^s(R, \theta )=\sum _{n\in {\mathbb {Z}}}{\varvec{u}}^s_n(R) e^{\mathrm{i}n\theta }=\sum _{n\in {\mathbb {Z}}}\big (u_n^{s, r}(R)\varvec{e}_r + u_n^{s, \theta }(R){\varvec{e}}_\theta \big ) e^{\mathrm{i}n\theta } \end{aligned}$$

and $M_n$ is a $2\times 2$ matrix defined by

$$\begin{aligned} M_n=\begin{bmatrix} M_{11}^{(n)} &{} M_{12}^{(n)}\\ M_{21}^{(n)} &{} M_{22}^{(n)} \end{bmatrix} =\frac{1}{\Lambda _n(R)}\begin{bmatrix} N_{11}^{(n)} &{} N_{12}^{(n)}\\ N_{21}^{(n)} &{} N_{22}^{(n)} \end{bmatrix}. \end{aligned}$$

(A.4)

Here

$$\begin{aligned} \Lambda _n(R)=\left( \frac{n}{R}\right) ^2-\alpha _{1n}(R)\alpha _{2n}(R), \quad \alpha _{jn}(R)=\frac{\kappa _j H_n^{(1)'}(\kappa _j R)}{H_{n}^{(1)}(\kappa _j R)}, \end{aligned}$$

(A.5)

and

$$\begin{aligned} N_{11}^{(n)} =&\mu \left( \frac{n}{R}\right) ^2\Big (\alpha _{2n}(R)-\frac{1}{R}\Big ) -\alpha _{2n}(R)\bigg [(\lambda +2\mu )\frac{\kappa _1^2 H_{n}^{(1)''}(\kappa _1 R)}{H_{n}^{(1)}(\kappa _1 R))}\\&\qquad +(\lambda +\mu )\Big (\frac{1}{R}\alpha _{1n}(R)-\left( \frac{n}{R} \right) ^2\Big )\bigg ], \\ N_{12}^{(n)}=&\mu \frac{\mathrm{i}n}{R}\alpha _{1n}(R)\Big (\alpha _{2n}(R)-\frac{1}{R}\Big ) -\frac{\mathrm{i}n}{R} \bigg [(\lambda +2\mu )\frac{\kappa _1^2 H_{n}^{(1)''}(\kappa _1 R)}{H_{n}^{(1)}(\kappa _1 R))}\\&\qquad +(\lambda +\mu )\Big (\frac{1}{R}\alpha _{1n}(R)-\left( \frac{n}{R} \right) ^2\Big )\bigg ],\\ N_{21}^{(n)} =&-\mu \frac{\mathrm{i}n}{R}\alpha _{2n}(R)\Big (\alpha _{1n}(R)-\frac{1}{R}\Big ) +\mu \frac{\mathrm{i}n}{R}\frac{\kappa _2^2 H_{n}^{(1)''}(\kappa _2 R)}{H_{n}^{(1)}(\kappa _2 R)},\\ N_{22}^{(n)} =&\mu \left( \frac{n}{R}\right) ^2 \Big (\alpha _{1n}(R)-\frac{1}{R}\Big ) -\mu \alpha _{1n}(R)\frac{\kappa _2^2 H_{n}^{(1)''}(\kappa _2 R)}{H_{n}^{(1)}(\kappa _2 R)}. \end{aligned}$$

The matrix entries $N_{ij}^{(n)}, i,j=1, 2$ can be further simplified. Recall that the Hankel function $H_n^{(1)}(z)$ satisfies the Bessel differential equation

$$\begin{aligned} z^2 H_n^{(1)''}(z)+z H_n^{(1)'} (z)+(z^2-n^2) H_{n}^{(1)}(z)=0. \end{aligned}$$

We have from straightforward calculations that

$$\begin{aligned} N_{11}^{(n)}&=-\alpha _{2n}(R)\Bigg [(\lambda +2\mu )\bigg [-\frac{1}{R^2} \Big (\kappa _j R \frac{H^{(1)'}_n(\kappa _j R)}{H^{(1)}_n(\kappa _j R)}+\left( (\kappa _j R)^2-n^2\right) \Big )\bigg ]\\&\qquad +(\lambda +\mu )\left( \frac{1}{R}\alpha _{1n}(R)-\left( \frac{n}{R} \right) ^2\right) \Bigg ] +\mu \left( \frac{n}{R}\right) ^2\left( \alpha _{2n} (R)-\frac{1}{R}\right) \\&\quad =-\alpha _{2n}(R)\Bigg [-\left( \frac{\lambda +2\mu }{R}\right) \alpha _{1n} (R)-(\lambda +2\mu )\kappa _1^2\\&\qquad +(\lambda +2\mu )\left( \frac{n}{R}\right) ^2 +\left( \frac{\lambda +\mu }{R}\right) \alpha _{1n}(R)\\&\qquad -(\lambda +\mu )\left( \frac{n}{R}\right) ^2\Bigg ] +\mu \left( \frac{n}{R}\right) ^2\left( \alpha _{2n}(R)-\frac{1}{R}\right) \\&=-\frac{\mu }{R}\left[ \left( \frac{n}{R}\right) ^2-\alpha _{1n}(R)\alpha _{2n} (R)\right] +\alpha _{2n}(R)\omega ^2\\&= -\frac{\mu }{R}\Lambda _n(R)+\alpha _{2n}(R)\omega ^2,\\ N_{12}^{(n)}&= -\frac{\mathrm{i}n}{R}\Bigg [(\lambda +2\mu )\bigg [-\frac{1}{R^2}\Big (\kappa _j R \frac{H^{(1)'}_n(\kappa _j R)}{H^{(1)}_n(\kappa _j R)}+\left( (\kappa _j R)^2-n^2\right) \Big )\bigg ]\\&\qquad +(\lambda +\mu )\left( \frac{1}{R}\alpha _{1n}(R)-\left( \frac{n}{R} \right) ^2\right) \Bigg ] \\&\qquad +\frac{\mathrm{i}n\mu }{R}\alpha _{1n}(R)\alpha _{2n}(R)-\mu \frac{\mathrm{i}n}{R^2}\alpha _{1n}(R)\\&= -\frac{\mathrm{i}n}{R}\left[ -\frac{\mu }{R}\alpha _{1n}(R)+\mu \left( \frac{n}{R}\right) ^2 -(\lambda +2\mu )\kappa _1^2\right] \\&\qquad +\frac{\mathrm{i}n\mu }{R}\alpha _{1n}(R)\alpha _{2n}(R)-\frac{\mathrm{i}n}{R^2}\mu \alpha _{1n}(R)\\&= -\frac{\mathrm{i}n\mu }{R}\Lambda _n(R)+\frac{\mathrm{i}n}{R}\omega ^2,\\ N_{21}^{(n)}&= -\mu \frac{\mathrm{i}n}{R}\alpha _{2n}(R)\alpha _{1n}(R)+\frac{\mathrm{i}n\mu }{R^2}\alpha _{2n}(R) \\&\qquad +\mu \frac{\mathrm{i}n}{R}\left( \frac{-1}{R^2}\right) \left( R\alpha _{2n}(R)+(\kappa _2 R)^2-n^2\right) \\&= -\mu \frac{\mathrm{i}n}{R}\alpha _{1n}(R)\alpha _{2n}(R)+\frac{\mathrm{i}n\mu }{R^2}\alpha _{2n}(R) -\mu \frac{\mathrm{i}n}{R^2}\alpha _{2n}(R)-\frac{\mathrm{i}n\mu }{R}\kappa _2^2+\mathrm{i}\mu \left( \frac{n}{R}\right) ^3\\&= \frac{\mathrm{i}\mu n}{R}\Lambda _n(R)-\frac{\mathrm{i}n}{R}\omega ^2,\\ N_{22}^{(n)}&= \mu \left( \frac{n}{R}\right) ^2\alpha _{1n}(R)-\frac{\mu }{R}\left( \frac{n}{R} \right) ^2-\mu \alpha _{1n}(R)\frac{-1}{R^2}\left( R\alpha _{2n}(R)+(\kappa _2 R)^2-n^2\right) \\&=\mu \left( \frac{n}{R}\right) ^2\alpha _{1n}(R)-\frac{\mu }{R}\left( \frac{n}{R }\right) ^2+\frac{\mu }{R}\alpha _{1n}(R)\alpha _{2n}(R)\\&\qquad +\alpha _{1n}(R)\mu \kappa _2^2-\mu \left( \frac{n}{R}\right) ^2\alpha _{1n}(R)\\&=-\frac{\mu }{R}\left( \left( \frac{n}{R}\right) ^2-\alpha _{1n}(R)\alpha _{2n} (R)\right) +\alpha _{1n}(R)\omega ^2\\&=-\frac{\mu }{R}\Lambda _n(R)+\alpha _{1n}(R)\omega ^2. \end{aligned}$$

Substituting the above into (A.3), we obtain

$$\begin{aligned} {\mathscr {B}}{\varvec{u}}^s&= {\mathscr {T}}{\varvec{u}}^s =\sum \limits _{n\in {\mathbb {Z}}}\frac{1}{\Lambda _n}\bigg \{ \Big [\Big (-\frac{\mu }{R}\Lambda _n(R)+\alpha _{2n}(R)\omega ^2\Big ) u_n^{s, r}(R) \nonumber \\&\quad +\Big (-\frac{\mathrm{i}n\mu }{R}\Lambda _n(R)+\frac{\mathrm{i}n}{R}\omega ^2\Big ) u_n^{s, \theta }(R)\Big ] {\varvec{e}}_r \nonumber \\&\quad + \Big [\Big (\frac{\mathrm{i}\mu n}{R}\Lambda _n(R)-\frac{\mathrm{i}n}{R}\omega ^2 \Big )u_n^{s, r}(R) \nonumber \\&\quad +\Big (-\frac{\mu }{R}\Lambda _n(R)+\alpha _{1n}(R)\omega ^2 \Big )u_n^{s, \theta }(R)\Big ]{\varvec{e}}_{\theta } \bigg \} e^{\mathrm{i}n\theta }. \end{aligned}$$

(A.6)

Lemma A.1

Let $z>0$. For sufficiently large |n|, $\Lambda _n(z)$ admits the following asymptotic property

$$\begin{aligned} \Lambda _n(z)=\frac{1}{2}(\kappa _1^2+\kappa _2^2)+{\mathcal {O}}\Big (\frac{1}{|n|}\Big ). \end{aligned}$$

Proof

Using the asymptotic expansions of the Hankel functions [46]

$$\begin{aligned} \frac{H_n^{(1)'}(z)}{H_n^{(1)}(z)}=-\frac{|n|}{z}+\frac{z}{2|n|}+{\mathcal {O}} \Big (\frac{1}{|n|^2}\Big ), \end{aligned}$$

we have

$$\begin{aligned} \alpha _{jn}(z)=\frac{\kappa _j H_n^{(1)'}(\kappa _j z)}{H_{n}^{(1)}(\kappa _j z)}=-\frac{|n|}{z}+\frac{\kappa ^2_j z}{2|n|}+{\mathcal {O}} \Big (\frac{1}{|n|^2}\Big ). \end{aligned}$$

(A.7)

A simple calculation yields that

$$\begin{aligned} \Lambda _n(z)=\left( \frac{n}{z}\right) ^2-\alpha _{1n}(z)\alpha _{2n}(z)=\frac{1}{2 }(\kappa _1^2+\kappa _2^2)+{\mathcal {O}}\Big (\frac{1}{|n|}\Big ), \end{aligned}$$

which completes the proof.$\square$