Consider the problem of scattering of acoustic waves by an elastic body immersed in a compressible, inviscid fluid (air or water) in a half-space with an unbounded rough boundary. This problem is also refereed to as a fluid-solid interaction problem which can be mathematically formulated as an initial-boundary transmission problem and has been widely studied (see, e.g. [25, 23, 32, 30, 24, 26, 21, 1] and the references quoted there). This problem can also be categorized into the class of unbounded rough surface scattering problems, which is the subject of intensive studies in the engineering and mathematics communities. For the rough surface scattering problems, the usual Sommerfeld radiation condition and Silver-Müller radiation condition is not valid anymore due to the unbounded structure. We refer to [9, 8, 6, 5] for the mathematical analysis of the time-harmonic case using both the integral equation method and the variational method.
In most of real-world problems, the model setting not only depends on the space, but also depends on the time. Recently, this class of problems has attracted much attention due to their capability of capturing wide-band signals and modeling more general material and nonlinearity (see, e.g. [28, 10, 37, 36] and the references quoted there). In particular, the analysis of time-dependent scattering problems can be found in [11, 36] for the acoustic case, in [12, 29, 19, 20] for the electromagnetic case including the cases with bounded obstacles, diffraction gratings and unbounded surfaces, and in [1, 21, 26, 38] for the time-dependent fluid-solid interaction problems including the cases with bounded elastic bodies [1, 26], locally rough surfaces  and unbounded layered structures .
The perfectly matched layer (PML) method is a fast and effective method for solving unbounded scattering problems which was originally proposed by Bérenger in 1994 for electromagnetic scattering problems . A large amount of work have been done since then to construct various PML absorption layers [35, 27, 33, 13, 7, 11]. The key idea of the PML method is to surround the computational domain with a specially designed medium containing a finite thickness layer in which the scattered waves decay rapidly regardless of the frequencies and incident angles, thereby greatly reducing the computational complexity of the scattering problems. This makes the PML method a popular approach to solve a variety of wave scattering problems [35, 18, 2, 4].
The convergence of the PML method has always been a topic of interest to mathematicians. There are a lot of works on the convergence of the time-harmonic PML method, most of which focus on the exponential convergence of the PML method in terms of the thickness of the PML layer for the case of bounded scatterers (see, e.g., [27, 22, 14, 2, 16]). In 2009, Chandler-Wilde and Monk extend the PML method to time-harmonic scattering problems by unbounded rough surfaces in , where only the linear convergence of the PML method was established in terms of the PML layer thickness.
Compared with the time-harmonic case, only several results are available for the rigorous convergence analysis of the PML method for time-domain wave scattering problems. For the case of time-domain acoustic scattering by a bounded scatterer, the exponential convergence in terms of the thickness and parameter of the PML layer was proved in  for a circular PML method and in  for an uniaxial PML method. The method used in [11, 15] is based on the Laplace transform and complex coordinate stretching technique. For the case of time-domain electromagnetic scattering by bounded scatterers, the exponential convergence of a spherical PML method was recently shown in  in terms of the thickness and parameter of the PML layer, based on a real coordinate stretching technique associated with in the Laplace domain, where is the Laplace transform variable. Recently, a time-domain PML method was studied in  for the transient acoustic-elastic interaction problem, where a bounded elastic body is immersed in a homogeneous, compressible, inviscid fluid (air or water) in . The well-posedness and stability estimate of the PML solution have been established, but no convergence analysis of the PML method is given in .
In this paper, we study the time-domain PML method for the transient acoustic-elastic interaction problem associated with a bounded elastic body immersed in a homogeneous, compressible, inviscid fluid (air or water) above an unbounded rough surface. Our purpose is to introduce a time-domain PML layer to truncate the unbounded domain of the interaction problem above a finite layer in the direction containing the elastic body, leading to a PML problem in a finite strip domain. The idea used in [11, 15] to construct the PML layer seems difficult to apply to the transient acoustic-elastic interaction problem considered in this paper. Motivated by , we make use of the real coordinate stretching technique associated with in the Laplace domain with the Laplace transform variable . The well-posedness and stability estimate of the PML problem are then established, by employing the Laplace transform and the energy method. Further, we establish the error estimate between the Dirichlet-to-Neumann (DtN) operators of the original problem and the PML problem, which is then used to prove the exponential convergence of the PML method in terms of the thickness and parameters of the PML layer.
The outline of this paper is as follows. In Section 2, we first formulate the transient interaction problem and then use the exact transparent boundary condition (TBC) to reduce the unbounded interaction problem into an equivalent initial-boundary transmission problem in a finite strip domain. In addition, the well-posedness and stability are also studied for the reduced problem. In Section 3, we first propose the time-domain PML method for the acoustic-elastic interaction problem, based on the real coordinate stretching technique, and then establish its exponential convergence in terms of the thickness and parameters of the PML layer. Conclusions are given in Section 4.
2 The acoustic-elastic interaction problem
In this section, we formulate the mathematical formulation of the interaction problem for acoustic and elastic waves with appropriate transmission conditions on the interface between the elastic body and the acoustic medium. In addition, an exact time-domain transparent boundary condition (TBC) is proposed to reformulate the unbounded interaction problem into an initial-boundary value problem in a finite strip domain. We finally establish the well-posedness and stability of solutions to the reduced problem.
We first introduce some basic notions to be used in this paper. Throughout, let , where . Denote by the bounded homogeneous, isotropic elastic body with a Lipschitz boundary immersed in the unbounded domain , where with the boundary described by the smooth function . We assume that lies between the planes and , where and are two constants. Suppose the elastic body is described by a constant mass density . Let be connected and occupied by a compressible fluid with constant density . Define , where the positive constant is assumed to be large enough such that is over , and let . See Figure 1 for the geometric setting of the problem. Finally, define .
Elastic domain. In the elastic body , the elastic displacement is governed by the linear elastodynamic equation:
where is the Lamé operator defined as
are called the stress and strain tensors, respectively, given by
is the identity matrix anddenotes the displacement gradient tensor:
Further, Lamé constants and are assumed to satisfy the condition that and .
Fluid domain. In the unbounded fluid domain , the pressure and the velocity are governed by the conservation and dynamic equations in the time-domain:
Eliminating the velocity from (2.2), we get the wave equation for the pressure :
where is the sound speed and is the acoustic source which is assumed to be supported in and . We assume that satisfies the Dirichlet boundary condition on :
In addition, we impose the Upward Angular Spectrum Representation (UASR) condition on proposed in :
for , where is the inverse Laplace transform,
denotes the Fourier transform of(the Laplace transform of with respect to ) restricted on (the definition and relationship of the Fourier and Laplace transforms are given in Appendix A), with and .
Further, we have the following transmission conditions on the interface between the elastic and fluid media (see ):
(i) The kinematic interface condition
(ii) The dynamic interface condition
where is the unit normal on directed into the exterior of the domain .
To be more precise, the acoustic-elastic interaction problem we consider is that a time-dependent acoustic wave propagates in a fluid domain above a rough surface in which a bounded elastic body is immersed. The problem is to determine the scattered pressure in the fluid domain and the displacement field in the elastic domain at any time. The time-dependent scattering problem can be now modelled by combining (2.1) for the elastic displacement field and (2.3) for the pressure field together with the transmission conditions (2.6)-(2.7), the Dirichlet boundary conditions (2.4) on as well as the homogeneous initial conditions
which can be formulated mathematically as follows:
To study the well-posedness of the scattering problem (2.9), we reformulate it into a transmission problem in the strip domain by using the transparent boundary condition (TBC) on the plane proposed in :
Then (2.9) can be equivalently reduced to the transmission problem (TP) in :
In the remaining part of this section, we establish the well-posedness and stability of the reduced problem (2.11) by using the Laplace transform. The proof is similar to that used in , and so we only present the main results without detailed proofs. To this end, we take the Laplace transform of and , respectively, in (2.11) with respect to and write . Then (2.11) can be reduced to the problem in -domain:
where and is the Dirichlet-to-Neumann (DtN) operator in -domain satisfying .
For any function defined on , the DtN operator is defined by
For with the DtN operator is bounded from to , that is,
where is a constant depending only on and .
For any we have
where denotes the dual product on between and .
To study the -domain problem (2.12) we introduce the Hilbert space , where . The norm of the product space is defined by
where denotes the usual -norm and is defined by
with the Frobenius norm
It is easy to verify that
Hereafter, the expression or means or , respectively, for a generic positive constant which does not depend on any function and important parameters in our model.
The variational formulation of (2.12) can be obtained as follows: Find a solution such that
where the sesquilinear form is defined as
with denoting the Frobenius inner product of the square matrices and .
we obtain that
where use has been made of Lemma 2 (ii) to get the first inequality, , and . This means that the sesquilinear form is uniformly coercive in . By Lemma 2 (i), the trace theorem (see [21, Lemma 2.2]) and the Lax-Milgram theorem, we can obtain the following result on the well-posedness of the -domain problem (2.12) or equivalently its variational formulation (2.17).
For each , the variational problem has a unique solution satisfying that
To prove the well-posedness of the reduced problem (2.11), and establish the convergence of the PML method, we need the following assumptions on the inhomogeneous term :
Further, we always assume that can be extended to with respect to such that
The reduced problem has a unique solution such that
with the estimates
3 The time-domain PML problem
In this section, we shall derive the time-domain PML formulation for the acoustic-elastic interaction problem (2.9). The well-posedness and stability of the PML problem can be established based on a similar method as used in the proof of Theorem 3.2 in . Finally, we prove the exponential convergence of the time-domain PML method via constructing a special PML layer in the -direction, based on the real coordinate stretching technique.
3.1 The PML problem and its well-posedness
Let us first introduce the PML geometry which is presented in Figure 2. Let denote the truncated PML domain and let denote the PML layer with the exterior boundary , where is the thickness of the PML layer. Now, let be an arbitrarily fixed parameter and let us introduce the PML medium property :
where is a positive constant, is a given integer. In what follows, we will take the real part of the Laplace transform variable to be , that is, .
We now derive the PML equation by the technique of change of variables, starting with the real stretched coordinate with
Taking the Laplace transform of the wave equation (2.3) with respect to gives
Denote by the PML extension of the pressure satisfying (3.2). Formally, the technique of change of variables requires to satisfy
Then, by the chain rule and using the fact that, we obtain the PML equation
with the diagonal matrix .
where the unbounded domain is truncated into the finite strip layer by imposing the homogeneous Dirichlet boundary condition on , in view of the exponential decay of the transformed pressure field .
We now prove the well-posedness of the truncated PML problem (3.4a)-(3.4f) by the variational method in the Hilbert space , where and the norm of is defined similarly as that of in (2.15) with replaced by . To this end, use Green’s and Betti’s formulas as well as the transmission conditions (3.4c)-(3.4d) to obtain the following variational formulation of the PML problem (3.4a)-(3.4f): find a solution such that
where the sesquilinear form is defined as
Noting that for , combining the Korn’s inequality, we have
which means that is uniformly coercive in .
Arguing similarly as in the proof of Lemma 2 (noting that the TBC in the -domain is now replaced with the Dirichlet boundary condition), we can obtain the following theorem.
The truncated PML variational problem (3.5) has a unique solution for each with . Further, we have the following estimates