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1. Introduction
The phenomena of electromagnetic scattering by open cavities have attracted much attention due to the significant industrial and military applications in such areas as antenna synthesis and stealth design. The underlying scattering problems have been extensively studied by many researchers in the engineering and applied mathematics communities. We refer to the survey [20] and the references cited therein for a comprehensive account on analysis, computation, and optimal design of the cavity scattering problems.
In applications, one of particular interests is the radar cross section (RCS) analysis, which aims at how to mitigate or amplify a signal. The RCS is a quantity which measures the detectability of a target by radar system. Deliberate control in the form of enhancement or reduction of the RCS of a target is of high importance in the electromagnetic interference, especially in the aircraft detection and the stealth design. Since the problems are imposed in open domains and the solutions may have singularities, it presents challenging and significant mathematical and computational questions on precise modeling and accurate computing for the cavity scattering problems in order to successfully implement any desired control of the RCS. This paper concerns the numerical solutions of the open cavity scattering problems. We intend to develop an adaptive finite element method with the perfect matched layer (PML) technique to overcome the difficulties.
The PML technique was first proposed by Bérenger for solving the time-dependent Maxwell equations [7]. Due to its effectiveness, simplicity and flexibility, the PML technique is widely used in computational wave propagation [15, 24, 25, 14]. It has been recognized as one of the most important and popular approaches for the domain truncation. Under the assumption that the exterior solution is composed of outgoing waves only, the basic idea of the PML technique is to surround the domain of interest with a layer of finite thickness of a special medium, which is designed to either slow down or attenuate all the waves propagating into the PML layer from inside of the computational domain. As either the PML parameter or the thickness of the PML layer tends to infinity, the exponential convergence error estimate was obtained in [17, 19] between the solution of the PML problem and the solution of the Helmholtz-type scattering problem. The convergence analysis of the PML problems for the three-dimensional electromagnetic scattering was stuided in [6, 8, 9, 21].
In practice, if we use a very thick PML layer and a uniform finite element mesh, it requires very excessive grids points and hence involves more computational cost. In contrast, if we choose a thin PML layer, it is inevitable to have a rapid variation of the PML medium property, which renders a very fine mesh in order to reach the desired accuracy. On the other hand, the solutions of the open cavity scattering problem may have singularities due to the existence of corners of cavities or the discontinuity of the dielectric coefficient for the filling medium. These singularities slow down the speed of convergence if uniform mesh refinements are applied. The a posteriori error estimate based adaptive finite element method is an ideal tool to handle these issues.
A posteriori error estimators are computable quantities in terms of numerical solutions and data. They measure the error between the numerical solution and the exact solution without requiring any a priori information of the exact solution. A reliable a posteriori error estimator plays a crucial role in an adaptive procedure for mesh modification such as refinement or coarsening. Since the work of Babuška and Rheinboldt [4], the study of adaptive method based on a posteriori error estimator has become an active research topic in scientific computing. Some relevant work can be found in [5, 2, 1, 11, 10] on the adaptive finite element method. We refer to [22, 23, 12, 13] for studies on the scattering problems by using the a posteriori error estimate based adaptive finite element method.
Motivated by the work of Chen and Liu [12], we develop an adaptive procedure, which combines the finite element method and the PML technique, to solve the open cavity scattering problems. Specifically, we consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity embedded in an infinite ground plane. The ground plane and the cavity wall are assumed to be perfect electric conductors. The cavity is assumed to be filled with some inhomogeneous medium, which may protrude out of the cavity to the upper half-space in a finite extend. The upper half-space above the ground plane and the protruding part of the cavity is assumed to be filled with some homogeneous medium. By assuming invariance of the cavity in the direction, we consider two fundamental polarizations: transverse magnetic (TM) and transverse electric (TE) polarizations, where the three-dimensional Maxwell equations can be reduced to the two-dimensional Helmholtz equation. We restrict our attention to the numerical solutions of the TM and TE polarizations. In each polarization, the scattering problem is reduced equivalently into a boundary value problem of the two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition. Computationally, the PML technique is utilized to truncate the infinite half-space above the ground plane and the homogeneous Dirichlet boundary condition is imposed on the outer boundary of the PML layer. The a posteriori error estimate is deduced between the solution of the original scattering problem and the finite element solution of the truncated PML problem. The a posteriori error estimate takes account both of the finite element discretization error and the truncation error of the PML method. The PML truncation error has a nice feature of exponential decay in terms of the PML medium parameter and the thickness of the layer. Based on this property, the proper PML medium parameter and the thickness of the layer can be chosen to make the PML error negligible compared with the finite element discretization error. Once the PML region and the medium property are fixed, the finite element discretization error is used to design the adaptive strategy.
We point out a closely related work [28], where an adaptive finite element method with transparent boundary condition (TBC) was developed for solving the open cavity scattering problems. Since the nonlocal TBC is directly used to truncate the open domain, it does not require a layer of artificially designed absorbing medium to enclose the domain of interest, which makes the TBC method different from the PML approach. But the TBC is given as an infinite series and needs to be truncated into a sum of finitely many terms in computation. Due to the simplicity in the implementation of the PML method, this work provides a viable alternative to the adaptive finite element TBC method for solving the open cavity scattering problems. Numerical experiments are presented and compared with the adaptive finite element TBC method for both polarizations to illustrate the competitive behavior of the adaptive finite element PML method.
The outline of this paper is as follows. In Section 2, we introduce the problem formulation, where the governing equations are given for the TM and TE polarizations. Sections 3 and 4 are devoted to the analysis of the TM and TE polarizations, respectively. Topics are organized to address the variational problem, the PML problem and its convergence, the finite element approximation, the a posteriori error analysis for the discrete truncated PML problem, and the adaptive finite element algorithm. In Section 5, some numerical examples are presented to illustrate the performance of the proposed method. The paper is concluded with some general remarks in Section 6.
2. Problem formulation
Let us first specify the problem geometry which is shown in Figure 1. Denote by the cross section of an -invariant cavity with a Lipschitz continuous boundary , where refers to as the cavity wall and is the opening of the cavity. We assume that the cavity wall is a perfect electric conductor and the opening is aligned with the perfectly electrically conducting infinite ground plane . The cavity may be filled with some inhomogeneous medium, which can be characterized by the dielectric permittivity and the magnetic permeability . Moreover, the medium may protrude from the cavity into the upper half-space. In this case, the cavity is called an overfilled cavity. Let and be the upper half-discs with radii and , where . Denote by and the upper semi-circles. The radius can be chosen large enough such that the upper half-disc can enclose the possibly protruding inhomogeneous medium from the cavity. The infinite exterior domain is assumed to be filled with some homogeneous medium with a constant dielectric permittivity and a constant magnetic permeability .
Since the structure is invariant the -axis, we consider two fundamental polarizations: transverse magnetic (TM) polarization and transverse electric (TE) polarization. The three-dimensional Maxwell equations can be reduced to the two-dimensional Helmholtz equation under these two modes. In the TM polarization, the magnetic field is transverse to the -axis and the electric field has the form , where the scalar function satisfies
| (2.1) |
where is the wave number and is the angular frequency. In the TE polarization, the electric field is transverse to the -axis and the magnetic field takes the form , where satisfies
| (2.2) |
where
is the unit outward normal vector to
.Consider the incidence of a plane wave
which is sent from the above to impinge the cavity. Here is the angle of the incidence, and is the wavenumber in the free space. Due to the perfectly electrically conducting ground plane, the reflected field in the TM polarization is
while the reflected field in the TE polarization is
Let the reference field be the superposition of the incident field and the reflected field, i.e., . The total field consists of the reference field and the scattered field , i.e.,
In addition, the scattered field is required to satisfy the Sommerfeld radiation condition
| (2.3) |
3. TM polarization
In this section, we consider the TM polarization. First the transparent boundary condition is introduced to reduce the open cavity problem into a boundary value problem in a bounded domain. Next the variational problem is described, and the PML problem and its convergence are discussed. Then the finite element approximation and the a posteriori error estimate are studied. Finally the adaptive finite element method with PML is presented for solving the discrete PML problem.
3.1. The variational problem
It can be verified from (2.1) that the scattered field satisfies the Helmholtz equation
| (3.1) |
Based on the radiation condition (2.3), we know that the solution of (3.1) has the Fourier series expansion
| (3.2) |
where is the Hankel function of the first kind with order . Noting the fact and on , we have , which implies and (3.2) reduces to
| (3.3) |
Taking the partial derivative of (3.3) with respect to and evaluating it at yields
| (3.4) |
Let . For any , it has the Fourier series expansion
Define the trace function space , where the norm is given by
It is clear that the dual space of is with respect to the scalar product in given by
Introduce the DtN operator
| (3.5) |
It is shown in [27] that the boundary operator is continuous. Using (3.4)–(3.5), we obtain the transparent boundary condition for the scattered field :
which can be equivalently imposed for the total field :
where .
Let . The open cavity scattering problem can be reduced to the following boundary value problem:
which has the variational formulation: find such that
| (3.6) |
where the sesquilinear form is defined by
| (3.7) |
The following result states the well-posedness of the variational problem (3.6). The proof can be found in [27].
Theorem 3.1.
The variational problem (3.6) has a unique weak solution in , which satisfies the estimate
where is a constant.
It follows from the general theory in Babuka and Aziz [3, Chapter 5] that there exists a constant such that the following inf-sup condition holds:
| (3.8) |
3.2. The PML problem
Let be the PML region which encloses the bounded domain in the upper half-space. Denote by the computational domain in which the truncated PML problem is formulated.
Define the PML parameters by using the complex coordinate stretching
| (3.9) |
where . In practice, is usually taken as a power function
where is a positive constant and is an integer. It can be seen from (3.9) that
In the polar coordinates, the gradient and divergence operators can be written as
| (3.10) |
where and
. By the chain rule and (
3.9), a simple calculation yields| (3.11) |
Combining (3.10) and (3.11), we introduce the modified gradient operator
It is easy to verify
where
Hence we obtain the PML equation for the scattered field :
where is required to be uniformly bounded as . In practice, the open domain needs to be truncated into a bounded domain. Replacing with in (3.3) and noting the exponential decay of the Hankel functions with a complex argument, we can observe that the scattered field decays exponentially in . Hence it is reasonable to impose the Dirichlet boundary condition
We obtain the truncated PML problem
| (3.12) |
where
Introduce another DtN operator which defined as follows: given ,
where satisfies
Using the boundary condition
and noting in , we reformulate (3.12) equivalently into the following boundary value problem:
| (3.13) |
where . The weak formulation of the problem (3.13) is to find such that
| (3.14) |
where the sesquilinear form is defined as
3.3. Convergence of the PML problem
Consider a boundary value problem of the PML equation in :
| (3.15) |
Define . Given , the weak formulation of (3.15) is to find such that and
| (3.16) |
where the sesquilinear form is
As is discussed in [16], in general, the uniqueness of (3.16
) can not be guaranteed due to the possible existence of eigenvalues which form a discrete set. Since our focus is on the convergence analysis, we simply assume that the PML problem (
3.16) has a unique solution in the PML region.For any , define
It is easy to show that the norm is equivalent to the usual -norm. An application of the general theory in [3, Chapter 5] implies that there exists a positive constant depending on and such that
| (3.17) |
The following results play an important role in the convergence analysis. The proof is similar to that of [12, Theorem 2.4] for solving the obstacle scattering problem and is omitted here for brevity.
Theorem 3.2.
There exists a constant independent of , and such that the following estimates are satisfied:
| (3.18) | |||||
| (3.19) |
where is given in (3.17) and .
Following the idea in [19], for any function , we introduce the propagation operator defined by
As shown in [12], the operator is well defined and satisfies the estimate
| (3.20) |
Lemma 3.3.
For any , we have
Proof.
Theorem 3.4.
For sufficiently large , the PML problem (3.14) has a unique solution . Moreover, we have the following estimate:
3.4. Finite element approximation
Define . The weak formulation of (3.12) is to find and such that
| (3.21) |
where and the sesquilinear form is given by
| (3.22) |
Let be a regular triangulation of , where denotes the maximum diameter of all the elements in . To avoid being distracted from the main focus of the a posteriori error analysis, we assume for simplicity that is polygonal to keep from using the isoparametric finite element space and deriving the approximation error of the boundary .
Let be the a conforming finite element space, i.e.,
where is a positive integer and denotes the set of all polynomials of degree no more than . The finite element approximation to the variational problem (3.21) is to find with such that
| (3.23) |
where .
For sufficiently small , the discrete inf-sup condition of the sesquilinear form can be established by an argument of Schatz [26]. It follows from the general theory in [3] that the truncated variational problem (3.23) admits a unique solution. Since our focus is the a posteriori error analysis and the associated adaptive algorithm, we assume that the discrete problem (3.23) has a unique solution .
3.5. A posteriori error analysis
For any triangular element , denote by its diameter. Let denote the set of all the edges that do not lie on . For any , denotes its length. For any , we introduce the residual
For any interior edge , which is the common side of triangular elements , we define the jump residual across as
where is the unit outward normal vector on the boundary of . Let
For any triangle , denote by the local error estimator as follows:
where the rescaling function
For any , let be its extension in such that
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