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A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition
... In practice, h ≤ Ck −1 is acceptable in the practical computation, which requires about 2π/C degrees of freedom per wave-length and is consistent with the requirement to control the interpolation error. Recently, a variety of FEMs have been developed to reduce the pollution error, including the hp-FEM [2,10,11,13,14], the CIP-FEM [5,[15][16][17][18][19][20][21], the discontinuous Galerkin method [22][23][24][25][26][27][28][29][30], the weak Galerkin method [31][32][33], the Trefftz methods [34][35][36][37][38][39][40], and the multiscale mathods [41][42][43][44]. For other approaches to reduce the pollution error for Helmholtz equation, such as the RF-bubble method, GLS-FEM, GFEM, QSFEM, etc., we refer to [9,10,[45][46][47][48][49][50]. ...
... The CIP-FEM, which was first proposed by Douglas and Dupont [51] for elliptic and parabolic problems in the 1970s, has recently shown great potential in significantly reducing the pollution errors for the Helmholtz equation with high wave number; see [5,[16][17][18]. The CIP-FEM uses the same approximation space as the FEM but modifies the bilinear form of the FEM by adding a continuous interior penalty term (1.3) at mesh interfaces: , where E I h is the set of all interior edges/faces of the mesh and γ e,j denotes the penalty parameters to be determined and may be tuned to greatly reduce the pollution error. ...
... As an alternative to the adaptive finite element PML method, the adaptive finite element DtN method (DtN-FEM) has also been proposed to solve the obstacle scattering problems [8,11,12,36,37,45], the diffraction grating problems [5,41,43,49,53], and the open cavity scattering problem [52], where the transparent boundary conditions are used to truncate the unbounded domain. In this new approach, the layer of artificial medium is no longer needed to enclose the domain of interest, which is different from the PML method. ...
Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable and isotropic elastic solid, which is immersed in a homogeneous compressible air/fluid. By the Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the model is formulated as a boundary value problem of acoustic-elastic interaction. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated DtN boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN boundary operator, where the latter decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction 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.
... However, the solutions obtained by this method exhibit oscillatory behavior at the cavity boundary. Motivated by recent works [2,15,35,46], we introduce the interior penalty term or the boundary penalty term to the variational formulations of the Helmholtz and modified Helmholtz equations, effectively suppressing the oscillation of the bending moment of the solution at the cavity boundary. ...
Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate-wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate-wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions at the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution at the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem.
... These volumetric discretization approaches can treat problems in general geometries and including spatially varying media. As is well known, however, these methods typically suffer from spatial and temporal numerical dispersion errors (also known as pollution errors [6,38]), and they therefore require use of fine spatial and temporal meshes-and thus, large computer-memory and run-times-to achieve accurate solutions in applications involving high frequencies and/or long time simulations. ...
This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, 1) A multiple scattering strategy that decomposes a given time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open-arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems; 2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point 1); 3) A smooth "Time-windowing and recentering" methodology that enables both treatment of incident signals of long duration and long time simulation; and, 4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem-which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology.
... As a viable alternative, the finite element DtN method has been proposed to solve the obstacle scattering problems [29,32,40], the diffraction grating problems [31,45], and the open cavity scattering problem [47], respectively, where the transparent boundary conditions are used to truncate the domains. In this new approach, the layer of artificial medium is not needed to enclose the domain of interest, which makes is different from the PML method. ...
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.
... The DtN operator truncation error needs to be integrated into the a posteriori error estimate and the truncation number can be determined automatically through the estimate. The adaptive finite element DtN method has been successfully applied to solve many scattering problems, including acoustic waves [5,18,20,27], electromagnetic waves [19,34], and elastic waves [3,25,26]. ...
This paper is concerned with a numerical solution to the scattering of a time-harmonic electromagnetic wave by a bounded and impenetrable obstacle in three dimensions. The electromagnetic wave propagation is modeled by a boundary value problem of Maxwell's equations in the exterior domain of the obstacle. Based on the Dirichlet-to-Neumann (DtN) operator, which is defined by an infinite series, an exact transparent boundary condition is introduced and the scattering problem is reduced equivalently into a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is developed to solve the discrete variational problem, where the DtN operator is truncated into a sum of finitely many terms. The a posteriori error estimate takes into account both the finite element approximation error and the truncation error of the DtN operator. The latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.
... Therefore, the DtN operator for the elastic wave equation can be obtained from the well-studied DtN operators for the Helmholtz and Maxwell equations. Since the TBC is exact, the artificial boundary could be put as close as possible to the obstacle in order to reduce the computational complexity [19,24]. ...
Consider the elastic scattering of an incident wave by a rigid obstacle in three dimensions, which is formulated as an exterior problem for the Navier equation. By constructing a Dirichlet-to-Neumann (DtN) operator and introducing a transparent boundary condition, the scattering problem is reduced equivalently to a boundary value problem in a bounded domain. The discrete problem with the truncated DtN operator is solved by using the a posteriori error estimate based adaptive finite element method. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to converge exponentially with respect to the truncation parameter. Moreover, the generalized Woodbury matrix identity is utilized to solve the resulting linear system efficiently. Numerical experiments are presented to demonstrate the superior performance of the proposed method.
... The numerical results show that the adaptive finite element DtN method is competitive with the adaptive finite element PML method. We refer to [27] for a continuous interior penalty finite element method (CIP-FEM) for solving high frequency scattering problems with the truncated DtN boundary condition. ...
Scattering problems have wide applications in the medical and military fields. In this paper, the weighted least-squares (WLS) collocation method based on radial basis functions (RBFs) is developed to solve elastic wave scattering problems, which are governed by the Navier equation and the Helmholtz equations with coupled boundary conditions. The perfectly matched layer (PML) technique is used to truncate the unbounded domain into a bounded domain. The WLS method is constructed by setting the collocation points denser than the trial centers and imposing different weights on different types of boundary conditions. The WLS method can overcome the matrix singularity problem encountered in the Kansa method, and the convergence rate of WLS is [Formula: see text] for Sobolev kernel with kernel smoothness [Formula: see text]. Furthermore, compared with the finite element method (FEM) and the Kansa method, WLS can provide higher accuracy and more stable solutions for relatively large angular frequencies. The numerical example with a circular obstacle is used to verify the effectiveness and convergence behavior of the WLS. Besides, the proposed scheme can easily handle irregular obstacles and obtain stable results with high accuracy, which is validated through experiments with ellipse and kite-shaped obstacles.
Consider a time-harmonic acoustic plane wave incident onto an elastic body with an unbounded periodic surface. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid air/fluid of constant mass density, whereas the elastic body is assumed to be isotropic and linear. By the Dirichlet-to-Neumann (DtN) operators for the Helmholtz equation and the Navier equation, respectively, exact transparent boundary conditions are introduced and the model is formulated as an acoustic-elastic interaction boundary value problem. Based on a new duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated DtN boundary operators. The a posteriori error estimate consists of the finite element approximation errors and the truncation error of the DtN boundary operators, where the latter decays exponentially with respect to the truncation parameter. Based on the a posteriori error control, an adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction 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 adaptive algorithm.
This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains . The approach relies on four main elements, namely, (1) A multiple scattering strategy that decomposes a given interior time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems ; (2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point (1); (3) A smooth “Time-windowing and recentering” methodology that enables both treatment of incident signals of long duration and long time simulation; and, (4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem—which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology.
A new coupled perfectly matched layer (PML) method is proposed for the Helmholtz equation in the whole space with inhomogeneity concentrated on a nonconvex domain. Rigorous analysis is presented for the stability and convergence of the proposed coupled PML method, which shows that the PML solution converges to the solution of the original Helmholtz problem exponentially with respect to the product of the wave number and the width of the layer. An iterative algorithm and a continuous interior penalty finite element method (CIP-FEM) are also proposed for solving the system of equations associated to the coupled PML. Numerical experiments are presented to illustrate the convergence and performance of the proposed coupled PML method as well as the iterative algorithm and the CIP-FEM.
We consider the diffraction of an electromagnetic plane wave by a biperiodic structure. This paper is concerned with a numerical solution of the diffraction grating problem for three-dimensional Maxwell’s equations. Based on the Dirichlet-to-Neumann (DtN) operator, an equivalent boundary value problem is formulated in a bounded domain by using a transparent boundary condition. An a posteriori error estimate-based adaptive edge finite element method is developed for the variational problem with the truncated DtN operator. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the former is used for local mesh refinements and the latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to demonstrate the competitive behaviour of the proposed method.
The paper is concerned with the three-dimensional electromagnetic scattering from a large open rectangular cavity that is embedded in a perfectly electrically conducting infinite ground plane. By introducing a transparent boundary condition, the scattering problem is formulated into a boundary value problem in the bounded cavity. Based on the Fourier expansions of the electric field, the Maxwell equation is reduced to one-dimensional ordinary differential equations for the Fourier coefficients. A fast algorithm, employing the fast Fourier transform and the Gaussian elimination, is developed to solve the resulting linear system for the cavity which is filled with either a homogeneous or a layered medium. In addition, a novel scheme is designed to evaluate rapidly and accurately the Fourier transform of singular integrals. Numerical experiments are presented for large cavities to demonstrate the superior performance of the proposed method.
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