Transparent Boundary Conditions Research Articles

A highly efficient and accurate numerical method for the electromagnetic scattering problem with rectangular cavities

This paper presents a robust numerical solution to the electromagnetic scattering problem involving multiple multi-layered cavities in both transverse magnetic and electric polarizations. A transparent boundary condition is introduced at the open aperture of the cavity to transform the problem from an unbounded domain into that of bounded cavities. By employing Fourier series expansion of the solution, we reduce the original boundary value problem to a two-point boundary value problem, represented as an ordinary differential equation for the Fourier coefficients. The analytical derivation of the connection formula for the solution enables us to construct a small-scale system that includes solely the Fourier coefficients on the aperture, streamlining the solving process. Furthermore, we propose accurate numerical quadrature formulas designed to efficiently handle the weakly singular integrals that arise in the transparent boundary conditions. To demonstrate the effectiveness and versatility of our proposed method, a series of numerical experiments are conducted.

Numerical solution of the cavity scattering problem for flexural waves on thin plates: Linear finite element methods

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 on 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 on the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem.

An edge-based smoothed finite element method combined with Newmark method for time domain acoustic wave scattering problem

Time domain obstacle scattering widely exists in the scientific fields of geophysics, remote sensing, seismology, and medicine. However, it is difficult to solve the time domain scattering problems with arbitrarily shaped obstacles only within the fixed domain of interest. In this paper, an edge-based smoothed finite element method using linear triangular elements (ES-FEM-T3) combined with the Newmark method is proposed for the two-dimensional time domain acoustic scattering by an arbitrarily shaped obstacle. In the algorithm, a time domain transparent boundary condition (TBC) and the Newmark method are introduced to transform the time domain problem in an unbounded domain into a series of steady-state PDEs on a bounded domain, which is suitable for an arbitrary shaped obstacle scattering problem. In the numerical discretization, ES-FEM-T3 is used to construct a close-to-exact stiffness matrix by the smoothed gradient technique, making it space stable and obtaining more accurate results compared to FEM-T3. Especially, through in advance calculating the function σ(t), the convolution term in the time-domain TBC can be quickly calculated. Finally, through the numerical experiments, the effectiveness of the proposed method is demonstrated. Especially, ES-FEM-T3 can better improve the time convergence stability of the Newmark method and have better accuracy at γ = 0.5 compared to FEM-T3.

Stability of grating diffraction problems for plane wave incidence: Explicit dependence on wavenumbers and incident angles

Suppose a time-harmonic plane wave is incident onto an impenetrable grating profile of Dirichlet or Impedance kind in two dimensions. The grating profile is assumed to be given by a Lipschitz function. We derive a stability estimate of this grating diffraction problem using the variational method with a transparent boundary condition. An explicit dependence of solutions on the incident wavenumber and the incident angle is obtained, which carries over to transmission problems for penetrable gratings. Our approach relies heavily on using Rellich's identities in periodic structures.

Well-Posedness and Convergence Analysis of PML Method for Time-Dependent Acoustic Scattering Problems Over a Locally Rough Surface

Abstract We aim to analyze and calculate time-dependent acoustic wave scattering by a bounded obstacle and a locally perturbed non-selfintersecting curve. The scattering problem is equivalently reformulated as an initial-boundary value problem of the wave equation in a truncated bounded domain through a well-defined transparent boundary condition. Well-posedness and stability of the reduced problem are established. Numerically, we adopt the perfect matched layer (PML) scheme for simulating the propagation of perturbed waves. By designing a special absorbing medium in a semi-circular PML, we show the well-posedness and stability of the truncated initial-boundary value problem. Finally, we prove that the PML solution converges exponentially to the exact solution in the physical domain. Numerical results are reported to verify the exponential convergence with respect to absorbing medium parameters and thickness of the PML.

Design of dispersion-flattened photonic crystal fiber with 800 nm wide low-dispersion band

We propose a new design for a dispersion-flattened photonic crystal fiber with 800 nm wide low-dispersion band by employing the imaginary-distance beam propagation method and the transparent boundary condition using BeamPROP simulation software. The chromatic dispersion of the fiber is −1.4 to 4.8 ps/nm·km in an 800 nm wavelength range of 1.1–1.9 μm. The effective mode area at a wavelength of 1.55 μm is 12.2 μm2 and the fiber can be easily joined to conventional nonlinear optical fiber with a small core. The calculated effective index is extremely linear via the wavelength, and the decision coefficient with an approximately straight function is 0.999996, this means the waveguide dispersion of the proposed PCF completely compensates for the material dispersion curves.

The electromagnetic scattering from multiple arbitrarily shaped cavities with inhomogeneous anisotropic media

We study the electromagnetic scattering problem from multiple arbitrarily shaped cavities filled with inhomogeneous anisotropic media. A local-refined Petrov-Galerkin finite element interface (PGFEI) method is developed for solving the scattering problem. By introducing the transparent boundary condition, the unbounded scattering problem can be reduced into bounded domain problem with coupled boundary conditions on the apertures of multiple cavities. The multiple cavity scattering in transverse magnetic (TM) and transverse electric (TE) polarizations are both investigated based on uniform non-body-fitted meshes. We adopt the level-set functions to capture the irregular cavity wall and the interfaces of inhomogeneous media. The special basis is constructed to satisfy the jump conditions across the interfaces of different materials further. The local-refined algorithm is proposed to enhance the accuracy of the solutions around the discontinuous interfaces. The generation of the grid points facilitates a special numbering scheme which enables a stepwise Schur complement decomposition of the discrete linear system derived by the PGFEI method. It could be an alternative method for solving the scattering problems which have similar structures. The proposed method can handle matrix coefficients caused by the permittivity and permeability tensor of anisotropic media. Numerical experiments illustrate the validity and efficiency of the method for solving the scattering from multiple arbitrarily shaped cavities with inhomogeneous anisotropic media in TM and TE polarization.

Manakov system on metric graphs: Modeling the reflectionless propagation of vector solitons in networks

We consider the reflectionless transport of Manakov solitons in networks. The system is modeled in terms of the Manakov system on metric graphs subject to transparent boundary conditions at the branching points. Simple constraints combining the equivalent usual Kirchhoff vertex conditions with the transparent conditions are derived in terms of nonlinearity coefficients. Formulation of the problem, derivation of transparent boundary conditions and numerical experiment showing absence of the back scattering in the transmission of soliton through the vertex under the above constraints are presented for the case of metric star graph. Extension of the problem to other graph topologies is briefly discussed by considering the metric tree graph.

Eliminating Artificial Boundary Conditions in Time-Dependent Density Functional Theory Using Fourier Contour Deformation.

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrödinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators. We adapt FCD to time-dependent density functional theory (TDDFT), and describe a simple algorithm to smoothly and automatically truncate long-range Coulomb-like potentials to a time-dependent constant outside of a bounded domain of interest, so that FCD can be used. This approach eliminates errors originating from the use of artificial boundary conditions, leaving only the error of the potential truncation, which is controlled and can be systematically reduced. The method enables accurate simulations of ultrastrong nonlinear electronic processes in molecular complexes in which the interference between bound and continuum states is of paramount importance. We demonstrate results for many-electron TDDFT calculations of absorption and strong field photoelectron spectra for one and two-dimensional models, and observe a significant reduction in the size of the computational domain required to achieve high quality results, as compared with the popular method of complex absorbing potentials.

Scattering in a partially open waveguide: the forward problem

AbstractThis paper is devoted to an acoustic scattering problem in a 2D partially open waveguide, in the sense that the left part of the waveguide is closed, that is with a ‘bounded’ cross-section, while the right part is bounded in the transverse direction by some perfectly matched layers that mimic the situation of an open waveguide, that is with an ‘unbounded’ cross-section. We prove well-posedness of such scattering problem in the Fredholm sense (uniqueness implies existence) and exhibit the asymptotic behaviour of the solution in the longitudinal direction with the help of the Kondratiev approach. Having in mind the numerical computation of the solution, we also propose some transparent boundary conditions in such longitudinal direction, based on Dirichlet-to-Neumann operators. After proving such artificial conditions actually enable us to approximate the exact solution, some numerical experiments illustrate the quality of such approximation.

On transparent vertex boundary conditions for quantum graphs

On transparent vertex boundary conditions for quantum graphs

Transparent boundary conditions for the nonlocal nonlinear Schrödinger equation: A model for reflectionless propagation of PT-symmetric solitons

We consider the problem of reflectionless propagation of PT-symmetric solitons described by the nonlocal nonlinear Schrödinger equation on a line in the framework of the concept of transparent boundary conditions for evolution equations. Transparent boundary conditions for the nonlocal nonlinear Schrödinger equation are derived. The absence of backscattering at the artificial boundaries is confirmed by the numerical implementation of the transparent boundary conditions.

О численном моделировании трековых мембран, используемых в качестве коллиматоров рентгеновского излучения

In this work, we present the results of numerical modeling of distribution of the field amplitude inside micron-sized through cylindrical pores in polymer track membranes in the soft X-ray wavelength range 13.5–30.4 nm. The calculations were performed by numerically solving the 3D parabolic equation with a finite-difference method using an exact transparent boundary condition. The dependences of the X-ray transmittivity through a pore on the incidence angle as well as on the pore diameter were computed. For the membranes with some thicknesses and pore diameters, calculated angular dependences of the transmittance are compared with the measurements known from the literature. It is demonstrated that the calculations agree with the measurements if a transition layer on the inner surface of the pores, which accounts for the finite surface roughness, is introduced.

A diffraction problem for the biharmonic wave equation in one-dimensional periodic structures

This work concerns the propagation of flexural waves through one-dimensional periodic structures embedded in thin elastic plates. We show that the out-of-plane displacement of the plate only contains the Helmholtz wave component and the modified Helmholtz wave component is not supported when the Navier boundary condition is imposed. An adaptive finite element method with transparent boundary condition is developed for solving the associated boundary value problem. Numerical results show that the method is effective to solve the diffraction grating problem of the biharmonic wave equation.

The time‐domain scattering by the elastic shell in a two‐layered unbounded structure

The subject of this research is the time‐domain scattering problem for a three‐dimensional layered elastic shell submerged in a two‐layered fluid separated by an unbounded rough surface. The essence of the problem is to model the scattering interaction between the given elastic shell and some incident wave in a two‐layered medium. Using the exact transparent boundary condition (TBC), we reformulate the unbounded scattering problem into an equivalent initial‐boundary value problem. The well‐posedness is proved for the problem by the Laplace transform and variational method in the s‐domain. Moreover, we show that the reduced problem has a unique weak solution by the energy method. The priori estimates with explicit dependence on the time are derived for the acoustic pressure and the elastic displacement in the time domain.

Transparent boundary condition for simulating rogue wave solutions in the nonlinear Schrödinger equation.

This paper addresses the construction of numerical boundary conditions for simulating rogue wave solutions in the nonlinear Schrödinger equation. While three kinds of commonly used boundary conditions require a big enough computational domain to reproduce solutions faithfully in the central domain, we propose transparent boundary conditions for the Peregrine soliton and Kuznetsov-Ma breather solutions, respectively. For both solutions, these boundary conditions require a smaller computational domain than other boundary conditions to attain the best accuracy of the Crank-Nicolson scheme and selected mesh size, which will be referred to as the "acceptable accuracy" below. In particular, the computational domain with these boundary conditions is only 1/16 as small as others in the simulations of the Peregrine soliton solution. As a result, they reduce both the memory requirement and the computing time for the Peregrine soliton solution.

Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

Absorbing boundary conditions are important when one simulates the propagation of waves on a bounded numerical domain without creating artificial reflections. In this paper, we consider various hyperbolic–dispersive equations modeling water wave propagation. A typical example is the Korteweg–de Vries equation 1 $$\begin{aligned} \displaystyle u_t+u\,u_x+\varepsilon u_{xxx}=0,\quad \forall x\in {\mathbb {R}}, \quad \forall t>0. \end{aligned}$$ In the case of linearized equations, some progress was recently done for one dimensional scalar dispersive equations using discrete transparent boundary conditions. However, a generalization of this approach to multi-dimensional setting is not obvious. In this paper, we consider the alternative perfectly matched layer (PML) approach for the linearized Korteweg–de Vries equation: 2 $$\begin{aligned} \displaystyle u_t+U\,u_x+\varepsilon u_{xxx}=0\quad \forall x\in {\mathbb {R}}, \quad \forall t>0, \end{aligned}$$ where $$U\in {\mathbb {R}}$$ denotes a reference speed. We first propose a direct perfectly matched layer approach and study the stability of the modified system. These equations are not always stable, the main obstruction being the classical condition $$v_g(k)v_\phi (k)\ge 0$$ found in the literature on PML (Bécache et al in J Comput Phys 188(2):399–433, 2003) that we recover in our analysis. Then, we introduce a hyperbolic system with a source term that is an approximation of the Korteweg–de Vries equations. In this case, the complete PML equations are not, again, completely stable. However, a version of the PML equations for this system derived without the source term is found to be stable and can absorb outgoing waves although it may create reflections as it is not perfectly matched. Finally, we consider the BBM–Boussinesq system that models bi-directional waves at the surface of an inviscid fluid layer. The dispersive properties for a subclass of physically relevant models are better suited for PML techniques since the condition $$v_g(k)v_\phi (k)\ge 0$$ is always satisfied. We show that the PML equations are always stable in this case. We illustrate numerically the absorbing and stability properties of these PML models and provide also KdV type simulation by choosing properly initial data.

On the Treatment of Exterior Domains for the Time-Harmonic Equations of Stellar Oscillations

In a recent article we started to analyze the time-harmonic equations of stellar oscillations. As a first step we considered bounded domains together with an essential boundary condition and established the well-posedness of the equation. In this article we consider the physical relevant case of the domain being $\mathbb{R}^3$. We discuss the treatment of the exterior domain, and show how to couple the two parts to obtain a well-posedness result. Further, for the Cowling approximation (which neglects the Eulerian perturbation of gravity) we derive a scalar equation in the atmosphere, couple it to the vectorial interior equation, and prove the well-posedness of the new system. This coupled system has the big advantages that it simplifies the construction of approximating transparent boundary conditions and leads to significant less degrees of freedom for discretizations.

Fast High-Order Algorithms for Electromagnetic Scattering Problem from Finite Array of Cavities in TE Case with High Wave Numbers

In this paper, fast high-order finite difference algorithms for solving the electromagnetic scattering from the finite array of two-dimensional rectangular cavities are proposed in TE polarization. The scattering problem from the cavity array is described as coupled Helmholtz equations with transparent boundary conditions on open apertures. Second-order and fourth-order schemes for solving the coupled systems are developed in TE polarization respectively. A special technique is applied to construct a fourth-order scheme for transparent boundary conditions. Further, we propose fast algorithms which can simplify the larger global system to a small linear interface system on the apertures of cavities. Numerical experiments show the validity and efficiency of the proposed fast algorithms for solving the scattering problem with high wave numbers.

FEM–BEM coupling for the thermoelastic wave equation with transparent boundary conditions in 3D

We consider the thermoelastic wave equation in three dimensions with transparent boundary conditions on a bounded, not necessarily convex domain. In order to solve this problem numerically, we introduce a coupling of the thermoelastic wave equation in the interior domain with time-dependent boundary integral equations. Here, we want to highlight that this type of problem differs from other wave-type problems that dealt with FEM–BEM coupling so far, e.g., the acoustic as well as the elastic wave equation, since our problem consists of coupled partial differential equations involving a vector-valued displacement field and a scalar-valued temperature field. This constitutes a nontrivial challenge which is solved in this paper. Our main focus is on a coercivity property of a Calderón operator for the thermoelastic wave equation in the Laplace domain, which is valid for all complex frequencies in a half-plane. Combining Laplace transform and energy techniques, this coercivity in the frequency domain is used to prove the stability of a fully discrete numerical method in the time domain. The considered numerical method couples finite elements and the leapfrog time-stepping in the interior with boundary elements and convolution quadrature on the boundary. Finally, we present error estimates for the semi- and full discretization.