Transparent Boundary Conditions Research Articles

Analysis of guided and leaky modes of planar optical waveguides using transparent boundary conditions

We present a numerical approach to compute and characterize both guided and leaky modes in a multilayer planar optical waveguide made of any lossy and dispersive materials. Usually, in numerical calculations based on finite element methods, perfectly matched layers (PMLs) are used to truncate the unbounded substrate and cover layers. However, it is difficult to make such PMLs transparent for both guided and leaky modes at the same time, and often, different or even contradictory PML parameters are required for these two types of modes. In contrast, the transparent boundary conditions (TBCs) that we use in this work can terminate the unbounded waveguide and, simultaneously, provide perfect transparency for the modes. In addition, this type of boundary condition does not contaminate the solutions with non-existent modes, such as PML modes. More importantly, the TBC approach yields the nonlinear eigenvalue solutions that can be intrinsically mapped to the parameter space of transverse wavenumbers in the claddings. This allows us to uniquely determine the power flow properties of all the calculated modes. A finite element Python package is developed to treat a variety of planar waveguides in a robust and systematic way.

Learned infinite elements for helioseismology. Learning transparent boundary conditions for the solar atmosphere

Acoustic waves in the Sun are affected by the atmospheric layers, but this region is often ignored in forward models because it increases the computational cost. The purpose of this work is to take the solar atmosphere into account without significantly increasing the computational cost. We solved a scalar-wave equation that describes the propagation of acoustic modes inside the Sun using a finite-element method. The boundary conditions used to truncate the computational domain were learned from the Dirichlet-to-Neumann operator, that is, the relation between the solution and its normal derivative at the computational boundary. These boundary conditions may be applied at any height above which the background medium is assumed to be radially symmetric. We show that learned infinite elements lead to a numerical accuracy similar to the accuracy that is obtained for a traditional radiation boundary condition in a simple atmospheric model. The main advantage of learned infinite elements is that they reproduce the solution for any radially symmetric atmosphere to a very good accuracy at low computational cost. In particular, when the boundary condition is applied directly at the surface instead of at the end of the photosphere, the computational cost is reduced by 20<!PCT!> in 2D and by 60<!PCT!> in 3D. This reduction reaches 70<!PCT!> in 2D and 200<!PCT!> in 3D when the computational domain includes the atmosphere. We emphasize the importance of including atmospheric layers in helioseismology and propose a computationally efficient method to do this.

Transparent boundary condition and its effectively local approximation for the Schrödinger equation on a rectangular computational domain

The transparent boundary condition for the free Schrödinger equation on a rectangular computational domain requires implementation of an operator of the form ∂t−i△Γ where △Γ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Padé approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.

Thick interface coupling technique for weakly dispersive models of waves

The primary focus of this work is the coupling of dispersive free-surface flow models through the utilization of a thick interface coupling technique. The initial step involves introducing a comprehensive framework applicable to various dispersive models, demonstrating that classical weakly dispersive models are encompassed within this framework. Next, a thick interface coupling technique, well-established in hyperbolic framework, is applied. This technique enables the formulation of unified models across different subdomains, each corresponding to a specific dispersive model. The unified model preserves the conservation of mechanical energy, provided it holds for each initial dispersive model. We propose a numerical scheme that preserve the projection structure at the discrete level and as a consequence is entropy-satisfying when the continuous model conserve the mechanical energy. We perform a deep numerical analysis of the waves reflected by the interface. Finally, we illustrate the usefulness of the method with two applications known to pose problems for dispersive models, namely the imposition of a time signal as a boundary condition or the imposition of a transparent boundary condition, and wave propagation over a discontinuous bathymetry.

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.