Transmission Boundary Conditions Research Articles

Uniqueness and numerical method for phaseless inverse diffraction grating problem with known superposition of incident point sources

In this paper, we establish the uniqueness of identifying a smooth grating profile with a mixed boundary condition (MBC) or transmission boundary conditions (TBCs) from phaseless data. The existing uniqueness result requires the measured data to be in a bounded domain. To break this restriction, we design an incident system consisting of the superposition of point sources to reduce the measurement data from a bounded domain to a line above the grating profile. We derive reciprocity relations for point sources, diffracted fields, and total fields, respectively. Based on Rayleigh’s expansion and reciprocity relation of the total field, a grating profile with a MBC or TBCs can be uniquely determined from the phaseless total field data. An iterative algorithm is proposed to recover the Fourier modes of grating profiles at a fixed wavenumber. To implement this algorithm, we derive the Fréchet derivative of the total field operator and its adjoint operator. Some numerical examples are presented to verify the correctness of theoretical results and to show the effectiveness of our numerical algorithm.

Exact domain truncation for the Morse-ingard equations

Morse and Ingard [1] give a coupled system of time-harmonic equations for the temperature and pressure of an excited gas. These equations form a critical aspect of modeling trace gas sensors. Like other wave propagation problems, the computational problem must be closed with suitable far-field boundary conditions. Working in a scattered-field formulation, we adapt a nonlocal boundary condition proposed in [2] for the Helmholtz equation to this coupled system. This boundary condition uses a Green's formula for the true solution on the boundary, giving rise to a nonlocal perturbation of standard transmission boundary conditions. However, the boundary condition is exact and so Galerkin discretization of the resulting problem converges to the restriction of the exact solution to the computational domain. Numerical results demonstrate that accuracy can be obtained on relatively coarse meshes on small computational domains, and the resulting algebraic systems may be solved by GMRES using the local part of the operator as an effective preconditioner. These numerical results taken together combine several advanced techniques, including higher-order finite elements, geometric multigrid in curvilinear geometry, native use of complex arithmetic, and incorporation of nonlocal operators. These are tied together in a high-level simulation using the Firedrake library.

The boundary value problem for the two-dimensional Laplace equation with transmission conditions given on open Lipschitz curve

We consider boundary value problem for Laplace equation in bounded two-dimensional Lipschitz domain with transmission boundary condition given upon open curve. This conditions includes itself the jump of solution of boundary value problem and the meaning of boundary value of its normal derivative. We prove the equivalence of considered boundary value problem and obtained variational problem. As a result we prove existence and uniqueness of solution of the posed problems in appropriate functional spaces. Based on the integral representation of the solution the considered boundary value problem is reduced to the system of boundary integral equation which has unique solution.

Advanced Numerical Methods for Graphene Simulation with Equivalent Boundary Conditions: A Review

Since the discovery of graphene, due to its excellent optical, thermal, mechanical and electrical properties, it has a broad application prospect in energy, materials, biomedicine, electromagnetism and other fields. A great quantity of researches on the physical mechanism of graphene has been applied to engineering in electromagnetism and optics. To study the properties of graphene, different kinds of numerical methods such as the mixed finite element method (Mixed FEM), the mixed spectral element method (Mixed SEM), Method of Auxiliary Sources (MAS), discontinuous Galerkin time-domain method (DGTD) and interior penalty discontinuous Galerkin time domain (IPDG) have been developed for simulating the electromagnetic field effects of graphene and equivalent boundary conditions such as impedance transmission boundary condition (ITBC), surface current boundary condition (SCBC), impedance matrix boundary condition (IMBC) and surface impedance boundary condition (SIBC) have been employed to replace graphene in the computational domain. In this work, the numerical methods with equivalent boundary conditions are reviewed, and some examples are provided to illustrate their applicability.

The inverse scattering problem for an inhomogeneous two-layered cavity

Abstract In this paper, we consider the inverse scattering problem of identifying a two-layered cavity by internal acoustic measurements under the condition that the interior interface has a mixed transmission boundary condition. We focus on the mathematical analysis of recovering the shape of the interior interface by using the linear sampling method, including reconstructing the surface conductivity by the same measurements.

A Two-Step Lagrange–Galerkin Scheme for the Shallow Water Equations with a Transmission Boundary Condition and Its Application to the Bay of Bengal Region—Part I: Flat Bottom Topography

A two-step Lagrange–Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC) is presented. First, we show the experimental order of convergence to see the second-order accuracy in time realized by the two-step methods for conservative and non-conservative material derivatives along the trajectory of fluid particles. Second, we observe the effect of the TBC in a simple domain, and the artificial reflection is removed significantly when the wave touches the TBC. Third, we apply the scheme to a practical domain with islands, namely, the Bay of Bengal region, and observe the effect of the TBC again for the practical domain; the artificial reflections are removed significantly from the transmission boundaries on open sea boundaries. We also study the effect of a position of an open sea boundary with the TBC and reveal that it is sufficiently small to neglect. The numerical results in this study show that the scheme has the following properties: (i) the same advantages of Lagrange–Galerkin methods (the CFL-free robustness for convection-dominated problems and the symmetry of the matrices for the system of linear equations); (ii) second-order accuracy in time by the two-step methods; (iii) mass preservation of the function for the water level from the reference height (until the contact with the transmission boundaries of the wave); and (iv) no significant artificial reflection from the transmission boundaries. The numerical results by the scheme presented in this paper are for the flat bottom topography of the domain. In the next part of this work, Part II, the scheme will be applied to rapidly varying bottom surfaces and a real bottom topography of the Bay of Bengal region.

Mathematical problems of dynamical interaction of fluids and multiferroic solids

In the paper, we consider a three-dimensional mathematical problem of fluid-solid dynamical interaction, when an anisotropic elastic body occupying a bounded region is immersed in an inviscid fluid occupying an unbounded domain . In the solid region, we consider the generalized Green–Lindsay's model of the thermo-electro-magneto-elasticity theory. In this case, in the domain we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function), while we have a scalar acoustic pressure field in the unbounded domain . The physical kinematic and dynamical relations are described mathematically by the appropriate initial and boundary-transmission conditions. Using the Laplace transform, the dynamical interaction problem is reduced to the corresponding boundary-transmission problem for elliptic pseudo-oscillation equations containing a complex parameter τ. We derive the appropriate norm estimates with respect to the complex parameter τ and construct the solution of the original dynamical problem by the inverse Laplace transform. As a result, we prove the uniqueness, existence, and regularity theorems for the dynamical interaction problem. Actually, the present investigation is a continuation of the paper [Chkadua G, Natroshvili D. Mathematical aspects of fluid-multiferroic solid interaction problems. Math Meth Appl Sci. 2021;44(12):9727–9745], where the fluid-solid interaction problems for elliptic pseudo-oscillation equations associated with the above mentioned generalized thermo-electro-magneto-elasticity theory are studied by the potential method and the theory of pseudodifferential equations.

Global existence and energy decay for a transmission problem under a boundary fractional derivative type

<abstract><p>The paper considers the effects of fractional derivative with a high degree of accuracy in the boundary conditions for the transmission problem. It is shown that the existence and uniqueness of the solutions for the transmission problem in a bounded domain with a boundary condition given by a fractional term in the second equation are guaranteed by using the semigroup theory. Under an appropriate assumptions on the transmission conditions and boundary conditions, we also discuss the exponential and strong stability of solution by also introducing the theory of semigroups.</p></abstract>

New scheme of the discrete sources method for two-dimensional scattering problems by penetrable obstacles

A new computational scheme of the Discrete Sources Method (DSM) is developed for the numerical solution of two-dimensional acoustic and electromagnetic transmission boundary-value problems (BVPs) pertaining to the Helmholtz equation. To establish the new DSM scheme, we show that the matrix integral operator, corresponding to the transmission boundary conditions, has a dense range. The approximate solution of the BVP is constructed according to the new DSM scheme and it is proven that it converges uniformly to the exact solution of the BVP. An analytic representation is derived for the calculation of the scattering cross section in terms of the determined DSs amplitudes avoiding an integration over the unit circle. The numerical implementation procedure of the DSM is described and numerical results for the application of the DSM are presented. The new scheme is shown to be accurate and efficient even for highly-elongated scatterers.

On the direct and inverse electromagnetic scattering by chiral media

Consider the direct and inverse electromagnetic scattering problems by a penetrable cylindrical chiral medium in a homogeneous achiral background medium. The scattering problem can be modeled by a pair of the two-dimensional Helmholtz equations with coupled transmission boundary conditions. The boundary integral equation method is applied to solve the direct scattering problem in a suitable Sobolev space. Then we investigate the linear sampling method for recovering the unknown scatterer from far-field pattern of the scattered field. The coupled boundary conditions bring new challenges during the procedure, and some novel techniques are needed in the theoretical analysis. Finally, we perform several numerical experiments to demonstrate the efficiency and effectiveness of this method.

A well-conditioned weak coupling of boundary element and high-order finite element methods for time-harmonic electromagnetic scattering by inhomogeneous objects

In this work, we present efficient weak coupling formulations between the boundary element method and the high-order finite element method for solving time-harmonic electromagnetic scattering problems. The approach is based on the use of a non-overlapping domain decomposition method involving optimal transmission operators. The associated transmission boundary conditions are constructed through a localization process based on complex rational approximants of the nonlocal Magnetic- to- Electric operators.

The factorization method for the scattering by a mixed inhomogeneous medium

Abstract We use the classical factorization method proposed firstly by Kirsch to reconstruct the support of the mixed inhomogeneous medium associated with complex valued refractive indexes and different transmission boundary conditions. We will show that for well-chosen inhomogeneous backgrounds, one obtains a necessary and sufficient condition characterizing the support of the medium via the eigensystem of a self-adjoint operator, which is related to the far field operator. Moreover, for completeness of our problem, the variational method is applied to solve the direct scattering problem. And, we present a variant of numerical examples in 2D to verify the effectiveness and robustness of the proposed inverse algorithms.

Large-amplitude internal fronts in two-fluid systems

In this announcement, we report results on the existence of families of large-amplitude internal hydrodynamic bores. These are traveling front solutions of the full two-phase incompressible Euler equation in two dimensions. The fluids are bounded above and below by flat horizontal walls and acted upon by gravity. We obtain continuous curves of solutions to this system that bifurcate from the trivial solution where the interface is flat. Following these families to the their extreme, the internal interface either overturns, comes into contact with the upper wall, or develops a highly degenerate "double stagnation" point. Our construction is made possible by a new abstract machinery for global continuation of monotone front-type solutions to elliptic equations posed on infinite cylinders. This theory is quite robust and, in particular, can treat fully nonlinear equations as well as quasilinear problems with transmission boundary conditions.

Optimal control for nonlocal reaction‐diffusion system describing calcium dynamics in cardiac cell

The purpose of this paper is to introduce an optimal control for a nonlocal calcium dynamic model in a cardiac cell acting on ryanodine receptors. The optimal control problem is considered as a coupled nonlocal reaction‐diffusion system with a transmission boundary condition covering the sarcoplasmic reticulum and cytosolic domain. We establish the well‐posedness result of the adjoint problem using Faedo‐Galerkin approximation, a priori estimates, and compactness arguments. The numerical discretization of direct and adjoint problems is realized by using the implicit Euler method in time and the finite element for spatial discretization. Moreover, we obtain the stability result in the L2‐norm for the direct and the adjoint discrete problems. Finally, in order to illustrate the control of our calcium dynamic model, we present some numerical experiments devoted to constant and nonlocal diffusions using the proposed numerical scheme.

Upscaling the interplay between diffusion and polynomial drifts through a composite thin strip with periodic microstructure

We study the upscaling of a system of many interacting particles through a heterogenous thin elongated obstacle as modeled via a two-dimensional diffusion problem with a one-directional nonlinear convective drift.Assuming that the obstacle can be described well by a thin composite strip with periodically placed microstructures, we aim at deriving the upscaled model equations as well as the effective transport coefficients for suitable scalings in terms of both the inherent thickness at the strip and the typical length scales of the microscopic heterogeneities. Aiming at computable scenarios, we consider that the heterogeneity of the strip is made of an array of periodically arranged impenetrable solid rectangles and identify two scaling regimes what concerns the small asymptotics parameter for the upscaling procedure: the characteristic size of the microstructure is either significantly smaller than the thickness of the thin obstacle or it is of the same order of magnitude.We scale up the diffusion–polynomial drift model and list computable formulas for the effective diffusion and drift tensorial coefficients for both scaling regimes. Our upscaling procedure combines ideas of two-scale asymptotics homogenization with dimension reduction arguments. Consequences of these results for the construction of more general transmission boundary conditions are discussed. We illustrate numerically the concentration profile of the chemical species passing through the upscaled strip in the finite thickness regime and point out that trapping of concentration inside the strip is likely to occur in at least two conceptually different transport situations: (i) full diffusion/dispersion matrix and nonlinear horizontal drift, and (ii) diagonal diffusion matrix and oblique nonlinear drift.

Existence of local strong solutions for the incompressible viscous and non-resistive MHD-structure interaction model

The purpose of this paper is to study the local well-posedness problem on the magnetohydrodynamics (MHD)-structure interaction (MHDSI) systems. The fluid is represented by the incompressible viscous and non-resistive MHD equation in Euler coordinates while the structure is modeled by the elasticity equation with superconductor material in Lagrangian coordinates. The equations are coupled along the moving interface though transmission boundary conditions for velocity, stress and magnetic field. The local existence of at least one strong solution in time to the incompressible viscous and non-resistive MHD-structure interaction model was proved in the sense of one suitable Sobolev's space norm by using the careful energy method and fixed point theory combining with penalization and regularization techniques and by overcoming the coupling difficulties caused by the magnetic field.

Dispersion of waves in two and three-dimensional periodic media

We consider the propagation of acoustic time-harmonic waves in homogeneous media containing periodic lattices of spherical or cylindrical inclusions. It is assumed that the wavelength has the order of the periods of the lattice while the radius a of inclusions is small. A new approach is suggested to derive the complete asymptotic expansions of the dispersion relations in two- and three-dimensional cases as and first several terms of the expansions are evaluated explicitly. Our method is based on the reduction of the original singularly perturbed (by inclusions) problem to the regular one. The Dirichlet, Neumann, and transmission boundary conditions are considered. In the former case, we estimate the cutoff wavelength supported by the periodic medium in two and three dimensions. The effective wave speed is obtained as a function of the wave frequency, the filling fraction of the inclusions, and the physical properties of the constituents of the mixture. Dependence of the asymptotic formulas obtained in the paper on geometric and material parameters is illustrated by graphs.

Reconstruction of the Sturm-Liouville differential operators with discontinuity conditions and a constant delay

In this manuscript, we study second–order differential operators with a constant delay and transmission boundary conditions. We establish properties of the spectral characteristics and investigate the inverse problem of recovering operators from their spectra. Also, we construct the potential function by using the Fourier series and delay point of the Sturm–Liouville differential operator.

Frequency-dependent transmission boundary condition in the acoustic time-domain nodal discontinuous Galerkin model

Accurate modeling of boundary conditions is of critical importance for acoustic simulations. Recently, the time-domain nodal discontinuous Galerkin (TD-DG) method has emerged as a potential wave-based method for acoustic modeling. Although the acoustic reflection behavior of various time-domain impedance boundaries has been studied extensively, the modeling of the sound transmission across a locally-reacting layer of impedance discontinuity is far less developed. This paper presents a formulation of broadband time-domain transmission boundary conditions for locally-reacting surfaces in the framework of the TD-DG method. The formulation simulates the acoustic wave behavior at each of the boundary nodes using the plane-wave theory. Through the multi-pole model representation of the transmission coefficient, various types of transmission layers can be simulated. One-dimensional numerical examples demonstrate the capability of the proposed formulation to accurately simulate the reflection and transmission characteristics of the limp wall and the porous layer, where quantitative error behavior against analytical results is presented. Furthermore, to demonstrate the applicability, two scenarios of two-dimensional acoustic environment are considered. One is the sound transmission between two rooms partitioned by a limp panel and the other is the sound propagation through a transmissive noise barrier. Comparison of the predicted results from the proposed method against the results from the frequency-domain finite element simulations further verifies the formulation.

Manypoint Boundary Value Problems for Elliptic Differential-Operator Equations with Interior Singularities

In this work, a elliptic differential-operator equation together with boundary-transmission conditions is considered on two disjoint intervals. The boundary-transmission conditions may contain finite number interior points and abstract linear operators. We establish such important properties as coerciveness and Fredholmness of the problem under consideration. Moreover, we prove that the system of root functions forms an Abel basis in the corresponding Hilbert space.