Exact Nonreflecting Boundary Conditions Research Articles

Wave diffraction by floating bodies in water of finite depth using an exact DtN boundary condition

For the exact Non-Reflecting Boundary Conditions (NRBCs) or so-called Dirichlet-to-Neumann (DtN) boundary conditions, the artificial boundary is usually chosen as a spherical or ellipsoidal surface by which the computational domain is enclosed fully with finite volume. For the wave diffraction problem in finite-depth water unbounded horizontally, the free water boundary condition and wall boundary condition should be satisfied on still water level (SWL) and seabed, respectively. In this paper, the artificial boundary for DtN condition is chosen as only a lateral cylindrical surface by which it is impossible to envelope the objects and fluid domain fully, but possible by adding still water surface and seabed. A novel DtN condition on the artificial boundary is suggested and then Boundary Element Method (BEM) implementation is described to solve the three-dimensional wave diffraction problems by arbitrary-shaped floating bodies in water of finite depth. Upon successful verification for the case of a bottom-truncated cylinder, the scheme is applied to a single chamfer box and an array of chamfer boxes.

A numerically exact nonreflecting boundary condition applied to the acoustic Helmholtz equation

When modeling wave propagation, truncation of the computational domain to a manageable size requires nonreflecting boundaries. To construct such a boundary condition on one side of a rectangular domain for a finite-difference discretization of the acoustic wave equation in the frequency domain, the domain is extended on that one side to infinity. Constant extrapolation in the direction perpendicular to the boundary provides the material properties in the exterior. Domain decomposition can split the enlarged domain into the original one and its exterior. Because the boundary-value problem for the latter is translation-invariant, the boundary Green functions obey a quadratic matrix equation. Selection of the solvent that corresponds to the outgoing waves provides the input for the remaining problem in the interior. The result is a numerically exact nonreflecting boundary condition on one side of the domain. When two nonreflecting sides have a common corner, the translation invariance is lost. Treating each side independently in combination with a classic absorbing condition in the other direction restores the translation invariance and enables application of the method at the expense of numerical exactness. Solving the quadratic matrix equation with Newton’s method turns out to be more costly than solving the Helmholtz equation and may select unwanted incoming waves. A proposed direct method has a much lower cost and selects the correct branch. A test on a 2D acoustic marine seismic problem with a free surface at the top, a classic second-order Higdon condition at the bottom, and numerically exact boundaries at the two lateral sides demonstrates the capability of the method. Numerically exact boundaries on each side, each computed independently with a free-surface or Higdon condition, provide even better results.

Accurate Full-Vectorial Finite Element Method Combined with Exact Non-Reflecting Boundary Condition for Computing Guided Waves in Optical Fibers

Abstract The vector electromagnetic problem for eigenwaves of optical fibers, originally formulated on the whole plane, is equivalently reduced to a linear parametric eigenvalue problem posed in a circle, convenient for numerical solution. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Asymptotic properties of the dispersion curves and their smoothness are investigated for the new formulation of the problem. A numerical method based on finite element approximations combined with an exact non-reflecting boundary condition is developed. Error estimates for approximating eigenvalues and eigenfunctions are derived.

Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

The numerical computation of nonlocal Schrodinger equations (SEs) on the whole real axis is considered. Based on the artificial boundary method, we first derive the exact artificial nonreflecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458–3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonreflecting boundary condition and leads us to reformulate the original infinite discrete system into an equivalent finite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are finally provided to demonstrate the effectiveness of our approach.

EXACT NON-REFLECTING BOUNDARY CONDITIONS WITH AN FDTD SCHEME FOR THE SCALAR WAVE EQUATION IN WAVEGUIDE PROBLEMS

EXACT NON-REFLECTING BOUNDARY CONDITIONS WITH AN FDTD SCHEME FOR THE SCALAR WAVE EQUATION IN WAVEGUIDE PROBLEMS

Modeling and simulation of an acoustic well stimulation method

This paper presents a mathematical model and a numerical procedure to simulate an acoustic well stimulation (AWS) method for enhancing the permeability of the rock formation surrounding oil and gas wells. The AWS method considered herein aims to exploit the well-known permeability-enhancing effect of mechanical vibrations in acoustically porous materials, by transmitting time-harmonic sound waves from a sound source device—placed inside the well—to the well perforations made into the formation. The efficiency of the AWS is assessed by quantifying the amount of acoustic energy transmitted from the source device to the rock formation in terms of the emission frequency and the well configuration. A simple methodology to find optimal emission frequencies for a given well configuration is presented. The proposed model is based on the Helmholtz equation, a sound-hard boundary condition at the casing, and an impedance boundary condition that effectively accounts for the porous solid–fluid interaction at the interface between the rock formation and the well perforations. Exact non-reflecting boundary conditions derived from Dirichlet-to-Neumann maps are utilized to truncate the circular cylindrical waveguides considered in the model. The resulting boundary value problem is then numerically solved by means of the finite element method. A variety of numerical examples are presented in order to demonstrate the effectiveness of the proposed procedure for finding optimal emission frequencies.

Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition

Abstract The original problem for eigenwaves of weakly guiding optical fibers formulated on the plane is reduced to a convenient for numerical solution linear parametric eigenvalue problem posed in a disk. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Properties of dispersion curves are investigated for the new formulation of the problem. An efficient numerical method based on FEM approximations is developed. Error estimates for approximate solutions are derived. The rate of convergence for the presented algorithm is investigated numerically.

Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability

Exact non-reflecting boundary conditions for a linear incompletely parabolic system in one dimension have been studied. The system is a model for the linearized compressible Navier---Stokes equations, but is less complicated which allows for a detailed analysis without approximations. It is shown that well-posedness is a fundamental property of the exact non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, it is also shown that energy stability follows automatically.

Exact non-reflecting boundary condition for 3D time-dependent multiple scattering–multiple source problems

We consider some 3D wave equation problems defined in an unbounded domain, possibly with far field sources. For their solution, by means of standard finite element methods, we propose a Non Reflecting Boundary Condition (NRBC) on the chosen artificial boundary B, which is based on a known space–time integral equation defining a relationship between the solution of the differential problem and its normal derivative on B. Such a NRBC is exact, non local both in space and time. We discretize it by using a fast convolution quadrature technique in time and a collocation method in space. The computational complexity of the discrete convolution is of order NlogN, being N the total number of time steps performed. That of the fully discretized NRBC is O(NB2NlogN), where NB denotes the number of mesh points taken on B.Besides showing a good accuracy and numerical stability, the proposed NRBC has the property of being suitable for artificial boundaries of general shapes. It also allows the treatment of far field (multiple) sources, that do not have to be necessarily included in the finite computational domain, being transparent not only for outgoing waves but also for incoming ones. This approach is in particular used to solve multiple scattering problems.

Exact nonreflecting boundary conditions for exterior wave equation problems

We consider the classical wave equation problem defined on the exterior of a bounded 2D space domain, possibly having far field sources. We consider this problem in the time domain, but also in the frequency domain. For its solution we propose to associate with it a boundary integral equation (BIE) defined on an artificial boundary surrounding the region of interest. This boundary condition is nonreflecting (or transparent) for both outgoing and incoming waves and it does not have to include necessarily the problem datum supports. The problem physical domain can even be a multi-domain, defined by the union of several disjoint domains. These domains can be convex or nonconvex. This transparent boundary condition is imposed pointwise on the chosen artificial boundary; therefore, its (space collocation) discretization can be coupled with a (space) finite difference or finite element method for the associated PDE problem. In the time-domain case, a classical (explicit or implicit) time integrator is also used. We present a consistency result for the BIE discretization and a sample of the intensive numerical testing we have performed.

Exact nonreflecting boundary conditions for three dimensional poroelastic wave equations

Exact nonreflecting boundary conditions for three dimensional poroelastic wave equations

Fast and Accurate Computation of Time-Domain Acoustic Scattering Problems with Exact Nonreflecting Boundary Conditions

This paper is concerned with fast and accurate computation of exterior wave equations truncated via exact circular or spherical nonreflecting boundary conditions (NRBCs, known to be nonlocal in both time and space). We first derive analytic expressions for the underlying convolution kernels, which allow for a rapid and accurate evaluation of the convolution with $O(N_t)$ operations over $N_t$ successive time steps. To handle the nonlocality in space, we introduce the notion of boundary perturbation, which enables us to handle general bounded scatters by solving a sequence of wave equations in a regular domain. We propose an efficient spectral-Galerkin solver with Newmark's time integration for the truncated wave equation in the regular domain. We also provide ample numerical results to show high-order accuracy of NRBCs and efficiency of the proposed scheme.

Non-reflecting boundary conditions for plane waves in anisotropic elasticity and poroelasticity

The paper presents exact non-reflecting boundary conditions for transient plane waves in an anisotropic elastic solid for oblique incidence. The boundary conditions are expressed through the eigenvectors of the acoustic tensor and are written in impedance form as a relation between the velocity vector and the traction vector. The approach is extended to anisotropic fluid-saturated porous solids. Exact plane-wave non-reflecting boundary conditions are derived for transient non-dissipative waves in a medium with infinite or zero permeability, and for steady-state dissipative waves.

Time history kernel functions for square lattice

We propose an exact non-reflecting boundary condition for simulations of a square lattice in finite atomic sub-domain. The boundary condition is based on the series expansion of the single source kernel functions. The time history kernel Laplace transform is first derived in a closed form, where the Fourier transform inversion is performed analytically. An efficient numerical procedure is also developed for the inverse Laplace transform. Differences in the finite boundary condition and the infinite boundary approximation elucidate the cause of the long-lasting difficulty of corner effects. Numerical tests verify accuracy of the proposed approach.

Dynamics of steps along a martensitic phase boundary II: Numerical simulations

We investigate the dynamics of steps along a phase boundary in a cubic lattice undergoing antiplane shear deformation. The phase transition is modeled by assuming piecewise linear stress–strain law with respect to one component of the shear strain, while the material response to the other component is linear. In the first part of the paper we have constructed semi-analytical solutions featuring sequential propagation of steps. In this work we conduct a series of numerical simulations to investigate stability of these solutions and study other phenomena associated with step nucleation. We show that sequential propagation of sufficiently small number of steps can be stable, provided that the velocity of the steps is below a certain critical value that depends on the material parameters and the step configuration. Above this value we observe a cascade nucleation of multiple steps which then join sequentially moving groups. Depending on material anisotropy, the critical velocity can be either subsonic or supersonic, resulting in subsonic step nucleation in the first case and steady supersonic sequential motion in the second. The numerical simulations are facilitated with an exact non-reflecting boundary condition and a fast algorithm for its implementation, which are developed to eliminate the possible artificial wave reflection from the computational domain boundary.

Numerical Solution to the Sine-Gordon Equation Defined on the Whole Real Axis

Numerical simulation of the solution to the sine-Gordon equation on the whole real axis is considered in this paper. Based on nonlinear spectral analysis, exact nonreflecting boundary conditions are derived at two artificially introduced boundary points. Then a numerical scheme of second order is proposed to approximate the solution. In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed scheme.

Numerical simulation of a modified KdV equation on the whole real axis

The numerical simulation of the solution to a modified KdV equation on the whole real axis is considered in this paper. Based on the work of Fokas (Comm Pure Appl Math 58(5):639–670, 2005), a kind of exact nonreflecting boundary conditions which are suitable for numerical purposes are presented with the inverse scattering theory. With these boundary conditions imposed on the artificially introduced boundary points, a reduced problem defined on a finite computational interval is formulated. The discretization of the nonreflecting boundary conditions is studied in detail, and a dual-Petrov–Galerkin spectral method is proposed for the numerical solution to the reduced problem. Some numerical tests are given, which validate the effectiveness, and suggest the stability of the proposed scheme.

Nonreflecting boundary condition for time-dependent multiple scattering

An exact nonreflecting boundary condition (NBC) is derived for the numerical solution of time-dependent multiple scattering problems in three space dimensions, where the scatterer consists of several disjoint components. Because each sub-scatterer can be enclosed by a separate artificial boundary, the computational effort is greatly reduced and becomes independent of the relative distances between the different sub-domains. In fact, the computational work due to the NBC only requires a fraction of the computational work inside Ω, due to any standard finite difference or finite element method, independently of the mesh size or the desired overall accuracy. Therefore, the overall numerical scheme retains the rate of convergence of the interior scheme without increasing the complexity of the total computational work. Moreover, the extra storage required depends only on the geometry and not on the final time. Numerical examples show that the NBC for multiple scattering is as accurate as the NBC for a single convex artificial boundary [M.J. Grote, J.B. Keller, Nonreflecting boundary conditions for time-dependent scattering, J. Comput. Phys. 127(1) (1996), 52–65], while being more efficient due to the reduced size of the computational domain.

Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations

The numerical approximation of one-dimensional cubic nonlinear Schrödinger equations on the whole real axis is studied in this paper. Based on the work of A. Boutet de Monvel, A.S. Fokas and D. Shepelsky [Lett. Math. Phys., 65(3): 199–212, 2003], a kind of exact nonreflecting boundary conditions are derived on the artificially introduced boundary points. The related numerical issues are discussed in detail. Several numerical tests are performed to demonstrate the behaviour of the proposed scheme.

Parallel iterative solution for the Helmholtz equation with exact non-reflecting boundary conditions

Efficient and scalable parallel solution methods are presented for the Helmholtz equation with global non-reflecting DtN boundary conditions. The symmetric outer-product structure of the DtN operator is exploited to significantly reduce inter-processor communication required by the non-locality of the DtN to one collective communication per iteration with a vector size equal to the number of harmonics included in the DtN series expansion, independent of the grain size. Numerical studies show that in the context of iterative equation solvers, and for the same accuracy, the global DtN applied to a tightly fitting spheroidal boundary and implemented as a low-rank update with the multiplicative split is more cost-effective (both in wall-clock times and memory) compared to local approximate boundary conditions.