Matching boundary conditions for lattice dynamics

We design a class of accurate and efficient absorbing boundary conditions for molecular dynamics simulations of crystalline solids. In one space dimension, the proposed matching boundary conditions take the form of a linear constraint of displacement and velocity at atoms near the boundary, where the coefficients are determined by matching the dispersion relation with a minimal number of atoms involved. Bearing the nice features of compactness, locality, and high efficiency, the matching boundary conditions are then extended to treat the out‐of‐plane wave problems in the square lattice. We construct multidirectional absorbing boundary conditions via operator multiplications. Reflection coefficient analysis and numerical studies verify their effectiveness for spurious reflection suppression in all directions. Compact and local in both space and time, they are directly applicable to nonlinear lattices and multiscale simulations. Copyright © 2012 John Wiley & Sons, Ltd.     

Matching boundary conditions for diatomic chains

We propose a class of efficient matching boundary conditions to suppress spurious reflection for multiscale computations of one dimensional diatomic chains. This provides the first local effective treatment of both acoustic and optical phonons. Adopting the extended zone scheme of the dispersion relation, we design a class of force boundary conditions by enforcing perfect absorption at certain selected wave numbers. Reflection suppression is improved by involving more neighboring atoms in the condition. The effectiveness of the proposed matching boundary conditions is demonstrated by reflection coefficient analysis, numerical tests, and comparisons with the time history treatment.     

Transmitting boundary conditions for 1D peridynamics

The peridynamic theory reformulates the equations of continuum mechanics in terms of integro‐differential equations instead of partial differential equations. It is not straightforward to apply the available artificial boundary conditions for continua to peridynamic modeling. We therefore develop peridynamic transmitting boundary conditions (PTBCs) for 1D wave propagation. Differently from the previous method where the matching boundary condition is constructed for only one boundary material point, the PTBCs are established by considering the interaction and exchange of information between a group of boundary material points and another group of inner material points. The motion of the boundary material points is recursively constructed in terms of their locations and is determined through matching the peridynamic dispersion relation. The effectiveness of the PTBCs is examined by reflection analyses, numerical tests, and numerical convergent conditions. Furthermore, two‐way interfacial conditions are proposed. The PTBCs are then applied to simulations of wave propagation in a bar with a defect, a composite bar with interfaces, and a domain with a seismic source. All the analyses and applications demonstrate that the PTBCs can effectively remove undesired numerical reflections at artificial boundaries. The methodology may be applied to modeling of wave propagation by other nonlocal theories. Copyright © 2016 John Wiley & Sons, Ltd.     

Nondegenerate parametric down conversion in coherently prepared two-level atomic gas

We describe the parametric down conversion process in a two-level atomic gas, where the atoms are in a superposition state of relevant energy levels. This superposition results in splitting of the phase matching condition into three different conditions. Another, more important, peculiarity of the system under discussion is the nonsaturability of amplification coefficients with increasing pump wave intensity, under ‘sideband’ generation conditions.     

Scattering of obliquely incident surface acoustic waves at the boundary between metallized and free surfaces on anisotropic substrates

A method of theoretically analyzing the reflection and transmission of obliquely incident acoustic surface waves at the boundary between metallized and free surfaces on substrates with general anisotropy is presented. The analysis is performed under the assumption that the fields of the reflecting/transmitting wave can be expressed as a linear combination of three-dimensional surface-wave solutions for the unbounded metallized or free surface, having a common wave number in the direction along the boundary. From the approximate wave matching condition at the boundary, analytical expressions of the reflection and transmission coefficients are derived. Numerical results of field distributions for different kinds of substrates demonstrate the effectiveness of the present analysis.     

Transverse electric scattering by a dielectric biconvex cylinder loaded with a shallow circular trough in a ground plane

The scattering problem of transverse electric wave from a dielectric biconvex cylinder buried in a shallow circular trough of a ground plane is investigated and a rigorous series solution is also derived. Based on the region-matching method, the analysed region is decomposed into two subregions by introducing a semi-circular auxiliary boundary. The magnetic field of each subregion is expressed in terms of cylindrical wave functions with unknown expansion coefficients. After imposing the matching conditions and the boundary condition on the trough surface with the aid of Graf's addition theorem, the unknown coefficients are determined. Comparisons with published data for a dielectric circular cylinder case show very good agreement. Visible effects of depth-to-half-width ratios of a dielectric biconvex cylinder on echo width, far- field pattern and near-field distribution are illustrated in graphical form.     

Pressure-transmitting boundary conditions for molecular-dynamics simulations

A scheme for establishing boundary conditions in molecular-dynamics simulations that prevent pressure wave reflection out of the simulation volume is formulated. The algorithm is easily implemented for a one-dimensional geometry. Its efficiency is tested for compressive waves in Cu.     

Numerical aspects of wave forces on an artificial island

Abstract

The wave forces on an artificial island standing on the sea‐bed are investigated by the proposed hybrid method. A boundary integral equation is developed for the inner fluid domain encompassing the arbitrarily shaped island, while an eigenfunction expansion is used to represent a simple velocity field in the outer fluid domain. Applying the natural boundary condition on the matching surface, the wave force and moment on the wetted surface of the island are found according to the calculated velocity potentials. A gravity step circular dock is used as a test case to confirm the present computation. As a practical application of the present method, numerical results for an elliptical artificial island are presented.     

DDA方法中的两种无反射边界研究

在非连续变形分析(DDA)方法中研究并实现了叠加边界和粘性边界两种无反射边界条件,分别采用固定边界、叠加边界和粘性边界对一岩石细杆在飞块撞击下的动态响应进行了模拟。模拟结果表明:叠加边界和粘性边界都对反射应力波具有一定的衰减能力,但叠加边界的衰减能力更强,并且它对入射应力波的影响也比粘性边界小。     

Formulation and validation of Berenger's PML absorbing boundary forthe FDTD simulation of acoustic scattering

Berenger's perfectly matched layer (PML) absorbing boundary condition for electromagnetic (EM) waves is derived to absorb 2-D and 3-D acoustic waves in finite difference time domain (FDTD) simulation of acoustic wave propagation and scattering. A PML medium suitable for acoustic waves is constructed. Plane wave propagation in the PML medium is solved for both 2-D and 3-D cases and explicit FDTD boundary conditions are derived. The equations show that a matched PML medium is a perfect simulation of free space in that a plane wave does not change its direction of propagation or its speed when it propagates from free space into a matched PML medium. FDTD simulation of a pulsed point source propagating in two dimensions is carried out to test the performance of the PML boundary for acoustic waves. Results show that an eight layer PML boundary condition reduces the reflected error 40 dB over Mur's second order boundary condition     

A least squares finite element formulation for elastodynamic problems

A residual finite element formulation is developed in this paper to solve elastodynamic problems in which body wave potentials are primary unknowns. The formulation is based on minimizing the square of the residuals of governing equations as well as all boundary conditions. Since the boundary conditions in terms of wave potentials are neither Dirichlet nor Neumann type it is difficult to construct a functional to satisfy all governing equations and boundary conditions following the variational principle designed for conventional finite element formulation. That is why the least squares technique is sought. All boundary conditions are included in the functional expression so that the satisfaction of any boundary condition does not become a requirement of the trial functions, but they should satisfy some continuity conditions across the interelement boundary to guarantee proper convergence. In this paper it is demonstrated that the technique works well for elastodynamic problems; however, it is equally applicable to any other field problem.     

一种基于本征波匹配的EMD边界处理方法

希尔伯特-黄变换(Hilbert-Huang Transform,简称HHT)是上世纪末出现的一种处理非线性非平稳信号的新方法 ,EMD边界处理是其中的关键技术之一。着重介绍了一种基于本征波匹配的EMD边界处理方法——本证波匹配预测法,特别是介绍了将该方法和其他两种有代表性的EMD边界处理方法应用于仿真信号和和实测信号的算例对比,从而验证了本证波匹配预测法的先进性。本文工作使得相关研究得到较大程度地深化和完善。     

Simulation of acoustic wave propagation in dispersive media with relaxation losses by using FDTD method with PML absorbing boundary condition

We present a method to incorporate the relaxation dominated attenuation into the finite-difference time-domain (FDTD) simulation of acoustic wave propagation in complex media. A dispersive perfectly matched layer (DPML) boundary condition, which is suitable for boundary matching to such a dispersive media whole space, is also proposed to truncate the FDTD simulation domain. The numerical simulation of a Ricker wavelet propagating in a dispersive medium, described by second-order Debye model, shows that the Ricker wavelet is attenuated in amplitude and expanded in time in its course of propagation, as required by Kramers-Kronig relations. The numerical results also are compared to exact solution showing that the dispersive FDTD method is accurate and that the DPML boundary condition effectively dampens reflective waves. The method presented here is applicable to the simulation of ultrasonic instrumentation for medical imaging and other nondestructive testing problems with frequency dependent, attenuating media.     

Accurate radiation boundary conditions for the time‐dependent wave equation on unbounded domains

Asymptotic and exact local radiation boundary conditions (RBC) for the scalar time‐dependent wave equation, first derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear first‐order temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed efficiently and concurrently without altering the local character of the finite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related non‐reflecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form high‐order accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved efficiency over the exact condition. Copyright © 2000 John Wiley & Sons, Ltd.     

On the long time simulation of the Rayleigh–Taylor instability

We investigate the classical Rayleigh–Taylor instability with a phase‐field method. Despite of the long history of numerical simulations for the Rayleigh–Taylor instability, almost all results were relatively short time experiments. This is partly because of the way of treating the pressure boundary conditions. We implement a time‐dependent pressure boundary condition through a time‐dependent density field at the boundary. Owing to the pressure boundary treatment, we can perform long time evolutions resulting in an equilibrium state. In addition to the bubble and spike fronts, we have found that the width of sides is another important landmark on the interface of the Rayleigh–Taylor instability. Copyright © 2010 John Wiley & Sons, Ltd.     

Fully nonlinear wave-current interactions and kinematics by a BEM-based numerical wave tank

A numerical wave tank (NWT) with fully nonlinear free-surface boundary conditions is developed to investigate nonlinear wave–wave and wave–current interactions and the resulting kinematics. In the present paper, the variation of wave amplitude and wave length of a monochromatic wave under several different speeds of steady uniform currents is studied through direct numerical simulations in the time domain. The nonlinear wave–current interactions are solved using a boundary integral equation method (BIEM) and a Mixed Eulerian–Lagrangian (MEL) time marching scheme. Both a semi-Lagrangian approach and Lagrangian (material-node) approach are employed and their performance is compared. A regridding algorithm based on cubic spline fitting is devised for updating the free-surface moving boundary in a stable and accurate manner. The incident waves are generated by feeding prescribed analytical waves on the input boundary. An efficient artificial numerical beach is devised and applied to dissipate wave energy and minimize wave reflections from the downstream wall. Nonlinear wave kinematics as a result of nonlinear wave–current interactions is calculated and the results are compared with a multi-layer Boussinesq model. The spatial variation of nonlinear wave profiles and kinematics affected by currents are also addressed and discussed.     

A GPU‐accelerated nodal discontinuous Galerkin method with high‐order absorbing boundary conditions and corner/edge compatibility

Discontinuous Galerkin finite element schemes exhibit attractive features for accurate large‐scale wave‐propagation simulations on modern parallel architectures. For many applications, these schemes must be coupled with nonreflective boundary treatments to limit the size of the computational domain without losing accuracy or computational efficiency, which remains a challenging task. In this paper, we present a combination of a nodal discontinuous Galerkin method with high‐order absorbing boundary conditions for cuboidal computational domains. Compatibility conditions are derived for high‐order absorbing boundary conditions intersecting at the edges and the corners of a cuboidal domain. We propose a GPU implementation of the computational procedure, which results in a multidimensional solver with equations to be solved on 0D, 1D, 2D, and 3D spatial regions. Numerical results demonstrate both the accuracy and the computational efficiency of our approach.     

利用数值耗散层改进Boussinesq模型的无反射边界条件

Boussinesq 方程是近年来比较理想的模拟近岸波浪的一个模型, 它可以考虑波浪的色散、变浅和非线性等一系列因素, 其数值模型的吸收边界通常使用一种能在较大的频率范围内减弱波浪的数值海绵层。该文从理论上提出一种利用数值耗散模拟波浪无反射边界条件的方法。通过模拟规则波和随机波在水槽的传播表明该方法可以达到与海绵层相同的效果。由于该方法不需要修改模型代码, 仅需在开边界处增加一段网格逐渐增加的耗散带, 操作简便, 有很强的实用性。     

Study of radial basis collocation method for wave propagation

A numerical method based on radial basis functions and collocation method is proposed for wave propagation. Standard collocation and weighted boundary collocation approaches yield significant errors in wave problems. Therefore, a new method based on explicit time integration scheme that can correct the inaccuracy in the solutions and the errors accumulated in time integration is developed. This method can be easily applied for low and high dimensional wave problems. The stability conditions are obtained and the relationships between control parameters and stability are evaluated. Requirement of collocation points in numerical dispersion is studied and nondispersion condition is formulated. Eigenvalue analysis is investigated to evaluate the effectiveness of radial basis collocation method for solving wave problems. Eigenvalue study with and without imposing the boundary conditions are compared. The influences of shape parameters and distribution of collocation points and source points are presented. Numerical examples are simulated to examine and validate the proposed method.     

基于谱有限元的自由阻尼梁结构损耗因子分析

使用谱有限元分析了自由阻尼梁中扩散波类型及其损耗因子,通过对其特征波形的分析探讨了各个频率下每种波的激励条件,然后使用有限元对自由阻尼长梁做谐响应分析得到各个单元的应变能,根据应变能法计算得到该长梁分别在垂直和水平激励下的结构损耗因子,对结构损耗因子同激励起的波类型之间的关系进行了讨论,最后分析了边界反射对结构损耗因子的影响。研究表明：垂直激励主要激励起最高阶弯曲波或者某些低阶纵波,水平激励主要激励起最高阶纵波;边界反射对结构损耗因子的影响随着梁长增大以及激励点远离边界而变小;不同边界条件的结构损耗因子一般情况下比较接近,但在一些频率点处会存在一定的差异;自由阻尼梁的结构损耗因子主要由激励所能激励起的波类型所决定,在一定程度上受到边界反射影响。