透射边界条件在波动谱元模拟中的实现:一维波动

多次透射公式(multi-transmitting formula,MTF)是一种具有普适性的局部人工边界条件,但其在近场波动数值模拟中一般与有限元法结合.由于波动谱元模拟的数值格式与有限元格式有极大的不同,传统的MTF在谱元离散格式中无法直接实现.为了使物理概念清楚、精度可控的多次透射人工边界条件能够适应波动谱元模拟的需求,首先指出多次透射边界与谱元离散格式结合的基本问题,并分析了空间内插和时间内插两种方案的可行性.然后从空间内插角度出发,提出基于拉格朗日多项式插值模式的MTF谱元格式,并采用一种简单内插方法实现高阶MTF.最后通过一维波动数值试验检验这些MTF谱元格式的精度,并讨论其数值稳定性.结果表明:对于一、二阶MTF,几种格式的精度相当;对于三、四阶MTF,基于谱单元位移模式插值的格式精度最高.相反,随着插值多项式阶次的升高,不同MTF格式的稳定临界值逐步降低,但是所有格式均在人工波速大大超过物理波速时才可能发生失稳.     

透射边界高频失稳机理及其消除方法——SH波动

用有限元法求解近场波动问题,须选取人工边界条件以实现对无限域稳定、高效的数值模拟. 该文探讨了SH波导 有限元数值模拟中透射边界引发的高频失稳问题. 从离散模型出发,分析了内节点与人工边界节点运动方程频散曲线之间的匹配关系,揭示了高频失稳的一种机理,即二者相互耦合所 得计算方案支持自发从人工边界向计算区域内行进的高频波动. 提出通过调整内节点运动方程以改变这一匹配关系,从而消除失稳的措施. 理论分析与数值结果表明该措施能有效地消除高频振荡失稳.     

透射边界条件在波动谱元模拟中的实现:二维波动

将邢浩洁和李鸿晶提出的多次透射公式(multi-transmitting formula,MTF)的谱元格式应用于均匀介质中线弹性SH波动问题的谱元模拟.假定紧邻人工边界的一层谱单元为具有直线边界的四边形单元,以保证每个人工边界节点都唯一对应一条指向内域的离散网格线.人工边界节点在某时刻的位移由该离散网格线上的节点在前若干时刻的位移确定,按照MTF谱元格式进行计算.通过平面波以一定角度传播的外源问题算例和点源脉冲自由扩散的内源问题算例,验证了方法的可行性以及对实际复杂波动问题的适用性.通过不同类型初值问题算例,在时域内分析了插值多项式阶次、人工波速和透射阶次三个参数对反射误差的影响.结果表明:插值多项式阶次较高的格式会表现出更好的精度,但总体上对反射误差的影响较小;人工波速对反射误差具有显著影响,当人工波速小于介质物理波速时反射误差较大,而当人工波速等于或稍大于介质物理波速时反射误差处于较低水平;透射阶次对反射误差具有决定性影响,表现在不失稳的情形下提高透射阶次能够迅速降低反射误差,但内源问题从三阶MTF开始出现飘移失稳,外源问题从二阶MTF开始出现轻微的飘移失稳.     

局部人工边界稳定性的一种分析方法

采用时域局部人工边界同有限元或有限差分相结合的方法分析无界域中的波动问题时可能出现由边界引起的数值失稳现象．本文提出一种分析局部人工边界稳定性的实用方法．该方法从时域逐步积分格式出发，考虑了边界节点运动和内部节点运动的耦合以及离散化的影响，可以给出局部人工边界在多维波动数值模拟中的稳定性近似准则．文中对多次透射边界在一维及二维出平面波动模拟中稳定性的分析结果，表明了该方法的合理性     

多次透射公式飘移问题的控制方法

多次透射公式(multi-transmitting formula, MTF)是在近场波动数值模拟中一种广泛应用的人工边界条件, 具有形式简单、精度可控和通用性好的优点, 但高阶MTF与有限元方法相结合有时会出现飘移问题. 现有的几种MTF消飘方法往往会显著地影响边界精度, 为此本文提出一种新的消飘因子修正MTF格式, 能够在较高精度水平下实现对飘移问题的有效控制. 该方法保持MTF的一次透射项不变, 仅对各高次透射误差项进行修正, 从而大幅降低了因消飘因子造成的精度损失. 消飘因子设置格式确保在零频和零波数情形下能够满足GKS准则, 从理论上保证了消飘目标的实现. 进一步给出该方法的高阶统一形式, 并将传统的消飘因子修正MTF的方法归结为该统一形式的一个特例. 通过反射系数分析, 证明本文方法不仅具有精度优势, 而且消飘因子选取的适应性更强、取值范围更广. 数值算例表明, 本文的消飘因子对波动能量比较集中的法向和小角度透射波动的模拟精度影响很小, 在控制高阶MTF飘移问题和保持模拟精度方面, 均能够取得明显效果.      

多次透射公式在波动数值模拟中的一种实现方案

将一种常用的吸收边界条件─多次透射公式合并于紧接人工边界的内节点控制方程给出了该公式的一个新的实现方案.该方案较之原实现方案不仅收缩了计算区、改进了边界的模拟精度,而且揭示了该公式截断误差的阶数与大区域数值解之间的关系,阐明了提高吸收边界的精度阶对改进波动数值模拟精度的局限性.通过数值试验比较了新方案、原方案及Givoli-Neta吸收边界条件的在人工边界上的精度,结果表明前者优于后二者.     

一种新的高阶弹簧-阻尼-质量边界——-无限域圆柱对称波动问题

提出一种描述力-位移时间卷积关系的高阶弹簧-阻尼-质量模型,并将其作为人工边界条件直接应用于弹性动力学无限域圆柱对称运动问题的时域数值求解. 该人工边界条件不存在旁轴近似、多次透射等位移型外推人工边界条件普遍存在的高、低频失稳问题;与黏性、黏弹性边界等应力型人工边界条件相比,它具有高阶精度,且是严格高、低频双渐近的,也可以退化到黏性、黏弹性边界;该边界可以像黏性、黏弹性边界一样利用商用有限元软件中内置的并联弹簧-阻尼器、质量单元和时间积分求解器在商用软件中方便地实现,便于研究人员和工程师应用. 分析的几个简单数值算例也验证了该边界条件的上述优点.      

通用有限元系统实现无限域波动仿真方法

在有限元模型中加入人工边界,可以进行无限域波动仿真。本文将介绍一种改进的Higdon人工边界,并通过适当的变换形式,将其在通用有限元软件上实现。最后通过数值实验验证了这种方法具有良好的稳定性和计算精度,并且实施简单方便,其他人工边界条件可参照本文方法,通过变换形式在通用有限元程序中实现。     

成层半空间出平面自由波场的一维化时域算法

提出了一种计算出平面SH波斜入射时弹性水平成层半空间中自由波场时域计算的一维化有 限元方法. 在进行有限元网格划分时，竖向单元取满足有限元模拟精度的任意尺寸，水平向 网格尺寸由时间离散步长和水平视波速确定，并自动进行虚拟网格划分. 基底设置人工边界， 并将波动输入转化为等效荷载施加在边界节点上. 然后将集中质量有限元法和中心差分法相 结合建立节点运动方程，并将水平方向相邻节点的运动用该节点相邻时刻的运动表示，从而 将求解节点运动的二维方程组转化为一维方程组. 求解此方程组，即得到自由场中竖向一列 节点的运动. 最后根据行波传播的特点，可方便地确定全部自由波场. 理论分析和数值算例 表明，该方法具有较高的精度和良好的稳定性.     

波动问题中流体介质的动力人工边界

无限域流体介质的波动辐射效应是影响海域工程动力反应的重要因素,人工边界是实现此类开放系统近场波动问题数值分析的有效方法.基于位移格式的流体波动理论推导开放域流体介质的人工边界,分别给出一维、二维和三维空间中平面波、柱面波和球面波的流体介质动力人工边界条件,其中一维平面波动人工边界为经典的黏性边界,二维柱面波、三维球面波的人工边界处节点应力与节点速度和加速度成正比,可等效为由阻尼与质量系统构成的人工边界条件.讨论相应的数值模拟技术,给出流体介质动力人工边界在ANSYS软件平台的具体实现方法.近场流体介质动力反应问题的算例表明,所发展的流体动力人工边界对于轴对称波动与非轴对称波动在近场有限域截断处的透射吸收作用的模拟计算精度均较为良好,说明此流体介质人工边界具有较高的可靠性与实用性.所发展的流体介质动力人工边界可较为方便地与大型商用有限元软件结合,可为包括海域地形和海岛在内的海域工程的动力分析提供一定的方法借鉴.     

A Higdon-like non-reflecting boundary condition for the Klein–Gordon equation with evanescent waves

The non-reflecting boundary condition developed by Higdon and automated by Givoli and Neta is highly effective at absorbing propagating waves in a finite difference setting, but it does not absorb evanescent waves. In this paper, we augment the Higdon scheme with additional terms to absorb these evanescent waves in the context of the two-dimensional Klein–Gordon equation. Numerical examples illustrate the performance of this technique.     

Artificial boundary conditions for out-of-plane motion in penta-graphene

In this paper, artificial boundary conditions are designed for out-of-plane waves in penta-graphene, a newly proposed allotrope of carbon. By matching the dispersion relation for acoustic branch phonons in the long wave limit, we determine parameters in proposed linear constraints among displacements and velocities at the boundary and nearby atoms. Reflection analysis for normal incidences and a numerical test demonstrate the effectiveness of the artificial boundary conditions. These conditions may be used for studying mechanical behaviours of the novel complex lattice of penta-graphene.     

Elastic Waves in Swelling Porous Media

Time harmonic waves in a swelling porous elastic medium of infinite extent and consisting of solid, liquid and gas phases have been studied. Employing Eringen’s theory of swelling porous media, it has been shown that there exist three dilatational and two shear waves propagating with distinct velocities. The velocities of these waves are found to be frequency dependent and complex valued, showing that the waves are attenuating in nature. Here, the appearance of an additional shear wave is new and arises due to swelling phenomena of the medium, which disappears in the absence of swelling. The reflection phenomenon of an incident dilatational wave from a stress-free plane boundary of a porous elastic half-space has been investigated for two types of boundary surfaces: (i) surface having open pores and (ii) surface having sealed pores. Using appropriate boundary conditions for these boundary surfaces, the equations giving the reflection coefficients corresponding to various reflected waves are presented. Numerical computations are performed for a specific model consisting of sandstone, water and carbon dioxide as solid, liquid and gas phases, respectively, of the porous medium. The variations of phase speeds and their corresponding attenuation coefficients are depicted against frequency parameter for all the existing waves. The variations of reflection coefficients and corresponding energy ratios against the angle of incidence are also computed and depicted graphically. It has been shown that in a limiting case, Eringen’s theory of swelling porous media reduces to Tuncay and Corapcioglu theory of porous media containing two immiscible fluids. The various numerical results under these two theories have been compared graphically.     

Diffraction by a nonplanar screen

This paper is concerned with diffraction of short waves by a nonplanar screen (two-dimensional case, Dirichlet boundary condition). The high-frequency asymptotic approximation to the solution is obtained. First the wave field of the primary wave is found in a neighbourhood of the screen edge and then this field is continued along the boundary. Secondary waves arise here as the consequence of interaction between the edge and the primary wave. The secondary wave is diffracted by another edge of the screen, and a third order wave arises, and so on. This process gives the formulas for the wave field in a neighbourhood of the screen. Green's formula is used to continue the solution outside of this neighbourhood.     

On formulas for the Rayleigh wave velocity in pre-stressed compressible solids

In this paper, formulas for the velocity of Rayleigh waves in compressible isotropic solids subject to uniform initial deformations are derived using the theory of cubic equation. They are explicit, have simple algebraic forms, and hold for a general strain energy function. Unlike the previous investigations where the derived formulas for Rayleigh wave velocity are approximate and valid for only small enough values of pre-strains, this paper establishes exact formulas for Rayleigh wave velocity being valid for any range of pre-strains. When the prestresses are absent, the obtained formulas recover the Rayleigh wave velocity formula for compressible elastic solids. Since obtained formulas are explicit, exact and hold for any range of pre-strains, they are good tools for evaluating nondestructively prestresses of structures.