Boundary integral equation analysis for radiation and scattering of elastic waves in three dimensions

Abstract

In this paper, the boundary integral equation (BIE) method is employed to investigate the radiation and scattering of time‐harmonic elastic waves by obstacles of arbitrary shape embedded in an infinite medium. Based on the vector BIE, entirely free of Cauchy principal value integrals, an efficient numerical scheme using quadratic isoparametric boundary elements is proposed. Furthermore, the difficulty of non‐uniquess of a solution inherent with BIE formulations for exterior elastodynamic problems is studied numerically and analytically. The counterparts of the combined Helmholtz integral formulation method for elastodynamics together with the least‐square or Lagrange‐multiplier technique are derived and applied to overcome this difficulty successfully. In addition, the elastic‐wave fields radiated or scattered by either a spherical cavity or a rigid sphere in an infinite medium are calculated and the results are compared with the analytical solutions to demonstrate the accuracy and versatility of the proposed numerical scheme.     

Plane wave basis in Galerkin BEM for bidimensional wave scattering

This paper considers the problem of scattering of a time-harmonic acoustic incident wave by a bidimensional hard obstacle. The numerical solution to this problem is found using a Galerkin wave boundary integral formulation whereby the functional space is built as the product of conventional low order piecewise polynomials with a set of plane waves propagating in various directions. In this work we improve the original method by presenting new strategies when dealing with irregular meshes and corners. Numerical results clearly demonstrate that these improvements allow the handling of scatterers with complicated geometries while maintaining a low discretization level of 2.5–3 degrees of freedom per full wavelength. This makes the method a reliable tool for tackling high-frequency scattering problems.     

A 2D time-domain BEM for transient wave scattering analysis by a crack in anisotropic solids

A two-dimensional (2D) time-domain boundary element method (BEM) is presented in this paper for transient analysis of elastic wave scattering by a crack in homogeneous, anisotropic and linearly elastic solids. A traction boundary integral equation formulation is applied to solve the arising initial-boundary value problem. A numerical solution procedure is developed to solve the time-domain boundary integral equations. A collocation method is used for the temporal discretization, while a Galerkin-method is adopted for the spatial discretization of the boundary integral equations. Since the hypersingular boundary integral equations are first regularized to weakly singular ones, no special integration technique is needed in the present method. Special attention of the analysis is devoted to the computation of the scattered wave fields. Numerical examples are given to show the accuracy and the reliability of the present time-domain BEM. The effects of the material anisotropy on the transient wave scattering characteristics are investigated.     

层状饱和半空间中沉积谷地对斜入射平面P1波的三维散射

在作者给出层状饱和场地三维精确动力刚度矩阵和层状饱和半空间中移动荷载动力格林函数基础上,采用间接边界元方法在频域内求解了层状流体饱和场地中沉积谷地对斜入射平面P1波的三维散射问题。该方法的特点在于虚拟移动均布荷载和斜线孔隙水压可以直接施加在沉积与层状饱和半空间交界面而不存在奇异性。该文通过与已有结果的比较验证了方法的正确性,并以均匀饱和半空间和弹性基岩上单一饱和土层中沉积谷地为例进行了数值计算分析。研究表明,沉积谷地对平面P1波的三维散射与二维散射之间存在本质差别,入射角度、孔隙率、饱和土层刚度和饱和土层厚度等参数对沉积谷地附近动力响应有着显著影响。     

A scattering problem by means of the spectral representation of Green's function for a layered acoustic half-space

 The problem in this paper is for scattering waves caused by an object and a plane wave in a layered acoustic half space. The boundary integral equation method as well as the spectral representation of Green's function for a layered acoustic half space are introduced to the present analyses. The spectral form of Green's function developed here is expressed in terms of the eigenfunctions for the point and the continuous spectra, that is the extension form of Green's function expressed by Ewing, Jardetsky and Press (1957). The advantage of the spectral representation of Green's function is that it enables us to decompose the scattering waves into eigenfunctions for the layered medium. Several numerical calculations are carried out to examine the efficiency of the present method as well as the properties of the scattering waves. According to the numerical results, the spectral form of Green's function provides accurate values and is applicable to the boundary element analysis for a layered medium. The spectral structures of the scattering waves are also found to be able to explain their properties. Received 2 November 1999     

多层介质硬币形(Penny-Shaped)交界裂纹的弹性波散射

本文研究多层介质硬币形交界裂纹的弹性波散射.文中采用Hankel积分变换,得到了含有硬币形交界裂纹多层介质模型的散射波传递矩阵,并将散射问题为转化求解矩阵形式的对偶积分方程.作为特例,文中给出了单一弹性层与半空间的硬币形交界裂纹的弹性波散射远场模式,并计算了几组不同弹性常数组合情形下的远场模式的幅频特性曲线,其结果表明有共振峰存在.     

弹性波二维散射快速多极子间接边界元法求解

求解方程的稠密矩阵特征极大削弱了传统边界元法在求解大规模实际工程问题中的优势。为此,结合快速多极子展开技术,发展一种新的高精度快速间接边界元方法,用于求解大尺度或高频弹性波二维散射问题。以全空间孔洞周围SH波散射为例,给出了具体求解步骤。算例分析表明该方法具有很高的计算精度和求解效率,同时能够大幅度降低计算存储量,可在目前主流计算机上实现上百万自由度弹性波散射问题的快速求解。最后以半空间中凹陷场地对SH波的高频散射为例,讨论了凹陷周围高频波散射的基本特征,可为峡谷地形中大型工程抗震设计提供部分理论依据。     

Scattering of a plane harmonic SH wave by a completely embedded corrugated scatterer

Anti‐plane‐strain model for steady‐state scattering of elastic waves by a rough inclusion or a cavity embedded in a half space is considered by using a direct boundary integral equation method. The roughness of the scatterer is assumed to be periodic with arbitrary amplitude and period. Detailed testing of the numerical results is presented. The motion along the half‐space surface is evaluated for different corrugations, frequencies and impedance contrast of the materials. The importance of the scatterer roughness upon the displacement field is clearly demonstrated. It was shown that larger corrugation amplitudes, shorter corrugation periods and higher frequencies may produce significant change in the displacement field when compared with the corresponding smooth scatterer result. This effect strongly depends upon the impedance contrast of the materials. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element solution for electromagnetic scattering from two-dimensional bodies

The problem of scattering from bodies in free space is formulated using a differential equation approach. The finite element mesh extends outward into the far field region of the scattering body, where the outer boundary condition is evaluated using the asymptotic expression for the scattered field. Numerical results for two scattering bodies are presented and discussed. Non-physical, standing waves appear in the results due to the inadequacy of the outer boundary condition in fulfilling the radiation condition for the scattered field. The differential equation approach does not appear to be competitive with integral equation approaches for thin bodies, but seems promising for handling scattering from thick inhomogeneous bodies into which the field penetrates.     

A Hybrid T Matrix/Boundary Element Method for Elastic Wave Scattering from a Defect Near a Non-planar Surface

The in-plane P-SV scattering of elastic waves by a defect and a close non-planar surface is considered. A hybrid T matrix/boundary element approach is used, where a boundary integral equation is used for the non-planar surface and the Green’s tensor in this integral equation is chosen as the one for the defect and thus incorporates the transition (T) matrix of the defect. The integral equation is discretized by the boundary element method in a standard way. Also models of ultrasonic probes in transmission and reception are included. In the numerical examples the defect is for simplicity chosen as a circular cavity. This cavity is located close to a non-planar surface, which is planar except for a smooth transition between two planar parts. It is illustrated that the scattering by the cavity and the non-planar surface becomes quite complicated, and that shielding and masking may appear.     

Time domain modelling of thin wire arrays in the presence of a two media configuration using the boundary element analysis

Transient response of a multiple wire configuration in the presence of a two-media configuration excited by a voltage source (antenna mode) or illuminated by an incident field is analysed using the boundary element method (BEM). The analysis is based on the solution of the corresponding set of the coupled space-time Hallen integral equation and it is carried out directly in the time domain. The influence of a two media configuration is taken into account via the space time reflection coefficient. The corresponding integral equation set is handled via the time domain variant of the Galerkin–Bubnov indirect boundary element method (GB-IBEM). Some illustrative numerical results for both antenna and scattering mode are presented in the paper.     

Ultrasonic guided wave scattering in a plate overlap

Guided wave scattering in a plate overlap is investigated by numerical calculations and experimental measurements of transmission and reflection factors from the overlap region. In the numerical study, a hybrid boundary element-finite element method is used to calculate the guided wave scattered field from the overlap region. Transmission and reflection factors are calculated for incident A0 and S0 Lamb and n0 shear horizontal waves, including higher modes generated through mode conversion phenomena. In addition, parametric studies of transmission and reflection factors in this problem are performed numerically over various incident modes, frequencies, and overlap lengths. For verification and comparison with numerical results, experiments were conducted to measure the transmission and reflection factors for incident Lamb and shear horizontal waves in steel plates with two different overlap areas. The experimental results agree well with the numerical calculations. The numerical and experimental results show that it is highly feasible to carry out efficient Lamb wave nondestructive evaluation (NDE) in overlapped plates and in multilayer structures with various lap joints by selecting various modes and tuning frequency.     

Lagrangian finite element treatment of transient vibration/acoustics of biosolids immersed in fluids

Superposition principle is used to separate the incident acoustic wave from the scattered and radiated waves in a displacement‐based finite element model. An absorbing boundary condition is applied to the perturbation part of the displacement. Linear constitutive equation allows for inhomogeneous, anisotropic materials, both fluids and solids. Displacement‐based finite elements are used for all materials in the computational volume. Robust performance for materials with limited compressibility is achieved using assumed‐strain nodally integrated simplex elements or incompatible‐mode brick elements. A centered‐difference time‐stepping algorithm is formulated to handle general damping accurately and efficiently. Verification problems (response of empty steel cylinder immersed in water to a step plane wave, and scattering of harmonic plane waves from an elastic sphere) are discussed for assumed‐strain simplex and for voxel‐based brick finite element models. A voxel‐based modeling scheme for complex biological geometries is described, and two illustrative results are presented from the bioacoustics application domain: reception of sound by the human ear and simulation of biosonar in beaked whales. Copyright © 2007 John Wiley & Sons, Ltd.     

Scattering of inhomogeneous wave by viscoelastic interface crack

Summary The reflection, refraction and scattering of inhomogeneous plane waves of SH type by an interface crack between two dissimilar viscoelastic bodies are investigated. The singular integral equation method is used to reduce the scattering problem into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Then, the singular integral equation is solved numerically by Kurtz's piecewise continous function method. The crack opening displacement and dynamic stress intensity factor characterizing the scattered near-field are estimated for various incident angles, frequencies and relaxation times. The differences on crack opening displacement and stress intensity factor between elastic and viscoelastic interface crack are contrasted. And the effects of incident angle, incident frequency and relaxation time of the viscoelastic material are analyzed and explained by the features of phase lag and energy dissipation of the viscoelastic wave.     

直角域中椭圆形夹杂对SH波的散射与地震动

利用复变函数法和保角映射法研究了SH波入射到含有椭圆形夹杂的直角域中时的散射问题。首先,利用"镜像"叠加原理构造出满足直角域两个直线边界应力自由条件的等效入射及反射波场,通过将椭圆形夹杂的外域映射为单位圆外域构造出满足直角域自由表面应力自由条件的椭圆形夹杂的散射波场。其次,利用椭圆形夹杂边界处的应力和位移连续条件建立求解未知系数的积分方程组,并通过截断有限项求解。最后,给出了在不同参数条件下直角域水平边界处的地表位移幅值。通过算例可以看出:入射波数、入射角度、夹杂位置、介质参数等参数均对地表位移幅值有影响。     

Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method

In this paper, we investigate a numerical approach based on the partition of unity finite element method, for the time‐harmonic elastic wave equations. The aim of the proposed work is to accurately model two‐dimensional elastic wave problems with fewer elements, capable of containing many wavelengths per nodal spacing, and without refining the mesh at each frequency. The approximation of the displacement field is performed via the standard finite element shape functions, enriched by superimposing pressure and shear plane wave basis, which incorporate knowledge of the wave propagation. A variational framework able to handle mixed boundary conditions is described. Numerical examples dealing with the radiation and the scattering of elastic waves by a circular body are presented. The results show the performance of the proposed method in both accuracy and efficiency. Copyright © 2008 John Wiley & Sons, Ltd.     

基于TI介质模型的凸起地形对平面SH波的放大作用

在模拟地震波在介质中传播过程的研究中,正演模拟是一个重要的方面。该文采用间接边界元法（IBEM）对横观各向同性（TI）不同参数介质的凸起场地中入射SH波的散射问题进行了正演模拟,研究了TI凸起场地对SH波的放大效应。放大效应是通过地表位移与基岩露头位移幅值的比值来体现的。该间接边界元法结合了层状TI半空间精确动力刚度矩阵和均布斜线荷载动力格林函数,具有较高的精度。该文分别给出了TI介质凸起中SH波散射问题在频域和时域内的解答。频域内研究分析表明,土体TI介质参数的变化会对场地位移幅值的放大谱和空间分布产生显著的影响,即改变了凸起部分和土层的动力特性,使得两者动力相互作用发生改变。时域内研究表明,SH波在凸起周围的传播同时依赖于TI介质参数的具体取值和传播方向。时域位移幅值云图清晰地展示入射波、透射波、反射波和散射波的传播过程,凸起角点产生的散射波在TI介质中传播呈现"半椭圆"型,而在各向同性介质中呈现"半圆"型。     

Modified fundamental solutions for the scattering of elastic waves by a cavity: Numerical results

The boundary integral equation method is very often used to solve exterior problems of scattering of waves (elastic waves, acoustic waves, water waves and electromagnetic waves). It is known, however, that this method fails to provide a unique solution at the so-called irregular frequencies. This difficulty is inherent to the method used rather than the nature of the problem. In the context of elastodynamics. we proposed, in a recent work¹, two methods for eliminating these irregular frequencies. Both are based on modifying the fundamental solution. Here we present numerical results pertaining to the solutions of the modified and unmodified integral equations.     

Numerical resolution of the boundary integral equations for elastic scattering by a plane crack

The problem of wave scattering by a plane crack is solved, either in the case of acoustic waves or in the case of elastic waves incidence using the boundary integral equation method. A collocation method is often used to solve that equation, but here we will use a variational method, first writing the problem of Fourier variables, and then writing the associated integrals in the sesquilinear form with weak singularity kernels. This representation is used in the numerical approach, made with a finite element method in the surface of the crack. Numerical tests were made with circular and elliptical cracks, but this method can be extended to other shapes, with the same convergence profiles. Extensive results are given concerning the crack opening displacement, the scattering cross-section, the back-scattered amplitude and far-field patterns.     

Numerical algorithm for defect reconstruction in elastic media with a circular ultrasonic scanning

A numerical reconstruction method is proposed, which is applied to image identification of defects detected in elastic solid samples, in the case when a circular Ultrasonic scanning provides a measurement of the scattering pattern over full interval of the incident polar angle. The problem is first formulated as a system of respective boundary integral equations whose solution is used to calculate the far-field scattering diagram. Then the inverse reconstruction problem is reduced to a minimization of a certain strongly nonlinear functional. The proposed numerical algorithm is tested on some examples of volumetric flaw. It is also evaluated the influence of the error in the input data on precision of the reconstruction.