Three‐dimensional BEM for scattering of elastic waves in general anisotropic media

Based on the full‐space Green's functions, a three‐dimensional time‐harmonic boundary element method is presented for the scattering of elastic waves in a triclinic full space. The boundary integral equations for incident, scattered and total wave fields are given. An efficient numerical method is proposed to calculate the free terms for any geometry. The discretization of the boundary integral equation is achieved by using a linear triangular element. Applications are discussed for scattering of elastic waves by a spherical cavity in a 3D triclinic medium. The method has been tested by comparing the numerical results with the existing analytical solutions for an isotropic problem. The results show that, in addition to the frequency of the incident waves, the scattered waves strongly depend on the anisotropy of the media. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Time-domain BEM for 3-D transient elastodynamic problems with interacting rigid movable disc-shaped inclusions

Formulation of time-domain boundary element method for elastodynamic analysis of interaction between rigid massive disc-shaped inclusions subjected to impinging elastic waves is presented. Boundary integral equations (BIEs) with time-retarded kernels are obtained by using the integral representations of displacements in a matrix in terms of interfacial stress jumps across the inhomogeneities and satisfaction of linearity conditions at the inclusion domains. The equations of motion for each inclusion complete the problem formulation. The time-stepping/collocation scheme is implemented for the discretization of the BIEs by taking into account the traveling nature of the generated wave field and local structure of the solution at the inclusion edges. Numerical results concern normal incidence of longitudinal wave onto two coplanar circular inclusions. The inertial effects are revealed by the time dependencies of inclusions’ kinematic parameters and dynamic stress intensity factors in the inclusion vicinities for different mass ratios and distances between the interacting obstacles.     

Numerical solution of elastic inclusion problem using complex variable boundary integral equation

Based on a complex variable boundary integral equation (CVBIE) suggested previously, this paper provides a numerical solution for the elastic inclusion problem using CVBIE. A dissimilar elastic inclusion is embedded in the infinite matrix. The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for the elastic inclusion, while the other is an exterior BVP for the matrix with notch. Both problems are connected by conventional boundary integral equations (BIEs) in complex variables. After performing discretization for the coupled BIEs, the inverse matrix technique is suggested to solve the relevant algebraic equations. Based on the properties of some integral operators, three ways for the inverse matrix technique are suggested. Several numerical examples are carried out to prove the efficiency of the suggested method.     

An energy approach to space–time Galerkin BEM for wave propagation problems

In this paper we consider Dirichlet or Neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential. Starting from a natural energy identity, a space–time weak formulation for 1D integral problems is briefly introduced, and continuity and coerciveness properties of the related bilinear form are proved. Then, a theoretical analysis of an extension of the introduced formulation for 2D problems is proposed, pointing out the novelty with respect to existing literature results. At last, various numerical simulations will be presented and discussed, showing unconditional stability of the space–time Galerkin boundary element method applied to the energetic weak problem. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Cohesive-zone-model formulation and implementation using the symmetric Galerkin boundary element method for homogeneous solids

A new symmetric boundary integral formulation for cohesive cracks growing in the interior of homogeneous linear elastic isotropic media with a known crack path is developed and implemented in a numerical code. A crack path can be known due to some symmetry implications or the presence of a weak or bonded surface between two solids. The use of a two-dimensional exponential cohesive law and of a special technique for its inclusion in the symmetric Galerkin boundary element method allows us to develop a simple and efficient formulation and implementation of a cohesive zone model. This formulation is dependent on only one variable in the cohesive zone (relative displacement). The corresponding constitutive cohesive equations present a softening branch which induces to the problem a potential instability. The development and implementation of a suitable solution algorithm capable of following the growth of the cohesive zone and subsequent crack growth becomes an important issue. An arc-length control combined with a Newton–Raphson algorithm for iterative solution of nonlinear equations is developed. The boundary element method is very attractive for modeling cohesive crack problems as all nonlinearities are located along the boundaries (including the crack boundaries) of linear elastic domains. A Galerkin approximation scheme, applied to a suitable symmetric integral formulation, ensures an easy treatment of cracks in homogeneous media and excellent convergence behavior of the numerical solution. Numerical results for the wedge split and mixed-mode flexure tests are presented.     

Dynamic solution of unbounded domains using finite element method: discrete Green's functions in frequency domain

In this paper we present a new approach for finite element solution of time‐harmonic wave problems on unbounded domains. As representatives of the wave problems, discrete Green's functions are evaluated in finite element sense. The finite element mesh is considered to be of repeatable pattern (cell) constructed in rectangular co‐ordinates. The system of FE equations is therefore reduced to a set of well‐known dispersion equations by using a spectral solution approach. The spectral wave bases are constructed directly from the FE dispersion equations. Radiation condition is satisfied by selecting the wave bases so that the wave information is transmitted in appropriate directions at the cell level. Dirichlet/Neumann boundary conditions are defined at the edges of a quadrant of the main domain while using the axes of symmetry of the problem. A new discrete transformation method, recently proposed by the authors, is used to satisfy the boundary conditions. Comprehensive studies are made for showing the validity, accuracy and convergence of the solutions. The results of the benchmark problems indicate that the proposed method can be used to evaluate discrete Green's functions whose analytical forms are not available. Copyright © 2006 John Wiley & Sons, Ltd.     

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。     

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.     

Boundary-only element solutions of 2D and 3D nonlinear and nonhomogeneous elastic problems

This paper presents a robust boundary element method (BEM) that can be used to solve elastic problems with nonlinearly varying material parameters, such as the functionally graded material (FGM) and damage mechanics problems. The main feature of this method is that no internal cells are required to evaluate domain integrals appearing in the conventional integral equations derived for these problems, and very few internal points are needed to improve the computational accuracy. In addition, one of the basic field quantities used in the boundary integral equations is normalized by the material parameter. As a result, no gradients of the field quantities are involved in the integral equations. Another advantage of using the normalized quantities is that no material parameters are included in the boundary integrals, so that a unified equation form can be established for multi-region problems which have different material parameters. This is very efficient for solving composite structural problems.     

Source point isolation boundary element method for solving general anisotropic potential and elastic problems with varying material properties

This paper presents a new robust boundary element method, based on a source point isolation technique, for solving general anisotropic potential and elastic problems with varying coefficients. Different types of fundamental solutions can be used to derive the basic integral equations for specific anisotropic problems, although fundamental solutions corresponding to isotropic problems are recommended and adopted in the paper. The use of isotropic fundamental solutions for anisotropic and/or varying material property problems results in domain integrals in the basic integral equations. The radial integration method is employed to transform the domain integrals into boundary integrals, resulting in a pure boundary element analysis algorithm that does not need any internal cells. Numerical examples for 2D and 3D potential and elastic problems are given to demonstrate the correctness and robustness of the proposed method.     

Diffraction of shear waves on cylindrical inclusions and cavities in an elastic half space

Dynamic problems of the steady-state oscillations of a half space with different types of cylindrical homogeneities (cavities, rigid and stiff inclusions) are examined. The boundary of the half space is assumed to be fixed or free of forces. A harmonic shear wave radiating from infinity or a concentrated harmonic source may be radiators of the exciting wave field. Integral representations of displacement amplitudes, which automatically satisfy fixity conditions on the boundary of the half space and radiation conditions at infinity are constructed. The edge problems are reduced to Fredholm integral equations of the second kind and to singular integral equations. Selection of additional conditions for the latter is substantiated. Some computer-generated results are presented.Translated from Problemy Prochnosti, No. 11, pp. 90–94, November, 1990.     

A boundary element solution to elasto-plastic torsion of solids of revolution

A boundary element method for the solution of problems of elastic and elasto-plastic torsion of solids of revolution is proposed. The displacement and stress field produced by a circumferential ring load in an infinite elastic body are derived and used as the fundamental solution to establish the governing integral equations. For the elasto-plastic case, linear boundary elements and bilinear quadrilateral internal cells are used for discretization of these equations. In order to carry out Gaussian integrations along boundary elements and over internal cells, a method ensuring sufficient accuracy for the calculation of singular integrals is proposed. Numerical results for several problems are given.     

考虑热松弛的热弹耦合二维问题的有限元法

为了解决利用积分变换方法在求解Lord-Shulman (L-S)型广义热弹性耦合二维问题时由于数值反变换所引起的计算精度降低的问题,该文采用新近被应用的直接有限元方法,求解了基于L-S型广义热弹性理论的半无限大体受热冲击作用的动态响应问题,结果表明,该方法对求解L-S型广义热弹性耦合二维问题具有很高的精度。该文给出了L-S型广义热弹性理论下的热弹耦合的控制方程,建立了L-S型的广义热弹性问题的虚位移原理,推导得到了相应的有限元方程。经计算得到了半无限大体中无量纲温度、位移及应力的分布规律,从温度分布图上可以清晰地观察到热波波前的特有属性,即热波波前处存在温度突变。     

2D analysis for geometrically non-linear elastic problems using the BEM

A boundary element method (BEM) approach for the solution of the elastic problem with geometrical non-linearities is proposed. The geometrical non-linearities that are considered are both finite strains and large displacements. Material non-linearities are not considered in this paper, so the constitutive law employed is Hooke's elastic one and the fundamental solution introduced in the integral equations is the usual one for isotropic linear elasticity. In order to deal with the intricate non-linear equations that govern the problem, an incremental–iterative method is proposed. The equations are linearized and a Total Lagrangian Formulation is adopted. The integral equations of the BEM are developed in an incremental form. The iterative process is necessary in order to achieve a good approximation to the governing equations. The problem of a slab under homogeneous deformation is solved and the results obtained agree with the analytical solution. The problem of a hollow cylinder under internal pressure is also solved and its solution compared with that obtained by a standardized finite element method code.     

A circular inclusion in a finite domain I. The Dirichlet-Eshelby problem

Summary This is the first paper in a series concerned with the precise characterization of the elastic fields due to inclusions embedded in a finite elastic medium. A novel solution procedure has been developed to systematically solve a type of Fredholm integral equations based on symmetry, self-similarity, and invariant group arguments. In this paper, we consider a two-dimensional (2D) circular inclusion within a finite, circular representative volume element (RVE). The RVE is considered isotropic, linear elastic and is subjected to a displacement (Dirichlet) boundary condition. Starting from the 2D plane strain Navier equation and by using our new solution technique, we obtain the exact disturbance displacement and strain fields due to a prescribed constant eigenstrain field within the inclusion. The solution is characterized by the so-called Dirichlet-Eshelby tensor, which is provided in closed form for both the exterior and interior region of the inclusion. Some immediate applications of the Dirichlet-Eshelby tensor are discussed briefly.     

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.     

A robust formulation for solving fluid-structure interaction problems

 In this article, a coupled finite element and boundary element approach for the acoustic radiation and scattering from submerged elastic bodies of arbitrary shape is presented. An alternative to the direct boundary element method for acoustics is proposed. By taking an auxiliary source surface inside the radiating boundary and following the usual discretization and integration procedures employed in the boundary element method, both the singularities of the integrands and the nonuniqueness problems do not arise. In addition, the difficulty of slope discontinuity also can be overcome. This procedure is formulated in a similar fashion of wave superposition method, except that the direct boundary integral equations are adopted. The proposed formulation employ the surface Helmholtz integral equation and its normal gradient like that adopted in the Burton–Miller approach, but do not employ any coupling constant. Typical examples are presented that demonstrate the efficiency of the proposed technique. Received 9 April 2000     

A stable 3D energetic Galerkin BEM approach for wave propagation interior problems

We consider 3D interior wave propagation problems with vanishing initial and mixed boundary conditions, reformulated as a system of two boundary integral equations with retarded potentials. These latter are then set in a weak form, based on a natural energy identity satisfied by the solution of the differential problem, and discretized by the energetic Galerkin boundary element method. Numerical results are presented and discussed in order to show the stability and accuracy of the proposed technique.