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.     

Iteratively-improved Robin boundary conditions for the finite element solution of scattering problems in unbounded domains

An iterative procedure is described for the finite-element solution of scalar scattering problems in unbounded domains. The scattering objects may have multiple connectivity, may be of different materials or with different boundary conditions. A fictitious boundary enclosing all the objects involved is introduced. An appropriate Robin (mixed) condition is initially guessed on this boundary and is iteratively improved making use of Green's formula. It will be seen that the best choice for the Robin boundary condition is an absorbing-like one. A theorem about the theoretical convergence of the procedure is demonstrated. An analytical study of the special case of a circular cylindrical scatterer is made. Comparisons are made with other methods. Some numerical examples are provided in order to illustrate and validate the procedure and to show its applicability whatever the frequency of the incident wave. Although particular emphasis is laid in the paper on electromagnetic problems, the procedure is fully applicable to other kinds of physical phenomena such as acoustic ones. © 1998 John Wiley & Sons, Ltd.     

Application of the boundary element method to elastic wave scattering by irregular defects

A time-harmonic boundary element formulation for elastic wave scattering in 3D is adapted to ultrasonic NDE. Defect classes addressed are volumetric voids and inclusions, and crack-like elliptical voids. For axisymmetric flaws, comparisons are made with method of optimal truncation (MOOT) and transition-matrix calculations. Comparison to experiment is made for more general shapes. For crack-like voids, comparisons are made with the Kirchhoff, geometric theory of diffraction (GTD), and quasistatic asymptotic approximations. The efficiency and usefulness of the boundary element method (BEM) in finding the bounds of applicability of these approximate theories are demonstrated. An example of a flaw characterization technique based on intermediate frequency scattering data simulated by BEM is given. The ability of BEM to handle nonplanar incident fields, as described by a transducer beam model, is shown. Other computational and modeling efficiencies of the BEM are noted.     

Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems

A diagonal form fast multipole boundary element method (BEM) is presented in this paper for solving 3-D acoustic wave problems based on the Burton-Miller boundary integral equation (BIE) formulation. Analytical expressions of the moments in the diagonal fast multipole BEM are derived for constant elements, which are shown to be more accurate, stable and efficient than those using direct numerical integration. Numerical examples show that using the analytical moments can reduce the CPU time by a lot as compared with that using the direct numerical integration. The percentage of CPU time reduction largely depends on the proportion of the time used for moments calculation to the overall solution time. Several examples are studied to investigate the effectiveness and efficiency of the developed diagonal fast multipole BEM as compared with earlier p³ fast multipole method BEM, including a scattering problem of a dolphin modeled with 404,422 boundary elements and a radiation problem of a train wheel track modeled with 257,972 elements. These realistic, large-scale BEM models clearly demonstrate the effectiveness, efficiency and potential of the developed diagonal form fast multipole BEM for solving large-scale acoustic wave problems.     

Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems

A new adaptive fast multipole boundary element method (BEM) for solving 3-D half-space acoustic wave problems is presented in this paper. The half-space Green's function is employed explicitly in the boundary integral equation (BIE) formulation so that a tree structure of the boundary elements only for the boundaries of the real domain need to be applied, instead of using a tree structure that contains both the real domain and its mirror image. This procedure simplifies the implementation of the adaptive fast multipole BEM and reduces the CPU time and memory storage by about a half for large-scale half-space problems. An improved adaptive fast multipole BEM is presented for the half-space acoustic wave problems, based on the one developed recently for the full-space problems. This new fast multipole BEM is validated using several simple half-space models first, and then applied to model 3-D sound barriers and a large-scale windmill model with five turbines. The largest BEM model with 557470 elements was solved in about an hour on a desktop PC. The accuracy and efficiency of the BEM results clearly show the potential of the adaptive fast multipole BEM for solving large-scale half-space acoustic wave problems that are of practical significance.     

Boundary element method calculation for coherent electromagnetic scattering from two and three dielectric spheres

A quadratic, isoparametric boundary element formulation has been used to calculate the multiple scattering of electromagnetic waves from systems of two and three dielectric spheres. Extinction efficiency results for the scattering of a plane wave are presented for variations of the separation of the two spheres in three kinds of orientations of the system with respect to the incident wave. These have been verified against analytical calculations based on Mie's theory and calculations by other authors. The results demonstrate a large side scattering resonance (the so-called specular resonance). Agreement between the results establishes the boundary element method as a very powerful tool for solving multiple scattering problems because the method applies to arbitrarily shaped objects having a homogeneous dielectric constant in any configuration. To illustrate the versatility of the method, scattering from three spheres is calculated.     

An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation

The high solution costs and non-uniqueness difficulties in the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) formulation are two main weaknesses in the BEM for solving exterior acoustic wave problems. To tackle these two weaknesses, an adaptive fast multipole boundary element method (FMBEM) based on the Burton–Miller formulation for 3-D acoustics is presented in this paper. In this adaptive FMBEM, the Burton–Miller formulation using a linear combination of the CBIE and hypersingular BIE (HBIE) is applied to overcome the non-uniqueness difficulties. The iterative solver generalized minimal residual (GMRES) and fast multipole method (FMM) are adopted to improve the overall computational efficiency. This adaptive FMBEM for acoustics is an extension of the adaptive FMBEM for 3-D potential problems developed by the authors recently. Several examples on large-scale acoustic radiation and scattering problems are presented in this paper which show that the developed adaptive FMBEM can be several times faster than the non-adaptive FMBEM while maintaining the accuracies of the BEM.     

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.     

A new boundary element formulation for wave load analysis

A new boundary element (BEM) formulation is proposed for wave load analysis of submerged or floating bodies. The presented formulation, through establishing an impedance relation, permits the evaluation of the hydrodynamic coefficients (added mass and damping coefficients) and the coefficients of wave exciting forces systematically in terms of system matrices of BEM without solving any special problem, such as, unit velocity or unit excitation problem. It also eliminates the need for scattering analysis in the evaluation of wave exciting forces. The imaginary and real parts of impedance matrix give, respectively, added mass and damping matrices whose elements describe the fluid resistance against the motion of the body. The formulation is explained through the use of a simple fluid-solid system under wave excitations, which involves a uniform fluid layer containing a solid cylindrical body. In the formulation, the solid body is taken first as deformable, then, it is specialized when it is rigid. The validity of the proposed method is verified by comparing its result with those available in literature for rigid submerged or floating bodies.     

Elastic wave scattering by a rectangular crack near a non-planar back surface

A 3D model of non-destructive ultrasonic testing for cracks near a non-planar back surface is presented. The scattering by an interior rectangular crack in a thick-walled component with a back surface of general geometry is considered. The 3D wave scattering problem is solved using boundary integral equation methods (BIEMs): the boundary element method (BEM) for the back surface displacement is combined with an analytical technique for the hypersingular traction boundary integral equation for the crack opening displacement. The solution method generates many unknowns, but by applying a threshold criterion a sparse approximation of the system matrix is obtained such that a fast sparse solver may be used. The computations are accelerated further using the stationary phase approximation for the computation of probe field integrals. The action of ultrasonic probes in transmission and reception, calibration by side-drilled holes and effects of material damping are taken into account in the model, and a few numerical examples illustrate the influence of the back surface geometry.     

An iterative algorithm for singular Cauchy problems for the steady state anisotropic heat conduction equation

In this paper, we propose an alternating iterative algorithm to solve a singular Cauchy problem for the anisotropic heat conduction equation. The numerical algorithm is based on the boundary element method (BEM), modified to take into account the form of the singularity, without substantially increasing the amount of computation involved. Two test examples, the first with a singularity caused by an abrupt change in the boundary conditions and the second with a singularity caused by a sharp re-entrant corner, are investigated. The numerical results obtained confirm that provided an appropriate stopping regularization criterion is imposed, the iterative BEM is efficient in dealing with the difficulties arising from both the instabilities produced by the boundary condition formulation and the slow rate of convergence of standard numerical methods around the singular point.     

二维层状地基波阻板隔振分析

为了研究层状地基中波阻板的隔振效果，基于薄层法在研究层状介质中波的传播问题的高效性、边界单元法处理无限域问题的精确性，结合两种方法的优点，本文采用以薄层法层状半空间基本解答作为格林函数的边界元法，分别对上软下硬和上硬下软两种层状半空间地基中波阻板的隔振效果进行分析。研究表明：增加波阻板的厚度、提高波阻板的弹性模量可以显著提高隔振效果；分层土体厚度和土性对于水平和竖向的位移振幅衰减系数有较大影响。     

Boundary element method solutions for elastic wave scattering in 3D

Time-harmonic elastic wave scattering problems such as those encountered in ultrasonic non destructive evaluation are solved by the boundary element method (BEM). Selected results for spherical and spheroidal shaped voids and inclusions are compared with analytical and other numerical solutions. Results for ellipsoids, which require a full three-dimensional formulation, are provided as a benchmark for comparison when other numerical methods would be developed for this problem class in the future. The modelling of cracklike defects with this formulation is discussed. Recent theoretical findings regarding the fictitious eigenfrequency difficulty (FED) are confirmed by a numerical study.     

Three-dimensional localization and optical imaging of objects in turbid media with independent component analysis

A new approach for optical imaging and localization of objects in turbid media that makes use of the independent component analysis (ICA) from information theory is demonstrated. Experimental arrangement realizes a multisource illumination of a turbid medium with embedded objects and a multidetector acquisition of transmitted light on the medium boundary. The resulting spatial diversity and multiple angular observations provide robust data for three-dimensional localization and characterization of absorbing and scattering inhomogeneities embedded in a turbid medium. ICA of the perturbations in the spatial intensity distribution on the medium boundary sorts out the embedded objects, and their locations are obtained from Green's function analysis based on any appropriate light propagation model. Imaging experiments were carried out on two highly scattering samples of thickness approximately 50 times the transport mean-free path of the respective medium. One turbid medium had two embedded absorptive objects, and the other had four scattering objects. An independent component separation of the signal, in conjunction with diffusive photon migration theory, was used to locate the embedded inhomogeneities. In both cases, improved lateral and axial localizations of the objects over the result obtained by use of common photon migration reconstruction algorithms were achieved. The approach is applicable to different medium geometries, can be used with any suitable photon propagation model, and is amenable to near-real-time imaging applications.     

BEM for crack-hole problems in thermopiezoelectric materials

The boundary element formulation for analysing interaction between a hole and multiple cracks in piezoelectric materials is presented. Using Green's function for hole problems and variational principle, a boundary element model (BEM) for a 2-D thermopiezoelectric solid with cracks and holes has been developed and used to calculate stress intensity factors of the crack-hole problem. In BEM, the boundary condition on the hole circumference is satisfied a priori by Green's function, and is not involved in the boundary element equations. The method is applicable to multiple-crack problems in both finite and infinite solids. Numerical results for stress and electric displacement intensity factors at a particular crack tip in a crack-hole system of piezoelectric materials are presented to illustrate the application of the proposed formulation.     

A comparison of techniques for overcoming non‐uniqueness of boundary integral equations for the collocation partition of unity method in two‐dimensional acoustic scattering

The Partition of Unity Method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements. The non‐uniqueness of solution obtained from the conventional boundary integral equation (CBIE) is well known. The CBIE derived through Green's identities suffers from a problem of non‐uniqueness at certain characteristic frequencies. Two of the standard methods of overcoming this problem are the so‐called Combined Helmholtz Integral Equation Formulation (CHIEF) method and that of Burton and Miller. The latter method introduces a hypersingular integral, which may be treated in various ways. In this paper, we present the collocation partition of unity boundary element method (PUBEM) for the Helmholtz problem and compare the performance of CHIEF against a Burton–Miller formulation regularised using the approach of Li and Huang. Copyright © 2013 John Wiley & Sons, Ltd.     

Use of constant,linear and quadratic boundary elements in 3D wave diffraction analysis

The performance of the Boundary Element Method (BEM) depends on the size of the elements and the interpolation function used. However, improvements in accuracy and efficiency obtained with both expansion and grid refinement increases demand on the computational effort. This paper evaluates the performance of constant, linear and quadratic elements in the analysis of the three-dimensional scattering caused by a cylindrical cavity buried in an infinite homogeneous elastic medium subjected to a point load. A circular cylindrical cavity for which analytical solutions are known is used in the simulation analysis. First, the dominant BEM errors are identified in the frequency domain and related to the natural vibration modes of the inclusion. Comparisons of BEM errors are then made for different types of boundary elements, maintaining similar computational costs. Finally, the accuracy of the BEM solution is evaluated when the nodal points are moved inside linear and quadratic discontinuous elements.     

On the BEM lumped mass formulations of wave equation problems

This paper continues earlier research on the performance of several consistent mass matrices for the boundary element method (BEM) dynamic analysis (wave equation problems), examining now the applicability of lumped mass matrices for several BEM formulations. In the beginning, the lumped masses are distributed along the boundary and three relative BEM formulations are proposed. Finally, the lumped matrices are introduced into the Partial Differential Equation (PDE) as Dirac-type domain inertial terms.     

Shape optimization of acoustic scattering bodies

Shape optimization of acoustic scattering bodies is carried out using genetic algorithms (GA) coupled to a boundary element method for exterior acoustics. The BEM formulation relies on a modified Burton-Miller algorithm to resolve exterior acoustics and to address the uniqueness issue of the representation problem associated with the Helmholtz integral equation at the eigenvalues of the associated interior problem. The particular problem of interest considers an incident wave approaching an axisymmetric shaped body. The objective is to arrive at a geometric configuration that minimizes the acoustic intensity captured by a receiver located at a distance from the scattering body. In particular, the acoustic intensity is required to be minimum as measured proportional to the integral of the product of the potential and its complex conjugate over a volume of space which models the receiver. This is opposed to the more traditional measure of the potential at a single point in space.     

3D acoustic wave simulation using BEM formulations: Closed form integration of singular and hypersingular integrals

Among the obstacles to applying boundary element techniques to three-dimensional wave propagation problems is the difficulty of accurately representing the singular and hypersingular terms at the points of application of the virtual loads. This paper presents the analytical evaluation of the singular and hypersingular integrals for constant boundary elements. First, the singular integral results are compared with those evaluated by means of a Gaussian quadrature scheme, which uses an enormous amount of sampling points. In the case of hypersingular integrals the comparison makes use of the results provided by the method presented by Terai [T. Terai, On calculation of sound fields around three dimensional objects by integral equation methods, J Sound Vib 69 (1980) 71–100.]. An additional verification is performed by comparing the boundary element method (BEM) results with known analytical solutions for cylindrical inclusions.