A boundary integral equation method for radiation and scattering of elastic waves in three dimensions
01 Jan 1985-International Journal for Numerical Methods in Engineering (Wiley)-Vol. 21, Iss: 1, pp 115-129
TL;DR: Etude du rayonnement et de la diffusion d'ondes elastiques par des obstacles de forme arbitraire as discussed by the authors, a.k.a.
Abstract: Etude du rayonnement et de la diffusion d'ondes elastiques par des obstacles de forme arbitraire
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15 Jul 2007TL;DR: Important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral equation research.
Abstract: Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral equation research. Also, learning the fundamentals of linear elastic wave theory does not require a quantum leap for electromagnetic practitioners. Integral equation methods have been around for several decades, and their introduction to electromagnetics has been due to the seminal works of Richmond and Harrington in the 1960s. There was a surge in the interest in this topic in the 1980s (notably the work of Wilton and his coworkers) due to increased computing power. The interest in this area was on the wane when it was demonstrated that differential equation methods, with their sparse matrices, can solve many problems more efficiently than integral equation methods. Recently, due to the advent of fast algorithms, there has been a revival in integral equation methods in electromagnetics. Much of our work in recent years has been in fast algorithms for integral equations, which prompted our interest in integral equation methods. While previously, only tens of thousands of unknowns could be solved by integral equation methods, now, tens of millions of unknowns can be solved with fast algorithms. This has prompted new enthusiasm in integral equation methods.
473 citations
TL;DR: In this article, a general method for the direct evaluation of Cauchy principal value integrals in several dimensions is presented, which is an issue of major concern in any boundary element method analysis in applied mechanics.
Abstract: This paper presents a new general method for the direct evaluation of Cauchy principal value integrals in several dimensions, which is an issue of major concern in any boundary element method analysis in applied mechanics. It is shown that the original Cauchy principal value integral can be transformed into an element-by-element sum of regular integrals, each one expressed in terms of intrinsic (local) coordinates. The actual computation can be performed by standard quadrature formulae and can be easily included in any existing computer code. The numerical results demonstrate the accuracy and efficiency of the method, along with its insensitivity to the mesh pattern. This new method has full generality and, therefore, can be applied in any field of applied mechanics. Moreover, there are no restrictions on the numerical implementation, as the singular integrals may be defined on surface elements or internal cells of any order and type.
374 citations
TL;DR: In this article, the problem of structural isolation from ground transmitted vibrations by open or infilled trenches under conditions of plane strain is numerically studied, where the soil medium is assumed to be linear elastic or viscoelastic, homogeneous and isotropic.
Abstract: The problem of structural isolation from ground transmitted vibrations by open or infilled trenches under conditions of plane strain is numerically studied. The soil medium is assumed to be linear elastic or viscoelastic, homogeneous and isotropic. Horizontally propagating Rayleigh waves or waves generated by the motion of a rigid foundation or by surface blasting are considered in this work. The formulation and solution of the problem is accomplished by the boundary element method in the frequency domain for harmonic disturbances or in conjunction with Laplace transform for transient disturbances. The proposed method, which requires a discretisation of only the trench perimeter, the soil-foundation interface and some portion of the free soil surface on either side of the trench appears to be better than either finite element or finite difference techniques. Some parametric studies are also conducted to assess the importance of the various geometrical, material and dynamic input parameters and provide useful guidelines to the design engineer.
224 citations
TL;DR: In this paper, a 2.5D coupled finite element-boundary element methodology for the computation of the dynamic interaction between a layered soil and structures with a longitudinally invariant geometry, such as railway tracks, roads, tunnels, dams, and pipelines is presented.
200 citations
TL;DR: In this article, the isolation of structures from ground transmitted waves by open and infilled trenches in a 3D context is numerically studied, where the soil medium is assumed to be elastic or viscoelastic, homogeneous and isotropic.
Abstract: The isolation of structures from ground transmitted waves by open and infilled trenches in a three-dimensional context is numerically studied. The soil medium is assumed to be elastic or viscoelastic, homogeneous and isotropic. Waves generated by the harmonic motion of a surface rigid machine foundation are considered in this work. The formulation and solution of the problem is accomplished by the boundary element method in the frequency domain. The infinite space fundamental solution is used requiring discretization of the trench surface, the soil-foundation interface and some portion of the free soil surface. The proposed methodology is first tested for accuracy by solving three characteristic wave propagation problems with known solutions and then applied to several vibration isolation problems involving open and concrete infilled trenches. Three-dimensional graphic displays of the surface displacement pattern around the trenches are also presented.
159 citations
References
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01 Jan 1904-Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences
TL;DR: In this paper, the propagation of vibrations over the surface of a "semi-infinite" isotropic elastic solid, i.e., a solid bounded only by a plane, is considered.
Abstract: 1. This paper treats of the propagation of vibrations over the surface of a “semiinfinite” isotropic elastic solid, i. e. ,a solid bounded only by a plane. For purposes of description this plane may be conceived as horizontal, and the solid as lying below it, although gravity is not specially taken into account. The vibrations are supposed due to an arbitrary application of force at a point. In the problem most fully discussed this force consists of an impulse applied vertically to the surface; but some other cases, including that of an internal source of disturbance, are also (more briefly) considered. Owing to the complexity of the problem, it has been thought best to concentrate attention on the vibrations as they manifest themselves at the free surface. The modifications which the latter introduces into the character of the waves propagated into the interior of the solid are accordingly not examined minutely.
1,000 citations
TL;DR: In this article, a combined Helmholtz Integral Equation Formulation (CHIEF) was proposed to obtain an approximate solution of the exterior steadystate acoustic radiation problem for an arbitrary surface whose normal velocity is specified.
Abstract: Three different integral formulations have been used as a basis for obtaining approximate solutions of the exterior steady‐state acoustic radiation problem for an arbitrary surface whose normal velocity is specified: (1) the simple‐source formulation, adapted from potential theory; (2) the surface Helmholtz integral formulation, based on the integral expression for pressure in the field in terms of surface pressure and normal velocity; and (3) the interior Helmholtz integral formulation, in which the surface pressure is determined by making a certain integral vanish for all points interior to the radiating surface. For certain characteristic wavenumbers, it is shown that no solution of the simple‐source formulation exists in general and that there is no unique solution of the surface Helmholtz integral formulation. The interior Helmholtz integral formulation is subject to similar difficulties and has undesirable computational characteristics. A Combined Helmholtz Integral Equation Formulation (CHIEF) that overcomes the deficiencies of the first two methods and the undesirable computational characteristics of the third, is described. The significant improvement over the previous three methods, which is accomplished through the use of CHIEF, is illustrated by numerical examples involving spheres, finite cylinders, cubes, and a steerable array mounted in two different boxlike structures.
986 citations
TL;DR: An accurate method is presented for the numerical inversion of Laplace transform, which is a natural continuation to Dubner and Abate's method, and the error bound on the inverse f{t) becomes independent of t, instead of being exponential in t.
Abstract: An accurate method is presented for the numerical inversion of Laplace transform, which is a natural continuation to Dubner and Abate's method. (Dubner and Abate, 1968). The advantages of this modified procedure are twofold: first, the error bound on the inverse f{t) becomes independent of t, instead of being exponential in t; second, and consequently, the trigonometric series obtained for fit) in terms of F(s) is valid on the whole period 2T of the series. As it is proved, this error bound can be set arbitrarily small, and it is always possible to get good results, even in rather difficult cases. Particular implementations and numerical examples are presented.
953 citations
TL;DR: In this paper, a vector boundary formula relating the boundary values of displacement and traction for the general equilibrated stress state is derived, which is used to generate integral equations for the solution of the traction, displacement, and mixed boundary value problems of plane elasticity.
Abstract: The analogy between potential theory and classical elasticity suggests an extension of the powerful method of integral equations to the boundary value problems of elasticity. A vector boundary formula relating the boundary values of displacement and traction for the general equilibrated stress state is derived. The vector formula itself is shown to generate integral equations for the solution of the traction, displacement, and mixed boundary value problems of plane elasticity. However, an outstanding conceptual advantage of the formulation is that it is not restricted to two dimensions. This distinguishes it from the methods of Muskhelishvili and most other familiar integral equation methods. The presented approach is a real variable one and is applicable, without inherent restriction, to multiply connected domains. More precisely, no difficulty of the order of determining a mapping function is present and unwanted Volterra type dislocation solutions are eliminated a priori. An indication of techniques necessary to effect numerical solution of the resulting integral equations is presented with numerical data from a set of test problems.
699 citations
TL;DR: In this paper, the elastic body is divided into subregions, and the surface and interfaces are represented by quadrilateral and triangular elements with quadratic variation of geometry.
Abstract: The field equations of three-dimensional elastostatics are transformed to boundary integral equations. The elastic body is divided into subregions, and the surface and interfaces are represented by quadrilateral and triangular elements with quadratic variation of geometry and linear, quadratic or cubic variation of displacement and traction with respect to intrinsic co-ordinates. The integral equation is discretized for each subregion, and a system of banded form obtained. For the integration of kernel-shape function products, Gaussian quadrature formulae are chosen according to upper bounds for error in terms of derivatives of the integrands. Use of the integral formulation is illustrated by the analysis of a prestressed concrete nuclear reactor pressure vessel.
670 citations