D. Givoli, Numerical Methods for Problems in Infinite Domains (Elsevier, Amsterdam, 1992).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in both CFD and other areas of scientific computing that the treatment of ABC s may have a profound impact on the overall performance of numerical algorithms and interpretation of the results. The literature on various ABC s techniques is extensive, a detailed review can be found in work by Givoli[1, 2], as well as in a more recent paper by Tsynkov[3] The construction of ABC s based on the di#erence potentials method (DPM) by Ryaben kii [4, 5, 6] was an attempt to combine in one technique the advantages relevant to both local and global methodologies, see Refs. 7, 8, 9, 10, 11, 12, 13, 14, ....

Givoli, D., Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.

....boundary integrals (e.g. boundary element methods [59] to connect the problem within the finite computational domain to the unbounded exterior region. The use of infinite elements also allows a computational framework that incorporates the behav30 ior of the fields at infinity into the solution [47]. Both of these techniques have some advantages but may introduce other computational di#culties. For a detailed review of di#erent techniques used for the numerical solution of problems on unbounded domains, see [106] It is important to recognize that for the classes of geophysical applications ....

D. Givoli. Numerical Methods for Problems in Infinite Domains. Elsevier, 1992. 142

....which is very often encountered in the numerical treatment of problems in unbounded domains. That is, the problem of finding appropriate boundary conditions to be imposed on artificial boundary introduced to make the computational domain finite. There is a vast literature about this problem (see [11], for instance, and the many references cited therein) and it is well known that exact artificial boundary conditions can be formulated in terms of the Dirichlet to Neumann (DtN) operator. By means of the semi classical Green functions we constructed we are able to make this procedure completely ....

D. Givoli. Numerical Methods for Problems in Infinite Domains. Elsevier, Amsterdam, 1992.

....boundary conditions. First, and most important, are the Sommerfeld like boundary conditions based on Sommerfeld s famous asymptotic boundary condition, 3] r u Gamma in 0 k 0 u = o(r ) Then there are the nonlocal Dirichlet to Neumann boundary conditions for separable coordinate systems [4] and [5] the family of corresponding local and asymptotic approximations [6] infinite element methods [7] and the various kinds of boundary element methods [8] However, if the exterior domain becomes inhomogeneous, e.g. by an embedded waveguide (Fig. 1) these methods do not longer work. For ....

D. Givoli. Numerical methods for problems in infinite domains, volume 33 of Studies in Applies Mechanics. Elsevier, 1992.

....recent implementation of this scheme, finite elements were used to model the transient response of an acoustic fluid in an exterior domain coupled to a viscoelastic structure [125] Numerous other schemes have been proposed, many of which are surveyed in the exhaustive reviews [1, pp. 95116 ] and [40]. Once a boundary value problem is formulated in a bounded region, finite element methods may be employed for computation, taking advantage of the wide range of applicability and rich mathematical structure inherent in these techniques. In this work, a computational formulation is derived by the ....

D. Givoli. Numerical Methods for Problems in Infinite Domains. Elsevier, Amsterdam, 1992.

....subject has been concentrated on the development and analysis of fast and reliable numerical schemes combining effective methods for the interior problem with methods constructed to solve problems in exterior domains. The methods most often used are presented and analyzed in the books by Givoli [4] and by Ihlenburg [8] and can be grouped roughly into the following classes: 1. Combination of interior methods like the finite element method with boundary element methods to solve both the interior and the exterior problem. 2. Combination of interior methods with an infinite element method to ....

D. Givoli. Numerical methods for problems in infinite domains, volume 33 of Studies in Applies Mechanics. Elsevier, 1992.

....is certainly one of the most intensively researched topic in the area of the numerical wave propagation. The most popular techniques include local absorbing boundary conditions, see e.g. 1] various variations of B erenger s perfectly matched layer [3] the Dirichlet to Neumann (DtN) method [17, 18, 19], and many others, see e.g. 20] The literature on the subject is really overwhelming, see e.g. 31] This contribution focuses on the technique of Infinite Elements. The idea of coupled Finite Element (FE) Infinite Element (IE) approximations for exterior wave propagation problems dates back to ....

....The idea of coupled Finite Element (FE) Infinite Element (IE) approximations for exterior wave propagation problems dates back to the pioneering contributions of Bettess and Bettess and Zienkiewicz, see [4] and the literature cited therein. The works of Astley, Cremers, Coyette et al..l. Givoli [2, 9, 17] and many others recognized the spectral character of the approximation and pointed to the necessity of multipole expansions. Burnett [5] revolutionized the approach from the practical point of view, introducing prolate and oblate spheroidal elements, and formulating a new, symmetric unconjugated ....

D. Givoli, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.

....formula for the Neumann data on Gamma in terms of the Dirichlet data on the same curve, which is known as the Dirichlet to Neumann mapping. This has been done for several elliptic operators, including the Lam e system for elasticity, by using Fourier type series developments (see, e.g. 9] [17], 19] 21] Moreover, the mapping obtained in [21] together with the mixed finite element approach from [15] were utilized in [11] to study the weak solvability of an exterior hyperelastic interface problem. The same problem had been solved before in [1] by using displacement finite element ....

....from [19] for the corresponding linear problem) indicates that, with a sufficiently large circle Gamma , a small N may provide very good accuracy in the computations. This property has also been observed before with analogue DtN mappings for other exterior boundary value problems (see, e.g. [17], 21] 5 The a posteriori error estimator 5.1 Preliminaries Let fT h g h2I be a regular family of triangulations of Omega , as defined in Section 4.2, where I is a at most numerable set of indexes, say I : fh j g j2N , with h j h j 1 8 j 2 N. Here, regular means that the interior angles of ....

Givoli, D.: Numerical Methods for Problems in Infinite Domains. Elsevier Science Publishers B.V., Studies in Applied Mechanics 33. Gabriel N. Gatica, Luis F. Gatica and Ernst P. Stephan

....region to approximate the effects of the missing unbounded portion. Simple ideas often do not work well and reflect spurious information from the artificial exterior boundary back into the finite computational domain. A survey of the state of the art in this area has been compiled by Givoli [22]. A portion of the book deals with finite element methods but there is enough general material to construct a project on boundary conditions for problems on unbounded domains. Pay particular attention to material in Chapters 1, 3 8, 10. A project could concentrate on the very effective DtN ....

D. Givoli. Numerical Methods for Problems in Infinite Domains. Elsevier, Amsterdam, 1992.

....and constantly increasing. The following two sections do not attempt to reproduce a general survey of the field, but simply to present some of the previous work this dissertation and its conclusions are based upon. A survey and extensive bibliography can be found in the recent monograph by Givoli [17]. 1.1 Frequency Domain Methods The reduced wave equation or Helmholtz equation describes many diverse forms of scattering phenomena. These arise in such seemingly unrelated fields as acoustics, electromagnetics, hydrodynamics, elastodynamics, quantum mechanics, and medical imaging. This equation ....

....domain for all wave numbers, and in general improves CHAPTER 2. THE HELMHOLTZ EQUATION 39 upon the accuracy. Although we have presented the modified DtN condition only for the Helmholtz equation, our procedure applies to other equations for which DtN boundary conditions have been developed [17]. The modification procedure can also be applied to the exact nonreflecting boundary conditions that were obtained in [28] for the time dependent wave equation in three space dimensions. We have also derived the DtN and modified DtN boundary conditions for elliptic and prolate spheroidal ....

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D. Givoli. Numerical Methods for Problems in Infinite Domains, volume 33 of Studies in Applied Mechanics. Elsevier, 1992.

....condition. We will show some comparisons of the first two BGT conditions as well as the Sommerfeld condition versus the exact DtN condition in Chapter 2. An extensive survey of existing work on NRBC s can be found in the survey article [46] by Dan Givoli as well as his recent monograph [47]. 1.5.2 The DtN Boundary Condition The DtN boundary condition requires that, on the boundary under consideration, the Dirichlet and Neumann data of the solution of a differential equation be related by the Dirichlet to Neumann (DtN) mapping M of the problem at hand : e.g. in our case it requires ....

....2 Gamma art : hold. The DtN map can be computed once one has a suitable representation of the solution. The systematic development of the DtN boundary condition as a tool for CHAPTER 1. INTRODUCTION 32 solving boundary value problems on truncated domains is due to J.B. Keller and D. Givoli cf. [67, 47, 48] and the references therein. General Boundaries For smooth artificial boundaries of general shape, the DtN map is given by an integral operator which, for the Laplace or Helmholtz equations, can be expressed in terms of the classical layer potentials. Given an arbitrary (smooth, say, C 2 or ....

D. Givoli. Numerical Methods for Problems in Infinite Domains, volume 33 of Studies in Applied Mechanics. Elsevier, 1992.

....boundary value problem in the computational domain for all wave numbers, and in general improves upon the accuracy. Although we have presented the modified DtN condition only for the Helmholtz equation, our procedure applies to other equations for which DtN boundary conditions have been developed [19]. The modification procedure can also be applied to the exact nonreflecting boundary conditions that were obtained in [20] for the time dependent wave equation in three space dimensions. We have also derived the DtN and modified DtN boundary conditions for elliptic and prolate spheroidal ....

D. Givoli, Numerical Methods for Problems in Infinite Domains, Studies in Appl. Mech. 33, Elsevier, 1992.

....requirements when using such simple boundary procedures, one often has to choose the excessively large computational domains. A survey of methods for setting the ABC s in different areas of scientific computing can be found in our work [1] as well as in the comprehensive reviews by Givoli [2, 3]. These surveys give a comparative assessment of advantages and disadvantages of various global and local techniques, and also point out the relations between the global and local methods. 1.2. Methods and Objectives. This paper continues our work on constructing the ABC s that would combine the ....

D. Givoli, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.

....paper. When writing a review on different ABC s techniques, we have tried to make it as comprehensive as possible within the limits of a journal publication. At the same time we clearly realize that no survey paper of this kind can be completely thorough. Therefore, we mention here work by Givoli [204,205] that presents a detailed review of the state of the area that dates a few years back. We also mention that the previous, shorter version of this review was published as NASA paper [206] In addition, a survey of the ABC s 28 S. V. TSYNKOV techniques for CEM that apply to solving the ....

D. Givoli, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.

....finite elements. On this truncation boundary special boundary conditions need to be imposed that allow radiated waves to leave the computational domain without any reflection. Several non reflecting boundary conditions with varying accuracy and computational characteristics have been proposed, see [1] for a review of various boundary y Graduate research assistant, E mail: manish am sun2.stanford.edu z Professor of Mechanical Engineering, E mail: pinsky am sun2.stanford.edu conditions. In this paper we consider the Dirichlet to Neumann (DtN) radiation condition [2] for time harmonic ....

D. Givoli. Numerical methods for problems in infinite domains. Elsevier, 1992.

....i;2 Gamman i;1 ; 1.4.i) and ff i = s i 2 i ae i ; fi i = r i ae i ; 1.4.ii) where n i = n i;1 ; n i;2 ) is the outward normal to Gamma i . The equation (1. 3) is the first order absorbing boundary condition for elastic wave, the derivation of which can be found in [5] 6] [8] and [12] The conditions on the interfaces of two media are complicated. On a solid solid interface, the interface conditions should impose the vanishing of the relative movements (welded boundary condition) and the continuity of the normal stresses, that is, u i Gamma u j = 0; on Gamma ij ; ....

D. Givoli, Numerical methods for problems in infinite domains, Elsevie, Amsterdam/New York, 1992.

....always aim to obtaining a robust numerical procedure, therefore, the robustness becomes one of the most important experimental criteria for assessing and comparing different ABC s. Obviously, the review provided above is not complete. Referring the reader to two comprehensive works by Givoli [78, 79] for more survey information, we now proceed to describing the DPM [29, 30] and the DPM based ABC s, which are the focus of this presentation. 2. The Generalized Potentials and the Difference Potentials Method. Our ultimate goal is to construct ABC s that, to a certain extent, combine the ....

D. Givoli, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.

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D. Givoli, Numerical Methods for Problems in Infinite Domains (Elsevier, Amsterdam, 1992).

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D. Givoli, Numerical Methods for Problems in Infinite Domains (Elsevier, Amsterdam, 1992).

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Givoli, D., Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.

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