Browsing by Author "Nedelec, Jean Claude"

Now showing 1 - 11 of 11
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    An Efficient Galerkin BEM to Compute High Acoustic Eigenfrequencies
    (ASME-AMER SOC MECHANICAL ENG, 2009) Duran, Mario; Nedelec, Jean Claude; Ossandon, Sebastian
    An efficient numerical method, using integral equations, is developed to calculate precisely the acoustic eigenfrequencies and their associated eigenvectors, located in a given high frequency interval. It is currently known that the real symmetric matrices are well adapted to numerical treatment. However, we show that this is not the case when using integral representations to determine with high accuracy the spectrum of elliptic, and other related operators. Functions are evaluated only in the boundary of the domain, so very fine discretizations may be chosen to obtain high eigenfrequencies. We discuss the stability and convergence of the proposed method. Finally we show some examples.
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    Asymptotics for Helmholtz and Maxwell Solutions in 3-D Open Waveguides
    (GLOBAL SCIENCE PRESS, 2012) Jerez Hanckes, Carlos; Nedelec, Jean Claude
    We extend classic Sommerfeld and Silver-Muller radiation conditions for bounded scatterers to acoustic and electromagnetic fields propagating over three isotropic homogeneous layers in three dimensions. If x = (x(1), x(2), x(3)) is an element of R-3, with x(3) denoting the direction orthogonal to the layers, standard conditions only hold for the outer layers in the region |x(3) | > parallel to x parallel to(gamma), for gamma is an element of (1/4, 1/2) and x large. For |x(3)| < parallel to x parallel to(gamma). and inside the slab, asymptotic behavior depends on the presence of surface or guided modes given by the discrete spectrum of the associated operator.
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    Computing Green's function of elasticity in a half-plane with impedance boundary condition
    (ELSEVIER FRANCE-EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER, 2006) Duran, Mario; Godoy, Eduardo; Nedelec, Jean Claude
    This Note presents an effective and accurate method for numerical calculation of the Green's function G associated with the time harmonic elasticity system in a half-plane, where an impedance boundary condition is considered. The need to compute this function arises when studying wave propagation in underground mining and seismological engineering. To theoretically obtain this Green's function, we have drawn our inspiration from the paper by Duran et al. (2005), where the Green's function for the Helmholtz equation has been computed. The method consists in applying a partial Fourier transform, which allows an explicit calculation of the so-called spectral Green's function. In order to compute its inverse Fourier transform, we separate (G) over cap as a sum of two terms. The first is associated with the whole plane, whereas the second takes into account the half-plane and the boundary conditions. The first term corresponds to the Green's function of the well known time-harmonic elasticity system in R-2 (cf. J. Dompierre, Thesis). The second term is separated as a sum of three terms, where two of them contain singularities in the spectral variable (pseudo-poles and poles) and the other is regular and decreasing at infinity. The inverse Fourier transform of the singular terms are analytically computed, whereas the regular one is numerically obtained via an FFT algorithm. We present a numerical result. Moreover, we show that, under some conditions, a fourth additional slowness appears and which could produce a new surface wave. (c) 2006 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
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    EXPLICIT REPRESENTATION FOR THE INFINITE-DEPTH TWO-DIMENSIONAL FREE-SURFACE GREEN'S FUNCTION IN LINEAR WATER-WAVE THEORY
    (SIAM PUBLICATIONS, 2010) Hein, Ricardo; Duran, Mario; Nedelec, Jean Claude
    In this paper we derive an explicit representation for the two-dimensional free-surface Green's function in water of infinite depth, based on a finite combination of complex-valued exponential integrals and elementary functions. This representation can easily and accurately be evaluated in a numerical manner, and its main advantage over other representations lies in its simplicity and in the fact that it can be extended towards the complementary half-plane in a straightforward manner. It seems that this extension has not been studied rigorously until now, and it is required when boundary integral equations are extended in the same way. For the computation of the Green's function, the limiting absorption principle and a partial Fourier transform along the free surface are used. Some of its properties are also discussed, and an expression for its far field is developed, which allows us to state appropriately the involved radiation condition. This Green's function is then used to solve the two-dimensional infinite-depth water-wave problem by developing a corresponding boundary integral equation, whose solution is determined by means of the boundary element method. To validate the computations, a benchmark problem based on a half-circle is presented and solved numerically.
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    EXPLICIT VARIATIONAL FORMS FOR THE INVERSES OF INTEGRAL LOGARITHMIC OPERATORS OVER AN INTERVAL
    (SIAM PUBLICATIONS, 2012) Jerez Hanckes, Carlos; Nedelec, Jean Claude
    We introduce explicit and exact variational formulations for the weakly singular and hypersingular operators over an open interval as well as for their corresponding inverses. Contrary to the case of a closed curve, these operators no longer map fractional Sobolev spaces in a dual fashion but degenerate into different subspaces depending on their extensibility by zero. We show that an average and jump decomposition leads to precise coercivity results and characterize the mismatch occurring between associated functional spaces. Through this setting, we naturally define Calderon-type identities with their potential use as preconditioners. Moreover, we provide an interesting relation between the logarithmic operators and one-dimensional Laplace Dirichlet and Neumann problems. This work is a detailed and extended version of the article "Variational Forms for the Inverses of Integral Logarithmic Operators over an Interval" by Jerez-Hanckes and Nedelec [C. R. Acad. Sci. Paris Ser. I, 349 (2011), pp. 547-552].
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    On the existence of surface waves in an elastic half-space with impedance boundary conditions
    (ELSEVIER SCIENCE BV, 2012) Godoy, Eduardo; Duran, Mario; Nedelec, Jean Claude
    In this work, the problem of surface waves in an isotropic elastic half-space with impedance boundary conditions is investigated. It is assumed that the boundary is free of normal traction and the shear traction varies linearly with the tangential component of displacement multiplied by the frequency, where the impedance corresponds to the constant of proportionality. The standard traction-free boundary conditions are then retrieved for zero impedance. The secular equation for surface waves with impedance boundary conditions is derived in explicit form. The existence and uniqueness of the Rayleigh wave is properly established, and it is found that its velocity varies with the impedance. Moreover, we prove that an additional surface wave exists in a particular case, whose velocity lies between those of the longitudinal and the transverse waves. Numerical examples are presented to illustrate the obtained results. (C) 2012 Elsevier B.V. All rights reserved.
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    Radiation condition and uniqueness for the outgoing elastic wave in a half-plane with free boundary
    (ELSEVIER FRANCE-EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER, 2009) Duran, Mario; Muga, Ignacio; Nedelec, Jean Claude
    In this Note we deduce an explicit Sommerfeld-type radiation condition which is convenient to prove the uniqueness for the time-harmonic outgoing wave problem in an isotropic elastic half-plane with free boundary condition. The expression is obtained from a rigorous asymptotic analysis of the associated Green's function. The main difficulty is that the free boundary condition allows the propagation of a Rayleigh wave which cannot be neglected in the far field expansion. We also give the existence result for this problem. To cite this article: M. Duran et al., C R. Acad. Sci. Paris, Ser. I 347 (2009). (C) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
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    The Helmholtz equation in a locally perturbed half-plane with passive boundary
    (OXFORD UNIV PRESS, 2006) Duran, Mario; Muga, Ignacio; Nedelec, Jean Claude
    In this article, we study the existence and uniqueness of outgoing solutions for the Helmholtz equation in locally perturbed half-planes with passive boundary. We establish an explicit outgoing radiation condition which is somewhat different from the usual Sommerfeld's one due to the appearance of surface waves. We work with the help of Fourier analysis and a half-plane Green's function framework. This is an extended and detailed version of the previous article Duran et al.
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    THE OUTGOING TIME-HARMONIC ELASTIC WAVE IN A HALF-PLANE WITH FREE BOUNDARY
    (SIAM PUBLICATIONS, 2011) Duran, Mario; Muga, Ignacio; Nedelec, Jean Claude
    Under a time-harmonic assumption, we prove existence and uniqueness results for the outgoing elastic wave in an isotropic half-plane, where the source is given by a local normal stress excitation of the free boundary. This is the starting point for problems of elastic wave scattering by locally perturbed flat surfaces. The main difficulty is that the free boundary condition induces the propagation of a Rayleigh surface wave guided by the unbounded flat frontier. Due to the presence of this surface wave in the far field expansion, we need to impose a new radiation condition in order to describe both volume and surface outgoing wave behavior and to show uniqueness.
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    THEORETICAL ASPECTS AND NUMERICAL COMPUTATION OF THE TIME-HARMONIC GREEN'S FUNCTION FOR AN ISOTROPIC ELASTIC HALF-PLANE WITH AN IMPEDANCE BOUNDARY CONDITION
    (EDP SCIENCES S A, 2010) Duran, Mario; Godoy, Eduardo; Nedelec, Jean Claude
    This work presents an effective and accurate method for determining, from a theoretical and computational point of view, the time-harmonic Green's function of an isotropic elastic half-plane where an impedance boundary condition is considered. This method, based on the previous work done by Duran et al. (cf. [Numer. Math. 107 (2007) 295-314; IMA J. Appl. Math. 71 (2006) 853-876]) for the Helmholtz equation in a half-plane, combines appropriately analytical and numerical techniques, which has an important advantage because the obtention of explicit expressions for the surface waves. We show, in addition to the usual Rayleigh wave, another surface wave appearing in some special cases. Numerical results are given to illustrate that. This is an extended and detailed version of the previous article by Duran et al. [C. R. Acad. Sci. Paris, Ser. IIB 334 (2006) 725-731].
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    Variational forms for the inverses of integral logarithmic operators over an interval
    (ELSEVIER FRANCE-EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER, 2011) Jerez Hanckes, Carlos; Nedelec, Jean Claude
    We present explicit and exact variational formulations for the weakly singular and hypersingular operators over an interval as well as for their corresponding inverses. By decomposing the solutions in symmetric and antisymmetric parts, we precisely characterize the associated Sobolev spaces. Moreover, we are able to define novel Calderon-type identities in each case. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.