Browsing by Author "Hein, Ricardo"

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    Computing numerically the Green's function of the half-plane Helmholtz operator with impedance boundary conditions
    (2007) Duran, Mario; Hein, Ricardo; Nedelec, Jean-Claude
    In this article we compute numerically the Green's function of the half-plane Helmholtz operator with impedance boundary conditions. A compactly perturbed half-plane Helmholtz problem is used to motivate this calculation, by treating it through integral equation techniques. These require the knowledge of the calculated Green's function, and lead to a boundary element discretization. The Green's function is computed using the inverse Fourier operator of its spectral transform, applying an inverse FFT for the regular part, and removing the singularities analytically. Finally, some numerical results for the Green's function and for a benchmark resonance problem are shown.
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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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    Fast multipole boundary element method for the Laplace equation in a locally perturbed half-plane with a Robin boundary condition
    (2012) Perez-Arancibia, Carlos; Ramaciotti, Pedro; Hein, Ricardo; Duran, Mario
    A fast multipole boundary element method (FM-BEM) for solving large-scale potential problems ruled by the Laplace equation in a locally-perturbed 2-D half-plane with a Robin boundary condition is developed in this paper. These problems arise in a wide gamut of applications, being the most relevant one the scattering of water-waves by floating and submerged bodies in water of infinite depth. The method is based on a multipole expansion of an explicit representation of the associated Green's function, which depends on a combination of complex-valued exponential integrals and elementary functions. The resulting method exhibits a computational performance and memory requirements similar to the classic FM-BEM for full-plane potential problems. Numerical examples demonstrate the accuracy and efficiency of the method. (C) 2012 Elsevier B.V. All rights reserved.