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Domínguez Baguena, Víctor

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https://academica-e.unavarra.es/handle/2454/48906

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Domínguez Baguena

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Víctor

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Estadística, Informática y Matemáticas

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InaMat2. Instituto de Investigación en Materiales Avanzados y Matemáticas

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0000-0002-6095-619X

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88591

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2552

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2015 - 201922020 - 20246
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Now showing 1 - 8 of 8
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    PublicationOpen Access
    A fully discrete Calderón calculus for the two-dimensional elastic wave equation
    (Elsevier, 2015) Domínguez Baguena, Víctor; Sánchez Vizuet, Tonatiuh; Sayas, Francisco Javier; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
    In this paper we present a full discretization of the layer potentials and boundary integral operators for the elastic wave equation on a parametrizable smooth closed curve in the plane. The method can be understood as a non-conforming Petrov–Galerkin discretization, with a very precise choice of testing functions by symmetrically combining elements on two staggered grids, and using a look-around quadrature formula. Unlike in the acoustic counterpart of this work, the kernel of the elastic double layer operator includes a periodic Hilbert transform that requires a particular choice of the mixing parameters. We give mathematical justification of this fact. Finally, we test the method on some frequency domain and time domain problems, and demonstrate its applicability on smooth open arcs.
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    PublicationEmbargo
    Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions
    (Oxford University Press, 2024-11-09) Domínguez Baguena, Víctor; Turc, Catalin; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
    Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves.
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    PublicationOpen Access
    An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: formulation, analysis, algorithm, and simulation
    (Elsevier, 2020) Domínguez Baguena, Víctor; Ganesh, M.; Sayas, Francisco Javier; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
    A natural medium for wave propagation comprises a coupled bounded heterogeneous region and an unbounded homogeneous free-space. Frequency-domain wave propagation models in the medium, such as the variable coefficient Helmholtz equation, include a faraway decay radiation condition (RC). It is desirable to develop algorithms that incorporate the full physics of the heterogeneous and unbounded medium wave propagation model, and avoid an approximation of the RC. In this work we first present and analyze an overlapping decomposition framework that is equivalent to the full-space heterogeneous-homogenous continuous model, governed by the Helmholtz equation with a spatially dependent refractive index and the RC. Our novel overlapping framework allows the user to choose two free boundaries, and gain the advantage of applying established high-order finite and boundary element methods (FEM and BEM) to simulate an equivalent coupled model. The coupled model comprises auxiliary interior bounded heterogeneous and exterior unbounded homogeneous Helmholtz problems. A smooth boundary can be chosen for simulating the exterior problem using a spectrally accurate BEM, and a simple boundary can be used to develop a high-order FEM for the interior problem. Thanks to the spectral accuracy of the exterior computational model, the resulting coupled system in the overlapping region is relatively very small. Using the decomposed equivalent framework, we develop a novel overlapping FEM-BEM algorithm for simulating the acoustic or electromagnetic wave propagation in two dimensions. Our FEM-BEM algorithm for the full-space model incorporates the RC exactly. Numerical experiments demonstrate the efficiency of the FEM-BEM approach for simulating smooth and non-smooth wave fields, with the latter induced by a complex heterogeneous medium and a discontinuous refractive index.
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    PublicationOpen Access
    High order Nyström methods for transmission problems for Helmholtz equation
    (Springer, 2016) Domínguez Baguena, Víctor; Turc, Catalin; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
    We present super-algebraic compatible Nyström discretizations for the four Helmholtz boundary operators of Calderón’s calculus on smooth closed curves in 2D. These discretizations are based on appropriate splitting of the kernels combined with very accurate product-quadrature rules for the different singularities that such kernels present. A Fourier based analysis shows that the four discrete operators converge to the continuous ones in appropriate Sobolev norms. This proves that Nyström discretizations of many popular integral equation formulations for Helmholtz equations are stable and convergent. The convergence is actually super-algebraic for smooth solutions.
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    PublicationEmbargo
    Robust boundary integral equations for the solution of elastic scattering problems via Helmholtz decompositions
    (Elsevier, 2024-12-01) Turc, Catalin; Domínguez Baguena, Víctor; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2
    Helmholtz decompositions of the elastic fields open up new avenues for the solution of linear elastic scattering problems via boundary integral equations (BIE) (Dong et al. (2021) [20]). The main appeal of this approach is that the ensuing systems of BIE feature only integral operators associated with the Helmholtz equation. However, these BIE involve non standard boundary integral operators that do not result after the application of either the Dirichlet or the Neumann trace to Helmholtz single and double layer potentials. Rather, the Helmholtz decomposition approach leads to BIE formulations of elastic scattering problems with Neumann boundary conditions that involve boundary traces of the Hessians of Helmholtz layer potential. As a consequence, the classical combined field approach applied in the framework of the Helmholtz decompositions leads to BIE formulations which, although robust, are not of the second kind. Following the regularizing methodology introduced in Boubendir et al. (2015) [6] we design and analyze novel robust Helmholtz decomposition BIE for the solution of elastic scattering that are of the second kind in the case of smooth scatterers in two dimensions. We present a variety of numerical results based on Nyström discretizations that illustrate the good performance of the second kind regularized formulations in connections to iterative solvers.
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    PublicationEmbargo
    Nyström discretizations of boundary integral equations for the solution of 2D elastic scattering problems
    (Elsevier, 2024) Domínguez Baguena, Víctor; Turc, Catalin; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
    We present three high-order Nyström discretization strategies of various boundary integral equation formulations of the impenetrable time-harmonic Navier equations in two dimensions. One class of such formulations is based on the four classical Boundary Integral Operators (BIOs) associated with the Green’s function of the Navier operator. We consider two types of Nyström discretizations of these operators, one that relies on Kussmaul–Martensen logarithmic splittings (Chapko et al., 2000; Domínguez and Turc, 2000), and the other on Alpert quadratures (Alpert, 1999). In addition, we consider an alternative formulation of Navier scattering problems based on Helmholtz decompositions of the elastic fields (Dong et al., 2021), which can be solved via a system of boundary integral equations that feature integral operators associated with the Helmholtz equation. Owing to the fact that some of the BIOs that are featured in those formulations are non-standard, we use Quadrature by Expansion (QBX) methods for their high order Nyström discretization. Alternatively, we use Maue integration by parts techniques to recast those non-standard operators in terms of single and double layer Helmholtz BIOs whose Nyström discretizations is amenable to the Kussmaul– Martensen methodology. We present a variety of numerical results concerning the high order accuracy that our Nyström discretization elastic scattering solvers achieve for both smooth and Lipschitz boundaries. We also present extensive comparisons regarding the iterative behavior of solvers based on different integral equations in the high frequency regime. Finally, we illustrate how some of the Nyström discretizations we considered can be incorporated seamlessly into the Convolution Quadrature (CQ) methodology to deliver high-order solutions of the time domain elastic scattering problems.
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    PublicationOpen Access
    Boundary integral equation methods for the solution of scattering and transmission 2D elastodynamic problems
    (Oxford University Press, 2022) Domínguez Baguena, Víctor; Turc, Catalin; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
    We introduce and analyse various regularized combined field integral equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyse OS methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators, which is also the basis of high-order Nystrom quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.
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    PublicationOpen Access
    Analysis and application of an overlapped FEM-BEM for wave propagation in unbounded and heterogeneous media
    (Elsevier, 2022) Domínguez Baguena, Víctor; Ganesh, M.; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
    An overlapped continuous model framework, for the Helmholtz wave propagation problem in unbounded regions comprising bounded heterogeneous media, was recently introduced and analyzed by the authors (2020) [10]. The continuous Helmholtz system incorporates a radiation condition (RC) and our equivalent hybrid framework facilitates application of widely used finite element methods (FEM) and boundary element methods (BEM), and the resulting discrete systems retain the RC exactly. The FEM and BEM discretizations, respectively, applied to the designed interior heterogeneous and exterior homogeneous media Helmholtz systems include the FEM and BEM solutions matching in artificial interface domains, and allow for computations of the exact ansatz based far-fields. In this article we present rigorous numerical analysis of a discrete two-dimensional FEM-BEM overlapped coupling implementation of the algorithm. We also demonstrate the efficiency of our discrete FEM-BEM framework and analysis using numerical experiments, including applications to non-convex heterogeneous multiple particle Janus configurations. Simulations of the far-field induced differential scattering cross sections (DSCS) of heterogeneous configurations and orientation-averaged (OA) counterparts are important for several applications, including inverse wave problems. Our robust FEM-BEM framework facilitates computations of such quantities of interest, without boundedness or homogeneity or shape restrictions on the wave propagation model. © 2021 IMACS