5,389 research outputs found

    Performance assessment of a new class of local absorbing boundary conditions for elliptical- and prolate spheroidal-shaped boundaries

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    International audienceNew approximate local DtN boundary conditions are proposed to be applied on elliptical- or prolate-spheroid exterior boundaries when solving respectively two- or three-dimensional acoustic scattering problems by elongated obstacles. These new absorbing conditions are designed to be exact for the first modes. They can be easily incorporated in any finite element parallel code while preserving the local structure of the algebraic system. Unlike the standard approximate local DtN boundary conditions that are restricted to circular- or spherical-shaped boundaries, the proposed conditions are applicable to exterior elliptical-shaped boundaries that are more suitable for surrounding elongated scatterers because they yield to smaller computational domains. The mathematical and numerical analysis of the effect of the frequency and the eccentricity values of the boundary on the accuracy of these conditions, when applied for solving radiating and scattering problems, reveals - in particular - that the new second-order DtN boundary condition retains a good level of accuracy, in the low frequency regime, regardless of the slenderness of the boundary

    TWO-DIMENSIONAL APPROXIMATE LOCAL DtN BOUNDARY CONDITIONS FOR ELLIPTICAL-SHAPED BOUNDARIES

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    International audienceWe propose a new class of approximate local DtN boundary conditions to be applied on elliptical-shaped exterior boundaries when solving acoustic scattering problems by elongated obstacles. These conditions are : (a) exact for the first modes, (b) easy to implement and to parallelize, (c) compatible with the local structure of the computational finite element scheme, and (d) applicable to exterior elliptical-shaped boundaries that are more suitable in terms of cost-effectiveness for surrounding elongated scatterers. We investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions. We also compare their performance to the second order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel when expressed in elliptical coordinates. The analysis reveals that, in the low frequency regime, the new second order DtN condition (DtN2) retains a good level of accuracy regardless of the slenderness of the boundary. In addition, the DtN2 boundary condition outperforms the BGT2 condition. Such superiority is clearly noticeable for large eccentricity values

    Construction and performance analysis of local DtN absorbing boundary conditions for exterior Helmholtz problems. Part II : Prolate spheroid boundaries

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    We propose a new class of approximate {\it local} DtN boundary conditions to be applied on prolate spheroid-shaped exterior boundaries when solving acoustic scattering problems by elongated obstacles. These conditions are : (a) exact for the first modes, (b) easy to implement and to parallelize, (c) compatible with the local structure of the computational finite element scheme, and (d) applicable to exterior elliptical-shaped boundaries that are more suitable in terms of cost-effectiveness for surrounding elongated scatterers. Moreover, these conditions coincide with the classical local DtN condition designed for spherical-shaped boundaries. We investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions when applied for solving radiators and scattering problems. We also compare their performance to the second order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel when expressed in prolate spheroidal coordinates. The analysis reveals that, in the low frequency regime, the new second order DtN condition (DtN2) retains a good level of accuracy {\it regardless} of the slenderness of the boundary. In addition, the DtN2 boundary condition outperforms the BGT2 condition. Such superiority is clearly noticeable for large eccentricity values

    Construction and performance analysis of local DtN absorbing boundary conditions for exterior Helmholtz problems. Part I : Elliptical shaped boundaries

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    We propose a new class of approximate {\it local} DtN boundary conditions to be applied on prolate spheroid-shaped exterior boundaries when solving acoustic scattering problems by elongated obstacles. These conditions are : (a) exact for the first modes, (b) easy to implement and to parallelize, (c) compatible with the local structure of the computational finite element scheme, and (d) applicable to exterior elliptical-shaped boundaries that are more suitable in terms of cost-effectiveness for surrounding elongated scatterers. Moreover, these conditions coincide with the classical local DtN condition designed for spherical-shaped boundaries. We investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions when applied for solving radiators and scattering problems. We also compare their performance to the second order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel when expressed in prolate spheroidal coordinates. The analysis reveals that, in the low frequency regime, the new second order DtN condition (DtN2) retains a good level of accuracy regardless of the slenderness of the boundary. In addition, the DtN2 boundary condition outperforms the BGT2 condition. Such superiority is clearly noticeable for large eccentricity values

    High-frequency analysis of the efficiency of a local approximate DtN2 boundary condition for prolate spheroidal-shaped boundaries

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    The performance of the second-order local approximate DtN boundary condition suggested in [4] is investigated analytically when employed for solving high-frequency exterior Helmholtz problems with elongated scatterers. This study is performed using a domain-based formulation and assuming the scatterer and the exterior artificial boundary to be prolate spheroid. The analysis proves that, in the high-frequency regime, the reflected waves at the artificial boundary decay faster than 1/(ka)15/8, where k is the wavenumber and a is the semi-major axis of this boundary. Numerical results are presented to illustrate the accuracy and the efficiency of the proposed absorbing boundary condition, and to provide guidelines for satisfactory performance

    The simulation of elastic wave propagation in presence of void in the subsurface

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    Underground voids, whether man-made (e.g., mines and tunnels) or naturally occurring (e.g., karst terrain), can cause a variety of threats to surface activity. Therefore, it is important to be able to locate and characterize a potential void in the subsurface so that mitigating measures can be taken. In real-world environments, the subsurface properties and the existence of a void is not known, so the problem is challenging to solve. The numerical analysis conducted in this study takes a step toward understanding the seismic response with and without a void in various types of domains. The Finite Difference Method (FDM) and Finite Element Method (FEM) numerical techniques were used to analyze 1-D and 2-D seismic wave propagation for homogenous domains, layered domains, and with voids in the domain. The outputs of each numerical method were compared via their results and computational efficiency, which has not been completed in the current literature. Additionally, different void shapes were placed in the computational models to analyze each method’s void detection ability. For 1-D wave propagation, both methods produced identical results at different loading frequencies and Courant numbers. Computationally, both methods have similar run times, while FDM had a simpler implementation than FEM. In a 2-D simulation, COMSOL was used for the FEM, and the staggered-grid technique was used for the FDM. Slight dispersion was observed in all the FDM solutions, where this was attributed to the step size; however, using a smaller step size significantly increased the computational time. For a homogenous model, both methods produced similar vertical particle velocity contours and surface time histories. Computationally, FDM outperformed FEM, and due to its ease of implementation, it was recommended for homogenous wave propagation. A three-layered domain was analyzed that featured a silty clay upper layer, and two lower rock layers. Contours of vertical particle velocity displayed that the majority of the wave remained in the upper third of the domain because of the harsh difference in material properties between the first and second layers. Additionally, a numerical model was created that consisted of the material properties obtained by ultrasonic testing. Reflections were seen in the generated seismograms but were not as visible as the ones seen in the three-layered case because the measured properties are alike and allow the wave to travel easily through the domain. After analyzing the wave propagation in a domain without a void, three void shapes were placed at the center of the domain (ellipse, circle, and square), and the resulting wave propagation was analyzed. There was minimal noise near the interface of rounded shapes in the FDM results, which was attributed to the staircase approximation used to define the shape. The surface time histories displayed reflections due to the void that were not seen in homogeneous cases. The elliptical void produced slightly more pronounced reflections because the length of the shape was larger than the circle and square. The reflections were also more easily seen in the rock domain than in soil. It was difficult locating voids in the three-layer case, but plots that computed the difference between the no-void and void case revealed that the voids did affect wave propagation. The elliptical void had the largest maximum difference of the seismograms, which occurred at the receiver closest to the void. There were differences between the subtracted plots from each method, where this was attributed to the different source incorporation. However, future studies will need to be completed to fully analyze why these plots differed between each method. Reflections from the void were more easily seen in the domain featuring the results from ultrasonic testing because of the similar rock properties that the samples shared. The elliptical void had the most perturbations compared to the square and circular voids. Overall, the FEM had longer computational times than the FDM, but both methods can successfully analyze wave propagation in the studied domains

    On-surface radiation condition for multiple scattering of waves

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    The formulation of the on-surface radiation condition (OSRC) is extended to handle wave scattering problems in the presence of multiple obstacles. The new multiple-OSRC simultaneously accounts for the outgoing behavior of the wave fields, as well as, the multiple wave reflections between the obstacles. Like boundary integral equations (BIE), this method leads to a reduction in dimensionality (from volume to surface) of the discretization region. However, as opposed to BIE, the proposed technique leads to boundary integral equations with smooth kernels. Hence, these Fredholm integral equations can be handled accurately and robustly with standard numerical approaches without the need to remove singularities. Moreover, under weak scattering conditions, this approach renders a convergent iterative method which bypasses the need to solve single scattering problems at each iteration. Inherited from the original OSRC, the proposed multiple-OSRC is generally a crude approximate method. If accuracy is not satisfactory, this approach may serve as a good initial guess or as an inexpensive pre-conditioner for Krylov iterative solutions of BIE

    Exponential decay of high-order spurious prolate spheroidal modes induced by a local approximate dtn exterior boundary condition

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    We investigate analytically the asymptotic behavior of high-order spurious prolate spheroidal modes induced by a second-order local approximate DtN absorbing boundary condition (DtN2) when employed for solving high-frequency acoustic scattering problems. We prove that these reflected modes decay exponentially in the high frequency regime. This theoretical result demonstrates the great potential of the considered absorbing boundary condition for solving efficiently exterior high-frequency Helmholtz problems. In addition, this exponential decay proves the superiority of DtN2 over the widely used Bayliss-Gunsburger-Turkel absorbing boundary condition
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