A New Family of Mixed Finite Elements for the Linear Elastodynamic Problem

Bécache, Éliane; Joly, Patrick; Tsogka, Chrysoula

doi:10.1137/s0036142999359189

Cited by 79 publications

(80 citation statements)

References 26 publications

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“…It is then straightforward to obtain the PML model for the elastodynamics system (9), that can be rewritten as…”

Section: The Pml Model For Elastodynamicsmentioning

confidence: 99%

“…The numerical method used to solve the elastodynamic equations is based on a first-order original mixed formulation of the equations, described in [14], where the unknowns are the displacement searched in H 1 and some new vectorial unknowns searched in ðL 2 Þ 2 . Since we are not interested here in the effects of the numerical scheme, but in the properties of the continuous model, we have also checked these results with another method, developed in [8,9] and still based on the velocity-stress formulation, but with v in L 2 and r in H ðdivÞ.…”

Section: Some Instructive Numerical Simulationsmentioning

confidence: 99%

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Stability of perfectly matched layers, group velocities and anisotropic waves

Bécache

Fauqueux

Joly

2003

Journal of Computational Physics

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306

355

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Perfectly matched layers (PML) are a recent technique for simulating the absorption of waves in open domains. They have been introduced for electromagnetic waves and extended, since then, to other models of wave propagation, including waves in elastic anisotropic media. In this last case, some numerical experiments have shown that the PMLs are not always stable. In this paper, we investigate this question from a theoretical point of view. In the first part, we derive a necessary condition for the stability of the PML model for a general hyperbolic system. This condition can be interpreted in terms of geometrical properties of the slowness diagrams and used for explaining instabilities observed with elastic waves but also with other propagation models (anisotropic MaxwellÕs equations, linearized Euler equations). In the second part, we specialize our analysis to orthotropic elastic waves and obtain separately a necessary stability condition and a sufficient stability condition that can be expressed in terms of inequalities on the elasticity coefficients of the model.

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“…It is then straightforward to obtain the PML model for the elastodynamics system (9), that can be rewritten as…”

Section: The Pml Model For Elastodynamicsmentioning

confidence: 99%

Section: Some Instructive Numerical Simulationsmentioning

confidence: 99%

Stability of perfectly matched layers, group velocities and anisotropic waves

Bécache

Fauqueux

Joly

2003

Journal of Computational Physics

Self Cite

306

355

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“…Details of this mixed formulation may be found in [6]. The spatial discretization is compatible with mass-lumping, which leads to diagonal mass matrices, so that explicit time discretization schemes are obtained.…”

Section: Numerical Implementationmentioning

confidence: 99%

“…The solid contains a number of N r sensors which can act as sources as well. Our data is the response matrix of the scattered field, that is, the difference between the total field obtained in the damaged structure and the incident field corresponding to the response in the healthy structure.Numerical solution of the wave propagation problem is performed using a mixed finite element formulation in terms of the velocity and stress fields [6]. In order to dissociate the response caused by N d different defects, we apply the singular value decomposition (SVD) of the response matrix, while back-propagation of the projection of each singular vector corresponding to a non-zero singular value is performed in order to highlight each defect separately.…”

mentioning

confidence: 99%

“…Numerical solution of the wave propagation problem is performed using a mixed finite element formulation in terms of the velocity and stress fields [6]. In order to dissociate the response caused by N d different defects, we apply the singular value decomposition (SVD) of the response matrix, while back-propagation of the projection of each singular vector corresponding to a non-zero singular value is performed in order to highlight each defect separately.…”

mentioning

confidence: 99%

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Time Reversal in Elastodynamics and Applications to Structural Health Monitoring

Panagiotopoulos¹,

Petromichelakis²,

Tsogka³

2015

Proceedings of the 5th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. In structural health monitoring, detection of localized damage can be achieved by exploiting recorded signals at a limited number of sensors within the structure. Here we propose the localization of damage using an array of sensors as a computational time-reversal mirror (TRM). Time reversal (TR) is a physical process that exploits the time reversibility of wave equations and achieves re-focusing of the wave on the source of its origin by sending back, reversed in time, the signals recorded on an array of transducers. TR was originally introduced by Mathias Fink and his group and has several applications ranging from medical imaging to telecommunications [14].In the present work, we perform time reversal numerically in order to effectively detect and localize defects in a bounded two-dimensional elastic domain. This is a generalization of a respective time-reversal implementation in an acoustic medium [12]. The solid contains a number of N r sensors which can act as sources as well. Our data is the response matrix of the scattered field, that is, the difference between the total field obtained in the damaged structure and the incident field corresponding to the response in the healthy structure.Numerical solution of the wave propagation problem is performed using a mixed finite element formulation in terms of the velocity and stress fields [6]. In order to dissociate the response caused by N d different defects, we apply the singular value decomposition (SVD) of the response matrix, while back-propagation of the projection of each singular vector corresponding to a non-zero singular value is performed in order to highlight each defect separately.

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Residual‐based stabilization of the finite element approximation to the acoustic perturbation equations for low Mach number aeroacoustics

Guasch

Sánchez-Martín

Pont

et al. 2016

Numerical Methods in Fluids

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SUMMARYThe acoustic perturbation equations (APE) are suitable to predict aerodynamic noise in the presence of a non-uniform mean flow. As for any hybrid computational aeroacoustics approach, a first computational fluid dynamics simulation is carried out from which the mean flow characteristics and acoustic sources are obtained. In a second step, the APE are solved to get the acoustic pressure and particle velocity fields. However, resorting to the finite element method (FEM) for that purpose is not straightforward. Whereas mixed finite elements satisfying an appropriate inf-sup compatibility condition can be built in the case of no mean flow, i.e., for the standard wave equation in mixed form, these are difficult to implement and their good performance is yet to be checked for more complex wave operators. As a consequence, strong simplifying assumptions are usually considered when solving the APE with FEM. It is possible to avoid them by resorting to stabilized formulations. In this work, a residual-based stabilized FEM is presented for the APE at low Mach numbers, which allows one to deal with the APE convective and reaction terms in its full extent. The key of the approach resides in the design of the matrix of stabilization parameters. The performance of the formulation and the contributions of the different terms in the equations are tested for an acoustic pulse propagating in sheared solenoidal mean flow, and for the aeolian tone generated by flow past a two-dimensional cylinder.

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A New Family of Mixed Finite Elements for the Linear Elastodynamic Problem

Cited by 79 publications

References 26 publications

Stability of perfectly matched layers, group velocities and anisotropic waves

Stability of perfectly matched layers, group velocities and anisotropic waves

Time Reversal in Elastodynamics and Applications to Structural Health Monitoring

Residual‐based stabilization of the finite element approximation to the acoustic perturbation equations for low Mach number aeroacoustics

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