67 research outputs found
Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures
A theoretical foundation is developed for active seismic reconstruction of
fractures endowed with spatially-varying interfacial condition
(e.g.~partially-closed fractures, hydraulic fractures). The proposed indicator
functional carries a superior localization property with no significant
sensitivity to the fracture's contact condition, measurement errors, and
illumination frequency. This is accomplished through the paradigm of the
-factorization technique and the recently developed Generalized
Linear Sampling Method (GLSM) applied to elastodynamics. The direct scattering
problem is formulated in the frequency domain where the fracture surface is
illuminated by a set of incident plane waves, while monitoring the induced
scattered field in the form of (elastic) far-field patterns. The analysis of
the well-posedness of the forward problem leads to an admissibility condition
on the fracture's (linearized) contact parameters. This in turn contributes
toward establishing the applicability of the -factorization method,
and consequently aids the formulation of a convex GLSM cost functional whose
minimizer can be computed without iterations. Such minimizer is then used to
construct a robust fracture indicator function, whose performance is
illustrated through a set of numerical experiments. For completeness, the
results of the GLSM reconstruction are compared to those obtained by the
classical linear sampling method (LSM)
On the topological sensitivity of transient acoustic fields
The concept of topological sensitivity has been successfully employed as an imaging tool to obtain the correct initial topology and preliminary geometry of hidden obstacles for a variety of inverse scattering problems. In this paper, we extend these ideas to acoustic scattering involving transient waveforms and penetrable obstacles. Through a boundary integral equation framework, we present a derivation of the topological sensitivity for the featured class of problems and illustrate numerically the utility of the proposed method for preliminary geometric reconstruction of penetrable obstacles. For generality, we also cast the topological sensitivity in the so-called adjoint field setting that is amenable to a generic computational treatment using, for example, finite element or finite difference methods
Second-order homogenization of boundary and transmission conditions for one-dimensional waves in periodic media
International audienceWe consider the homogenized boundary and transmission conditions governing the mean-field approximations of 1D waves in finite periodic media within the framework of two-scale analysis. We establish the homogenization ansatz (up to the second order of approximation), for both types of problems, by obtaining the relevant boundary correctors and exposing the enriched boundary and transmission conditions as those of Robin type. Rigorous asymptotic analysis is performed for boundary conditions, while the applicability to transmission conditions is demonstrated via numerical simulations. Within this framework, we also propose an optimized second-order model of the homogenized wave equation for 1D periodic media, that follows more accurately the exact dispersion relationship and generally enhances the performance of second-order approximation. The proposed analysis is applied toward the long-wavelength approximation of waves in finite periodic bilaminates, subject to both boundary and transmission conditions. A set of numerical simulations is included to support the mathematical analysis and illustrate the effectiveness of the homogenization scheme
Error-in-constitutive-relation (ECR) framework for the characterization of linear viscoelastic solids
We develop an error-in-constitutive-relation (ECR) approach toward the
full-field characterization of linear viscoelastic solids described within the
framework of standard generalized materials. To this end, we formulate the
viscoelastic behavior in terms of the (Helmholtz) free energy potential and a
dissipation potential. Assuming the availability of full-field interior
kinematic data, the constitutive mismatch between the kinematic quantities
(strains and internal thermodynamic variables) and their ``stress''
counterparts (Cauchy stress tensor and that of thermodynamic tensions),
commonly referred to as the ECR functional, is established with the aid of
Legendre-Fenchel gap functionals linking the thermodynamic potentials to their
energetic conjugates. We then proceed by introducing the modified ECR (MECR)
functional as a linear combination between its ECR parent and the kinematic
data misfit, computed for a trial set of constitutive parameters. The
affiliated stationarity conditions then yield two coupled evolution problems,
namely (i) the forward evolution problem for the (trial) displacement field
driven by the constitutive mismatch, and (ii) the backward evolution problem
for the adjoint field driven by the data mismatch. This allows us to establish
compact expressions for the MECR functional and its gradient with respect to
the viscoelastic constitutive parameters. For generality, the formulation is
established assuming both time-domain (i.e. transient) and frequency-domain
data. We illustrate the developments in a two-dimensional setting by pursuing
the multi-frequency MECR reconstruction of (i) piecewise-homogeneous standard
linear solid, and (b) smoothly-varying Jeffreys viscoelastic material
Effective wave motion in periodic discontinua near spectral singularities at finite frequencies and wavenumbers
International audienceWe consider the effective wave motion, at spectral singularities such as corners of the Brillouin zone and Dirac points, in periodic continua intercepted by compliant interfaces that pertain to e.g. masonry and fractured materials. We assume the Bloch-wave form of the scalar wave equation (describing anti-plane shear waves) as a point of departure, and we seek an asymptotic expansion about a reference point in the wavenumber-frequency space-deploying wavenumber separation as the perturbation parameter. Using the concept of broken Sobolev spaces to cater for the presence of kinematic discontinuities, we next define the "mean" wave motion via inner product between the Bloch wave and an eigenfunction (at specified wavenumber and frequency) for the unit cell of periodicity. With such projection-expansion approach, we obtain an effective field equation, for an arbitrary dispersion branch, near apexes of "wavenumber quadrants" featured by the first Brillouin zone. For completeness, we investigate asymptotic configurations featuring both (a) isolated, (b) repeated, and (c) nearby eigenvalues. In the case of repeated eigenvalues, we find that the "mean" wave motion is governed by a system of wave equations and Dirac equations, whose size is given by the eigenvalue multiplicity, and whose structure is determined by the participating eigenfunctions, the affiliated cell functions, and the direction of wavenumber perturbation. One of these structures is shown to describe the so-called Dirac points-apexes of locally conical dispersion surfaces-that are relevant to the generation of topologically protected waves. In situations featuring clusters of tightly spaced eigenvalues, the effective model is found to entail a Diraclike system of equations that generates "blunted" conical dispersion surfaces. We illustrate the analysis by numerical simulations for two periodic configurations in R 2 that showcase the asymptotic developments in terms of (i) wave dispersion, (ii) forced wave motion, and (iii) frequency-and wavenumber-dependent phonon behavior
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