Qualitative identification of cracks using 3D transient elastodynamic topological derivative: Formulation and FE implementation

Abstract

To cite this version:Cédric Bellis, Marc Bonnet. Abstract A time-domain topological derivative (TD) approach is developed for transient elasticwave imaging of buried cracks. The TD, which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal trial crack, is expressed in terms of the time convolution of the free field and an adjoint field as a function of that specified location and of the trial crack shape. Following previous studies on cavity identification in similar conditions, the TD field is here considered as a natural and computationally efficient approach for defining a crack location indicator function. This study emphasizes the implementation and exploitation of TD fields using the standard displacement-based FEM, a straightforward exploitation of the relevant sensitivity formulation established here. Results on several numerical experiments on 3D elastodynamic and acoustic configurations are reported and discussed, allowing to assess and highlight many features of the proposed TD-based fast qualitative crack identification, including its ability to identify multiple cracks and its robustness against data noise.

Inverse acoustic scattering by solid obstacles: topological sensitivity and its preliminary application

“…Owning to the linearity of the forward scattering problem with respect to the excitation, (31) and (32) …”

“…The concept of topological sensitivity (TS), since its inception in the context of structural shape optimization, [9,10] has been generalized and applied to deal with inverse scattering problems in acoustics, [7,[11][12][13][14][15][16][17] electromagnetism, [18][19][20][21][22][23] and elastodynamics. [24][25][26][27][28][29][30][31][32][33] In the reconstruction approach, the TS indicator function is defined as the sensitivity of a given cost functional (that would commonly be used as a platform for non-linear minimization) with respect to the nucleation of an infinitesimal obstacle at a prescribed sampling point in the reference (background) medium. Accordingly, the support of hidden obstacles is exposed through the spatial distribution of topological sensitivity, namely the regions where TS attains pronounced negative values.…”

A synoptic approach to the seismic sensing of heterogeneous fractures: From geometric reconstruction to interfacial characterization

“…where Φ ∞ L is the far-field pattern of a test radiating field, generated by a small trial fracture L ⊂ R 3 with prescribed FOD a ∈H 1/2 (L) 3 (see [47] for details). Without loss of generality, L can be taken as a vanishing penny-shaped fracture at z ∈ R 3 with normal n ∈ Ω and constant (mode I) FOD profile a ∝ n, in which case (8) yields…”

On the elastic-wave imaging and characterization of fractures with specific stiffness

“…where Γ trial contains the origin, the topological sensitivity (TS) of the featured cost functional can be defined as the leading-order term in the expansion of J(Γ ε ) with respect to the vanishing (trial) fracture size, ε → 0 [10]. In what follows, Γ trial is taken as a penny-shaped fracture of unit radius with unit normal n , shear specific stiffness κ t , and normal specific stiffness κ n .…”