Differential Tomography of Micromechanical Evolution in Elastic Materials of Unknown Micro/Macrostructure

Abstract

Differential evolution indicators are introduced for 3D spatiotemporal imaging of micromechanical processes in complex materials where progressive variations due to manufacturing and/or aging are housed in a highly scattering background of a-priori unknown or uncertain structure. In this vein, a three-tier imaging platform is established where: (1) the domain is periodically (or continuously) subject to illumination and sensing in an arbitrary configuration; (2) sequential sets of measured data are deployed to distill segment-wise scattering signatures of the domain's internal structure through carefully constructed, non-iterative solutions to the scattering equation; and (3) the resulting solution sequence is then used to rigorously construct an imaging functional carrying appropriate invariance with respect to the unknown stationary components of the background e.g., pre-existing interstitial boundaries and bubbles. This gives birth to differential indicators that specifically recover the 3D support of micromechanical evolution within a network of unknown scatterers. The direct scattering problem is formulated in the frequency domain where the background is comprised of a random distribution of monolithic fragments. The constituents are connected via highly heterogeneous interfaces of unknown elasticity and dissipation which are subject to spatiotemporal evolution. The support of internal boundaries are sequentially illuminated by a set of incident waves and thusinduced scattered fields are captured over a generic observation surface. The performance of the proposed imaging indicator is illustrated through a set of numerical experiments for spatiotemporal reconstruction of progressive damage zones featuring randomly distributed cracks and bubbles.

Poroelastic near-field inverse scattering

“…which can be computed without iterations [23]. Within this framework, the following theorem rigorously establishes the relation between the range of operator S * and the norm of penalty term in (38).…”

“…In this case, the poroelastic trial pattern Φ L ∈ L 2 (G ) 3 × L 2 (G ) is given by Definition 1 indicating that (a) the right hand side is not only a function of the dislocation geometry L but also a function of the trial density a ∈ H1/2 (L) 3 × H1/2 (L) × H−1/2 (L), and (b) computing Φ L generally requires an integration process at every sampling point x • . Conventionally, one may dispense with the integration process by considering a sufficiently localized (trial) density function e.g., see [41,23].…”

“…In this vein, recent advances in rigorous design of sampling-based waveform tomography solutions [20,21,22,23,24] seem particularly relevant owing to their robustness and newfound capabilities pertinent to imaging in complex and unknown domains [21,23,25,26]. So far, these developments mostly reside in the context of self-adjoint and energy preserving systems such as Helmholtz and Navier equations.…”

Laboratory application of sampling approaches to inverse scattering

“…Existing optimization-based approaches to waveform inversion typically incur high computational cost as a crucial obstacle to real-time sensing. Lately, non-iterative inverse scattering solutions [3,4,1] have been brought under the spotlight for their capabilities pertinent to fast imaging in highly scattering media [5]. Spurred by the early study in [6], such developments include: (i) the Factorization Method (FM) [7,8], (ii) the Linear Sampling Method (LSM) [2,3], (iii) MUSIC algorithms [9], (iv) the method of Topological Sensitivity (TS) [10,11], and (v) the Generalized Linear Sampling Method (GLSM) [12,1].…”

“…On the verification side, the effectiveness of sampling methods for elastic waveform tomography has been extensively examined by numerical simulations, see e.g., [3,12,1,5]. A systematic experimental investigation of these imaging tools, however, is still lacking.…”