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 $F_\sharp$-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 $F_\sharp$-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)

Sounding of finite solid bodies by way of topological derivative

This paper is concerned with an application of the concept of topological derivative to elastic-wave imaging of finite solid bodies containing cavities. Building on the approach originally proposed in the (elastostatic) theory of shape optimization, the topological derivative, which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity-free (reference) body, is used as a void indicator through an assembly of sampling points where it attains negative values. The computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary-value problems for the reference body, which are here formulated as boundary integral equations. For a comprehensive treatment of the subject, formulas for topological sensitivity are obtained using three alternative methodologies, namely (i) direct differentiation approach, (ii) adjoint field method, and (iii) limiting form of the shape sensitivity analysis. The competing techniques are further shown to lead to distinct computational procedures. Methodologies (i) and (ii) are implemented within a BEM-based platform and validated against an analytical solution. A set of numerical results is included to illustrate the utility of topological derivative for 3D elastic-wave sounding of solid bodies; an approach that may perform best when used as a pre-conditioning tool for more accurate, gradient-based imaging algorithms. Despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution, the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques

Elastic-wave identification of penetrable obstacles using shape-material sensitivity framework

This study deals with elastic-wave identification of discrete heterogeneities (inclusions) in an otherwise homogeneous ``reference'' solid from limited-aperture waveform measurements taken on its surface. On adopting the boundary integral equation (BIE) framework for elastodynamic scattering, the inverse query is cast as a minimization problem involving experimental observations and their simulations for a trial inclusion that is defined through its boundary, elastic moduli, and mass density. For an optimal performance of the gradient-based search methods suited to solve the problem, explicit expressions for the shape (i.e. boundary) and material sensitivities of the misfit functional are obtained via the adjoint field approach and direct differentiation of the governing BIEs. Making use of the message-passing interface, the proposed sensitivity formulas are implemented in a data-parallel code and integrated into a nonlinear optimization framework based on the direct BIE method and an augmented Lagrangian whose inequality constraints are employed to avoid solving forward scattering problems for physically inadmissible (or overly distorted) trial inclusion configurations. Numerical results for the reconstruction of an ellipsoidal defect in a semi-infinite solid show the effectiveness of the proposed shape-material sensitivity formulation, which constitutes an essential computational component of the defect identification algorithm

Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics

The aim of this study is an extension and employment of the concept of topological derivative as it pertains to the nucleation of infinitesimal inclusions in a reference (i.e. background) acoustic medium. The developments are motivated by the need to develop a preliminary indicator functional that would aid the solution of inverse scattering problems in terms of a rational initial 'guess' about the geometry and material characteristics of a hidden (finite) obstacle; an information that is often required by iterative minimization algorithms. To this end the customary definition of topological derivative, which quantifies the sensitivity of a given cost functional with respect to the creation of an infinitesimal hole, is adapted to permit the nucleation of a dissimilar acoustic medium. On employing the Green's function for the background domain, computation of topological sensitivity for the three-dimensional Helmholtz equation is reduced to the solution of a reference, Laplace transmission problem. Explicit formulae are given for the nucleating inclusions of spherical and ellipsoidal shapes. For generality the developments are also presented in an alternative, adjoint-field setting that permits nucleation of inclusions in an infinite, semi-infinite or finite background medium. Through numerical examples, it is shown that the featured topological sensitivity could be used, in the context of inverse scattering, as an effective obstacle indicator through an assembly of sampling points where it attains pronounced negative values. On varying a material characteristic (density) of the nucleating obstacle, it is also shown that the proposed methodology can be used as a preparatory tool for both geometric and material identification

Topological derivative for the inverse scattering of elastic waves

To establish an alternative analytical framework for the elastic-wave imaging of underground cavities, the focus of this study is an extension of the concept of topological derivative, rooted in elastostatics and shape optimization, to three-dimensional elastodynamics involving semi-infinite and infinite solids. The main result of the proposed boundary integral approach is a formula for topological derivative, explicit in terms of the elastodynamic fundamental solution, obtained by an asymptotic expansion of the misfit-type cost functional with respect to the creation of an infinitesimal hole in an otherwise intact (semi-infinite or infinite) elastic medium. Valid for an arbitrary shape of the infinitesimal cavity, the formula involves the solution of six canonical exterior elastostatic problems, and becomes fully explicit when the vanishing cavity is spherical. A set of numerical results is included to illustrate the potential of topological derivative as a computationally efficient tool for exposing an approximate cavity topology, location, and shape via a grid-type exploration of the host solid. For a comprehensive solution to three-dimensional inverse scattering problems involving elastic waves, the proposed approach can be used most effectively as a pre-conditioning tool for more refined, albeit computationally intensive minimization-based imaging algorithms. To the authors' knowledge, an application of topological derivative to inverse scattering problems has not been attempted before; the methodology proposed in this paper could also be extended to acoustic problems

Fast non-iterative methods for defect identification

This communication summarizes recent investigations on the identification of defects (cavities, inclusions) of unknown geometry and topology by means of the concept of topological sensitivity. This approach leads to the fast computation (equivalent to performing a few direct solutions), by means of ordinary numerical solution methods such as the BEM (used here), the FEM or the FDM, of defect indicator functions. Substantial further acceleration is obtained by using fast multipole accelerated BEMs. Possibilities afforded by this approach are demonstrated on numerical examples. The paper concludes with a discussion of further research on theoretical and numerical issues

On the stress-wave imaging of cavities in a semi-infinite solid

The problem of mapping underground cavities from surface seismic measurements is investigated within the framework of a regularized boundary integral equation (BIE) method. With the ground modeled as a uniform elastic half-space, the inverse analysis of elastic waves scattered by a three-dimensional void is formulated as a task of minimizing the misfit between experimental observations and theoretical predictions for an assumed void geometry. For an accurate treatment of the gradient search technique employed to solve the inverse problem, sensitivities of the predictive BIE model with respect to cavity parameters are evaluated semi-analytically using an adjoint problem approach and a continuum kinematics description. Several key features of the formulation, including the rigorous treatment of the radiation condition for semi-infinite solids, modeling of an illuminating seismic wave field, and treatment of the prior information, are highlighted. A set of numerical examples with spherical and ellipsoidal cavity geometries is included to illustrate the performance of the method. It is shown that the featured adjoint problem approach reduces the computational requirements by an order of magnitude relative to conventional finite-difference estimates, thus rendering the three-dimensional elastic-wave imaging of solids tractable for engineering applications

A computational basis for elastodynamic cavity identification in a semi-infinite solid

The focus of this paper is a computational platform for the non-intrusive, active seismic imaging of subterranean openings by means of an elastodynamic boundary integral equation (BIE) method. On simulating the ground response to steady-state seismic excitation as that of a uniform, semi-infinite elastic solid, solution to the 3D inverse scattering problem is contrived as a task of minimizing the misfit between experimental observations and BIE predictions of the surface ground motion. The forward elastodynamic solution revolves around the use of the half-space Greenrsquos functions, which analytically incorporate the traction-free boundary condition at the ground surface and thus allow the discretization and imaging effort to be focused on the surface of a hidden cavity. For a rigorous approach to the gradient-based minimization employed to resolve the cavity, sensitivities of the trial boundary element model with respect to (geometric) void parameters are evaluated using an adjoint field approach. Details of the computational treatment, including the regularized (i.e. Cauchy principal value-free) boundary integral equations for the primary and adjoint problem, the necessary evaluation of surface displacement gradients and their implementation into a parallel code, are highlighted. Through a suite of numerical examples involving the identification of an ellipsoidal cavity, a parametric study is presented which illustrates the importance of several key parameters on the imaging procedure including the prior information, ldquomeasurementrdquo noise, and the amount of experimental input