Topological Sensitivity Research Articles

Topological sensitivity and FMM-accelerated BEM applied to 3D acoustic inverse scattering

This study is set in the framework of inverse scattering of scalar (e.g. acoustic) waves. A qualitative probing technique based on the distribution of topological sensitivity of the cost functional associated with the inverse problem with respect to the nucleation of an infinitesimally small hard obstacle is formulated. The sensitivity distribution is expressed as a bilinear formula involving the free field and an adjoint field associated with the cost function. These fields are computed by means of a boundary element formulation accelerated by the fast multipole method. A computationally fast approach for performing a global preliminary search based on the available overspecified boundary data is thus defined. Its usefulness is demonstrated through results of numerical experiments on the qualitative identification of hard obstacles in a bounded 3D acoustic domain, for configurations featuring O ( 10 5 ) nodal unknowns and O ( 10 6 ) sampling points, based on exact or noisy synthetic data.

Topological sensitivity analysis in the context of ultrasonic non-destructive testing

This paper deals with the use of the topological derivative in detection problems involving waves. In the first part, a framework to carry out the topological sensitivity analysis in this context is proposed. Arbitrarily shaped holes and cracks with Neumann boundary condition in 2 and 3 space dimensions are considered. In the second part, a numerical example concerning the treatment of ultrasonic probing data in metallic plates is presented. With moderate noise in the measurements, the defects (air bubbles) are detected and satisfactorily localized by means of a single sensitivity computation.

Topological sensitivity analysis of inclusion in two-dimensional linear elasticity

The topological derivative gives the sensitivity of the problem when the domain under consideration is perturbed by the introduction of a hole. Alternatively, this same concept can also be used to calculate the sensitivity of the problem when, instead of a hole, a small inclusion is introduced at a point in the domain. In the present paper we apply the Topological-Shape Sensitivity Method to obtain the topological derivative of inclusion in two-dimensional linear elasticity, adopting the total potential energy as the cost function and the equilibrium equation as a constraint. For the sake of completeness, initially we present a brief description of the Topological-Shape Sensitivity Method. Then, we calculate the topological derivative for the problem under consideration in two steps: firstly we perform the shape derivative and next we calculate the limit when the perturbation vanishes using classical asymptotic analysis around a circular inclusion. In addition, we use this information as a descent direction in a topology design algorithm which allows to simultaneously remove and insert material. Finally, we explore this feature showing some numerical experiments of structural topology design within the context of two-dimensional linear elasticity problem.

Topological sensitivity analysis in large deformation problems

The aim of the present work is to apply the topological sensitivity analysis (TSA) to large-deformation elasticity based on the total Lagrangian formulation. The TSA results in a scalar function, denominated topological derivative, that gives for each point of the domain the sensitivity of a given cost function when a small hole is created. An approximated expression for the topological derivative is obtained by numerical asymptotic analysis. Numerical results of the presented approach are considered for elastic plane problems.

Asymptotic Models for the Topological Sensitivity Versus the Topological Derivative

An approach, based on the refined method of matched asymptotic expansions, is proposed for the construction of asymptotic models for the topological sensitivity of the energy functional with respect to the creation of a small hole in the geometrical domain. It is shown that the asymptotic model provides more information for calculations than the topo- logical derivative.

Asymptotic models for the topological sensitivity of the energy functional

The refined method of matched asymptotic expansions is used for constructing asymptotic models for the topological sensitivity of the energy functional with respect to the creation of several small holes in the geometrical domain.

A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows

AbstractTopology optimization of fluid dynamic systems is a comparatively young optimal design technique. Its central ingredient is the computation of topological sensitivity maps. Whereas, for finite element solvers, implementations of such sensitivity maps have been accomplished in the past, this study focuses on providing this functionality within a professional finite volume computational fluid dynamics solver. On the basis of a continuous adjoint formulation, we derive the adjoint equations and the boundary conditions for typical cost functions of ducted flows and present first results for two‐ and three‐dimensional geometries. Emphasis is placed on the versatility of our approach with respect to changes in the objective function. We further demonstrate that surface sensitivity maps can also be computed with the implemented functionality and establish their connection with topological sensitivities. Copyright © 2008 John Wiley & Sons, Ltd.

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.

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.

Topological sensitivity analysis for time-dependent problems

The topological sensitivity analysis consists in studying the behavior of a given shape functional when the topology of the domain is perturbed, typically by the nucleation of a small hole. This notion forms the basic ingredient of different topology optimization/reconstruction algorithms. From the theoretical viewpoint, the expression of the topological sensitivity is well-established in many situations where the governing p.d.e. system is of elliptic type. This paper focuses on the derivation of such formulas for parabolic and hyperbolic problems. Different kinds of cost functionals are considered.

Response to the “Discussion of second order topological sensitivity analysis” by M. Bonnet

Response to the “Discussion of second order topological sensitivity analysis” by M. Bonnet

Discussion of “Second order topological sensitivity analysis” by J. Rocha de Faria et al.

The aim of this discussion is to expose incorrect results in a previous IJSS article.

A linear sampling approach to inverse elastic scattering in piecewise-homogeneous domains

The focus of this study is a 3D inverse scattering problem underlying non-invasive reconstruction of piecewise-homogeneous (PH) defects in a layered semi-infinite solid from near-field, surface elastic waveforms. The solution approach revolves around the use of Green's function for the layered reference domain and a generalization of the linear sampling method to deal with the featured class of PH configurations. For a rigorous treatment of the full-waveform integral equation that is used as a basis for obstacle reconstruction, the developments include an extension of the Holmgren's uniqueness theorem to piecewise-homogeneous domains and an in-depth analysis of the situation when the sampling point is outside the support of the obstacle that employs the method of topological sensitivity. Owing to the ill-posed nature of the featured integral equation, a stable approximate solution is sought via Tikhonov regularization. A set of numerical examples is included to demonstrate the feasibility of 3D obstacle reconstruction when the defects are buried in a multi-layered elastic solid.

Topological sensitivity analysis for three-dimensional linear elasticity problem

In this work we use the Topological-Shape Sensitivity Method to obtain the topological derivative for three-dimensional linear elasticity problems, adopting the total potential energy as cost function and the equilibrium equation as constraint. This method, based on classical shape sensitivity analysis, leads to a systematic procedure to calculate the topological derivative. In particular, firstly we present the mechanical model, later we perform the shape derivative of the corresponding cost function and, finally, we calculate the final expression for the topological derivative using the Topological-Shape Sensitivity Method and results from classical asymptotic analysis around spherical cavities. In order to point out the applicability of the topological derivative in the context of topology optimization problems, we use this information as a descent direction to solve a three-dimensional topology design problem. Furthermore, through this example we also show that the topological derivative together with an appropriate mesh refinement strategy are able to capture high quality shapes even using a very simple topology algorithm.

Combining topological sensitivity and genetic algorithms for identification inverse problems in anisotropic materials

The identification inverse problem is solved here for flaw detection in anisotropic materials by means of an innovative approach: the combination of Genetic Algorithm and the Topological Sensitivity in anisotropic elasticity. The Topological Sensitivity provides a measure of the susceptibility of a defect being at a given location. This is based on a linearized topological expansion, applying Boundary Integral Equations and using solely information of the non-damaged state. It is proved that the Topological Sensitivity provides an accurate tool for estimating the location and size of defects. First, it is shown that the minimum of the residual (cost function) topological sensitivity pinpoints the location and size of the actual flaws, and secondly, the minimization of the residual topological sensitivity is carried out using Genetic Algorithm. When the Genetic Algorithm is applied to the residual Topological Sensitivity instead of to the full residual, the applicability of this method is enhanced since the computational effort, which is the major drawback of this type of search methods, is drastically reduced. In this paper, the formulation for linearly anisotropic elastic media is composed for the case of circular flaws, although the procedure is extensible to other kinds of defects like elliptical cavities, elastic or rigid inclusions or cracks.

Functional Network Topology Learning and Sensitivity Analysis Based on ANOVA Decomposition

A new methodology for learning the topology of a functional network from data, based on the ANOVA decomposition technique, is presented. The method determines sensitivity (importance) indices that allow a decision to be made as to which set of interactions among variables is relevant and which is irrelevant to the problem under study. This immediately suggests the network topology to be used in a given problem. Moreover, local sensitivities to small changes in the data can be easily calculated. In this way, the dual optimization problem gives the local sensitivities. The methods are illustrated by their application to artificial and real examples.

Topological sensitivity to diagonal member flips of two-layered statically determinate trusses under worst loading

The paper deals with the topological sensitivity of free, unsupported, statically determinate plane trusses whose horizontal and vertical members form two horizontal layers of square cells and two or more vertical layers. The topology of a truss is decomposed into a form vector – the placement of cells containing diagonal members – and a binary vector describing the slopes of the diagonals. The construction of complete form and slope spaces is provided for any number of vertical layers. Using exhaustive search, forms with minimum and maximum sensitivity to slope change are found for trusses with 2 × 2 through 2 × 8 layers under worst static load condition, represented by the lowest eigenvalue of the least-squares equilibrium matrix. Typical features of the least and most sensitive forms and associated loads and internal forces are shown. Changes of absolute and relative topological sensitivities with increasing number of vertical layers are discussed.

Second order topological sensitivity analysis

The topological derivative provides the sensitivity of a given cost function with respect to the insertion of a hole at an arbitrary point of the domain. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal holes. However, for practical applications, we need to insert holes of finite size. Therefore, we consider one more term in the expansion which is defined as the second order topological derivative. In order to present these ideas, in this work we apply the topological-shape sensitivity method as a systematic approach to calculate first as well as second order topological derivative for the Poisson’s equations, taking the total potential energy as cost function and the state equation as constraint. Furthermore, we also study the effects of different boundary conditions on the hole: Neumann and Dirichlet (both homogeneous). Finally, we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion, which has two main features: it allows us to deal with hole of finite size and provides a better descent direction in optimization process.

From imaging to material identification: A generalized concept of topological sensitivity

To establish a compact analytical framework for the preliminary stress-wave identification of material defects, the focus of this study is an extension of the concept of topological derivative, rooted in elastostatics and the idea of cavity nucleation, to 3D elastodynamics involving germination of solid obstacles. The main result of the proposed generalization is an expression for topological sensitivity, explicit in terms of the elastodynamic Green's function, obtained by an asymptotic expansion of a misfit-type cost functional with respect to the nucleation of a dissimilar elastic inclusion in a defect-free “reference” solid. The featured formula, consisting of an inertial-contrast monopole term and an elasticity-contrast dipole term, is shown to be applicable to a variety of reference solids (semi-infinite and infinite domains with constant or functionally graded elastic properties) for which the Green's functions are available. To deal with situations when the latter is not the case (e.g. finite reference bodies or those with pre-existing defects), an adjoint field approach is employed to derive an alternative expression for topological sensitivity that involves the contraction of two (numerically computed) elastodynamic states. A set of numerical results is included to demonstrate the potential of generalized topological derivative as an efficient tool for exposing not only the geometry, but also material characteristics of subsurface material defects through a local, point-wise identification of “optimal” inclusion properties that minimize the topological sensitivity at sampling location. Beyond the realm of non-invasive characterization of engineered materials, the proposed developments may be relevant to medical diagnosis and in particular to breast cancer detection where focused ultrasound waves show a promise of superseding manual palpation.

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.