Topological Derivative Concept Research Articles

A source sensitivity approach for source localization in steady-state linear systems

Localizing sources in physical systems represents a class of inverse problems with broad scientific and engineering applications. This paper is concerned with the development of a non-iterative source sensitivity approach for the localization of sources in linear systems under steady-state. We show that our proposed approach can be applied to a broad class of physical problems, ranging from source localization in elastodynamics and acoustics to source detection in heat/mass transport problems. The source sensitivity field introduced in this paper represents the change of a cost functional caused by the appearance of an infinitesimal (or point) source in a given domain (or its boundary). In order to extract macroscopic inferences, we apply a threshold to the source sensitivity field in a way that parallels the application of the topological derivative concept in shape identification. We establish precise formulas for the source sensitivity field using a direct approach and a Lagrangian formulation. We show that computing the source sensitivity field entails just obtaining the solution of a single adjoint problem. Hence, the computational expense of obtaining the source sensitivity is of the same order as that of solving one forward problem. We illustrate the performance of the method through numerical examples drawn from the areas of elastodynamics, acoustics, and heat/mass transport. Our results show that our proposed approach could be used on its own as a source detection tool or to obtain initial guesses for more quantitative iterative gradient-based minimization strategies.

Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals

Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. This approach remains, however, largely based on a heuristic interpretation of the topological derivative, whereas most other qualitative approaches to inverse scattering are backed by a mathematical justification. As an effort toward bridging this gap, this study focuses on a topological derivative approach applied to the L2-norm of the misfit between far-field measurements. Either an inhomogeneous medium or a finite number of point-like scatterers are considered, using either the Born approximation or a full-scattering model. Topological derivative-based imaging functionals are analyzed using a suitable factorization of the far-field operator, for each of the considered cases, in order to characterize their behavior and assess their ability to reconstruct the unknown scatterer(s). Results include the justification of the usual sign heuristic underpinning the method for (i) the Born approximation and (ii) full-scattering models limited to moderately strong scatterers. Semi-analytical and numerical examples are presented. Within the chosen framework, the topological derivative approach is finally discussed and compared to other well-known qualitative methods.

Shape Reconstruction of Thin Electromagnetic Inclusions via Boundary Measurements: Level-Set Method Combined with the Topological Derivative

An inverse problem for reconstructing arbitrary-shaped thin penetrable electromagnetic inclusions concealed in a homogeneous material is considered in this paper. For this purpose, the level-set evolution method is adopted. The topological derivative concept is incorporated in order to evaluate the evolution speed of the level-set functions. The results of the corresponding numerical simulations with and without noise are presented in this paper.

Analysis of a level set method for topology optimization

A level set method for the topological optimization of domains is proposed and analysed in this paper. The framework of the analysis is that of elliptic optimal control problems with a binary control acting as a source term of the state equation. The main ingredient of the algorithm is the concept of topological derivative. Convergence results are proven and illustrated by some numerical experiments.

On the topological sensitivity of cellular automata

Ever since the conceptualization of cellular automata (CA), much attention has been paid to the dynamical properties of these discrete dynamical systems, and, more in particular, to their sensitivity to the initial condition from which they are evolved. Yet, the sensitivity of CA to the topology upon which they are based has received only minor attention, such that a clear insight in this dependence is still lacking and, furthermore, a quantification of this so-called topological sensitivity has not yet been proposed. The lack of attention for this issue is rather surprising since CA are spatially explicit, which means that their dynamics is directly affected by their topology. To overcome these shortcomings, we propose topological Lyapunov exponents that measure the divergence of two close trajectories in phase space originating from a topological perturbation, and we relate them to a measure grasping the sensitivity of CA to their topology that relies on the concept of topological derivatives, which is introduced in this paper. The validity of the proposed methodology is illustrated for the 256 elementary CA and for a family of two-state irregular totalistic CA.

Topological Derivative for Fast Imaging of Two-Dimensional Thin Dielectric Inclusions in The Wave Propagation Environment

In this paper, we consider the topological derivative concept for developing a fast imaging algorithm of thin inclusions with dielectric contrast with respect to an embedding homogeneous domain with a smooth boundary. The topological derivative is evaluated by applying asymptotic expansion formulas in the presence of small, perfectly conducting cracks. Through the careful derivation, we can design a one-iteration imaging algorithm by solving an adjoint problem. Numerical experiments verify that this algorithm is fast, effective, and stable.

Topological derivative for multi‐scale linear elasticity models applied to the synthesis of microstructures

AbstractThis paper proposes an algorithm for the synthesis/optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation. The macroscopic elasticity tensor is estimated by a standard multi‐scale constitutive theory where the strain and stress tensors are volume averages of their microscopic counterparts over a representative volume element. The algorithm is of simple computational implementation. In particular, it does not require artificial algorithmic parameters or strategies. This is in sharp contrast with existing microstructural optimization procedures and follows as a natural consequence of the use of the topological derivative concept. This concept provides the correct mathematical framework to treat topology changes such as those characterizing microstuctural optimization problems. The effectiveness of the proposed methodology is illustrated in a set of finite element‐based numerical examples.Copyright © 2010 John Wiley & Sons, Ltd.

Crack nucleation sensitivity analysis

A simple analytical expression for crack nucleation sensitivity analysis is proposed relying on the concept of topological derivative and applied within a two-dimensional linear elastic fracture mechanics theory (LEFM). In particular, the topological asymptotic expansion of the total potential energy together with a Griffith-type energy of an elastic cracked body is calculated. As a main result, we derive a crack nucleation criterion based on the topological derivative and a criterion for determining the direction of crack growth based on the topological gradient. The proposed methodology leads to an axiomatic approach of crack nucleation sensitivity analysis. Copyright © 2010 John Wiley & Sons, Ltd.

Topological sensitivity analysis of a multi-scale constitutive model considering a cracked microstructure

This paper deals with the sensitivity analysis of the macroscopic elasticity tensor to topological microstructural changes of the underlying material. In particular, the microstucture is topologicaly perturbed by the nucleation of a small circular inclusion. The derivation of the proposed sensitivity relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. We consider that the RVE can contain a number of voids, inclusions and/or cracks. It is assumed that non-penetration conditions are imposed at the crack faces, which do not allow the opposite crack faces to penetrate each other. The derived sensitivity leads to a symmetric fourth-order tensor field over the unperturbed RVE domain, which measures how the macroscopic elasticity parameters estimated within the multi-scale framework changes when a small circular inclusion is introduced at the micro-scale level. Copyright © 2009 John Wiley & Sons, Ltd.

Augmented Lagrangian for cone constrained topology optimization

Algorithmic aspects for the solution of topological shape optimization problems subject to a cone constraint are addressed in this paper. In this framework, an augmented Lagrangian method based on the concept of topological derivative is proposed. It is illustrated by some numerical experiments in structural optimization with compliance and eigenfrequency constraints and multiple loads.

Higher-order topological sensitivity for 2-D potential problems. Application to fast identification of inclusions

This article concerns an extension of the topological derivative concept for 2-D potential problems involving penetrable inclusions, whereby a cost function J is expanded in powers of the characteristic size ε of a small inclusion. The O( ε 4) approximation of J is established for a small inclusion of given location, shape and conductivity embedded in a 2-D region of arbitrary shape and conductivity, and then generalized to several such inclusions. Simpler and more explicit versions of this result are obtained for a centrally-symmetric inclusion and a circular inclusion. Numerical tests are performed on a sample configuration, for (i) the O( ε 4) expansion of potential energy, and (ii) the identification of a hidden inclusion. For the latter problem, a simple approximate global search procedure based on minimizing the O( ε 4) approximation of J over a dense search grid is proposed and demonstrated.

On Topological Derivatives for Elastic Solids with Uncertain Input Data

In this paper, a new approach to the derivation of the worst scenario and the maximum range scenario methods is proposed. The derivation is based on the topological derivative concept for the boundary-value problems of elasticity in two and three spatial dimensions. It is shown that the topological derivatives can be applied to the shape and topology optimization problems within a certain range of input data including the Lame coefficients and the boundary tractions. In other words, the topological derivatives are stable functions and the concept of topological sensitivity is robust with respect to the imperfections caused by uncertain data. Two classes of integral shape functionals are considered, the first for the displacement field and the second for the stresses. For such classes, the form of the topological derivatives is given and, for the second class, some restrictions on the shape functionals are introduced in order to assure the existence of topological derivatives. The results on topological derivatives are used for the mathematical analysis of the worst scenario and the maximum range scenario methods. The presented results can be extended to more realistic methods for some uncertain material parameters and with the optimality criteria including the shape and topological derivatives for a broad class of shape functionals.

Sensitivity of the macroscopic elasticity tensor to topological microstructural changes

A remarkably simple analytical expression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topological microstructural changes of the underlying material is proposed. The derivation of the proposed formula relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. The derived sensitivity—a symmetric fourth order tensor field over the RVE domain—measures how the estimated two-dimensional macroscopic elasticity tensor changes when a small circular hole is introduced at the microscale level. This information has potential use in the design and optimisation of microstructures.

Topological sensitivity derivative and finite topology modifications: application to optimization of plates in bending

The concept of topological sensitivity derivative is introduced and applied to study the problem of optimal design of structures. It is assumed, that virtual topology variation is described by topological parameters. The topological derivative provides the gradients of objective functional and constraints with respect to these parameters. This derivative enables formulation of the conditions of topology transformation. In this paper formulas for the topological sensitivity derivative for bending plates are derived. Next, the topological derivative is used in the optimization process in order to formulate conditions of finite topology modifications and in order to localize positions of the modifications. In the case of plates they are related to introduction of holes and introduction of stiffeners. The theoretical considerations are illustrated by some numerical examples.

Sensitivity of the macroscopic thermal conductivity tensor to topological microstructural changes

This paper proposes a closed form expression for the sensitivity of the macroscopic heat conductivity tensor for two-dimensional problems to topological microstructural changes of the underlying material. The sensitivity formula is remarkably simple. It is derived by applying the concept of topological derivative within a variational multi-scale framework for steady-state heat conduction where the macroscopic temperature gradient and heat flux are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material. The classical Fourier law is assumed to hold at the scale referred to as microscopic (the RVE). The derived sensitivity – a symmetric second order tensor field over the RVE domain – measures how the estimated macroscopic conductivity tensor changes when a small circular inclusion is introduced at the micro-scale. The proposed formula finds potential application in the design and optimisation of heat conducting materials.

Inverse acoustic scattering by small-obstacle expansion of a misfit function

This paper concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems, whereby the featured cost function J is expanded in powers of the characteristic size ε of a sound-hard scatterer about ε = 0. The O(ε6) approximation of J is established for a small scatterer of arbitrary shape of given location embedded in an arbitrary acoustic domain, and generalized to several such scatterers. Simpler and more explicit versions of this result are obtained for a centrally-symmetric scatterer and a spherical scatterer. An approximate and computationally fast global search procedure is proposed, where the location and size of the unknown scatterer is estimated by minimizing the O(ε6) approximation of J over a search grid. Its usefulness is demonstrated on numerical experiments, where the identification of a spherical, ellipsoidal or banana-shaped scatterer embedded in a acoustic half-space from known acoustic pressure on the surface is considered.

Topological derivative: A tool for image processing

The increasing complexity in several fields of science and technology has motivated the use of techniques originally conceived in other areas of applications. An illustrative example of this is given by the topological derivative which quantifies the sensitivity of a problem when the domain under consideration is perturbed by changing its topology. This concept, initially conceived to deal with topology optimization problems, has also been successfully applied to inverse problems and material properties characterization. Our aim in this paper is to present an other field of application for the topological derivative: image processing. An appropriate functional and a variational problem are associated to the cost endowed to an specific image processing application. Thus, the corresponding topological derivative can be used as an indicator function that leads, through a minimization process, to the processed image. We focus our attention on two image processing application. In the first, the topological derivative is used in image restoration, i.e. to restore an image that was somehow degraded (acquisition process, transmission, storage, etc.). Moreover, a novel fully discrete algorithm based on the topological derivative concept is presented. In the second application, we use the topological derivative to derive a “continuous” and a fully discrete novel image segmentation algorithms, i.e. for objects identification in an image. Finally and in order to show the performance of these algorithms, several numerical examples are also presented in this work.

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.

Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain

Building on previous work for 3D inverse scattering in the frequency domain, this article develops the concept of topological derivative for 3D elastic and acoustic-wave imaging of media of arbitrary geometry using data in the time domain. The topological derivative, which quantifies the sensitivity of the cost functional associated with the inverse scattering problem due to the creation at a specified location of an infinitesimal hole (for the elastodynamic case) or rigid inclusion (for the acoustic case), is found to be expressed in terms of the time convolution of the free field and a supplementary adjoint field. The derivation of the topological derivative follows the generic pattern proposed in previous studies, which is transposable to a variety of other physical problems. A numerical example, where the featured cost function is defined in terms of synthetic data arising from the scattering of plane acoustic waves by a rigid spherical inclusion, illustrates the utility of the topological derivative concept for defect identification using time-varying data.