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PurposeThe purpose of this paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic acoustic waves scattering by sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions).Design/methodology/approachThe elliptic boundary value problems in the singularly perturbed domains are equivalently reduced to couples of boundary integral equations with unknown densities given by boundary traces. In the case of circular or spherical holes, the spectral Fourier and Mie series expansions of the potential operators are used to derive the first-order term in the asymptotic expansion of the boundary traces for the solution to the two- and three-dimensional perturbed problems.FindingsAs the shape gradients of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function in the iterated numerical solution of any shape optimization or imaging problem relying on time-harmonic acoustic waves propagation. When coupled with converging Gauss−Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.

PurposeIn this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.Design/methodology/approachFrom the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.FindingsAs the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

A new sampling technique for the application of proper orthogonal decomposition to a set of snapshots has been recently developed by the authors to facilitate a variety of data processing tasks (J. Comput. Phys. 335, 2017). According to it, robust modal expansions result from performing the decomposition on a limited number of relevant snapshots and a limited number of discretization mesh points, which are selected via Gauss elimination with double pivoting on the original snapshot matrix containing the given data. In the present work, the sampling method is adapted and combined with low-dimensional modeling. This combination yields a novel adaptive algorithm for the simulation of time-dependent non-linear dynamics in pattern-forming systems. Convenient snapshot sets, computed on demand over the evolution, are stored to record local temporal events whose underlying mechanisms are essential for the approximations. Also, a collection of sparse grid points, which are used to construct the mode basis and the reduced system of equations, is adaptively sampled according to unlinked spatial structures. The outcome is a reduced order model of the problem that (i) yields reliable approximations of the dynamical transitions, (ii) is well-suited to describe localized spatio-temporal complexity, and (iii) provides fast computations. Robustness, accuracy, and computational efficiency of the proposed algorithm are illustrated for some relevant pattern-forming systems, in both one and two spatial dimensions, exhibiting solutions with a rich spatio-temporal structure.

This paper deals with the acceleration of time-dependent solvers for nonlinear dissipative systems. The governing equations are Galerkin-projected onto a set of modes, which are obtained by applying proper orthogonal decomposition (POD) to a set of snapshots calculated by a standard numerical solver. The advantage of this approach is that the online operation of the resulting Galerkin system should be much faster than the standard numerical solver. The basic version of such reduced order model uses snapshots computed in a preprocess that is usually very computationally expensive. This difficulty can be overcome by an adaptive combination of the standard numerical solver and the Galerkin system along the simulation, using a method called POD on the Fly, which will be illustrated in a representative application. In addition, Galerkin projection can be performed using only a suitable set of collocation points, which decreases even further the computational cost. In this context, an efficient collocation method called LUPOD will be described and tested in various applications, including its combination with Galerkin projection.

We propose an automatic algorithm for 3D inverse electromagnetic scattering based on the combination of topological derivatives and regularized Gauss-Newton iterations. The algorithm is adapted to decoding digital holograms. A hologram is a two-dimensional light interference pattern that encodes information about three-dimensional shapes and their optical properties. The formation of the hologram is modeled using Maxwell theory for light scattering by particles. We then seek shapes optimizing error functionals which measure the deviation from the recorded holograms. Their topological derivatives provide initial guesses of the objects. Next, we correct these predictions by regularized Gauss-Newton techniques. In contrast to standard Gauss-Newton methods, in our implementation the number of objects can be automatically updated during the iterative procedure by new topological derivative computations. We show that the combined use of topological derivative based optimization and iteratively regularized Gauss-Newton methods produces fast and accurate descriptions of the geometry of objects formed by multiple components with nanoscale resolution, even for a small number of detectors and non convex components aligned in the incidence direction. The method could be applied in general imaging set-ups involving other waves (microwave imaging, elastography...) provided closed-form expressions for the topological and Frechet derivatives are determined.

In this paper we compute closed-form expressions for the topological derivative for three-dimensional time-harmonic electromagnetic waves for perfect conductors (Dirichlet condition), electromagnetic cavities (Neumann condition), absorbing obstacles (impedance condition), and dielectric inclusions (transmission conditions). The proofs are based on the computation of shape derivatives followed by asymptotic expansions using Mie series when infinitesimal spheres are considered. An exhaustive gallery of numerical experiments is presented, which demonstrate that the topological derivative is a very powerful tool for the detection of multiple electromagnetic scatterers without a priori information about their number, size, shape, or location. Numerical examples include highly demanding configurations where only a few incident directions and a few observation points (for near-field data) or a few far-field observation directions (for far-field data) are considered.

In this work we consider a scattering problem governed by the two-dimensional Helmholtz equation, where some objects of different nature (sound-hard, sound-soft and penetrable) are present in the background medium. First we propose and analyze a system of boundary integral equations to solve the direct problem. After that, we propose a numerical method based on the computation of a multifrequency topological energy based imaging functional to find the shape of the objects (without knowing their nature) from measurements of the total field at a set of observation points. Numerical examples show that the proposed indicator function is able to detect objects of different nature and/or shape and size when processing noisy data for a rich enough range of frequencies.

This paper deals with the use of steady active thermographic inspection as a structural health monitoring method to detect defects and inclusions in a heat conductive medium. The proposed postprocessing method has the advantage of taking into account the whole physics of the problem instead of a reduced set of physical information, as most of the traditional methods do. In comparison with other non-destructive testing techniques like ultrasonics, thermographic inspection is non-contact, more non-intrusive, and safer. However, processing thermographic data is a fairly demanding problem because heat conduction is short range and exhibits a poor signal-to-noise ratio. Numerical experiments illustrate that the present postprocessing technique detects damage fairly well even in very challenging scenarios where the size of the defects is rather small and/or thermograms are highly noisy.

The aim of this paper is to survey the research activity of the author during the years prior to the XIII SEMA Young Researcher Award. In this paper we present an overall view of numerical techniques for the simulation of acoustic and thermal waves from two complete opposite points of view: the fields of direct and inverse problems.

In this work we propose and analyze numerical methods for the approximation of the solution of Helmholtz transmission problems in two or three dimensions. This kind of problems arises in many applications related to scattering of acoustic, thermal and electromagnetic waves. Formulations based on boundary integral methods are powerful tools to deal with transmission problems in unbounded media. Different formulations using boundary integral equations can be found in the literature. We propose here new symmetric formulations based on a paper by Martin Costabel and Ernst P. Stephan (1985), that uses the Calderon projector for the interior and exterior problems to develop closed expressions for the interior and exterior Dirichlet-to-Neumann operators. These operators are then matched to obtain an integral system that is equivalent to the Helmholtz transmission problem and uses Cauchy data on the transmission boundary as unknowns. We show how to simplify the aspect and analysis of the method by employing an additional mortar unknown with respect to the ones used in the original paper, writing it in an appropriate way to devise Krylov type iterations based on the separate Dirichlet-to-Neumann operators.

An iterative strategy for the reconstruction of objects buried in a medium and the identification of their material parameters is analysed. The algorithm alternates guesses of the domains using topological derivatives with corrections of the parameters obtained by descent techniques. Numerical experiments in geometries with multiple scatterers show that our scheme predicts the number, location and shape of objects, together with their physical parameters, with reasonable accuracy in a few steps.

We present topological derivative and energy based procedures for the imaging of micro and nano structures using one beam of visible light of a single wavelength. Objects with diameters as small as 10 nm can be located and their position tracked with nanometer precision. Multiple objects dis-tributed either on planes perpendicular to the incidence direction or along axial lines in the incidence direction are distinguishable. More precisely, the shape and size of plane sections perpendicular to the incidence direction can be clearly determined, even for asymmetric and nonconvex scatterers. Axial resolution improves as the size of the objects decreases. Initial reconstructions may proceed by gluing together two-dimensional horizontal slices between axial peaks or by locating objects at three-dimensional peaks of topological energies, depending on the effective wavenumber. Below a threshold size, topological derivative based iterative schemes improve initial predictions of the lo-cation, size, and shape of objects by postprocessing fixed measured data. For larger sizes, tracking the peaks of topological energy fields that average information from additional incident light beams seems to be more effective.

In this work we address the inverse problem of reconstructing inclusions and their thermal parameters given temperature measurements at the accessible side of a material. We describe an iterative descent method that combines topological derivative computations to reconstruct the geometry of the defects with gradient iterations to approximate the material parameters. A numerical experiment showing the ability of the method to obtain reasonable reconstructions in a few iterations is presented.

In this paper we survey computational techniques based on boundary integral formulations for the simulation of thermal waves. Time-harmonic solutions to diffusion problems appear in many physical situations of interest and give rise to many interesting problems related to material characterization, parameter assessment or detection of defects. We review the main direct, indirect and mixed integral numerical methods for a model of scattering of thermal waves by many obstacles and discuss how multiple scattering techniques and other physical tools can be understood as iterative methods or used as preconditioners. We also deal with some transient problems that can be solved with boundary element methods using the Laplace transform and with coupled finite and boundary element schemes for non-homogeneous obstacles

This paper deals with the inverse problem of detecting the level of corrosion at the interface of an inclusion given thermal measurements at the accessible boundary of the sample. This leads to a transmission problem for the heat equation with an unknown coefficient in a transmission condition. We consider both time-harmonic and delta-pulse excitations. In both cases we prove uniqueness results for the inverse problem. To reconstruct the unknown level of corrosion numerically, we study a non-iterative method and a regularized Newton method and compare their performances in a number of numerical experiments. Photothermal techniques are suitable means of inspecting composite materials with nondestructive tests. We are interested in a technique that consists in heating the accessible side of the material by a defocused laser beam. The goal is to reconstruct internal properties of the material (to detect structural defects, reconstruct the size, depth, orientation of the inclusions and/or physical properties of them) from measurements of the temperature at the side that has been thermically excited. Some recent papers on physical experiments with this kind of techniques are [12, 27, 33, 34]. In this work we study the detection of the level of corrosion at the interface separating the internal inhomogeneities from the matrix material. We explore two different kinds of thermal excitement, produced by time-harmonic incident heating fields and by delta-pulse excitations. In the first one we deal with a steady-state problem. In the corresponding forward problem one looks for time-harmonic solutions of an elliptic conductive-transmission problem. This kind of solution is commonly referred to as thermal waves. For a detailed study of these waves and their uses we refer to [3, 22–24] and references therein. The numerical solution of the forward

Summary An adaptive low-dimensional model is considered to simulate time-dependent dynamics in nonlinear dissipative systems governed by PDEs. The method combines an inexpensive POD-based Galerkin system with short runs of a standard numerical solver that provides the snapshots necessary to first construct and then update the POD modes. Switching between the numerical solver and the Galerkin system is decided ‘on the fly’ by monitoring (i) a truncation error estimate and (ii) a residual estimate. The latter estimate is used to control the mode truncation instability and highly improves former adaptive strategies that detected this instability by monitoring consistency with a second instrumental Galerkin system based on a larger number of POD modes. The most computationally expensive run of the numerical solver occurs at the outset, when the whole set of POD modes is calculated. This step is improved by using mode libraries, which may either be generic or result from former applications of the method. The outcome is a flexible, robust, computationally inexpensive procedure that adapts itself to the local dynamics by using the faster Galerkin system for the majority of the time and few, on demand, short runs of a numerical solver. The method is illustrated considering the complex Ginzburg–Landau equation in one and two space dimensions. Copyright © 2015 John Wiley & Sons, Ltd.

This paper is devoted to the reconstruction of objects buried in a medium and their material properties by hybrid topological derivative-gradient based methods. After illustrating the techniques in time-harmonic acoustic problems with different boundary conditions and in electrical impedance tomography problems with continuous Neumann conditions, we extend the hybrid method for a realistic model in tomography where the boundary conditions are given at a discrete set of electrodes.

We present a technique to reconstruct the electromagnetic properties of a medium or a set of objects buried inside it from boundary measurements when applying electric currents through a set of electrodes. The electromagnetic parameters may be recovered by means of a gradient method without a priori information on the background. The shape, location and size of objects, when present, are determined by a topological derivative-based iterative procedure. The combination of both strategies allows improved reconstructions of the objects and their properties, assuming a known background.

A method is presented to accelerate numerical simulations on parabolic problems using a numerical code and a Galerkin system (obtained via POD plus Galerkin projection) on a sequence of interspersed intervals. The lengths of these intervals are chosen according to several basic ideas that include an a priori estimate of the error of the Galerkin approximation. Several improvements are introduced that reduce computational complexity and deal with: (a) updating the POD manifold (instead of calculating it) at the end of each Galerkin interval; (b) using only a limited number of mesh points to calculate the right hand side of the Galerkin system; and (c) introducing a second error estimate based on a second Galerkin system to account for situations in which qualitative changes in the dynamics occur during the application of the Galerkin system. The resulting method, called local POD plus Galerkin projection method, turns out to be both robust and efficient. For illustration, we consider a time-dependent Fisher-like equation and a complex Ginzburg-Landau equation.

We propose a numerical strategy to reconstruct scatterers buried in a medium when the incident radiation (electromagnetic, thermal, acoustic) is governed by Helmholtz transmission problems. The scattering problem is recast as a shape optimization problem with the Helmholtz equation as a constraint and the scatterer as a design variable. Our method is based on the (successive) computation of topological derivatives of the associated shape functional for updated guesses of the scatterer. We present an efficient scheme to compute the required topological derivatives at each step. The scheme combines explicit expressions for the topological derivatives in terms of the solutions of forward and adjoint transmission problems with BEM–FEM approximations. Our technique applies in either spatially homogeneous or inhomogeneous media. Finally, a two dimensional numerical test illustrates the ability of the method to reconstruct buried shapes.

Topological derivative methods are used to solve constrained optimization reformulations of inverse scattering problems. The constraints take the form of Helmholtz or elasticity problems with different boundary conditions at the interface between the surrounding medium and the scatterers. Formulae for the topological derivatives are found by first computing shape derivatives and then performing suitable asymptotic expansions in domains with vanishing holes. We discuss integral methods for the numerical approximation of the scatterers using topological derivatives and implement a fast iterative procedure to improve the description of their number, size, location and shape.

In this work we propose and analyse numerical methods for Helmholtz transmission problems in two and three dimensions. The methods we analyse use combined single and double layer potentials to represent the interior and exterior solution of the transmission problem. The corresponding boundary integral system includes weakly singular and hypersingular boundary integral operators on the interface. Its invertibility is equivalent to the unique solvability of the transmission problem, since the use of the above mentioned potentials does not introduce spurious eigenmodes in the formulation. We give necessary and sufficient conditions for the convergence of general Petrov–Galerkin schemes for solving the resulting system, providing some concrete methods for the two dimensional case. Some numerical experiments are shown.

This paper proposes and analyzes a BEM-FEM scheme to approximate a time-harmonic diffusion problem in the plane with non-constant coefficients in a bounded area. The model is set as a Helmholtz transmission problem with adsorption and with non-constant coefficients in a bounded domain. We reformulate the problem as a four-field system. For the temperature and the heat flux we use piecewise constant functions and lowest order Raviart-Thomas elements associated to a triangulation approximating the bounded domain. For the boundary unknowns we take spaces of periodic splines. We show how to transmit information from the approximate boundary to the exact one in an efficient way and prove well-posedness of the Galerkin method. Error estimates are provided and experimentally corroborated at the end of the work. The authors are partially supported by MEC/FEDER Project MTM2004-01905, Gobierno de Aragón (Grupo Consolidado PDIE) and by Gobierno de Navarra Ref. 18/2005.

Let Ω1,Ω..., Ωd denote a finite number of simply connected domains strictly contained in ℝ _ 2 := |(x 1, x 2) | x 2 < 0} with nonintersecting closures and such that \( \bar \Omega _k \cap \Pi = \not 0 \) for all k, being ∏: = |(x 1), 0) | x 1 ∈ ℝ}. The boundaries Γ k := ∂Ω k are assumed to be parameterizable C 2-curves. Normals are directed towards the exterior of Ω k for each k and the normal derivative on ∏ is directed towards the exterior of Ω (see Fig. 1).

We introduce new iterative schemes to reconstruct scatterers buried in a medium and their physical properties. The inverse scattering problem is reformulated as a constrained optimization problem involving transmission boundary value problems in heterogeneous media. Our first step consists in developing a reconstruction scheme assuming that the properties of the objects are known. In a second step, we combine iterations to reconstruct the objects with iterations to recover the material parameters. This hybrid method provides reasonable guesses of the parameter values and the number of scatterers, their location and size. Our schemes to reconstruct objects knowing their nature rely on an extended notion of topological derivative. Explicit expressions for the topological derivatives of the corresponding shape functionals are computed in general exterior domains. Small objects, shapes with cavities and poorly illuminated obstacles are easily recovered. To improve the predictions of the parameters in the successive guesses of the domains we use a gradient method.

In this paper we propose a hybrid between direct and indirect boundary integral methods to solve a transmission problem for the Helmholtz equation in Lipschitz and smooth domains. We present an exhaustive abstract study of the numerical approximation of the resulting system of boundary integral equations by means of Galerkin methods. Some particular examples of convergent schemes in the smooth case in two dimensions are given. Finally, we extend the results to a thermal scattering problem in a half plane with several obstacles and provide numerical results that illustrate the accuracy of our methods depending on the regularity of the interface.