Antoine Laurain

University of Duisburg-Essen | uni-due · Faculty of Mathematics

The problem of covering a two-dimensional bounded set with a fixed number of minimum-radius identical disks is studied in the present work. Bounds on the optimal radius are obtained for a certain class of nonsmooth domains, and an asymptotic expansion of the bounds as the number of disks goes to infinity is provided. The proof is based on the appro...

The problem of minimizing the first eigenvalue of the Dirichlet Laplacian with respect to a union of m balls with fixed identical radii and variable centers in the plane is investigated in the present work. The existence of a minimizer is shown and the shape sensitivity analysis of the eigenvalue with respect to the centers’ positions is presented....

In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints in Banach spaces. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the deriv...

Minimization diagrams encompass a large class of diagrams of interest in the literature, such as generalized Voronoi diagrams. We develop an abstract perturbation theory and perform a sensitivity analysis for functions depending on sets defined through intersections of smooth sets, and formulate precise conditions to avoid singular situations. This...

We introduce and analyze a lower envelope method (LEM) for the tracking of interfaces motion in multiphase problems. The main idea of the method is to define the phases as the regions where the lower envelope of a set of functions coincides with exactly one of the functions. We show that a variety of complex lower-dimensional interfaces naturally a...

Purpose Motivated by the acoustics of motor vehicles, a coupled fluid–solid system is considered. The air pressure is modeled by the Helmholtz equation, and the structure displacement is described by elastodynamic equations. The acoustic–structure interaction is modeled by coupling conditions on the common interface. First, the existence and unique...

The problem of covering a region of the plane with a fixed number of minimum-radius identical balls is studied in the present work. An explicit construction of bi-Lipschitz mappings is provided to model small perturbations of the union of balls. This allows us to obtain analytical expressions for first- and second-order derivatives using nonsmooth...

We investigate the problem of covering a region in the plane with the union of m identical balls of minimum radius. The region to be covered may be disconnected, be nonconvex, have Lipschitz boundary, and in particular have corners. Nullifying the area of the complement of the union of balls with respect to the region to be covered is considered as...

Velocity models presenting sharp interfaces are highly relevant in seismic imaging, e.g., for imaging the subsurface of the Earth in the presence of salt bodies. In order to mitigate the oversmoothing of classical regularization strategies such as the Tikhonov regularization, we propose a shape optimization approach for the sharp-interface reconstr...

We develop a framework and numerical method for controlling the full space-time tube of a geometrically driven flow. We consider an optimal control problem for the mean curvature flow of a curve or surface with a volume constraint, where the control parameter acts as a forcing term in the motion law. The control of the trajectory of the flow is ach...

Velocity models presenting sharp interfaces are highly relevant in seismic imaging, for instance for imaging the subsurface of the Earth in the presence of salt bodies. In order to mitigate the oversmoothing of classical regularization strategies such as the Tikhonov regularization, we propose a shape optimization approach for the sharp-interface r...

In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the derivative of a shape...

Working within the class of piecewise constant conductivities, the inverse problem of electrical impedance tomography can be recast as a shape optimization problem where the discontinuity interface is the unknown. Using Gröger’s W p 1 -estimates for mixed boundary value problems, the averaged adjoint method is extended to the case of Banach spaces,...

This paper is devoted to the theoretical and numerical study of an optimal design problem in high-temperature superconductivity (HTS). The shape optimization problem is to find an optimal superconductor shape which minimizes a certain cost functional under a given target on the electric field over a specific domain of interest. For the governing PD...

Working within the class of piecewise constant conductivities, the inverse problem of electrical impedance tomography can be recast as a shape optimization problem where the discontinuity interface is the unknown. Using Gr\"oger's $W^{1}_p$-estimates for mixed boundary value problems, the averaged adjoint method is extended to the case of Banach sp...

We study distributed and boundary integral expressions of Eulerian and Fréchet shape derivatives for several classes of nonsmooth domains such as open sets, Lipschitz domains, polygons and curvilinear polygons, semiconvex and convex domains. For general shape functionals, we establish relations between distributed Eulerian and Fréchet shape derivat...

In the standard level set method, the evolution of the level set function is determined by solving the Hamilton{Jacobi equation, which is derived by considering smooth boundary perturbations of the zero level set. The converse approach is to consider smooth perturbations of the level set function and to find the corresponding perturbations of the z...

In this paper we present an educational code written using FEniCS, based on the level set method, to perform compliance minimization in structural optimization. We use the concept of distributed shape derivative to compute a descent direction for the compliance, which is defined as a shape functional. The use of the distributed shape derivative is...

In this paper we analyze the reconstruction step of the Voronoi implicit interface method (VIIM) introduced by Saye and Sethian. The VIIM is a powerful method to track multiple interfaces with a single function. The central idea of the VIIM is to use an unsigned distance function to represent the multiphase system, unlike the level set method which...

In this paper we consider the inverse problem of simultaneously reconstructing the interface where the jump of the conductivity occurs and the Robin parameter for a transmission problem with piecewise constant conductivity and Robin-type transmission conditions on the interface. We propose a reconstruction method based on a shape optimization appro...

In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite w...

The purpose of this article is to propose a deterministic method for optimizing a structure considering its worst possible behaviour when a small uncertainty exists over its Lamé parameters. The idea is to take advantage of the small parameter to derive an asymptotic expansion of the displacement and of the compliance with respect to the contrast i...

The structure theorem of Hadamard-Zol\'esio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. Actually the domain representation, also known as distributed shape derivative, is more general than the boundary expression as it is we...

The inverse source problem consists of reconstructing a mass distribution in a geometrical domain from boundary measurements of the associated potential and its normal derivative. In this paper the inverse source problem is reformulated as a topology optimization problem, where the support of the mass distribution is the unknown variable. The Kohn–...

In this paper the shape derivative of an objective depending on the solution of an eddy current approximation of Maxwell’s equations is obtained. Using a Lagrangian approach in the spirit of Delfour and Zolésio, the computation of the shape derivative of the solution of the state equation is bypassed. This theoretical result is applied to magnetic...

Controlling droplet shape via surface tension has numerous technological applications, such as droplet lenses and lab-on-a-chip. This motivates a partial differential equationconstrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate. We use shape differential...

The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation...

Recent advances in the analytical as well as numerical treatment of classes of elliptic mathematical programs with equilibrium constraints (MPECs) in function space are discussed. In particular, stationarity conditions for control problems with point tracking objectives and subject to the obstacle problem as well as for optimization problems with v...

The problem of distributing two conducting materials with a prescribed volume ratio in a ball so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions is considered in two and three dimensions. The gap ε between the two conductivities is assumed to be small (low contrast regime). The main result of the paper is to sh...

A bilevel shape optimization problem with the exterior Bernoulli free boundary problem as lower-level problem and the control of the free boundary as the upper-level problem is considered. Using the shape of the inner boundary as the control, we aim at reaching a specific shape for the free boundary. A rigorous sensitivity analysis of the bilevel s...

In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0≤u≤1 and ∫u=m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation i...

Fluorescence tomography is a non-invasive imaging modality that reconstructs fluorophore distributions inside a small animal from boundary measurements of the fluorescence light. The associated inverse problem is stabilized by a priori properties or information. In this paper, cases are considered where the fluorescent inclusions are well separated...

The inverse potential problem consists in reconstructing an unknown measure with support in a geometrical domain from a single boundary measurement. In order to deal with this severely ill-posed inverse problem, we rewrite it as an optimization problem where a Kohn-Vogelius-type functional measuring the misfit between the solutions of two auxiliary...

In this article we consider the problem of the optimal distribution of two conducting materials with given volume inside a fixed domain, in order to minimize the first eigenvalue (the ground state) of a Dirichlet operator. It is known, when the domain is a ball, that the solution is radial, and it was conjectured that the optimal distribution of th...

Second-order topological expansions in electrical impedance tomography problems with piecewise constant conductivities are considered. First-order expansions usually consist of local terms typically involving the state and the adjoint solutions and their gradients estimated at the point where the topological perturbation is performed. In the case o...

This paper focuses on the study of a linear eigenvalue problem with indefinite weight and Robin type boundary conditions. We investigate the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization proble...

In this article we consider the problem of the optimal distribution of two conducting materials with given volume inside a fixed domain, in order to minimize the first eigenvalue (the ground state) of a Dirichlet operator. It is known, when the domain is a ball, that the solution is radial, and it was conjectured that the optimal distribution of th...

Magnetic resonance images which are corrupted by noise and by smooth modulations are corrected using a variational formulation incorporating a total variation like penalty for the image and a high order penalty for the modulation. The optimality system is derived and numerically discretized. The cost functional used is non-convex, but it possesses...

The Cartesian parallel magnetic imaging problem is formulated variationally using a high-order penalty for coil sensitivities and a total variation like penalty for the reconstructed image. Then the optimality system is derived and numerically discretized. The objective function used is non-convex, but it possesses a bilinear structure that allows...

Fluorescence tomography aims at the reconstruction of the concentration and life-time of fluorescent inclusions from boundary measurements of light emitted. The underlying ill-posed problem is often solved with gradient descent of Gauss-Newton methods, for example. Unfortunately, these approaches don't allow to assess the quality of the reconstruct...

p>Fluorescence tomography aims at the reconstruction of the concentration and life-time of fluorescent inclusions from boundary measurements of light emitted. The underlying ill-posed problem is often solved with gradient descent of Gauss-Newton methods, for example. Unfortunately, these approaches don't allow to assess the quality of the reconstru...

The shape of the free boundary arising from the solution of a variational inequality is controlled by the shape of the domain where the variational inequality is defined. Shape and topological sensitivity analysis is performed for the obstacle problem and for a regularized version of its primal-dual formulation. The shape derivative for the regular...

We calculate the main asymptotic terms for eigenvalues, both simple and multiple, and eigenfunctions of the Neumann Laplacian in a three-dimensional domain $\Om(h)$ perturbed by a small (with diameter $O(h)$) Lipschitz cavern $\overline{\om_h}$ in a smooth boundary $\partial\Om=\partial\Om(0)$. The case of the hole $\overline{\om_h}$ inside the dom...

A Bernoulli free boundary problem with geometrical constraints is studied. The domain $\Om$ is constrained to lie in the half space determined by $x_1\geq 0$ and its boundary to contain a segment of the hyperplane $\{x_1=0\}$ where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equat...

A shape and topology optimization driven solution technique for a class of linear complementarity problems (LCPs) in function space is considered. The main motivating application is given by obstacle problems. Based on the LCP together with its corresponding interface conditions on the boundary between the coincidence or active set and the inactive...

Topological sensitivity analysis is performed for the piecewise constant Mumford-Shah functional. Topological and shape derivatives are combined in order to derive an algorithm for image segmentation with fully automatized initialization. Segmentation of 2D and 3D data is presented. Further, a generalized Mumford-Shah functional is proposed and num...

Problems involving cracks are of particular importance in structural mechanics, and gave rise to many interesting mathematical techniques to treat them. The difficulties stem from the singularities of domains, which yield lower regularity of solutions. Of particular interest are techniques which allow us to identify cracks and defects from the mech...

Two approaches are proposed for solving inverse problems in shape optimization. We are looking for the unknown position of a small hole in a domain Ω. First, the asymptotic analysis of the underlying p.d.e. defined in a perturbed domain is performed and the so-called topological derivative is defined. Then, in the first approach, the self-adjoint e...

A class of shape optimization problems is solved numerically by the level set method combined with the topological derivatives for topology optimization. Actually, the topology variations are introduced on the basis of asymptotic analysis, by an evaluation of extremal points (local maxima for the specific problem) of the so-called topological deriv...

A level set based shape and topology optimization approach to Electrical Impedance Tomography (EIT) problems with piecewise constant conductivities is introduced. The proposed solution algorithm is initialized by using topological sensitivity analysis. Then it relies on the notion of shape derivatives to update the shape of the domains where conduc...

A Level Set Method in Shape and Topology Optimization for Variational Inequalities The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect...

Self-adjoint extensions of elliptic operators are used to model the solution of a partial differential equation defined in a singularly perturbed domain. The asymptotic expansion of the solution of a Laplacian with respect to a small parameter ε is first performed in a domain perturbed by the creation of a small hole. The resulting singular perturb...

In shape optimization, the main results concerning the case of domains with smooth boundaries and smooth perturbations of these domains are well-known, whereas the study of non-smooth domains, such as domains with cracks for instance, and the study of singular perturbations such as the creation of a hole in a domain is more recent and complex. Th...

En optimisation de formes, de nombreux résultats ont déjà été obtenus dans le cas de domaines à frontière régulière et pour des perturbations régulières de ces domaines. Par contre, l'étude de domaines non-réguliers, tels que des domaines fissurés par exemple, et l'étude de perturbations singulières telles que la création d'un trou dans un domai...

The structure of the derivative of shape functionals, which is well known for smooth domains, is studied in the case of domains with a cut. The derivative contains two terms, the first depending on the normal variations of the crack and the second depending on the tangential domain perturbations at the tips of the crack. The structure of the first...

The topological derivative is a new tool introduced by Sokolowski and Zochowski in shape optimization. It allows to measure the variation of a functional depending on the geometrical domain when a small cavity is created inside the domain. It is possible to define the topological derivative for energy functionals of obstacle problems, and for conta...