arXiv subjects

Antoine Laurain

Publications and source records attributed to Antoine Laurain.

13 recordsLinked to original sources

arXiv2026

A Numerical Investigation of the Rayleigh$-$Faber$-$Krahn Inequality for Polyhedral Domains

We present numerical results for the problem of minimizing the first Dirichlet eigenvalue of the Laplacian among three-dimensional convex polyhedra with a prescribed number of facets. This problem can be viewed as a polyhedral version of the classical Rayleigh$-$Faber$-$Krahn inequality. Using a parameterization of polytopes by supporting hyperplanes, the problem is first reformulated as an equivalent finite-dimensional constrained minimization problem. A Lagrangian framework is employed for the numerical solution. Under the assumption that each vertex is incident to exactly $d$ facets, the sensitivity analysis of the Lagrangian combines finite-dimensional perturbations of the facets of $d$-dimensional polyhedra with infinite-dimensional shape derivatives of both the eigenvalue and the volume. Depending on the number of facets, the numerical results yield well-known polyhedra, such as Platonic solids, prisms, and the tetrakaidecahedron, or reveal nonregular optimal shapes exhibiting several symmetries.

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arXiv2026

FormOpt: A FEniCSx toolbox for level set-based shape optimization supporting parallel computing

This article presents the toolbox FormOpt for two- and three-dimensional shape optimization with parallel computing capabilities, built on the FEniCSx software framework. We introduce fundamental concepts of shape sensitivity analysis and their numerical applications, mainly for educational purposes, while also emphasizing computational efficiency via parallelism for practitioners. We adopt an optimize-then-discretize strategy based on the distributed shape derivative and its tensor representation, following the approach of \cite{MR3843884} and extending it in several directions. The numerical shape modeling relies on a level set method, whose evolution is driven by a descent direction computed from the shape derivative. Geometric constraints are treated accurately through a Proximal-Perturbed Lagrangian approach. FormOpt leverages the powerful features of FEniCSx, particularly its support for weak formulations of partial differential equations, diverse finite element types, and scalable parallelism. The implementation supports three different parallel computing modes: data parallelism, task parallelism, and a mixed mode. Data parallelism exploits FEniCSx's mesh partitioning features, and we implement a task parallelism mode which is useful for problems governed by a set of partial differential equations with varying parameters. The mixed mode conveniently combines both strategies to achieve efficient utilization of computational resources.

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arXiv2025

Optimization of centroidal Voronoi tessellations

In this paper, we investigate the optimization of Centroidal Voronoi Tessellations (CVT) under geometric constraints. For this purpose, we minimize a linear combination of the standard CVT energy functional with terms involving geometric attributes such as area and perimeter. The derivative of the objective functional with respect to the position of the generators is computed using techniques of shape calculus and sensitivity analysis of minimization diagrams. Several numerical experiments are presented to explore the geometric constraints of cells with identical areas, cells without small edges, and density-based distributions of cells.

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arXiv2022

A shape optimization approach for electrical impedance tomography with point measurements

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 $W^{1}_p$-estimates for mixed boundary value problems, the averaged adjoint method is extended to the case of Banach spaces, which allows to compute the derivative of shape functionals involving point evaluations. We compute the corresponding distributed expression of the shape derivative and show that it may contain Dirac measures in addition to the usual domain integrals. We use this distributed shape derivative to devise a numerical algorithm, show various numerical results supporting the method, and based on these results we discuss the influence of the point measurements patterns on the quality of the reconstructions.

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arXiv2021

Sensitivity analysis and tailored design of minimization diagrams

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 allows us to define a general framework for solving optimization problems depending on minimization diagrams. The particular case of Voronoi diagrams is discussed to illustrate the general theory. A variety of numerical experiments is presented. The experiments include constructing Voronoi diagrams with cells of equal size, cells satisfying conditions on the relative size of their edges or their internal angles, cells with the midpoints of pairs of Voronoi and Delaunay edges as close as possible, or cells of varying sizes governed by a given function. Overall, the experiments show that the proposed methodology allows the construction of customized Voronoi diagrams using off-the-shelf well-established optimization algorithms.

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arXiv2021

Analysis and application of a lower envelope method for sharp-interface multiphase problems

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 appear in the process. The phases evolution is then achieved by solving a set of transport equations. In the first part of the paper, we show several theoretical properties, give conditions to obtain a well-posed behaviour, and show that the level set method is a particular case of the LEM. In the second part, we propose a LEM-based numerical algorithm for multiphase shape optimization problems. We apply this algorithm to an inverse conductivity problem with three phases and present several numerical results.

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arXiv2021

A Shape-Newton approach to the problem of covering with identical balls

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 shape optimization techniques under appropriate regularity assumptions. Singular cases are also studied using asymptotic analysis. For the case of regions given by the union of disjoint convex polygons, algorithms based on Voronoi diagrams that do not rely on approximations are given to compute the derivatives. Extensive numerical experiments illustrate the capabilities and limitations of the introduced approach.

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arXiv2020

An abstract lagrangian framework for computing shape derivatives

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 functional, the material derivative of the state, and the adjoint state. Furthermore, it allows to gain insight on the duality between the material derivative of the state and the adjoint state. We show several applications of our main result to the computation of distributed shape derivatives for problems involving linear elliptic, nonlinear elliptic, parabolic PDEs and distributions. We compare our approach with other techniques for computing shape derivatives including the material derivative method and the averaged adjoint method.

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arXiv2020

Shape optimization for superconductors governed by H(curl)-elliptic variational inequalities

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 PDE-model, we consider an elliptic curl-curl variational inequality (VI) of the second kind with an L1-type nonlinearity. In particular, the non-smooth VI character and the involved H(curl)-structure make the corresponding shape sensitivity analysis challenging. To tackle the non-smoothness, a penalized dual VI formulation is proposed, leading to the G{â}teaux differentiability of the corresponding dual variable mapping. This property allows us to derive the distributed shape derivative of the cost functional through rigorous shape calculus on the basis of the averaged adjoint method. The developed shape derivative turns out to be uniformly stable with respect to the penalization parameter, and strong convergence of the penalized problem is guaranteed. Based on the achieved theoretical findings, we propose 3D numerical solutions, realized using a level set algorithm and a Newton method with the Nedelec edge element discretization. Numerical results indicate a favourable and efficient performance of the proposed approach for a specific HTS application in superconducting shielding.

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arXiv2016

Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

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 weight linear eigenvalueproblem in a fixed box $Ω$ and we consider the general case of Robin boundary conditions on $\partialΩ$. It is well known that it suffices to consider {\it bang-bang} weights taking two values of different signs, that can be parametrized by the characteristic function of the subset $E$ of $Ω$ on which resources are located. Therefore, the optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to $E$, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. Namely, we completely solve the problem in dimension 1, we prove the counter-intuitive result that the ball is almost never a solution in dimension 2 or higher, despite what suggest the numerical simulations. We also introduce a new rearrangement in the ball allowing to get a better candidate than the ball for optimality when Neumann boundary conditions are imposed. We also provide numerical illustrations of our results and of the optimal configurations.

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arXiv2015

Distributed shape derivative via averaged adjoint method and applications

The structure theorem of Hadamard-Zolésio 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 well-defined for shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithms. In this paper we describe several advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We identify a tensor representation of the distributed shape derivative, study its properties and show how it allows to recover the boundary expression directly. We use a novel Lagrangian approach, which is applicable to a large class of shape optimization problems, to compute the distributed shape derivative. We also apply the technique to retrieve the distributed shape derivative for electrical impedance tomography. Finally we explain how to adapt the level set method to the distributed shape derivative framework and present numerical results.

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arXiv2015

Shape optimization of an electric motor subject to nonlinear magnetostatics

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 pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

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arXiv2010

On a Bernoulli problem with geometric constraints

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 equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

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