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1. Small perturbations in the type of boundary conditions for an elliptic operator

Author: Bonnetier, Eric, Dapogny, Charles, Vogelius, Michael, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Department of Mathematics - Rutgers School of Arts and Sciences, Rutgers, The State University of New Jersey [New Brunswick] (RU), Rutgers University System (Rutgers)-Rutgers University System (Rutgers), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)
Subjects: Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Applied Mathematics, General Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract: In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this ``background'' situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a ``small'' subset $\omega_\varepsilon$ of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a ``small'' subset $\omega_\varepsilon $ of the Dirichlet boundary. The relevant quantity that measures the ``smallness'' of the subset $\omega_\varepsilon $ differs in the two cases: while it is the harmonic capacity of $\omega_\varepsilon $ in the former case, we introduce a notion of ``Neumann capacity'' to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general $\omega_\varepsilon $, under the sole assumption that it is ``small'' in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset $\omega_\varepsilon $ is a vanishing surfacic ball.
Published: 2022

2. Stability of optimal shapes and convergence of thresholding algorithms in linear and spectral optimal control problems: Supplementary material

Author: Chambolle, Antonin, Mazari-Fouquer, Idriss, Privat, Yannick, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Quantitative inequalities, 49M05, 49M41, 49N05, 49Q10, Thresholding scheme AMS classification: 49M05, Convergence analysis, Numerical algorithms in optimal control, PDE constrained optimisation, Shape optimisation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: We address the convergence of thresholding schemes for spectral optimisation problems and linear control problems.
Published: 2023

3. Stability of optimal shapes and convergence of thresholding algorithms in linear and spectral optimal control problems

Author: Chambolle, Antonin, Mazari-Fouquer, Idriss, Privat, Yannick, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Quantitative inequalities, 49M05, 49M41, 49N05, 49Q10, PDE constrained optimisation, Shape optimisation, Thresholding scheme, Mathematics - Analysis of PDEs, Convergence analysis, Optimization and Control (math.OC), Numerical algorithms in optimal control, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract: We prove the convergence of the fixed-point (also called thresholding) algorithm in three optimal control problems under large volume constraints. This algorithm was introduced by Céa, Gioan and Michel, and is of constant use in the simulation of $L^\infty$ − $L^1$ optimal control problems. In this paper we consider the optimisation of the Dirichlet energy, of Dirichlet eigenvalues and of certain non-energetic problems. Our proofs rely on new diagonalisation procedure for shape hessians in optimal control problems, which leads to local stability estimates.
Published: 2023
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4. Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate

Author: Idriss Mazari, Grégoire Nadin, Yannick Privat, Institute for Analysis and Scientific Computing [Wien], Vienna University of Technology (TU Wien), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), TOkamaks and NUmerical Simulations (TONUS), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), I Mazari was partially supported by the Austrian Science Fund (FWF) projects I4052-N32 and F65. I. Mazari, G. Nadin and Y. Privat were partially supported by the Project 'Analysis and simulation of optimal shapes - application to lifesciences' of the Paris City Hall., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Université Paris sciences et lettres (PSL), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut Universitaire de France (IUF), and Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.)
Subjects: Applied Mathematics, shape optimization, 010102 general mathematics, calculus of variations, bilinear optimal control, 01 natural sciences, 010101 applied mathematics, optimal control, 35Q92,49J99,34B15, Mathematics - Analysis of PDEs, Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], AMS: 35Q92,49J99,34B15, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, diffusive logistic equation, Mathematics - Optimization and Control, Analysis, Analysis of PDEs (math.AP)
Abstract: International audience; In this article, we give an in-depth analysis of the problem of optimising the total population size for a standard logistic-diffusive model. This optimisation problem stems from the study of spatial ecology and amounts to the following question: assuming a species evolves in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? {In recent years, many authors contributed to this topic.} We settle here the proof of two fundamental properties of optimisers: the bang-bang one which had so far only been proved under several strong assumptions, and the other one is the fragmentation of maximisers. Here, we prove the bang-bang property in all generality using a new spectral method. The technique introduced to demonstrate the bang-bang character of optimizers can be adapted and generalized to many optimization problems with other classes of bilinear optimal control problems where the state equation is semilinear and elliptic. We comment on it in a conclusion section.Regarding the geometry of maximisers, we exhibit a blow-up rate for the $BV$-norm of maximisers as the diffusivity gets smaller: if $\Omega$ is an orthotope and if $m_\mu$ is an optimal control, then $\Vert m_\mu\Vert_{BV}\gtrsim \sqrt{\mu}$. The proof of this results relies on a very fine energy argument.
Published: 2021

5. A deformation theorem for tensor flat chains and applications (complement to 'Tensor rectifiable G-flat chains')

Author: Goldman, Michael, Merlet, Benoît, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Reliable numerical approximations of dissipative systems (RAPSODI ), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), INRIA RAPSODI, ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract: In this note we extend White's deformation theorem for G-flat chains to the setting of G-flat tensor chains. As a corollary we obtain that the groups of normal tensor chains identify with some subgroups of normal chains. Moreover the corresponding natural group isomorphisms are isometric with respect to norms based on the coordinate slicing mass. The coordinate slicing mass of a k-chain is the integral of the mass of its 0-slices along all coordinate-planes of codimension k. The fact that this quantity is equivalent to the usual mass is not straightforward. To prove it, we use the deformation theorem and a partial extension of the restriction operator defined for all chains (not only of finite mass). On the contrary, except in some limit or degenerate cases, the whole groups of tensor chains and of finite mass tensor chains do not identify naturally with subgroups of chains.
Published: 2022

6. Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates

Author: Michael Goldman, Benoît Merlet, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Michael Goldman is partially supported by the ANR SHAPO. Benoît Merlet is partially supported by the INRIA team RAPSODI and the Labex CEMPI (ANR-11-LABX-0007-01)., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Paul Painlevé (LPP), and Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe
Subjects: Physics, Smoothness (probability theory), Regular polygon, Mathematics::General Topology, Rigidity (psychology), Space (mathematics), Omega, Theoretical Computer Science, Combinatorics, Mathematics::Logic, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Converse implication, Analysis of PDEs (math.AP)
Abstract: We study a family of non-convex functionals $\{\mathcal{E}\}$ on the space of measurable functions$u: \Omega_1\times\Omega_2 \subset \mathbb{R}^{n_1}\times\mathbb{R}^{n_2} \to \mathbb{R}$. These functionals vanish on the non-convex subset $S(\Omega_1\times\Omega_2)$ formed by functions of the form $u(x_1,x_2)=u_1(x_1)$ or $u(x_1,x_2)=u_2(x_2)$. We investigate under which conditions the converse implication "$\mathcal{E}(u) = 0 \Rightarrow u \in S(\Omega_1\times\Omega_2)$" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) $\mathcal{E}(u)$ controls in a strong sense the distance of $u$ to $S(\Omega_1\times\Omega_2)$.
Published: 2021

7. Vector-borne disease outbreak control via instant vector releases

Author: Almeida, Luis, Bellver Arnau, Jesús, Privat, yannick, Rebelo, Carlota, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Universidade de Lisboa = University of Lisbon (ULISBOA), Also the support of Programa Pessoa 2020 (44567YH-5460). C. Rebelo has been supported by FCT projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT at FC-Universidade de Lisboa.The first three authors were partially supported by the Project 'Analysis and simulation of optimal shapes - application to lifesciences' of the Paris City Hall., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020), European Project: 754362,MathInParis(2017), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPC), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPC)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPC), and Universidade de Lisboa (ULISBOA)
Subjects: Dengue, Vector-borne diseases, Quantitative Biology::Populations and Evolution, Sterile insect technique, [SDV.SPEE]Life Sciences [q-bio]/Santé publique et épidémiologie, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Wolbachia, Optimal epidemic vector control, Arboviruses
Abstract: This paper is devoted to the study of optimal vector release strategies to control vector-borne diseases, such as dengue, Zika, chikungunya and malaria. Two techniques are considered: the sterile insect one (SIT), which consists in releasing sterilized male vectors among wild ones in order to perturb their reproduction, and the Wolbachia one (presently used mainly for mosquitoes), which consists in releasing vectors that are infected with a bacterium limiting their vector capacity and then releasing them among wild vectors. In each case, the time dynamics of the vector population is modeled by a system of ordinary differential equations in which the releases are represented by linear combinations of Dirac measures with positive coefficients determining their intensity. We introduce optimal control problems that we solve numerically using ad-hoc algorithms, based on writing first-order optimality conditions characterizing the best combination of Dirac measures. We then discuss the results obtained, focusing in particular on the complexity and efficiency of optimal controls and comparing the strategies obtained.
Published: 2022

8. Uniform $C^{1,\alpha}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter

Author: Goldman, Michael, Merlet, Benoît, Pegon, Marc, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Institut Polytechnique de Paris (IP Paris), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Lille-Centre National de la Recherche Scientifique (CNRS), M. Goldman was partially supported by the ANR grant SHAPO. B. Merlet and M. Pegon are partially supported by the Labex CEMPI (ANR-11-LABX-0007-01)., ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Laboratoire Paul Painlevé - UMR 8524 (LPP)
Subjects: nonlocal perimeters, Mathematics - Analysis of PDEs, geometric variational problems, regularity, liquid drop model, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 28A75, 49Q05, 49Q10, 49Q20, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], nonlocal isoperimetric problems, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Mathematical Physics
Abstract: In this paper, we establish a $C^{1,\alpha}$-regularity theorem for almost-minimizers of a functional $\mathcal{F}_{\varepsilon,\gamma}=P-\gamma P_{\varepsilon}$, where $\gamma\in (0,1)$ and $P_{\varepsilon}$ is a nonlocal energy converging to the perimeter as $\varepsilon$ vanishes. Our theorem provides a criterion for $C^{1,\alpha}$-regularity at a point of the boundary, which is uniform as the parameter $\varepsilon$ goes to $0$. Building upon previous work by the last two authors, as an application of this theorem we obtain that volume-constrained global minimizers of $\mathcal{F}_{\varepsilon,\gamma}$ are balls for any $\varepsilon$ small enough. For small $\varepsilon$, this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable kernel $G$ with sufficiently fast decay at infinity.
Published: 2022

9. Shape and topology optimization for maximum probability domains in quantum chemistry

Author: Braida, Benoît, Dalphin, Jérémy, Dapogny, Charles, Frey, Pascal, Privat, Yannick, Laboratoire de chimie théorique (LCT), Institut de Chimie du CNRS (INC)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut des Sciences du Calcul et des Données (ISCD), Sorbonne Université (SU), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), TOkamaks and NUmerical Simulations (TONUS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), The work of Charles Dapogny and Yannick Privat is partly supported by the project ANR-18-CE40-0013 SHAPO, financed by the French Agence Nationale de la Recherche (ANR)., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Calcul des Variations, Géométrie, Image (CVGI), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)
Subjects: Computational Mathematics, Maximum Probability Domains, AMS classification: 35P20, 93B07, 58J51, 49K20, Applied Mathematics, Fixed Point Algorithm, Shape Optimization, Level Set Method, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Quantum Chemistry
Abstract: International audience; This article is devoted to the mathematical and numerical treatments of a shape optimization problem emanating from the desire to reconcile quantum theories of chemistry and classical heuristic models: we aim to identify Maximum Probability Domains (MPDs), that is, domains $\Omega$ of the 3d space where the probability $\mathbb{P}_\nu(\Omega)$ to find exactly $\nu$ among the $n$ constituent electrons of a given molecule is maximum. In the Hartree-Fock framework, the shape functional $\mathbb{P}_\nu(\Omega)$ arises as the integral over $\nu$ copies of $\Omega$ and $(n-\nu)$ copies of the complement $\mathbb{R}^3 \setminus \Omega$ of an analytic function defined over the space $\mathbb{R}^{3n}$ of all the spatial configurations of the $n$ electron system. Our first task is to explore the mathematical well-posedness of the shape optimization problem: under mild hypotheses, we prove that global maximizers of the probability functions $\mathbb{P}_\nu(\Omega)$ do exist as open subsets of $\mathbb{R}^3$; meanwhile, we identify the associated necessary first-order optimality condition. We then turn to the numerical calculation of MPDs, for which we resort to a level set based mesh evolution strategy: the latter allows for the robust tracking of complex evolutions of shapes, while leaving the room for accurate chemical computations, carried out on high-resolution meshes of the optimized shapes. The efficiency of this procedure is enhanced thanks to the addition of a fixed-point strategy inspired from the first-order optimality conditions resulting from our theoretical considerations. Several three-dimensional examples are presented and discussed to appraise the efficiency of our algorithms.
Published: 2022

10. Tensor rectifiable G-flat chains

Author: Goldman, Michael, Merlet, Benoît, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Lille-Centre National de la Recherche Scientifique (CNRS), INRIA RAPSODI, ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), and ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)
Subjects: Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract: A rigidity result for normal rectifiable $k$-chains in $\mathbb{R}^n$ with coefficients in an Abelian normed group is established. Given some decompositions $k=k_1+k_2$, $n=n_1+n_2$ and some rectifiable $k$-chain $A$ in $\mathbb{R}^n$, we consider the properties:(1) The tangent planes to $\mu_A$ split as $T_x\mu_A=L^1(x)\times L^2(x)$ for some $k_1$-plane $L^1(x)\subset\mathbb{R}^{n_1}$ and some $k_2$-plane $L^2(x)\subset\mathbb{R}^{n_2}$.(2) $A=A_{\vert\Sigma^1\times\Sigma^2}$ for some sets $\Sigma^1\subset\mathbb{R}^{n_1}$, $\Sigma^2\subset\mathbb{R}^{n_2}$ such that $\Sigma^1$ is $k_1$-rectifiable and $\Sigma^2$ is $k_2$-rectifiable (we say that $A$ is $(k_1,k_2)$-rectifiable).The main result is that for normal chains, (1) implies (2), the converse is immediate. In the proof we introduce the new groups of tensor flat chains (or $(k_1,k_2)$-chains) in $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ which generalize Fleming's $G$-flat chains. The other main tool is White's rectifiable slices theorem. We show that on the one hand any normal rectifiable chain satisfying~(1) identifies with a normal rectifiable $(k_1,k_2)$-chain and that on the other hand any normal rectifiable $(k_1,k_2)$-chain is $(k_1,k_2)$-rectifiable.
Published: 2022
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11. Optimal control strategies for bistable ODE equations: Application to mosquito population replacement

Author: Luis Almeida, Jesús Bellver Arnau, Yannick Privat, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), TOkamaks and NUmerical Simulations (TONUS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), This research was supported by the Project 'Analysis and simulation of optimal shapes - application to life science' of the Paris City Hall. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement N° 754362. The third author were partially supported by the ANR Projects 'SHAPe Optimization - SHAPO' and 'TRECOS'., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020), and European Project: 754362,MathInParis(2017)
Subjects: Control and Optimization, Applied Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ordinary differential systems, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Wolbachia, Optimal control, Epidemic vector control
Abstract: International audience; Vector-borne diseases, in particular arboviruses, represent a major threat to human health. In the fight against these viruses, the endosymbiotic bacterium Wolbachia has become in recent years a promising tool as it has been shown to prevent the transmission of some of these viruses between mosquitoes and humans. In this work, we investigate optimal population replacement strategies, which consists in replacing optimally the wild population by a population carrying the aforementioned bacterium, making less likely the appearance of outbreaks of these diseases. We consider a two species model taking into account both wild and Wolbachia infected mosquitoes. To control the system, we introduce a term representing an artificial introduction of Wolbachia-infected mosquitoes. Assuming a high birth rate, we reduce the model to a simpler one regarding the proportion of infected mosquitoes. We study strategies optimizing a convex combination either of cost and time or cost and final proportion of mosquitoes in the population. We fully analyze each of the introduced problem families, proving a time monotonicity property on the proportion of infected mosquitoes and using a reformulation of the problem based on a suitable change of variable. Our results are useful in considerably more general contexts which we present.
Published: 2022

12. CFD-based geometrical shape optimization of a packed-bed reactor combining multi-objective and adjoint system methods

Author: Alexis Courtais, François Lesage, Yannick Privat, Cyril Pelaingre, Abderrazak M. Latifi, Laboratoire Réactions et Génie des Procédés (LRGP), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre Européen de Prototypage et d'Outillage Rapide (CIRTES), Université Henri Poincaré - Nancy 1 (UHP), ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Multi-objective shape optimization, OpenFOAM environment, Adjoint system method, Multi-criteria decision-making, Additive manufacturing, Applied Mathematics, General Chemical Engineering, Packed-bed reactor, Multi-objective shape optimization Adjoint system method OpenFOAM environment Packed-bed reactor Additive manufacturing Multi-criteria decision-making, [SPI.GPROC]Engineering Sciences [physics]/Chemical and Process Engineering, General Chemistry, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Industrial and Manufacturing Engineering
Abstract: This paper presents the development of a geometric shape optimization methodology based on the so-called "Hadamard boundary variation" method for performing very general domain deformations, and the related concept of domain differentiation. The resulting method is used to determine the optimal configuration of a twodimensional packed-bed reactor that simultaneously optimizes its conversion rate and fluid energy dissipation, and where a homogeneous first-order reaction or a catalytic surface reaction takes place. The considered multi-objective optimization problem is subjected to four constraints: the process model constraints consisting of the Navier-Stokes, continuity and mass balance equations, an iso-volume and two manufacturing constraints. The approach to solve the problem is based on the linear scalarization method which converts the multi-objective problem into a single objective problem. The adjoint system method is used to compute the gradient of the performance indices and constraints. Since the indices are conflicting, the solution of the problem is a set of solutions, called Pareto front. Each optimal solution is evaluated using multiattribute utility theory (MAUT) to determine the best optimal shape of the reactor. Finally, the resulting shape is fabricated using a 3D printing technique and will be experimentally validated.
Published: 2022

13. Optimal shape of stellarators for magnetic confinement fusion

Author: Yannick Privat, Rémi Robin, Mario Sigalotti, Université de Strasbourg (UNISTRA), TOkamaks and NUmerical Simulations (TONUS), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), AEX StellaCage, ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: Biot and Savart operator, Plasma Physics, Applied Mathematics, General Mathematics, FOS: Physical sciences, Physics - Plasma Physics, Plasma Physics (physics.plasm-ph), Riemannian manifolds, AMS classification (MSC 2020): 49Q10, 49Q12, 78A25, 65T40, Shape optimization, Optimization and Control (math.OC), [PHYS.PHYS.PHYS-PLASM-PH]Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control
Abstract: International audience; We are interested in the design of stellarators, devices for the production of controlled nuclear fusion reactions alternative to tokamaks. The confinement of the plasma is entirely achieved by a helical magnetic field created by the complex arrangement of coils fed by high currents around a toroidal domain. Such coils describe a surface called "coil winding surface" (CWS). In this paper, we model the design of the CWS as a shape optimization problem, so that the cost functional reflects both optimal plasma confinement properties, through a least square discrepancy, and also manufacturability, thanks to geometrical terms involving the lateral surface or the curvature of the CWS. We completely analyze the resulting problem: on the one hand, we establish the existence of an optimal shape, prove the shape differentiability of the criterion, and provide the expression of the differential in a workable form. On the other hand, we propose a numerical method and perform simulations of optimal stellarator shapes. We discuss the efficiency of our approach with respect to the literature in this area.
Published: 2021

14. Compactness and structure of zero-states for unoriented Aviles-Giga functionals

Author: Goldman, Michael, Merlet, Benoît, Pegon, Marc, Serfaty, Sylvia, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Courant Institute of Mathematical Sciences [New York] (CIMS), New York University [New York] (NYU), NYU System (NYU)-NYU System (NYU), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, and Centre National de la Recherche Scientifique (CNRS)-Université de Lille
Subjects: entropies, vortices, FOS: Physical sciences, Aviles-Giga functional, Mathematical Physics (math-ph), zero-states, Mathematics - Analysis of PDEs, rigidity, 35B36, 35B65, 35J60, 35L65, 49Q20, 49S05, 74G65, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], compactness, Mathematical Physics, Analysis of PDEs (math.AP), disclinations
Abstract: International audience; Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional.We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call "trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy.Our methods provide alternative proofs in the classical Aviles-Giga context.
Published: 2021

15. The diagram (λ 1 , µ 1 )

Author: Ftouhi, Ilias, Henrot, Antoine, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Blaschke-Santaló diagrams, convex sets, complete systems of inequalities, Mathematics::Spectral Theory, [MATH]Mathematics [math], sharp spectral inequalities
Abstract: In this paper we are interested in the possible values taken by the pair (λ1(Ω), µ1(Ω) the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain Ω. We prove that, without any particular assumption on the class of open sets Ω, the two classical inequalities (the Faber-Krahn inequality and the Weinberger inequality) provide a complete system of inequalities. Then we consider the case of convex plane domains for which we give new inequalities for the product λ1µ1. We plot the so-called Blaschke-Santaló diagram and give some conjectures.
Published: 2021

16. The diagram (λ 1 , µ 1 )

Author: Ftouhi, Ilias, Henrot, Antoine, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Blaschke-Santaló diagrams, convex sets, complete systems of inequalities, Mathematics::Spectral Theory, [MATH]Mathematics [math], sharp spectral inequalities
Abstract: In this paper we are interested in the possible values taken by the pair (λ1(Ω), µ1(Ω) the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain Ω. We prove that, without any particular assumption on the class of open sets Ω, the two classical inequalities (the Faber-Krahn inequality and the Weinberger inequality) provide a complete system of inequalities. Then we consider the case of convex plane domains for which we give new inequalities for the product λ1µ1. We plot the so-called Blaschke-Santaló diagram and give some conjectures.
Published: 2021

17. On the quadratic random matching problem in two-dimensional domains

Author: Luigi Ambrosio, Michael Goldman, Dario Trevisan, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Ambrosio, Luigi, Goldman, Michael, and Trevisan, Dario
Subjects: Statistics and Probability, Optimization, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Analysis of PDEs, Matching problem, Settore MAT/05 - Analisi Matematica, Optimal Transport, Probability (math.PR), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistics, Probability and Uncertainty, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract: We investigate the average minimum cost of a bipartite matching, with respect to the squared Euclidean distance, between two samples of n i.i.d. random points on a bounded Lipschitz domain in the Euclidean plane, whose common law is absolutely continuous with strictly positive H{\"o}lder continuous density. We confirm in particular the validity of a conjecture by D. Benedetto and E. Caglioti stating that the asymptotic cost as n grows is given by the logarithm of n multiplied by an explicit constant times the volume of the domain. Our proof relies on a reduction to the optimal transport problem between the associated empirical measures and a Whitney-type decomposition of the domain, together with suitable upper and lower bounds for local and global contributions, both ultimately based on PDE tools. We further show how to extend our results to more general settings, including Riemannian manifolds, and also give an application to the asymptotic cost of the random quadratic bipartite travelling salesperson problem.
Published: 2021

18. TETRAHEDRAL REMESHING IN THE CONTEXT OF LARGE-SCALE NUMERICAL SIMULATION AND HIGH PERFORMANCE COMPUTING

Author: G. Balarac, F. Basile, P. Bénard, F. Bordeu, J.-B. Chapelier, L. Cirrottola, G. Caumon, C. Dapogny, P. Frey, A. Froehly, G. Ghigliotti, R. Laraufie, G. Lartigue, C. Legentil, R. Mercier, V. Moureau, C. Nardoni, S. Pertant, M. Zakari, Institut des Sciences du Calcul et des Données (ISCD), Sorbonne Université (SU), Laboratoire des Écoulements Géophysiques et Industriels [Grenoble] (LEGI), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Airbus Operation S.A.S., Airbus [France], DAAA, ONERA, Université Paris-Saclay [Châtillon], ONERA-Université Paris-Saclay, Complexe de recherche interprofessionnel en aérothermochimie (CORIA), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Safran Tech, ONERA, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), GeoRessources, Institut national des sciences de l'Univers (INSU - CNRS)-Centre de recherches sur la géologie des matières premières minérales et énergétiques (CREGU)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria), Direction générale déléguée à l'innovation (DGD-I), IRT SystemX, and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], ComputingMethodologies_COMPUTERGRAPHICS
Abstract: International audience; The purpose of this article is to discuss several modern aspects of remeshing, which is the task of modifying an ill-shaped tetrahedral mesh with bad size elements so that it features an appropriate density of high-quality elements. After a brief sketch of classical stakes about meshes and local mesh operations, we notably expose (i) how the local size of the elements of a mesh can be adapted to a user-defined prescription (guided, e.g., by an error estimate attached to a numerical simulation), (ii) how a mesh can be deformed to efficiently track the motion of the underlying domain, (iii) how to construct a mesh of an implicitlydefined domain, and (iv) how remeshing procedures can be conducted in a parallel fashion when large-scale applications are targeted. These ideas are illustrated with several applications involving high-performance computing. In particular, we show how mesh adaptation and parallel remeshing strategies make it possible to achieve a high accuracy in large-scale simulations of complex flows, and how the aforementioned methods for meshing implicitly defined surfaces allow to represent faithfully intricate geophysical interfaces, and to account for the dramatic evolutions of shapes featured by shape optimization processes.
Published: 2021

19. Existence and stability results for an isoperimetric problem with a non-local interaction of Wasserstein type

Author: Jules Candau-Tilh, Michael Goldman, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Computational Mathematics, Control and Optimization, Mathematics - Analysis of PDEs, Control and Systems Engineering, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics::Metric Geometry, Analysis of PDEs (math.AP)
Abstract: The aim of this paper is to prove the existence of minimizers for a variational problem involving the minimization under volume constraint of the sum of the perimeter and a non-local energy of Wasserstein type. This extends previous partial results to the full range of parameters. We also show that in the regime where the perimeter is dominant, the energy is uniquely minimized by balls.
Published: 2021

20. An introduction to the topological derivative

Author: Samuel Amstutz, EA2151 Laboratoire de Mathématiques d'Avignon (LMA), Avignon Université (AU), The author benefited from the support of the chair 'Modeling advanced polymers for innovative material solutions' led by the Ecole polytechnique and the Fondation de l'Ecole polytechnique and sponsored by Arkema., and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Field (physics), Relation (database), 010102 general mathematics, Topology optimization, General Engineering, 01 natural sciences, Computer Science Applications, 010101 applied mathematics, topological derivative, Computational Theory and Mathematics, Simple (abstract algebra), shape optimization, Applied mathematics, Shape optimization, Topological derivative, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Software, topology optimization, Mathematics
Abstract: PurposeThis paper provides a self-contained introduction to the mathematical aspects of the topological derivative.Design/methodology/approachFull justifications are given on simple model problems following a modern approach based on the averaged adjoint state technique. Extensions are discussed in relation with the literature on the field.FindingsClosed expressions of topological derivatives are obtained and commented.Originality/valueSeveral cases are covered in a unified and didactic presentation. Some elements of proof are novel.
Published: 2021

21. A consistent approximation of the total perimeter functional for topology optimization algorithms

Author: Samuel Amstutz, Charles Dapogny, Alex Ferrer, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), This work is partly supported by the project ANR-18-CE40-0013 SHAPO, financed by the French Agence Nationale de la Recherche (ANR)., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Universitat Politècnica de Catalunya [Barcelona] (UPC), Universitat Politècnica de Catalunya. Departament de Física, and Universitat Politècnica de Catalunya. ANiComp - Anàlisi numèrica i computació científica
Subjects: Control and Optimization, Computer Science::Information Retrieval, Asymptotic analysis, Optimització matemàtica, Mathematical optimization, calculus of variations, Matemàtiques i estadística [Àrees temàtiques de la UPC], Differential equations, Partial, Equacions diferencials parcials, Topology, Partial differential equations, Topologia, Shape and topology optimization, Computational Mathematics, Perimeter function, asymptotic analysis, scientific computing, Control and Systems Engineering, perimeter function, partial differential equations, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH]Mathematics [math], Scientific computing, Calculus of variations
Abstract: This article revolves around the total perimeter functional, one particular version of the perimeter of a shape O contained in a fixed computational domain D measuring the total area of its boundary ¿O, as opposed to its relative perimeter, which only takes into account the regions of ¿O strictly inside D. We construct and analyze approximate versions of the total perimeter which make sense for general “density functions” u, as generalized characteristic functions of shapes. Their use in the context of density-based topology optimization is particularly convenient insofar as they do not involve the gradient of the optimized function u. Two different constructions are proposed: while the first one involves the convolution of the function u with a smooth mollifier, the second one is based on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The “consistency” of these approximations with the original notion of total perimeter is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to 0, when the considered density function u is the characteristic function of a “regular enough” shape O ¿ D. Then, we focus on the G-convergence of the second type of approximate total perimeter functional, that based on elliptic regularization. Several numerical examples are eventually presented in two and three space dimensions to validate our theoretical findings and demonstrate the efficiency of the proposed functionals in the context of structural optimization. This work is partly supported by the project ANR-18-CE40-0013 SHAPO, financed by the French Agence Nationale de la Recherche (ANR). S.A. benefitted from the support of the chair "Modeling advanced polymers for innovative material solutions" led by the École Polytechnique and the Fondation de l'École Polytechnique, and sponsored by Arkema.
Published: 2021

22. Optimal release strategies for mosquito population replacement

Author: Almeida, Luis, Bellver Arnau, Jesús, Privat, Yannick, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), TOkamaks and NUmerical Simulations (TONUS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), This research was supported by the Project 'Analysis and simulation of optimal shapes - application to life science' of the Paris City Hall. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement N° 754362. The third author were partially supported by the ANR Projects 'SHAPe Optimization - SHAPO' and 'TRECOS'., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020), European Project: 754362,MathInParis(2017), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)
Subjects: parasitic diseases, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ordinary differential systems, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Wolbachia, Optimal control, Epidemic vector control
Abstract: Vector-borne diseases, in particular arboviruses, represent a major threat to human health. In the fight against these viruses, the endosymbiotic bacterium Wolbachia has become in recent years a promising tool as it has been shown to prevent the transmission of some of these viruses between mosquitoes and humans. In this work, we investigate optimal population replacement strategies, which consists in replacing optimally the wild population by a population carrying the aforementioned bacterium, making less likely the appearance of outbreaks of these diseases. We consider a two species model taking into account both wild and Wolbachia infected mosquitoes. To control the system, we introduce a term representing an artificial introduction of Wolbachia-infected mosquitoes. Assuming a high birth rate, we reduce the model to a simpler one regarding the proportion of infected mosquitoes. We study strategies optimizing a convex combination either of cost and time or cost and final proportion of mosquitoes in the population. We fully analyze each of the introduced problem families, proving a time monotonicity property on the proportion of infected mosquitoes and using a reformulation of the problem based on a suitable change of variable.
Published: 2021

23. Longest minimal length partitions

Author: Bogosel, Beniamin, Oudet, Edouard, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Optimization and Control (math.OC), Applied Mathematics, TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY, FOS: Mathematics, Mathematics::Metric Geometry, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Computer Science::Computational Geometry, Mathematics - Optimization and Control, MathematicsofComputing_DISCRETEMATHEMATICS
Abstract: International audience; This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimizations steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase the minimal perimeter. The initialization of the optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A new algorithm is proposed to identify such diagrams, by computing the gradients of areas and perimeters for the Voronoi cells with respect to the Voronoi points.
Published: 2021

24. BODY OF CONSTANT WIDTH WITH MINIMAL AREA IN A GIVEN ANNULUS

Author: Antoine Henrot, Ilaria Lucardesi, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Work (thermodynamics), General Mathematics, constant width, inradius constraint, 52A10, 49Q10, 49Q12, 52A38, Characterization (mathematics), 01 natural sciences, Domain (mathematical analysis), Incircle and excircles of a triangle, Combinatorics, Planar, Mathematics - Metric Geometry, 2010 MSC: 52A10, 49Q10, 49Q12, 52A38, 0103 physical sciences, FOS: Mathematics, Mathematics::Metric Geometry, Shape optimization problem, 0101 mathematics, Reuleaux polygons, Mathematics - Optimization and Control, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics, area minimization, 010102 general mathematics, Annulus (mathematics), Metric Geometry (math.MG), Optimization and Control (math.OC), 010307 mathematical physics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Constant (mathematics)
Abstract: In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius. In the literature, the problem is ascribed to Bonnesen, who proposed it in \cite{BF}. In the present work, we give a complete answer to the problem, providing an explicit characterization of optimal sets for every choice of width and inradius. These optimal sets are particular Reuleaux polygons., Comment: to appear in Journal Ecole Polytechnique
Published: 2021

25. Partial regularity for the optimal $p$-compliance problem with length penalization

Author: Bohdan Bulanyi, Université de Paris - UFR Mathématiques [Sciences] (UP - UFR Mathématiques), Université de Paris (UP), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), and UCL - SST/IRMP - Institut de recherche en mathématique et physique
Subjects: Dimension (graph theory), Boundary (topology), Type (model theory), 01 natural sciences, compliance, Combinatorics, Mathematics - Analysis of PDEs, 49Q20, 35J92, FOS: Mathematics, Point (geometry), 0101 mathematics, [MATH]Mathematics [math], Mathematics - Optimization and Control, Mathematics, Applied Mathematics, 010102 general mathematics, 49Q20, Compliance problem, Codimension, minimizers, regularity theory, 010101 applied mathematics, Optimization and Control (math.OC), shape optimization, p-Laplace, Closed loop, Analysis, Analysis of PDEs (math.AP)
Abstract: We establish a partial $C^{1,\alpha}$ regularity result for minimizers of the optimal $p$-compliance problem with length penalization in any spatial dimension $N\geq 2$, extending some of the results obtained in [Chambolle-Lamboley-Lemenant-Stepanov 17], [Bulanyi-Lemenant 20]. The key feature is that the $C^{1,\alpha}$ regularity of minimizers for some free boundary type problem is investigated with a free boundary set of codimension $N-1$. We prove that every optimal set cannot contain closed loops, and it is $C^{1,\alpha}$ regular at $\mathcal{H}^{1}$-a.e. point for every $p\in (N-1,+\infty)$., Comment: 42 pages, 2 figures. arXiv admin note: text overlap with arXiv:1911.09240
Published: 2021
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26. A comparison between Neumann and Steklov eigenvalues

Author: Henrot, Antoine, Michetti, Marco, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Mathematics - Analysis of PDEs, FOS: Mathematics, Statistical and Nonlinear Physics, Geometry and Topology, [MATH]Mathematics [math], Mathematics::Spectral Theory, Mathematical Physics, Analysis of PDEs (math.AP)
Abstract: This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue $|\Omega| \mu_1(\Omega)$ for a Lipschitz open set $\Omega$ in the plane, and the normalized first (non-trivial) Steklov eigenvalue $P(\Omega) \sigma_1(\Omega)$. More precisely, we study the ratio $F(\Omega):=|\Omega| \mu_1(\Omega)/P(\Omega) \sigma_1(\Omega)$. We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets $\Omega$. Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. The paper finishes with the plot of the corresponding Blaschke-Santal\'o diagrams $(x,y)=\left(|\Omega| \mu_1(\Omega), P(\Omega) \sigma_1(\Omega) \right)$., Comment: In this version Lemma 3.2 and Lemma 3.5 (this Lemma was Lemma 3.4 in the previous version) are extended and we state in a correct way Conjecture 1
Published: 2021
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27. Optimization of spatial control strategies for population replacement, application to Wolbachia

Author: Yannick Privat, Romane Hélie, Michel Duprez, Nicolas Vauchelet, Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie (ICube), Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Réseau nanophotonique et optique, Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Matériaux et nanosciences d'Alsace, Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Université de Strasbourg (UNISTRA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université de Strasbourg (UNISTRA), Computational Anatomy and Simulation for Medicine (MIMESIS), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), TOkamaks and NUmerical Simulations (TONUS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, Project 'Analysis and simulation of optimal shapes - application to lifesciences' of the Paris City Hall, ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Matériaux et nanosciences d'Alsace (FMNGE), Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Université de Strasbourg (UNISTRA)-Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et nanosciences d'Alsace, Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Université de Strasbourg (UNISTRA)-Réseau nanophotonique et optique, Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Université de Strasbourg (UNISTRA), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Les Hôpitaux Universitaires de Strasbourg (HUS)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et Nanosciences Grand-Est (MNGE), Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Réseau nanophotonique et optique, Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Privat, Yannick, APPEL À PROJETS GÉNÉRIQUE 2018 - Optimisation de forme - - SHAPO2018 - ANR-18-CE40-0013 - AAPG2018 - VALID, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie (ICube), Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Université de Strasbourg (UNISTRA)-Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), and Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Les Hôpitaux Universitaires de Strasbourg (HUS)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et Nanosciences Grand-Est (MNGE)
Subjects: Mathematical optimization, Control and Optimization, Control (management), Population, Mosquito population, Population Replacement, 92D25, 49K15, 65K10, 01 natural sciences, 03 medical and health sciences, optimal control, Mathematics - Analysis of PDEs, parasitic diseases, Spatial strategy, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, education, Mathematics - Optimization and Control, 030304 developmental biology, Mathematics, 0303 health sciences, education.field_of_study, biology, fungi, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Optimal control, Fecundity, biology.organism_classification, 010101 applied mathematics, Computational Mathematics, Control and Systems Engineering, Optimization and Control (math.OC), Wolbachia, reaction-diffusion equation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis of PDEs (math.AP), second order optimality conditions
Abstract: In this article, we are interested in the analysis and simulation of solutions to an optimal control problem motivated by population dynamics issues. In order to control the spread of mosquito-borne arboviruses, the population replacement technique consists in releasing into the environment mosquitoes infected with the Wolbachia bacterium, which greatly reduces the transmission of the virus to the humans. Spatial releases are then sought in such a way that the infected mosquito population invades the uninfected mosquito population. Assuming very high mosquito fecundity rates, we first introduce an asymptotic model on the proportion of infected mosquitoes and then an optimal control problem to determine the best spatial strategy to achieve these releases. We then analyze this problem, including the optimality of natural candidates and carry out first numerical simulations in one dimension of space to illustrate the relevance of our approach., Comment: ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, In press
Published: 2020

28. Body-fitted topology optimization of 2D and 3D fluid-to-fluid heat exchangers

Author: Pierre Jolivet, Grégoire Allaire, Florian Feppon, Charles Dapogny, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Calcul des Variations, Géométrie, Image (CVGI), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....]), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées, Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Centre National de la Recherche Scientifique (CNRS), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)
Subjects: Convective heat transfer, Discretization, geometric constraints, Computer science, Computational Mechanics, General Physics and Astronomy, 010103 numerical & computational mathematics, 01 natural sciences, Shape and topology optimization, Heat exchanger, heat exchangers, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Shape optimization, 0101 mathematics, Mechanical Engineering, Numerical analysis, Topology optimization, Mechanics, non-mixing constraint, Computer Science Applications, 010101 applied mathematics, Test case, convective heat transfer, Mechanics of Materials, Solid body, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; We present a topology optimization approach for the design of fluid-to-fluid heat exchangers which rests on an explicit meshed discretization of the phases at stake, at every iteration of the optimization process. The considered physical situations involve a weak coupling between the Navier-Stokes equations for the velocity and the pressure in the fluid, and the convection-diffusion equation for the temperature field. The proposed framework combines several recent techniques from the field of shape and topology optimization, and notably a level-set based mesh evolution algorithm for tracking shapes and their deformations , an efficient optimization algorithm for constrained shape optimization problems, and a numerical method to handle a wide variety of geometric constraints such as thickness constraints and non-penetration constraints. Our strategy is applied to the optimization of various types of heat exchangers. At first, we consider a simplified 2D cross-flow model where the optimized boundary is the section of the hot fluid phase flowing in the transverse direction, which is naturally composed of multiple holes. A minimum thickness constraint is imposed on the cross-section so as to account for manufacturing and maximum pressure drop constraints. In a second part, we optimize the design of 2D and 3D heat exchangers composed of two types of fluid channels (hot and cold), which are separated by a solid body. A non-mixing constraint between the fluid components containing the hot and cold phases is enforced by prescribing a minimum distance between them. Numerical results are presented on a variety of test cases, demonstrating the efficiency of our approach in generating new, realistic, and unconventional heat exchanger designs.
Published: 2020

29. Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Author: Edouard Oudet, Chiu-Yen Kao, Braxton Osting, Calcul des Variations, Géométrie, Image (CVGI), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), University of California [Los Angeles] (UCLA), University of California, University of California (UC), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Mathematics - Differential Geometry, Control and Optimization, Boundary (topology), 01 natural sciences, eigenvalue optimization, Mathematics - Spectral Theory, Steklov eigenvalue, Genus (mathematics), free boundary minimal surface, FOS: Mathematics, Ball (mathematics), 0101 mathematics, [MATH]Mathematics [math], Spectral Theory (math.SP), Mathematics - Optimization and Control, Eigenvalues and eigenvectors, Mathematics, Minimal surface, 010102 general mathematics, Degenerate energy levels, Mathematical analysis, Eigenfunction, Mathematics::Spectral Theory, 16. Peace & justice, Surface (topology), 010101 applied mathematics, Computational Mathematics, Differential Geometry (math.DG), Optimization and Control (math.OC), Control and Systems Engineering, 35P05, 35P15, 49Q05, 65N25
Abstract: Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, $\Sigma$, with genus $\gamma$ and $b$ boundary components, we maximize $\sigma_j(\Sigma,g) \ L(\partial \Sigma, g)$ over a class of smooth metrics, $g$, where $\sigma_j(\Sigma,g)$ is the $j$-th nonzero Steklov eigenvalue and $L(\partial \Sigma, g)$ is the length of $\partial \Sigma$. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus $\gamma =0$ and $b=2,\dots, 9, 12, 15, 20$ boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with $b=2$ boundary components and for the third and fifth eigenvalues with $b=3$ boundary components., Comment: 29 pages, 18 figures
Published: 2020

30. Propagation for KPP bulk-surface systems in a general cylindrical domain

Author: Thomas Giletti, Andrea Tellini, Beniamin Bogosel, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Universidad Politécnica de Madrid (UPM), B.B. was partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR). T.G. was partially supported by the project ANR-14-CE25-0013 NONLOCAL financed by the French Agence National de la Recherche (ANR). A.T. was partially supported by: the European Research Council, under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC GrantAgreement number 321186 - ReaDi 'Reaction-Diffusion Equations, Propagation and Modelling' led by Henri Berestycki, the Institute of Complex Systems of Paris ˆIle-de-France, under DIM 2014 'Diffusion heterogeneities in different spatial dimensions and applications to ecology and medicine', the Ministry of Science, Innovation and Universities of the Government of Spain through projects PGC2018-097104-B-100 and contract Juan de la CiervaIncorporacion IJCI-2015-25084., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), ANR-14-CE25-0013,NONLOCAL,Phénomènes de propagation et équations non locales(2014), and European Project: ERC FP/2007‐2013
Subjects: Surface (mathematics), Applied Mathematics, 010102 general mathematics, Mathematical analysis, Scalar (physics), 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, Section (fiber bundle), Elliptic operator, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Shape optimization, 0101 mathematics, Diffusion (business), Analysis, Eigenvalues and eigenvectors, Mathematics, Analysis of PDEs (math.AP)
Abstract: International audience; In this paper, we investigate propagation phenomena for KPP bulk-surface systems in a cylindrical domain with general section and heterogeneous coefficients. As for the scalar KPP equation, we show that the asymptotic spreading speed of solutions can be computed in terms of the principal eigenvalues of a family of self-adjoint elliptic operators. Using this characterization, we analyze the dependence of the spreading speed on various parameters, including diffusion rates and the size and shape of the section of the domain. In particular, we provide new theoretical results on several asymptotic regimes like small and high diffusion rates and sections with small and large sizes. These results generalize earlier ones which were available in the radial homogeneous case. Finally, we numerically investigate the issue of shape optimization of the spreading speed. By computing its shape derivative, we observe, in the case of homogeneous coefficients, that a disk either maximizes or minimizes the speed, depending on the parameters of the problem, both with or without constraints. We also show the results of numerical shape optimization with non homogeneous coefficients, when the disk is no longer an optimizer.
Published: 2020

31. Numerical computation of the cut locus via a variational approximation of the distance function

Author: François Générau, Edouard Oudet, Bozhidar Velichkov, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Dipartimento di Matematica [Pisa], University of Pisa - Università di Pisa, and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: cut locus, Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), relaxation, manifold, Calculus of variation, FOS: Mathematics, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, 49J45, 49Q10, 35R35, 49M05, 35J25, [MATH]Mathematics [math], Analysis of PDEs (math.AP)
Abstract: We propose a new method for the numerical computation of the cut locus of a compact submanifold of ℝ3 without boundary. This method is based on a convex variational problem with conic constraints, with proven convergence. We illustrate the versatility of our approach by the approximation of Voronoi cells on embedded surfaces of ℝ3.
Published: 2020

32. The missing (A, D, r) diagram

Author: Delyon, Alexandre, Henrot, Antoine, Privat, Yannick, Institut Élie Cartan de Lorraine (IECL), Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), All three authors were partially supported by the ANR Project 'SHAPe Optimization - SHAPO'. The third author was partially supported by the Project 'Analysis and simulation of optimal shapes - application to lifesciences' of the Paris City Hall., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est
Subjects: convex geometry, 49Q10, 52A40, 28A75, 49K15, inradius, Metric Geometry (math.MG), 2-cap bodies, AMS classification: 49Q10, 52A40, 28A75, 49K15, Mathematics - Metric Geometry, Optimization and Control (math.OC), shape optimization, FOS: Mathematics, Mathematics::Metric Geometry, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], diameter, Blaschke-Santaló diagram, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics - Optimization and Control
Abstract: International audience; In this paper we are interested in "optimal" universal geometric inequalities involving the area, diameter and inradius of convex bodies. The term "optimal" is to be understood in the following sense: we tackle the issue of minimizing/maximizing the Lebesgue measure of a convex body among all convex sets of given diameter and inradius. The minimization problem in the two- dimensional case has been solved in a previous work, by M. Hernandez-Cifre and G. Salinas. In this article, we provide a generalization to the n-dimensional case based on a different approach, as well as the complete solving of the maximization problem in the two-dimensional case. This allows us to completely determine the so-called 2-dimensional Blaschke-Santaló diagram for planar convex bodies with respect to the three magnitudes area, diameter and inradius in euclidean spaces, denoted (A, D, r). Such a diagram is used to determine the range of possible values of the area of convex sets depending on their diameter and inradius. Although this question of convex geometry appears quite elementary, it had not been answered until now. This is likely related to the fact that the diagram description uses unexpected particular convex sets, such as a kind of smoothed nonagon inscribed in an equilateral triangle.
Published: 2020

33. Lipschitz continuity of the eigenfunctions on optimal sets for functionals with variable coefficients

Author: Baptiste Trey, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Calcul des Variations, Géométrie, Image (CVGI), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Trey, Baptiste, Laboratoire de Géologie de Lyon - Terre, Planètes, Environnement [Lyon] (LGL-TPE), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Géologie de Lyon - Terre, Planètes, Environnement (LGL-TPE), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut national des sciences de l'Univers (INSU - CNRS)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Control and Optimization, Spectral optimization problem, the two-phase problem, Open set, Hölder condition, [MATH] Mathematics [math], 01 natural sciences, Combinatorics, symbols.namesake, Mathematics - Analysis of PDEs, 0103 physical sciences, FOS: Mathematics, Free boundary problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, [MATH]Mathematics [math], Eigenvalues and eigenvectors, Mathematics, 010102 general mathematics, Mathematics::Spectral Theory, Eigenfunction, Lipschitz continuity, free boundary problem, Computational Mathematics, almost-minimizer, Control and Systems Engineering, Dirichlet boundary condition, Bounded function, symbols, 010307 mathematical physics, Analysis of PDEs (math.AP)
Abstract: This paper is dedicated to the spectral optimization problemmin{λ1(Ω)+⋯+λk(Ω)+Λ|Ω| : Ω⊂Dquasi-open}whereD⊂ ℝdis a bounded open set and 0 <λ1(Ω) ≤⋯ ≤λk(Ω) are the firstkeigenvalues onΩof an operator in divergence form with Dirichlet boundary condition and Hölder continuous coefficients. We prove that the firstkeigenfunctions on an optimal set for this problem are locally Lipschtiz continuous inDand, as a consequence, that the optimal sets are open sets. We also prove the Lipschitz continuity of vector-valued functions that are almost-minimizers of a two-phase functional with variable coefficients.
Published: 2020

34. The double queen Dido's problem

Author: Shigeru Sakaguchi, Antoine Henrot, Lorenzo Cavallina, Research Center for Pure and Applied Mathematics, GSIS, Tohoku University (GSIS Mathematics), Tohoku University [Sendai], Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), This research was partially supported by the Grants-in-Aid for Scientific Research (B) No.18H01126 and JSPS Fellows No.18J11430 of Japan Society for the Promotion of Science., and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Mathematics - Differential Geometry, Dido's problem, Dimension (graph theory), AMS subject classifications : 49Q20, two-phase, 01 natural sciences, Omega, constrained minimization problem, Combinatorics, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics, 010102 general mathematics, 49Q20, Constraint (information theory), Discontinuity (linguistics), Differential geometry, Hyperplane, Differential Geometry (math.DG), Isoperimetric problem, Piecewise, weighted manifold, 010307 mathematical physics, Geometry and Topology, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Isoperimetric inequality
Abstract: This paper deals with a variation of the classical isoperimetric problem in dimension $N\ge 2$ for a two-phase piecewise constant density whose discontinuity interface is a given hyperplane. We introduce a weighted perimeter functional with three different weights, one for the hyperplane and one for each of the two open half-spaces in which $\mathbb{R}^N$ gets partitioned. We then consider the problem of characterizing the sets $\Omega$ that minimize this weighted perimeter functional under the additional constraint that the volumes of the portions of $\Omega$ in the two half-spaces are given. It is shown that the problem admits two kinds of minimizers, which will be called type I and type II, respectively. These minimizers are made of the union of two spherical domes whose angle of incidence satisfies some kind of \textquotedblleft Snell's law\textquotedblright. Finally, we provide a complete classification of the minimizers depending on the various parameters of the problem., Comment: 20 pages, 4 figures
Published: 2020

35. Shape optimization of a weighted two-phase Dirichlet eigenvalue

Author: Idriss Mazari, Yannick Privat, Grégoire Nadin, Sorbonne Université, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), TOkamaks and NUmerical Simulations (TONUS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), The authors were partially supported by the Project ``Analysis and simulation of optimal shapes - application to lifesciences' of the Paris City Hall., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: drifted Laplacian, 0209 industrial biotechnology, Population, 35K57, 35P99, 49J20, 49J50, 49Q10, homogenization, spectral optimization, 02 engineering and technology, shape derivatives, 01 natural sciences, Domain (mathematical analysis), Combinatorics, 020901 industrial engineering & automation, Mathematics (miscellaneous), Dirichlet eigenvalue, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Nabla symbol, Ball (mathematics), 0101 mathematics, education, Mathematics - Optimization and Control, Mathematics, Pointwise, education.field_of_study, Mechanical Engineering, 010102 general mathematics, Order (ring theory), bang-bang functions, reaction-diffusion equations, Optimization and Control (math.OC), Bounded function, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis, Analysis of PDEs (math.AP)
Abstract: This article is concerned with a spectral optimization problem: in a smooth bounded domain $${\Omega }$$ , for a bounded function m and a nonnegative parameter $$\alpha $$ , consider the first eigenvalue $$\lambda _\alpha (m)$$ of the operator $${\mathcal {L}}_m$$ given by $${\mathcal {L}}_m(u)= -{\text {div}} \left( 1+\alpha m)\nabla u\right) -mu$$ . Assuming uniform pointwise and integral bounds on m, we investigate the issue of minimizing $$\lambda _\alpha (m)$$ with respect to m. Such a problem is related to the so-called “two phase extremal eigenvalue problem” and arises naturally, for instance in population dynamics where it is related to the survival ability of a species in a domain. We prove that unless the domain is a ball, this problem has no “regular” solution. We then provide a careful analysis in the case of a ball by: (1) characterizing the solution among all radially symmetric resources distributions, with the help of a new method involving a homogenized version of the problem; (2) proving in a more general setting a stability result for the centered distribution of resources with the help of a monotonicity principle for second order shape derivatives which significantly simplifies the analysis.
Published: 2020

36. Quantitative linearization results for the Monge-Ampère equation

Author: Michael Goldman, Martin Huesmann, Felix Otto, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Rheinische Friedrich-Wilhelms-Universität Bonn, Max Planck Institute for Mathematics in the Sciences (MPI-MiS), Max-Planck-Gesellschaft, ANR-18-CE40-0013,SHAPO,SHAPE OPTIMIZATION(2018), University of Münster, ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Goldman, Michael, and APPEL À PROJETS GÉNÉRIQUE 2018 - Optimisation de forme - - SHAPO2018 - ANR-18-CE40-0013 - AAPG2018 - VALID
Subjects: [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]
Abstract: This paper replaces and supersedes the first (deterministic) part of the preprint 'A large-scale regularity theory for the Monge-Ampère equation with rough data and application to the optimal matching problem' (which will therefore not be submitted).The first main difference is that we are able here to directly relate the displacements to the flux given by the Poisson equation and thus confirming the linearization ansatz by Caracciolo and al.The second main difference is in the statement (which is more general) and the proofs of the harmonic approximation result.; International audience; This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale regularity theory and prove that if a measure µ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.
Published: 2020

37. Maximization of the Steklov eigenvalues with a diameter constraint

Author: Florent Nacry, Abdelkader Al Sayed, Antoine Henrot, Beniamin Bogosel, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), LAboratoire de Mathématiques et PhySique (LAMPS), Université de Perpignan Via Domitia (UPVD), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Steklov eigenvalues, Mathematics - Spectral Theory, Shape optimization, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Optimization and Control, spectral geometry, Spectral Theory (math.SP), Eigenvalues and eigenvectors, Mathematics, Applied Mathematics, Mathematical analysis, Spectral geometry, Maximization, Mathematics::Spectral Theory, diameter constraint, Constraint (information theory), Computational Mathematics, Optimization and Control (math.OC), shape derivative, 2010 MSC : 35P15, 49Q10, 35J05, Convex domain, Analysis, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP], 49Q10, 35P15, 35J05
Abstract: International audience; In this paper, we address the problem of maximizing the Steklov eigenvalues with a diameter constraint. We provide an estimate of the Steklov eigenvalues for a convex domain in terms of its diameter and volume and we show the existence of an optimal convex domain. We establish that balls are never maximizers, even for the first non-trivial eigenvalue that contrasts with the case of volume or perimeter constraints. Under an additional regularity assumption, we are able to prove that the Steklov eigenvalue is multiple for the optimal domain. We illustrate our theoretical results by giving some optimal domains in the plane thanks to a numerical algorithm.
Published: 2020
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38. Null space gradient flows for constrained optimization with applications to shape optimization

Author: Florian Feppon, Grégoire Allaire, Charles Dapogny, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Calcul des Variations, Géométrie, Image (CVGI), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes [2020-....] (UGA [2020-....]), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Safran Tech, and Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)
Subjects: Mathematical optimization, Control and Optimization, Optimization problem, Nonlinear constrained optimization, Ode, Constrained optimization, Boundary (topology), 010103 numerical & computational mathematics, 01 natural sciences, Projection (linear algebra), Slack variable, null space method, 010101 applied mathematics, Computational Mathematics, shape and topology optimization, Control and Systems Engineering, gradient flows, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Shape optimization, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Quadratic programming, 0101 mathematics, [MATH]Mathematics [math], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: The purpose of this article is to introduce a gradient-flow algorithm for solving generic equality or inequality constrained optimization problems, which is suited for shape optimization applications. We rely on a variant of the Ordinary Differential Equation (ODE) approach proposed by Yamashita for equality constrained problems: the search direction is a combination of a null space step and a range space step, which are aimed to reduce the value of the minimized objective function and the violation of the constraints, respectively. Our first contribution is to propose an extension of this ODE approach to optimization problems featuring both equality and inequality constraints. In the literature, a common practice consists in reducing inequality constraints to equality constraints by the introduction of additional slack variables. Here, we rather solve their local combinatorial character by computing the projection of the gradient of the objective function onto the cone of feasible directions. This is achieved by solving a dual quadratic programming subproblem whose size equals the number of active or violated constraints, and which allows to identify the inequality constraints which should remain tangent to the optimization trajectory. Our second contribution is a formulation of our gradient flow in the context of-infinite-dimensional-Hilbert space settings. This allows to extend the method to quite general optimization sets equipped with a suitable manifold structure, and notably to sets of shapes as it occurs in shape optimization with the framework of Hadamard's boundary variation method. The cornerstone of this latter setting is the classical operation of extension and regularization of shape derivatives. Some numerical comparisons on simple academic examples are performed to illustrate the behavior of our algorithm. Its numerical efficiency and ease of implementation are finally demonstrated on more realistic shape optimization problems.
Published: 2020

39. Stability in shape optimization with second variation

Author: Dambrine, Marc, Lamboley, Jimmy, Dambrine-J, M, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), and ANR-18-CE40-0013,SHAPO,SHAPE OPTIMIZATION(2018)
Subjects: Hessian matrix, second order sensitivity, isoperimetric inequalities, 01 natural sciences, Perimeter, symbols.namesake, Dirichlet eigenvalue, FOS: Mathematics, Applied mathematics, Shape optimization, 0101 mathematics, Mathematics - Optimization and Control, Torsional rigidity, Mathematics, Applied Mathematics, stability in shape optimization, 010102 general mathematics, A domain, 49K20, 49Q10, 010101 applied mathematics, Optimization and Control (math.OC), Norm (mathematics), shape optimization, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Laplace operator, Analysis
Abstract: International audience; We are interested in the question of stability in the field of shape optimization, with focus on the strategy using second order shape derivative. More precisely, we identify structural hypotheses on the hessian of the considered shape function, so that critical stable domains (i.e. such that the first order derivative vanishes and the second order one is positive) are local minima for smooth perturbations; as we are in an infinite dimensional framework, and that in most applications there is a norm-discrepancy phenomenon, this type of result require a lot of work. We show that these hypotheses are satisfied by classical functionals, involving the perimeter, the Dirichlet energy or the first Laplace-Dirichlet eigenvalue. We also explain how we can easily deal with constraints and/or invariance of the functionals. As an application, we retrieve or improve previous results from the existing literature, and provide new local stability results. We finally test the sharpness of our results by showing that the local minimality is in general not valid for non-smooth perturbations.
Published: 2019

40. A variational approach to regularity theory in optimal transportation

Author: Goldman, Michael, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Abstract: International audience; This paper describes recent results obtained in collaboration with M. Huesmann and F. Otto on the regularity of optimal transport maps. The main result is a quantitative version of the well-known fact that the linearization of the Monge-Ampère equation around the identity is the Poisson equation. We present two applications of this result. The first one is a variational proof of the partial regularity theorem of Figalli and Kim and the second is the rigorous validation of some predictions made by Carraciolo and al. on the structure of the optimal transport maps in matching problems.
Published: 2019

41. Optimal location of resources maximizing the total population size in logistic models

Author: Yannick Privat, Grégoire Nadin, Idriss Mazari, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), TOkamaks and NUmerical Simulations (TONUS), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), The authors were partially supported by the Project ``Analysis and simulation of optimal shapes - application to lifesciences' of the Paris City Hall., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
Subjects: Constant coefficients, General Mathematics, Admissible set, 01 natural sciences, Domain (mathematical analysis), optimal control, Mathematics - Analysis of PDEs, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Shape optimization, 0101 mathematics, Logistic function, Extreme point, diffusive logistic equation, Mathematics, Pointwise, symmetrization, optimality conditions, Applied Mathematics, 010102 general mathematics, Optimal control, 010101 applied mathematics, shape optimization, rearrangement inequalities, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis of PDEs (math.AP)
Abstract: International audience; In this article, we consider a species whose population density solves the steady diffusive logistic equation in a heterogeneous environment modeled with the help of a spatially non constant coefficient standing for a resources distribution. We address the issue of maximizing the total population size with respect to the resources distribution, considering some uniform pointwise bounds as well as prescribing the total amount of resources. By assuming the diffusion rate of the species large enough, we prove that any optimal configuration is bang-bang (in other words an extreme point of the admissible set) meaning that this problem can be recast as a shape optimization problem, the unknown domain standing for the resources location. In the one-dimensional case, this problem is deeply analyzed, and for large diffusion rates, all optimal configurations are exhibited. This study is completed by several numerical simulations in the one dimensional case.
Published: 2019

42. Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations

Author: Idriss Mazari, Antoine Henrot, Yannick Privat, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), TOkamaks and NUmerical Simulations (TONUS), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Yannick Privat and Idriss Mazari were partially supported by the Project 'Analysis and simulation of optimal shapes - application to life sciences' of the Paris City Hall., Antoine Henrot, Idriss Mazari, and Yannick Privat were partially supported by the ANR Project ANR-18-CE40-0013 - SHAPO on Shape Optimization., ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Sorbonne Université (SU), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Y. Privat and I. Mazari were partially supported by the Project 'Analysisand simulation of optimal shapes - application to lifesciences' of the Paris City Hall, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Universités
Subjects: Control and Optimization, Stability (learning theory), 01 natural sciences, Domain (mathematical analysis), Dirichlet distribution, symbols.namesake, Shape optimization, MSC : 49J45, 49K20, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Ball (mathematics), [MATH]Mathematics [math], 0101 mathematics, Mathematics - Optimization and Control, existence/stability of optimal shapes AMS classification: 49J45, Mathematics, Dirichlet energy, 010102 general mathematics, Dirichlet's energy, 49J45, 49K20, 010101 applied mathematics, Constraint (information theory), Computational Mathematics, AMS classification: 49J45, 49K20, Elliptic partial differential equation, Optimization and Control (math.OC), existence/stability of optimal shapes, Control and Systems Engineering, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: International audience; Minimizing the so-called “Dirichlet energy” with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.
Published: 2021

43. Topology optimization of thermal fluid–structure systems using body-fitted meshes and parallel computing

Author: Florian Feppon, Grégoire Allaire, Pierre Jolivet, Charles Dapogny, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Calcul des Variations, Géométrie, Image (CVGI), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Physics and Astronomy (miscellaneous), fluid-structure interaction, Boundary (topology), Context (language use), 010103 numerical & computational mathematics, 01 natural sciences, Shape and topology optimization, Computational science, distributed computing, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Polygon mesh, Point (geometry), Shape optimization, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Numerical Analysis, Applied Mathematics, Topology optimization, Constrained optimization, mesh adaptation, aerodynamic design, Finite element method, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, convective heat transfer, Modeling and Simulation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; An efficient framework is described for the shape and topology optimization of realistic three-dimensional, weakly-coupled fluid-thermal-mechanical systems. At the theoretical level, the proposed methodology relies on the boundary variation of Hadamard for describing the sensitivity of functions with respect to the domain. From the numerical point of view, three key ingredients are used: (i) a level set based mesh evolution method allowing to describe large deformations of the shape while maintaining an adapted, high-quality mesh of the latter at every stage of the optimization process; (ii) an efficient constrained optimization algorithm which is very well adapted to the infinite-dimensional shape optimization context; (iii) efficient preconditioning techniques for the solution of large finite element systems in a reasonable computational time. The performance of our strategy is illustrated with two examples of coupled physics: respectively fluid-structure interaction and convective heat transfer. Before that, we perform three other test cases, involving a single physics (structural, thermal and aerodynamic design), for comparison purposes and for assessing our various tools: in particular, they prove the ability of the mesh evolution technique to capture very thin bodies or shells in 3D.
Published: 2020

44. A fluctuation result for the displacement in the optimal matching problem

Author: Michael Goldman, Martin Huesmann, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), University of Münster, and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Statistics and Probability, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Probability (math.PR), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistics, Probability and Uncertainty, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract: The aim of this paper is to justify in dimensions two and three the ansatz of Caracciolo et al. stating that the displacement in the optimal matching problem is essentially given by the solution to the linearized equation i.e. the Poisson equation. Moreover, we prove that at all mesoscopic scales, this displacement is close in suitable negative Sobolev spaces to a curl-free Gaussian free field. For this we combine a quantitative estimate on the difference between the displacement and the linearized object, which is based on the large-scale regularity theory recently developed in collaboration with F. Otto, together with a qualitative convergence result for the linearized problem., Comment: Comments very welcome!

45. Qualitative analysis of optimisation problems with respect to non-constant Robin coefficients

Author: Idriss Mazari, yannick privat, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), Privat, Yannick, and APPEL À PROJETS GÉNÉRIQUE 2018 - Optimisation de forme - - SHAPO2018 - ANR-18-CE40-0013 - AAPG2018 - VALID
Subjects: Shape optimisation, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Robin boundary conditions, Elliptic boundary values problems, Theoretical Computer Science, Qualitative analysis of optimisation problems, Mathematics (miscellaneous), Optimization and Control (math.OC), Bilinear optimal control problems, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Optimization and Control, Calculus of variations
Abstract: Following recent interest in the qualitative analysis of some optimal control and shape optimisation problems, we provide in this article a detailed study of the optimisation of Robin boundary conditions in PDE constrained calculus of variations. Our main model consists of an elliptic PDE of the form $-\Delta u_\beta=f(x,u_\beta)$ endowed with the Robin boundary conditions $\partial_\nu u_\beta+\beta(x)u_\beta=0$. The optimisation variable is the function $\beta$, which is assumed to take values between 0 and 1 and to have a fixed integral. Two types of criteria are under consideration: the first one is non-energetic criteria. In other words, we aim at optimising functionals of the form $\mathcal J(\beta)=\int_{\Omega\text{ or }\partial \Omega}j(u_\beta)$. We prove that, depending on the monotonicity of the function $j$, the optimisers may be of \emph{bang-bang} type (in other words, the optimisers write $1_\Gamma$ for some measurable subset $\Gamma$ of $\partial \ Omega $) or, on the contrary, that they may only take values strictly between 0 and 1. This has consequence for a related shape optimisation problem, in which one tries to find where on the boundary Neumann ($\partial_\nu u=0$ ) and constant Robin conditions ($\partial_\nu u+u=0$) should be placed in order to optimise criteria. The proofs for this first case rely on new fine oscillatory techniques, used in combination with optimality conditions. We then investigate the case of compliance-type functionals. For such energetic functionals, we give an in-depth analysis and even some explicit characterisation of optimal $\beta^*$.

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