Publications

2020

• A metric interpretation of the geodesic curvature in the Heisenberg group
  
  • Kohli Mathieu
  Journal of Dynamical and Control Systems, Springer Verlag, 2020, 26 (1), pp.159–174. In this paper we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc lenght in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this notion of geodesic curvature as the first corrective term in the Taylor expansion of the distance between two close points of the curve. (10.1007/s10883-019-09444-7)
  DOI : 10.1007/s10883-019-09444-7
• A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices
  
  • Gaubert Stéphane
  • Stott Nikolas
  Mathematical Control and Related Fields, AIMS, 2020, 10 (3), pp.573-590. We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of an entropy game, in which the state space consists of all switching sequences of a given length. We show that, by increasing this length, we arrive at a convergent approximation scheme to compute the joint spectral radius. The complexity of this method is exponential in the length of the switching sequences, but it is quite insensitive to the size of the matrices, allowing us to solve very large scale instances (several matrices in dimensions of order 1000 within a minute). An idea of this method is to replace a hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter eigenproblems, we introduce a projective version of Krasnoselskii-Mann iteration. This method is of independent interest as it applies more generally to the nonlinear eigenproblem for a monotone positively homogeneous map. Here, this method allows for scalability by avoiding the recourse to linear or semidefinite programming techniques. (10.3934/mcrf.2020011)
  DOI : 10.3934/mcrf.2020011
• Medical innovations to maintain the function in patients with chronic PJI for whom explantation is not desirable: a pathophysiology-, multidisciplinary-, and experience-based approach
  
  • Ferry Tristan
  • Batailler Cécile
  • Brosset Sophie
  • Kolenda Camille
  • Goutelle Sylvain
  • Sappey-Marinier Elliot
  • Josse Jérôme
  • Laurent Frédéric
  • Lustig Sébastien
  SICOT-J, EDP Open, 2020, 6, pp.26. Introduction: PJI is the most dramatic complication after joint arthroplasty. In patients with chronic infection, prosthesis exchange is in theory the rule. However, this surgical approach is sometimes not desirable especially in elderly patients with multiple comorbidities, as it could be associated with a dramatic loss of function, reduction of the bone stock, fracture, or peroperative death. We propose here to report different approaches that can help to maintain the function in such patients based on a pathophysiology-, multidisciplinary-, and an experience-based approach. Methods: We describe the different points that are needed to treat such patients: (i) the multidisciplinary care management; (ii) understanding the mechanism of bacterial persistence; (iii) optimization of the conservative surgical approach; (iv) use of suppressive antimicrobial therapy (SAT); (v) implementation of innovative agents that could be used locally to target the biofilm. Results: In France, a nation-wide network called CRIOAc has been created and funded by the French Health ministry to manage complex bone and joint infection. Based on the understanding of the complex pathophysiology of PJI, it seems to be feasible to propose conservative surgical treatment such as “debridement antibiotics and implant retention” (with or without soft-tissue coverage) followed by SAT to control the disease progression. Finally, there is a rational for the use of particular agents that have the ability to target the bacteria embedded in biofilm such as bacteriophages and phage lysins. Discussion: This multistep approach is probably a key determinant to propose innovative management in patients with complex PJI, to improve the outcome. Conclusion: Conservative treatment has a high potential in patients with chronic PJI for whom explantation is not desirable. The next step will be to evaluate such practices in nation-wide clinical trials. (10.1051/sicotj/2020021)
  DOI : 10.1051/sicotj/2020021
• Model Reduction for Large-Scale Earthquake Simulation in an Uncertain 3D Medium
  
  • Sochala Pierre
  • de Martin Florent
  • Le Maitre Olivier
  International Journal for Uncertainty Quantification, Begell House Publishers, 2020, 10 (2), pp.101-127. In this paper, we are interested in the seismic wave propagation into an uncertain medium. To this end, we performed an ensemble of 400 large-scale simulations that requires 4 million core-hours of CPU time. In addition to the large computational load of these simulations, solving the uncertainty propagation problem requires dedicated procedures to handle the complexities inherent to large data set size and the low number of samples. We focus on the peak ground motion at the free surface of the 3D domain, and our analysis utilizes a surrogate model combining two key ingredients for complexity mitigation: i) a dimension reduction technique using empirical orthogonal basis functions and ii) a functional approximation of the uncertain reduced coordinates by polynomial chaos expansions. We carefully validate the resulting surrogate model by estimating its predictive error using bootstrap, truncation, and cross-validation procedures. The surrogate model allows us to compute various statistical information of the uncertain prediction, including marginal and joint probability distributions, interval probability maps, and 2D fields of global sensitivity indices. (10.1615/Int.J.UncertaintyQuantification.2020031165)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2020031165
• A MOMENT CLOSURE BASED ON A PROJECTION ON THE BOUNDARY OF THE REALIZABILITY DOMAIN: 1D CASE
  
  • Pichard Teddy
  Kinetic and Related Models, AIMS, 2020, 13 (6), pp.1243-1280. This work aims to develop and test a projection technique for the construction of closing equations of moment systems. One possibility to define such a closure consists in reconstructing an underlying kinetic distribution from a vector of moments, then expressing the closure based on this reconstructed function. Exploiting the geometry of the realizability domain, i.e. the set of moments of positive distribution function, we decompose any realizable vectors into two parts, one corresponding to the moments of a chosen equilibrium function, and one obtain by a projection onto the boundary of the realizability domain in the direction of equilibrium function. A realizable closure of both of these parts are computed with standard techniques providing a realizable closure for the full system. This technique is tested for the reduction of a radiative transfer equation in slab geometry. (10.3934/xx.xx.xx.xx)
  DOI : 10.3934/xx.xx.xx.xx
• How a moving passive observer can perceive its environment ? The Unruh effect revisited
  
  • Fink Mathias
  • Garnier Josselin
  Wave Motion, Elsevier, 2020, 93, pp.102462. (10.1016/j.wavemoti.2019.102462)
  DOI : 10.1016/j.wavemoti.2019.102462
• A second order analysis of McKean-Vlasov semigroups
  
  • Arnaudon Marc
  • del Moral Pierre
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2020. We propose a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonlinear diffusions. Bismut-Elworthy-Li formulae for the gradient and the Hessian of the integro-differential operators associated with these expansions are also presented. The article also provides explicit Dyson-Phillips expansions and a refined analysis of the norm of these integro-differential operators. Under some natural and easily verifiable regularity conditions we derive a series of exponential decays inequalities with respect to the time horizon. We illustrate the impact of these results with a second order extension of the Alekseev-Gröbner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order perturbation analysis provides direct proofs of several uniform propagation of chaos properties w.r.t. the time parameter, including bias, fluctuation error estimate as well as exponential concentration inequalities. (10.1214/20-AAP1568)
  DOI : 10.1214/20-AAP1568
• Support optimization in additive manufacturing for geometric and thermo-mechanical constraints
  
  • Allaire Grégoire
  • Bihr Martin
  • Bogosel Beniamin
  Structural and Multidisciplinary Optimization, Springer Verlag, 2020, 61, pp.2377-2399. Supports are often required to safely complete the building of complicated structures by additive manufacturing technologies. In particular, supports are used as scaffoldings to reinforce overhanging regions of the structure and/or are necessary to mitigate the thermal deformations and residual stresses created by the intense heat flux produced by the source term (typically a laser beam). However, including supports increase the fabrication cost and their removal is not an easy matter. Therefore, it is crucial to minimize their volume while maintaining their efficiency. Based on earlier works, we propose here some new optimization criteria. First, simple geometric criteria are considered like the projected area and the volume of supports required for overhangs: they are minimized by varying the structure orientation with respect to the baseplate. In addition, an accessibility criterion is suggested for the removal of supports, which can be used to forbid some parts of the structure to be supported. Second, shape and topology optimization of supports for compliance minimization is performed. The novelty comes from the applied surface loads which are coming either from pseudo gravity loads on overhanging parts or from equivalent thermal loads arising from the layer by layer building process. Here, only the supports are optimized, with a given non-optimizable structure, but of course many generalizations are possible, including optimizing both the structure and its supports. Our optimization algorithm relies on the level set method and shape derivatives computed by the Hadamard method. Numerical examples are given in 2-d and 3-d.
• Multipoint formulas for scattered far field in multidimensions
  
  • Novikov Roman
  Inverse Problems, IOP Publishing, 2020, 36 (9), pp.095001(12 pp.). We give asymptotic formulas for finding the scattering amplitude at fixed frequency and angles (scattered far field) from the scattering wave function given at $n$ points in dimension $d\geq 2$. These formulas are explicit and their precision is proportional to $n$. To our knowledge these formulas are new already for $n\geq 2$. (10.1088/1361-6420/aba891)
  DOI : 10.1088/1361-6420/aba891
• Surface waves in a channel with thin tunnels and wells at the bottom: non-reflecting underwater tomography
  
  • Chesnel Lucas
  • Nazarov Sergei
  • Taskinen Jari
  Asymptotic Analysis, IOS Press, 2020. We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.
• Uncertainty Quantification for Stochastic Approximation Limits Using Chaos Expansion
  
  • Crépey Stéphane
  • Fort Gersende
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2020, 8 (3), pp.1061–1089. The uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm is analyzed. In our setup, this limit $f^*$ is defined as a zero of an intractable function and is modeled as uncertain through a parameter $\theta$. We aim at deriving the function $f^*$, as well as the probabilistic distribution of $f^*(\theta)$ given a probability distribution $\pi$ for $\theta$. We introduce the so-called Uncertainty Quantification for SA (UQSA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of $\theta \mapsto f^*(\theta)$ on an orthogonal basis of a suitable Hilbert space. UQSA, run with a finite number of iterations $K$, returns a finite set of coefficients, providing an approximation $\widehat{f^*_K}(\cdot)$ of $f^*$. We establish the almost-sure and $L^p$-convergences in the Hilbert space of the sequence of functions $\widehat{f^*_K}(\cdot)$ when the number of iterations $K$ tends to infinity. This is done under mild, tractable conditions, uncovered by the existing literature for convergence analysis of infinite dimensional SA algorithms. For a suitable choice of the Hilbert basis, the algorithm also provides an approximation of the expectation, of the variance-covariance matrix and of higher order moments of the quantity $\widehat{f^*_K}(\theta)$ when $\theta$ is random with distribution $\pi$. UQSA is illustrated and the role of its design parameters is discussed numerically. (10.1137/18M1178517)
  DOI : 10.1137/18M1178517
• Sparse recovery for inverse potential problems in divergence form
  
  • Baratchart Laurent
  • Villalobos Guillén Cristóbal
  • Hardin Douglas
  • Leblond Juliette
  , 2020, 1476, pp.012009. We discuss recent results on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure-theoretic, namely pure 1-unrectifiability of the support. The theory applies when a superset of the support is known to be slender, meaning it has measure zero and all connected components of its complement has infinite measure in R^3. We also discuss open issues in the non-slender case. (10.1088/1742-6596/1476/1/012009)
  DOI : 10.1088/1742-6596/1476/1/012009
• The tropicalization of the entropic barrier
  
  • Allamigeon Xavier
  • Aznag Abdellah
  • Gaubert Stéphane
  • Hamdi Yassine
  , 2020. The entropic barrier, studied by Bubeck and Eldan (Proc. Mach. Learn. Research, 2015), is a self-concordant barrier with asymptotically optimal self-concordance parameter. In this paper, we study the tropicalization of the central path associated with the entropic barrier, i.e., the logarithmic limit of this central path for a parametric family of linear programs defined over the field of Puiseux series. Our main result is that the tropicalization of the entropic central path is a piecewise linear curve which coincides with the tropicalization of the logarithmic central path studied by Allamigeon et al. (SIAM J. Applied Alg. Geom., 2018). One consequence is that the number of linear pieces in the tropical entropic central path can be exponential in the dimension and the number of inequalities defining the linear program.
• Problem and Non-Problem Gamblers: A Cross-Sectional Clustering Study by Gambling Characteristics
  
  • Guillou-Landréat Morgane
  • Chereau Boudet Isabelle
  • Perrot Bastien
  • Romo Lucia
  • Codina Irene
  • Magalon David
  • Fatseas Melina
  • Luquiens Amandine
  • Brousse Georges
  • Challet-Bouju Gaëlle
  • Grall-Bronnec Marie
  BMJ Open, BMJ Publishing Group, 2020, 10 (2), pp.e030424. OBJECTIVES: Gambling characteristics are factors that could influence problem gambling development. The aim of this study was to identify a typology of gamblers to frame risky behaviour based on gambling characteristics (age of initiation/of problem gambling, type of gambling: pure chance/chance with pseudoskills/chance with elements of skill, gambling online/offline, amount wagered monthly) and to investigate clinical factors associated with these different profiles in a large representative sample of gamblers. DESIGN AND SETTING: The study is a cross-sectional analysis to the baseline data of the french JEU cohort study (study protocol : Challet-Bouju et al, 2014). Recruitment (April 2009 to September 2011) involved clinicians and researchers from seven institutions that offer care for or conduct research on problem gamblers (PG). Participants were recruited in gambling places, and in care centres. Only participants who reported gambling in the previous year between 18 and 65 years old were included.Participants gave their written informed consent, it was approved by the French Research Ethics Committee. PARTICIPANTS: The participants were 628 gamblers : 256 non-problem gamblers (NPG), 169 problem gamblers without treatment (PGWT) and 203 problem gamblers seeking treatment (PGST). RESULTS: Six clustering models were tested, the one with three clusters displayed a lower classification error rate (7.92%) and was better suited to clinical interpretation : 'Early Onset and Short Course' (47.5%), 'Early Onset and Long Course' (35%) and 'Late Onset and Short Course' (17.5%). Gambling characteristics differed significantly between the three clusters. CONCLUSIONS: We defined clusters through the analysis of gambling variables, easy to identify, by psychiatrists or by physicians in primary care. Simple screening concerning these gambling characteristics could be constructed to prevent and to help PG identification. It is important to consider gambling characteristics : policy measures targeting gambling characteristics may reduce the risk of PG or minimise harm from gambling. TRIAL REGISTRATION NUMBER: NCT01207674 (ClinicalTrials.gov); Results. (10.1136/bmjopen-2019-030424)
  DOI : 10.1136/bmjopen-2019-030424
• 3D positive lattice walks and spherical triangles
  
  • Bogosel B
  • Perrollaz V
  • Raschel K.
  • Trotignon A
  Journal of Combinatorial Theory, Series A, Elsevier, 2020, 172, pp.105189. In this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard factorization, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with ten digits of precision. (10.1016/j.jcta.2019.105189)
  DOI : 10.1016/j.jcta.2019.105189
• The effect of the terminal penalty in receding horizon control for a class of stabilization problems
  
  • Kunisch Karl
  • Pfeiffer Laurent
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2020, 58. The Receding Horizon Control (RHC) strategy consists in replacing an infinite-horizon stabilization problem by a sequence of finite-horizon optimal control problems, which are numerically more tractable. The dynamic programming principle ensures that if the finite-horizon problems are formulated with the exact value function as a terminal penalty function, then the RHC method generates an optimal control. This article deals with the case where the terminal cost function is chosen as a cutoff Taylor approximation of the value function. The main result is an error rate estimate for the control generated by such a method, when compared with the optimal control. The obtained estimate is of the same order as the employed Taylor approximation and decreases at an exponential rate with respect to the prediction horizon. To illustrate the methodology, the article focuses on a class of bilinear optimal control problems in infinite-dimensional Hilbert spaces. (10.1051/cocv/2019037)
  DOI : 10.1051/cocv/2019037
• Semi-conical eigenvalue intersections and the ensemble controllability problem for quantum systems
  
  • Augier Nicolas
  • Boscain Ugo
  • Sigalotti Mario
  Mathematical Control and Related Fields, AIMS, 2020, 10, pp.877-911. We study one-parametric perturbations of finite dimensional real Hamiltonians depending on two controls, and we show that generically in the space of Hamiltonians, conical intersections of eigenvalues can degenerate into semi-conical intersections of eigenvalues. Then, through the use of normal forms, we study the problem of ensemble controllability between the eigenstates of a generic Hamiltonian. (10.3934/mcrf.2020023)
  DOI : 10.3934/mcrf.2020023
• EXISTENCE AND ASYMPTOTIC RESULTS FOR AN INTRINSIC MODEL OF LINEARIZED INCOMPATIBLE ELASTICITY
  
  • Amstutz Samuel
  • van Goethem Nicolas
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2020, 25 (10). A general model of incompatible linearized elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Elastic strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids.The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the limit case when the incompatibility modulus goes to infinity. Several examples are provided to illustrate this property and the physical meaning of the incompatibility modulus in connection with the dissipative nature of the processes under consideration.
• Spectral inequalities for nonnegative tensors and their tropical analogues
  
  • Friedland Shmuel
  • Gaubert Stéphane
  Vietnam Journal of Mathematics, Springer, 2020, 48 (4), pp.893-928. We extend some characterizations and inequalities for the eigenvalues of nonnegative matrices, such as Donsker–Varadhan, Friedland–Karlin, Karlin–Ost inequalities, to nonnegative tensors. Our approach involves a correspondence between nonnegative tensors, ergodic control and entropy maximization: we show in particular that the logarithm of the spectral radius of a tensor is given by en entropy maximization problem over a space of occupation measures. We study in particular the tropical analogue of the spectral radius, that we characterize as a limit of the classical spectral radius, and we give an explicit combinatorial formula for this tropical spectral radius. (10.1007/s10013-020-00432-0)
  DOI : 10.1007/s10013-020-00432-0
• Topology optimization of connections in mechanical systems
  
  • Rakotondrainibe Lalaina
  • Allaire Grégoire
  • Orval Patrick
  Structural and Multidisciplinary Optimization, Springer Verlag, 2020. One of the issues for the automotive industry is weight reduction. For this purpose, topology optimization is used for mechanical parts and usually involves a single part. Its connections to other parts are assumed to be fixed. This paper deals with a coupled topology optimization of both the structure of a part and its connections (location and number) to other parts. The present work focuses on two models of connections, namely rigid support and spring that prepares work for bolt connection. Rigid supports are modeled by Dirichlet boundary conditions while bolt-like connections are idealized and simplified as a non-local interaction to be representative enough at a low computational cost. On the other hand, the structure is modeled by the linearized elasticity system and its topology is represented by a level set function. A coupled optimization of the structure and the location of rigid supports is performed to minimize the volume of an engine accessories bracket under a compliance constraint. This coupled topology optimization (shape and connections) provides more satisfactory performance of a part than the one given by classical shape optimization alone. The approach presented in this work is therefore one step closer to the optimization of assembled mechanical systems. Thereafter, the concept of topological derivative is adapted to create an idealized bolt. The main idea is to add a small idealized bolt at the best location and to test the optimality of the solution with this new connection. The topological derivative is tested with a 3d academic test case for a problem of compliance minimization.
• Creation and annihilation of point-potentials using Moutard-type transform in spectral variable
  
  • Grinevich Piotr
  • Novikov Roman
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2020, 61 (9), pp.093501(9 pp.). We continue to develop the method for creation and annihilation of contour singularities in the d-bar-spectral data for the two-dimensional Schrödinger equation at fixed energy. Our method is based on the Moutard-type transforms for generalized analytic functions. In this note we show that this approach successfully works for point potentials . (10.1063/1.5143303)
  DOI : 10.1063/1.5143303
• Second-order analysis for the time crisis problem
  
  • Bayen Térence
  • Pfeiffer Laurent
  Journal of Convex Analysis, Heldermann, 2020, 27 (1), pp.139-163. In this article, we prove second-order necessary optimality conditions for the so-called time crisis problem that comes up within the context of viability theory. It consists in minimizing the time spent by solutions of a controlled dynamics outside a given subset K of the state space. One essential feature is the discontinuity of the characteristic function involved in the cost functional. Thanks to a change of time and an augmentation of the dynamics, we relate the time crisis problem to an auxiliary Mayer control problem. This allows us to use the classical tools of optimal control for obtaining optimality conditions. Going back to the original problem, we deduce that way second order optimality conditions for the time crisis problem.
• Markovian explorations of random planar maps are roundish
  
  • Curien Nicolas
  • Marzouk Cyril
  Bulletin de la société mathématique de France, Société Mathématique de France, 2020, 148 (4), pp.709-732. The infinite discrete stable Boltzmann maps are "heavy-tailed" generalisations of the well-known Uniform Infinite Planar Quadrangulation. Very efficient tools to study these objects are Markovian step-by-step explorations of the lattice called peeling processes. Such a process depends on an algorithm which selects at each step the next edge where the exploration continues. We prove here that, whatever this algorithm, a peeling process always reveals about the same portion of the map, thus growing roughly metric balls. Applied to well-designed algorithms, this easily enables us to compare distances in the map and in its dual, as well as to control the so-called pioneer points of the simple random walk, both on the map and on its dual. (10.24033/bsmf.2821)
  DOI : 10.24033/bsmf.2821
• Order antimorphisms of finite-dimensional cones
  
  • Walsh Cormac
  Selecta Mathematica (New Series), Springer Verlag, 2020, 26 (4), pp.paper number 53. We show that an order antimorphism on a finite-dimensional cone having no one-dimensional factors is homogeneous of degree −1. A consequence is that the existence of an order antimorphism on a finite-dimensional cone implies that the cone is a symmetric cone.
• 3-d topology optimization of modulated and oriented periodic microstructures by the homogenization method
  
  • Geoffroy-Donders Perle
  • Allaire Grégoire
  • Pantz Olivier
  Journal of Computational Physics, Elsevier, 2020, 401, pp.108994. This paper is motivated by the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. Generalizing our previous work in 2-d we propose a method for topology optimization of structures made of periodically perforated material , where the microscopic periodic cell can be macroscopically modulated and oriented. This method is made of three steps. The first step amounts to compute the homogenized properties of an adequately chosen parametrized mi-crostructure (here, a cubic lattice with varying bar thicknesses). The second step optimizes the homogenized formulation of the problem, which is a classical problem of parametric optimization. The third, and most delicate, step projects the optimal oriented microstructure at a desired length scale. Compared to the 2-d case where rotations are parametrized by a single angle, to which a confor-mality constraint can be applied, the 3-d case is more involved and requires new ingredients. In particular, the full rotation matrix is regularized (instead of just one angle in 2-d) and the projection map which deforms the square periodic lattice is computed component by component. Several numerical examples are presented for compliance minimization in 3-d.