An effective model for Lipschitz wrinkled arches

AbstractWithin the framework of the Koiter's linear elastic shell theory, we study the limit model of a Lipschitz curved arch whose mid-surface is periodically waved. The magnitude and the period of the wavings are of the same order. To achieve the asymptotic analysis, we consider a mixed formulation, for which we perform a two-scale homogenization technique. We prove the convergence of the displacements, the rotation of the normal, and the membrane strain. From the limit formulation, we derive an effective model for curved critically wrinkled arches. It introduces two membrane strain functions—instead of one in the classical case—and exhibits a corrector membrane term to the coupling between the rotation of the normal and the membrane strain

The Kalai-Smorodinski solution for many-objective Bayesian optimization

An ongoing aim of research in multiobjective Bayesian optimization is to extend its applicability to a large number of objectives. While coping with a limited budget of evaluations, recovering the set of optimal compromise solutions generally requires numerous observations and is less interpretable since this set tends to grow larger with the number of objectives. We thus propose to focus on a specific solution originating from game theory, the Kalai-Smorodinsky solution, which possesses attractive properties. In particular, it ensures equal marginal gains over all objectives. We further make it insensitive to a monotonic transformation of the objectives by considering the objectives in the copula space. A novel tailored algorithm is proposed to search for the solution, in the form of a Bayesian optimization algorithm: sequential sampling decisions are made based on acquisition functions that derive from an instrumental Gaussian process prior. Our approach is tested on four problems with respectively four, six, eight, and nine objectives. The method is available in the Rpackage GPGame available on CRAN at https://cran.r-project.org/package=GPGame

A Nash Topology Game for Tumoral Anti-angiogenesis

Tumoral angiogenesis and anti-angiogenesis are modeled as a Nash game. We consider the vessel-matrix-tumor system as a porous medium from the tumor viewpoint and as an elastic structural medium from the host tissue viewpoint. We define a competition between two density functions which are intended to represent respectively activators and inhibitors of angiogenesis. The activators want to minimize the pressure drop while the inhibitors intend to minimize the elastic compliance of the matrix or the drainage of the tumoral neovascularization. Numerical results illustrate how -theoretical- tumors develop multiple channels as an optimal response to optimal distribution of inhibitors

An effective model for critically wrinkled arches

Within the framework of the Koiter's linear elastic shell theory, we study the limit model of a plane arch whose mid-surface is periodically waved.The magnitude and the period of the wavings are of the same order. To achieve the asymptotic analysis, we consider a mixed formulation, for which we perform a two-scale homogenization technique. We prove the convergence of the displacements, the rotation of the normal and the membrane strain. From the limit formulation, we derive an effective model for critically wrinkled arches. It has a plane effective mid-surface, but exhibits a coupling between the rotation of the normal and the membrane strain

Modeling actin cable contraction

International audienceWe present a simple mathematical model for simulating the movement of an actin cable that is attached to an external epithelial tissue, like in the case of the boundary of an epidermal wound. We focus on the case of non homogeneous forces for wound healing and for dorsal closure of drosophila

Assessing the ability of the 2D Fisher-KPP equation to model cell-sheet wound closure

International audienceWe address in this paper the ability of the Fisher-KPP equations to render some of the dynamical features of epithelial cell-sheets during wound closure. Our approach is based on nonlinear parameter identification, in a two-dimensional setting, and using advanced 2D image processing of the video acquired sequences. As original contribution, we lead a detailed study of the profiles of the classically used cost functions, and we address the "wound constant speed" assumption, showing that it should be handled with care. We study five MDCK cell monolayer assays in a reference, activated and inhibited migration conditions. Modulo the inherent variability of biological assays, we show that in the assay where migration is not exogeneously activated or inhibited, the wound velocity is constant. The Fisher-KPP equation is able to accurately predict, until the final closure of the wound, the evolution of the wound area, the mean velocity of the cell front, and the time at which the closure occurred. We also show that for activated as well as for inhibited migration assays, many of the cell-sheet dynamics cannot be well captured by the Fisher-KPP model. Finally, we draw some conclusions related to the identified model parameters, and possible utilization of the model

NEUMANN-DIRICHLET NASH STRATEGIES FOR THE SOLUTION OF ELLIPTIC CAUCHY PROBLEMS

International audienceWe consider the Cauchy problem for an elliptic operator, formulated as a Nash game. The overspecified Cauchy data are split between two players: the first player solves the elliptic equation with the Dirichlet part of the Cauchy data prescribed over the accessible boundary and a variable Neumann condition (which we call first player's strategy) prescribed over the inaccessible part of the boundary. The second player makes use correspondingly of the Neumann part of the Cauchy data, with a variable Dirichlet condition prescribed over the inaccessible part of the boundary. The first player then minimizes the gap related to the nonused Neumann part of the Cauchy data, and so does the second player with a corresponding Dirichlet gap. The two costs are coupled through a difference term. We prove that there always exists a unique Nash equilibrium, which turns out to be the reconstructed data when the Cauchy problem has a solution. We also prove that the completion Nash game has a stable solution with respect to noisy data. Some numerical two- and three-dimensional experiments are provided to illustrate the efficiency and stability of our algorithm

Computational investigations of a two-class traffic flow model: mean-field and microscopic dynamics

We address a multi-class traffic model, for which we computationally assess the ability of mean-field games (MFG) to yield approximate Nash equilibria for traffic flow games of intractable large finite-players. We introduce a two-class traffic framework, following and extending the single-class lines of \cite{huang_game-theoretic_2020}. We extend the numerical methodologies, with recourse to techniques such as HPC and regularization of LGMRES solvers. The developed apparatus allows us to perform simulations at significantly larger space and time discretization scales. For three generic scenarios of cars and trucks, and three cost functionals, we provide numerous numerical results related to the autonomous vehicles (AVs) traffic dynamics, which corroborate for the multi-class case the effectiveness of the approach emphasized in \cite{huang_game-theoretic_2020}. We additionally provide several original comparisons of macroscopic Nash mean-field speeds with their microscopic versions, allowing us to computationally validate the so-called $\epsilon-$Nash approximation, with a rate slightly better than theoretically expected

Computational design of an automotive twist beam

International audienceIn recent years, the automotive industry has known a remarkable development in order to satisfy the customer requirements. In this paper, we will study one of the components of the automotive which is the twist beam. The study is focused on the multicriteria design of the automotive twist beam undergoing linear elastic deformation (Hooke's law). Indeed, for the design of this automotive part, there are some criteria to be considered as the rigidity (stiffness) and the resistance to fatigue. Those two criteria are known to be conflicting, therefore, our aim is to identify the Pareto front of this problem. To do this, we used a Normal Boundary Intersection (NBI) algorithm coupling with a radial basis function (RBF) metamodel in order to reduce the high calculation time needed for solving the multicriteria design problem. Otherwise, we used the free form deformation (FFD) technique for the generation of the 3D shapes of the automotive part studied during the optimization process

Determination of point-forces via extended boundary measurements using a game strategy approach

International audienceIn this work, we consider a game theory approach to deal with an inverse problem related to the Stokes system. The problem consists in detecting the unknown point-forces acting on the fluid from incomplete measurements on the boundary of a domain. The approach that we propose deals simultaneously with the reconstruction of the missing data and the determination of the unknown point-forces. The solution is interpreted in terms of Nash equilibrium between both problems. We develop a new point-force detection algorithm, and we present numerical results to illustrate the efficiency and robustness of the method