Analyse géométrique

Résumé

L'analyse géométrique est un champ mathématique à l'interface de la géométrie différentielle et des équations différentielles. Portée vignette| de surface minimale. Les surfaces minimales font partie des objets d'étude en analyse géométrique. L'analyse géométrique comprend à la fois l'utilisation de méthodes géométriques dans l'étude des équations aux dérivées partielles, et l'application de la théorie des équations aux dérivées partielles à la géométrie. Elle intègre des problèmes impliquant des courbes et des surfaces, ou des domaines avec des courbes limites, mais aussi l'étude de variétés riemanniennes en dimension arbitraire. Le calcul des variations est parfois considéré comme faisant partie de l'analyse géométrique, parce que les équations différentielles découlant de l'électromagnétisme ont un solide contenu géométrique. L'analyse géométrique inclut également l'analyse globale, qui concerne l'étude des équations différentielles sur des variétés, et la relation entr

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https://fr.wikipedia.org/wiki/Analyse_g%C3%A9om%C3%A9trique

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Some applications of symmetries in differential geometry and dynamical systems

Oana Mihaela Dragulete

This thesis deals with applications of Lie symmetries in differential geometry and dynamical systems. The first chapter of the thesis studies the singular reduction of symmetries of cosphere bundles, the conservation properties of contact systems and their reduction. We generalise the results of [15] to the singular case making a complete topological and geometrical analysis of the reduced space. Applying the general theory of contact reduction developed by Lerman and Willett in [33] and [57], one obtains contact stratified spaces that lose all information of the internal structure of the cosphere bundle. Based on the cotangent bundle reduction theorems, both in the regular and singular case, as well as regular cosphere bundle reduction, one expects additional bundle-like structure for the contact strata. The cosphere bundle projection to the base manifold descends to a continuous surjective map from the reduced space at zero to the orbit quotient of the configuration space, but it fails to be a morphism of stratified spaces if we endow the reduced space with its contact stratification and the base space with the customary orbit type stratification defined by the Lie group action. In this chapter we introduce a new stratification of the contact quotient at zero, called the C-L stratification (standing for the coisotropic or Legendrian nature of its pieces) which solves the above mentioned two problems. Its main features are the following. First, it is compatible with the contact stratification of the quotient and the orbit type stratification of the configuration orbit space. It is also finer than the contact stratification. Second, the natural projection of the C-L stratified quotient space to its base space, stratified by orbit types, is a morphism of stratified spaces. Third, each C-L stratum is a bundle over an orbit type stratum of the base and it can be seen as a union of C-L pieces, one of them being open and dense in its corresponding contact stratum and contactomorphic to a cosphere bundle. The other strata are coisotropic or Legendrian submanifolds in the contact components that contain them. We also describe the relation between contact vector fields and the time dependent Hamilton-Jacobi equation. The reduction of contact systems and time dependent Hamiltonians is mentioned. In the second chapter we study geometric properties of Sasakian and Kähler quotients. We construct a reduction procedure for symplectic and Kähler manifolds using the ray preimages of the momentum map. More precisely, instead of taking as in point reduction the preimage of a momentum value μ, we take the preimage of ℝ+μ, the positive ray of μ. We have two reasons to develop this construction. One is geometric: non zero Kähler point reduction is not always well defined. The problem is that the complex structure may not leave invariant the horizontal distribution of the Riemannian submersion πμ : J-1(μ) → Mμ. The solution proposed in the literature is correct only in the case of totally isotropic momentum (i.e. Gμ = G). The other reason is that it provides invariant submanifolds for conformal Hamiltonian systems. They are usually non-autonomous mechanical systems with friction whose integral curves preserve, in the case of symmetries, the ray pre-images of the momentum map. We extend the class of conformal Hamiltonian systems already studied and complete the existing Lie Poisson reduction with the general ray one. As examples of symplectic (Kähler) and contact (Sasakian) ray reductions we treat the case of cotangent and cosphere bundles and we show that they are universal for ray reductions. Using techniques of A. Futaki, we prove that, under appropriate hypothesis, ray quotients of Kähler-Einstein or Sasaki-Einstein manifolds remain Kähler or Sasaki-Einstein. Note that it suffices to prove the Kähler case and the compatibility of ray reduction with the Boothby-Wang fibration. In the last chapter, we prove a stratification theorem for proper groupoids. First we find an equivalent way of describing the same result for a proper Lie group action, way which uses the theory of foliations and can be adapted to the language of Lie groupoids. We treat separately the case of free and proper groupoids. The orbit foliation of a proper Lie groupoid is a singular Riemannian foliation and we show this explicitly.

EPFL2007

Chargement

In this thesis, we consider the numerical approximation of high order geometric Partial Differential Equations (PDEs). We first consider high order PDEs defined on surfaces in the 3D space that are represented by single-patch tensor product NURBS. Then, we spatially discretize the PDEs by means of NURBS-based Isogeometric Analysis (IGA) in the framework of the Galerkin method. With this aim, we consider the construction of periodic NURBS function spaces with high degree of global continuity, even on closed surfaces. As benchmark problems for the proposed discretization, we propose Laplace-Beltrami problems of the fourth and sixth orders, as well as the corresponding eigenvalue problems, and we analyze the impact of the continuity of the basis functions on the accuracy as well as on computational costs. The numerical solution of two high order phase field problems on both open and closed surfaces is also considered: the fourth order Cahn-Hilliard equation and the sixth order crystal equation, both discretized in time with the generalized-alpha method. We then consider the numerical approximation of geometric PDEs, derived, in particular, from the minimization of shape energy functionals by L^2-gradient flows. We analyze the mean curvature and the Willmore gradient flows, leading to second and fourth order PDEs, respectively. These nonlinear geometric PDEs are discretized in time with Backward Differentiation Formulas (BDF), with a semi-implicit formulation based on an extrapolation of the geometry, leading to a linear problem to be solved at each time step. Results about the numerical approximation of the two geometric flows on several geometries are analyzed. Then, we study how the proposed mathematical framework can be employed to numerically approximate the equilibrium shapes of lipid bilayer biomembranes, or vesicles, governed by the Canham-Helfrich curvature model. We propose two numerical schemes for enforcing the conservation of the area and volume of the vesicles, and report results on benchmark problems. Then, the approximation of the equilibrium shapes of biomembranes with different values of reduced volume is presented. Finally, we consider the dynamics of a vesicle, e.g. a red blood cell, immersed in a fluid, e.g. the plasma. In particular, we couple the curvature-driven model for the lipid membrane with the incompressible Navier-Stokes equations governing the fluid. We consider a segregated approach, with a formulation based on the Resistive Immersed Surface method applied to NURBS geometries. After analyzing benchmark fluid simulations with immersed NURBS objects, we report numerical results for the investigation of the dynamics of a vesicle under different flow conditions.

EPFL2017

Chargement

We consider the numerical approximation of lipid biomembranes at equilibrium described by the Canham-Helfrich model, according to which the bending energy is minimized under area and volume constraints. Energy minimization is performed via L-2-gradient flow of the Canham-Helfrich energy using two Lagrange multipliers to weakly enforce the constraints. This yields a highly nonlinear, high order, time dependent geometric Partial Differential Equation (PDE). We represent the biomembranes as single-patch NURBS closed surfaces. We discretize the geometric PDEs in space with NURBS-based Isogeometric Analysis and in time with Backward Differentiation Formulas. We tackle the nonlinearity in our formulation through a semi-implicit approach by extrapolating, at each time level, the geometric quantities of interest from previous time steps. We report the numerical results of the approximation of the Canham-Helfrich problem on ellipsoids of different aspect ratio, which leads to the classical biconcave shape of lipid vesicles at equilibrium. We show that this framework permits an accurate approximation of the Canham-Helfrich problem, while being computationally efficient. (C) 2019 Elsevier B.Y. All rights reserved.

ELSEVIER SCIENCE SA2019

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Some applications of symmetries in differential geometry and dynamical systems

Oana Mihaela Dragulete

EPFL2007

EPFL2017