Glosarium Matematika

273 p.; 24 cm

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

Numerical Methods for Optimal Transport and Elastic Shape Optimization

In this thesis, we consider a novel unbalanced optimal transport model incorporating singular sources, we develop a numerical computation scheme for an optimal transport distance on graphs, we propose a simultaneous elastic shape optimization problem for bone tissue engineering, and we investigate optimal material distributions on thin elastic objects. The by now classical theory of optimal transport admits a metric between measures of the same total mass. Various generalizations of this so-called Wasserstein distance have been recently studied in the literature. In particular, these have been motivated by imaging applications, where the mass-preserving condition is too restrictive. Based on the Benamou Brenier formulation we present a novel unbalanced optimal transport model by introducing a source term in the continuity equation, which is incorporated in the path energy by a squared L2-norm in time of a functional with linear growth in space. As a key advantage of our model, this source term functional allows singular sources in space. We demonstrate the existence of constant speed geodesics in the space of Radon measures. Furthermore, for a numerical computation scheme, we apply a proximal splitting algorithm for a finite element discretization. On discrete spaces, Maas introduced a Benamou Brenier formulation, where a kinetic energy is defined via an appropriate (e.g., logarithmic) averaging of mass on nodes and momentum on edges. Concerning a numerical optimization scheme, this, unfortunately, couples all these variables on the graph. We propose a conforming finite element discretization in time and prove convergence of corresponding path energy minimizing curves. To apply a proximal splitting algorithm, we introduce suitable auxiliary variables. Besides similar projections as for the classical optimal transport distance and additional simple operations, this allows us to separate the nonlinearity given by the averaging operator to projections onto three-dimensional convex sets, the associated (e.g., logarithmic) cones. In elastic shape optimization, we are usually concerned with finding a subdomain maximizing the mechanical stability w.r.t. given forces acting onto a larger domain of interest. Motivated by a biomechanical application in bone tissue engineering, where recently biologically degradable polymers have been explored as bone substitutes, we propose a simultaneous elastic shape optimization problem to guarantee stiffness of the polymer implant and of the complementary set where new bone tissue will grow first. Under the assumption that the microstructure of the scaffold is periodic, we optimize a single microcell. We define a novel cost functional depending on specific entries of the homogenized elasticity tensors of polymer and regrown bone. Additionally, the perimeter is penalized for regularizing the interface of the scaffold. For a numerical optimization scheme, we choose a phase-field model, which allows a diffuse approximation of the elastic objects and the perimeter by the Modica Mortola functional. We also incorporate further biomechanically relevant constraints like the diffusivity of the regrown bone. Finally, we investigate shape optimization problems for thin elastic objects. For a numerical discretization, we take into account the discrete Kirchhoff triangle (DKT) element for parametric surfaces and approximate the material distribution by a phase-field. To describe equilibrium deformations for a given force, we study different corresponding state equations. In particular, we consider nonlinear elasticity combining membrane and bending models. Furthermore, a special focus is on pure bending isometries, which can be efficiently approximated by the DKT element. We also analyze a one-dimensional model of nonlinear elastic planar beams, where our numerical simulations confirm and extend a theoretical classification result of the optimal design