A flow approach to Bartnik's static metric extension conjecture in axisymmetry

We investigate Bartnik's static metric extension conjecture under the additional assumption of axisymmetry of both the given Bartnik data and the desired static extensions. To do so, we suggest a geometric flow approach, coupled to the Weyl-Papapetrou formalism for axisymmetric static solutions to the Einstein vacuum equations. The elliptic Weyl-Papapetrou system becomes a free boundary value problem in our approach. We study this new flow and the coupled flow--free boundary value problem numerically and find axisymmetric static extensions for axisymmetric Bartnik data in many situations, including near round spheres in spatial Schwarzschild of positive mass.Comment: 60 pages, 13 figures. Expanded Section 3.3 to address longtime existence and uniqueness of solutions to the linearised flow equations. To appear in Pure and Applied Mathematics Quarterly, special issue in honour of Robert Bartni

Hofer's L∞L^{\infty}-geometry: energy and stability of Hamiltonian flows, part I

Consider the group \Ham^c(M) of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold (M,\om) with the Hofer $L^{\infty}$-norm. A path in \Ham^c(M) will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional \Ll. In this paper, we give a necessary condition for a path \ga to be a geodesic. We also develop a necessary condition for a geodesic to be stable, that is, a local minimum for \Ll. This condition is related to the existence of periodic orbits for the linearization of the path, and so extends Ustilovsky's work on the second variation formula. Using it, we construct a symplectomorphism of $S^2$ which cannot be reached from the identity by a shortest path. In later papers in this series, we will use holomorphic methods to prove the sufficiency of the condition given here for the characterisation of geodesics as well as the sufficiency of the condition for the stability of geodesics. We will also investigate conditions under which geodesics are absolutely length-minimizing

Non-collapsing in fully nonlinear curvature flows

We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the hy- persurface at each point is a subsolution of the linearized flow equation if the speed is concave. If the speed is convex then there is an analogous statement for exterior spheres. In particular, if the hypersurface moves with positive speed and the speed is concave in the principal curvatures, then the curvature of the largest touching inte- rior sphere is bounded by a multiple of the speed as long as the solution exists. The proof uses a maximum principle applied to a function of two points on the evolving hypersurface. We illustrate the techniques required for dealing with such functions in a proof of the known containment principle for flows of hypersurfaces

Large-deflection elasto-plastic analysis of discretely stiffened plates

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Active contraction of the left ventricle with cardiac tissue modelled as a micromorphic medium

The myocardium is composed of interconnected cardiac fibres which are responsible for contraction of the heart chambers. There are several challenges related to computational modelling of cardiac muscle tissue. This is due in part to the anisotropic, non-linear and time-dependent behaviour as well as the complex hierarchical material structure of biological tissues. In general, cardiac tissue is treated as a non-linear elastic and incompressible material. Most computational studies employ the theories of classical continuum mechanics to model the passive response of the myocardium and typically assume the myocardium to be either a transversely isotropic material or an orthotropic material. In this study, instead of a classical continuum formulation, we utilise a micromorphic continuum description for cardiac tissue. The use of a micromorphic model is motivated by the complex microstructure and deformations experienced by cardiac fibres during a heartbeat. The micromorphic theory may be viewed as an extension of the classical continuum theory. Within a micromorphic continuum, continuum particles are endowed with extra degrees of freedom by attaching additional vectors, referred to as directors, to the particles. In this study the directors are chosen such that they represent the deformation experienced by the cardiac fibres. In addition to the passive stresses, the myocardium experiences active stresses as a result of the active tension generated by cardiac fibres. The active tension in the heart is taken to be a function of the sarcomere length, intracellular calcium concentration and the time after the onset of contraction. Experimental studies show that the active behaviour of the myocardium is highly dependent on the tissue arrangement in the heart wall. With a classical continuum description, the sarcomere length is usually defined as a function of the stretch in the initial fibre direction. To allow for a more realistic description of the active behaviour, we define the sarcomere orientation, and consequently also the sarcomere stretch, as a function of the director field. Furthermore, we use the director field to describe the direction in which contraction takes place. The intent of this study is to use a micromorphic continuum formulation and an active-stress model to investigate the behaviour of the left ventricular myocardium during a heartbeat. The simulated results presented here correspond well with typical ventricular mechanics observed in clinical experiments. This work demonstrates the potential of a micromorphic formulation for analysing and better understanding ventricular mechanics

Parametric reduced-order aeroelastic modelling for analysis, dynamic system interpolation and control of flexible aircraft

This work presents an integral framework to derive aeroelastic models for very flexible aircraft that can be used in design routines, operational envelope analysis and control applications. Aircraft are modelled using a nonlinear geometrically-exact beam model coupled with an Unsteady Vortex-Lattice Method aerodynamic solver, capable of capturing important nonlinear couplings and effects that significantly impact the flight characteristics of very flexible aircraft. Then, complete linearised expressions of the aircraft system about trim reference conditions at possibly large deformations are presented. The nature of the aerodynamic models results in a high-dimensional system that requires of model reduction methods for efficient analysis and manipulation. Krylov-subspace model reduction methods are implemented to reduce the dimensionality of the multi-input multi-output linearised aerodynamic model and achieve a very significant reduction in the size of the size of the system. The reduced aerodynamic model is then coupled with a modal expression of the linearised beam model, resulting in a compact aeroelastic state-space that can be efficiently used on desktop hardware for linear analysis or as part of internal control models. These have been used to explore the design space of a very flexible wing with complex aeroelastic properties to determine the flutter boundaries, for which experimental data has become available that validates the methods presented herein. Additionally, they have been integrated in a model predictive control framework, where the reduced linear aerodynamic model is part of the control model, and the simulation plant is the nonlinear flight dynamic/aeroelastic model connected as a hardware-in-the-loop platform. Finally, in order to accelerate the design space exploration of very flexible structures, state-space interpolation methods are sought to obtain, with a few linearised models sampled across the domain, interpolated state-spaces anywhere in the parameter-space in a fast and accurate manner. The performance of the interpolation schemes is heavily dependent on the location of the sampling points on the design space, therefore, a novel adaptive Bayesian sampling scheme is presented to choose these points in an optimal approach that minimises the interpolation error function.Open Acces

Non-collapsing on Hypersurfaces of Prescribed Curvature

The establishment and development of a non-collapsing technique have recently received a great deal of attention in geometric analysis. The area of study conducted in this thesis is motivated by this technique which is based on the application of the maximum principle to a two-point function that is defined on manifolds or relies on some global information of partial differential equations PDEs. This approach is utilized in order to acquire some useful knowledge on the behaviour of geometric structures and properties of embedded hypersurfaces. Despite of that this technique is used in various settings in geometric analysis, it still has a similar machinery in general. Our first purpose of study concerns the uniqueness of a class of embedded Weingarten hypersurfaces in the higher dimensional sphere $\mathbb{S}^{n+1}$. In particular, we consider a more general class of embedded Weingarten hypersurfaces with two distinct principal curvatures into $\mathbb{S}^{n+1}$. Also, we assume that these hypersurfaces satisfying a PDE of the principal curvatures with assumptions on the number of multiplicities of these curvatures. As a result, we deduce that these principal curvatures are both constant and consequently our hypersurfaces are congruent to a Clifford torus. One of the key ingredients in this work is based on the smart deployment of the maximum principle argument to a function of two variables that is defined on our hypersurfaces. The second main object of study is the mean convex mean curvature flow in Minkowski space $\mathbb{L}^{n,1}$. Of particular interest is the study of co-compact mean convex embedded spacelike hypersurfaces evolving by the mean curvature flow into $\mathbb{L}^{n,1}$. In particular, we deduce a non-collapsing estimate by employing the maximum principal to a quantity that relies on two points of these spacelike hypersurfaces. More precisely, we compare the radius of the largest hyperbola which touches the spacelike hypersurface at a given point to the curvature at that point. Also, we assume that our spacelike hypersurfaces are asymptotic to the hyperplane in order to to control the behaviour of such hypersurfaces at infinity and ensure that the maximum principle is applicable to our setting