Finite element methods for fourth order axisymmetric geometric evolution equations

Fourth order curvature driven interface evolution equations frequently appear in the natural sciences. Often axisymmetric geometries are of interest, and in this situation numerical computations are much more efficient. We will introduce and analyze several new finite element schemes for fourth order geometric evolution equations in an axisymmetric setting, and for selected schemes we will show existence, uniqueness and stability results. The presented schemes have very good mesh and stability properties, as will be demonstrated by several numerical examples

Geometric integration on spheres and some interesting applications

Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility of these algorithms, we present representative calculations for reduced free rigid body motion (a conservative ODE) and a discretization of micromagnetics (a dissipative PDE). We emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms; this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions.Comment: This paper appeared in prin

Numerical evolution of axisymmetric, isolated systems in General Relativity

We describe in this article a new code for evolving axisymmetric isolated systems in general relativity. Such systems are described by asymptotically flat space-times which have the property that they admit a conformal extension. We are working directly in the extended `conformal' manifold and solve numerically Friedrich's conformal field equations, which state that Einstein's equations hold in the physical space-time. Because of the compactness of the conformal space-time the entire space-time can be calculated on a finite numerical grid. We describe in detail the numerical scheme, especially the treatment of the axisymmetry and the boundary.Comment: 10 pages, 8 figures, uses revtex4, replaced with revised versio

On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow

In this article, we study the axisymmetric surface diffusion flow (ASD), a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older regular surfaces of revolution embedded in R^3 and satisfying periodic boundary conditions. We also give conditions for global existence of solutions and prove that solutions are real analytic in time and space. Further, we investigate the geometric properties of solutions to ASD. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. Then, focusing on the family of cylinders, we establish results regarding stability, instability and bifurcation behavior, with the radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana

An axisymmetric evolution code for the Einstein equations on hyperboloidal slices

We present the first stable dynamical numerical evolutions of the Einstein equations in terms of a conformally rescaled metric on hyperboloidal hypersurfaces extending to future null infinity. Axisymmetry is imposed in order to reduce the computational cost. The formulation is based on an earlier axisymmetric evolution scheme, adapted to time slices of constant mean curvature. Ideas from a previous study by Moncrief and the author are applied in order to regularize the formally singular evolution equations at future null infinity. Long-term stable and convergent evolutions of Schwarzschild spacetime are obtained, including a gravitational perturbation. The Bondi news function is evaluated at future null infinity.Comment: 21 pages, 4 figures. Minor additions, updated to agree with journal versio

From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity

This article reviews some aspects in the current relationship between mathematical and numerical General Relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed on the light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity. Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24 November, 2006), part of the "General Relativity Trimester" at the Institut Henri Poincare (Fall 2006). Comments and references added. Typos corrected. Submitted to Classical and Quantum Gravit

An object-oriented programming of an explicit dynamics code: application to impact simulation

During the last fifty years, the development of better numerical methods and more powerful computers has been a major enterprise for the scientific community. Recent advances in computational softwares have lead to the possibility of solving more physical and complex problems (coupled problems, nonlinearities, high strain and high strain rate problems, etc.). The development of object-oriented programming leads to better structured codes for the finite element method and facilitates the development, the maintainability and the expandability of such codes. This paper presents an implementation in C++ of an explicit finite element program dedicated to the simulation of impacts. We first present a brief overview of the kinematics, the conservative and constitutive laws related to large deformation inelasticity. Then we present the design and the numerical implementation of some aspects developed with an emphasis on the object-oriented programming adopted. Finally, the efficiency and accuracy of the program are investigated through some benchmark tests

Supersonic quasi-axisymmetric vortex breakdown

An extensive computational study of supersonic quasi-axisymmetric vortex breakdown in a configured circular duct is presented. The unsteady, compressible, full Navier-Stokes (NS) equations are used. The NS equations are solved for the quasi-axisymmetric flows using an implicit, upwind, flux difference splitting, finite volume scheme. The quasi-axisymmetric solutions are time accurate and are obtained by forcing the components of the flowfield vector to be equal on two axial planes, which are in close proximity of each other. The effect of Reynolds number, for laminar flows, on the evolution and persistence of vortex breakdown, is studied. Finally, the effect of swirl ration at the duct inlet is investigated