Spectral Methods for Numerical Relativity

Version published online by Living Reviews in Relativity.International audienceEquations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the one-dimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit time-evolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers

The Combustion of Linear Droplet Arrays in a Coaxial Convective Flow.

As approximations for spray-combustion processes, a series of increasingly sophisticated numerical models has been developed to simulate the combustion of linear droplet arrays in a co-axial, convective flow. Common to all of the models is an embedded grid, developed to increase computational accuracy. The first and simplest model is potential flow model (for Re $\to$ $\infty$). The flow is assumed to be ideal and infinitely-fast kinetics (flame sheet assumption) represent the combustion. The results show that the instantaneous droplet burning rates are increased as the droplet spacing is increased, and the burning rates of droplets tend asymptotically to smaller values as the number of droplets in the array is increased. A second model, a Stokes flow model (for Re $\u3c$ 1), is developed by using the Stokes approximation for the flow field. The results show that, owing to the lack of strong convective flow, the temperature and species contours can penetrate deeper into the flow. The third model extends the analysis to heat transfer for linear arrays of spheres at any Reynolds number, i.e. the model is based on the steady-state Navier-Stokes equations, but no combustion is considered. A more accurate flow pattern around the arrays is obtained and the predicted heat transfer and drag agree well with the experimental data in the literature. The fourth model introduces combustion into the third model through a finite-rate, one-step chemical reaction approximation. The model predicts a thick flame layer, rather than a flame sheet. For large Reynolds numbers, the results show that the downstream droplets have a higher burning rate than the leading droplets. For small Reynolds numbers, the model predicts behavior similar to that predicted with the potential and Stokes flow models. Finally, an unsteady-state model, based on the full Navier-Stokes equations, is used to study the variation in burning behavior with time. The reduction in droplet size, velocity, and spacing is included. The results show that even when the droplet spacing is significantly reduced (from 14 to 6 radii) the burning behavior of droplets is not affected

Computational Scattering Models for Elastic and Electromagnetic Waves in Particulate Media

Numerical models were developed to simulate the propagation of elastic and electromagnetic waves in an arbitrary, dense dispersion of spherical particles. The scattering interactions were modeled with vector multipole fields using pure-orbital vector spherical harmonics, and solved using the full vector form of the boundary conditions. Multiple scattering was simulated by translating the scattered wave fields from one particle to another with the use of translational addition theorems, summing the multiple-scattering contributions, and recalculating the scattering in an iterative fashion to a convergent solution. The addition theorems were rederived in this work using an integral method, and were shown to be numerically equivalent to previously published theorems. Both ordered and disordered collections of up to 5,000 spherical particles were used to demonstrate the ability of the scattering models to predict the spatial and frequency distributions of the transmitted waves. The results of the models show they are qualitatively correct for many particle configurations and material properties, displaying predictable phenomena such as refractive focusing, mode conversion, and photonic band gaps. However, the elastic wave models failed to converge for specific frequency regions, possibly due to resonance effects. Additionally, comparison of the multiple-scattering simulations with those using only single-particle scattering showed the multiple-scattering computations are quantitatively inaccurate. The inaccuracies arise from nonconvergence of the translational addition theorems, introducing errors into the translated fields, which minimize the multiple-scattering contributions and bias the field amplitudes towards single-scattering contributions. The addition theorems are shown to converge very slowly, and to exhibit plateaus in convergence behavior that can lead to false indications of convergence. The theory and algorithms developed for the models are broad-based, and can accommodate a variety of structures, compositions, and wave modes. The generality of the approach also lends itself to the modeling of static fields and currents. Suggestions are presented for improving and implementing the models, including extension to nonspherical particles, efficiency improvements for the algorithms, and specific applications in a variety of fields

Electrostatic manipulation of piezoelectric fibres using a sharp probe electrode in a dielectric liquid : analysis of the electrohydrodynamic phenomena

Micro-assembly techniques have been identified as a major technology ‘pillar’ that will underpin further advancements in integrated micro-and nano-systems. In practice, there is a generic requirement for component parts that are often fragile, or that have been prepared by mutually incompatible processes, to be brought together to make a complete working system. This thesis discusses an electrostatic positioning technique for micro-scale elements that could form the basis of an industrial process. A highly non-uniform field generated between a needle-like upper electrode and a bottom flat electrode can be used to electrostatically capture, displace, and relocate elements into a predefined spatial configuration. The very intense field at the needle tip can facilitate the collection of the material at a precise point. However charge injection and local dielectric breakdown must also be considered as they can induce instability near the tip, and consequently interfere with any picking up action. The principal physical phenomena and potential benefits are analysed and discussed, considering three different configurations to achieve the pick and place operation for a micro-fibre in the needle-plane configuration. The first two are operated on an isolated single fibre lying on a flat bottom electrode, applying respectively a DC or an AC voltage. The third case is that of a group of fibres, and it exploits a dielectrophoretic chain structuring effect to assist in the micro-manipulation technique. Experimentation has focussed on the importance of the charge transfer mechanisms, leading to a model which provides good agreement with the observed behaviour. Moreover, an analysis of the forces exerted on the fibres showed that they arise not only from a polarisation effect, but that there is also an electrophoretic contribution. The viability of the proposed technique has been demonstrated using lead zirconate titanate (PZT rods and carbon fibres).EThOS - Electronic Theses Online ServiceGBUnited Kingdo