Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn–Hilliard equation.     

Evaluation of the effect of silicon on the carbonization process of Colombian semi-anthracites

The environmental impacts associated with the exploitation and transformation of fossil resources aggravate the planet's situation in terms of climate change. Due to this, this paper studies an alternative use of mineral coal as a precursor to obtaining new materials with different properties to the starting coals. The thermal degradation of two Colombian semi-anthracites is analyzed through the thermogravimetry (TG) technique coupled to a Fourier transform infrared spectroscopy (FTIR) equipment from room temperature (25 °C) to 900 °C, at a heating rate of 10 K min^?1 in an inert atmosphere. The catalytic effect of the addition of silicon to these samples before being subjected to a carbonization process is evaluated during this process. The results indicated that the primary reaction occurs in the temperature range between 400 and 680 °C, where the highest mass loss rate was observed. At the end of the heating process, the TG profile of the samples with silicon addition showed losses between 14.33 and 18.82% in mass, these values being slightly higher compared to the starting and demineralized samples. The release of water, light gases such as CO₂, CH₄, and species such as toluene, phenol and formic acid was identified in most of the samples. The presence of silica seems to favor the release of all these species, being more evident in one of the semi-anthracites studied. According to the results obtained, it is proved that the presence of silicon in samples subjected to carbonization processes has a catalytic effect that improves some characteristics of the new materials obtained, thus contributing to the use of carbon to get new materials.

     

Stars in five-dimensional Kaluza–Klein gravity

In the five-dimensional Kaluza–Klein (KK) theory there is a well known class of static and electromagnetic-free KK-equations characterized by a naked singularity behavior, namely the Generalized Schwarzschild solution (GSS). We present here a set of interior solutions of five-dimensional KK-equations. These equations have been numerically integrated to match the GSS in the vacuum. The solutions are candidates to describe the possible interior perfect fluid source of the exterior GSS metric and thus they can be models for stars for static, neutral astrophysical objects in the ordinary (four-dimensional) spacetime.     

Defining and optimizing algorithms for neighbouring particle identification in SPH fluid simulations

Lagrangian particle methods such as smoothed particle hydrodynamics (SPH) are very demanding in terms of computing time for large domains. Since the numerical integration of the governing equations is only carried out for each particle on a restricted number of neighbouring ones located inside a cut‐off radius r_c, a substantial part of the computational burden depends on the actual search procedure; it is therefore vital that efficient methods are adopted for such a search. The cut‐off radius is indeed much lower than the typical domain's size; hence, the number of neighbouring particles is only a little fraction of the total number. Straightforward determination of which particles are inside the interaction range requires the computation of all pair‐wise distances, a procedure whose computational time would be unpractical or totally impossible for large problems. Two main strategies have been developed in the past in order to reduce the unnecessary computation of distances: the first based on dynamically storing each particle's neighbourhood list (Verlet list) and the second based on a framework of fixed cells. The paper presents the results of a numerical sensitivity study on the efficiency of the two procedures as a function of such parameters as the Verlet size and the cell dimensions. An insight is given into the relative computational burden; a discussion of the relative merits of the different approaches is also given and some suggestions are provided on the computational and data structure of the neighbourhood search part of SPH codes. Copyright © 2008 John Wiley & Sons, Ltd.