PARA'04 State-of-the-Art
in Scientific Computing
June 20-23, 2004 (Home page)
Updated: February 1, 2004

High performance computing for a family of smooth trajectories using multinode cluster

Gianluca Argentini
Riello Group
New Technologies & Models Department
Verona, Italy

In this work I present a technique of construction and fast evaluation of a family of cubic polynomials for analytic smoothing and graphical rendering of particles trajectories in generic geometry fluid flows. The principal aims of the work are:

1.

the interpolation of 3D points by regular parametric curves; the improved technique permits to obtain smoothed geometric lines even in situations where there are few data-points or where the flow is turbulent;

2.

a fast and efficient evaluation of these polynomials in a set of suitable values of the parameter for a good resolution of graphic rendering; the method is based on parallel computing on a multiprocessor environment;

3.

the measure of speedup and efficiency for scientific and technical applications using cluster computing techniques.

The numerical approch is based on a cellular automaton evolving on a three-dimensional grid. This mechanism simulates in an adaptive manner the behavior of the flow to obtain the discrete set of data-points for every particle. After this first step, the smoothed curves are computed by interpolation of the points using a combination of Bezier method and piecewise cubic splines. These splines are polynomials where the coefficients are solutions of linear systems obtained imposing adequate conditions for slope and curvature. The functions so computed have the regular properties of Bezier curves, the simple algebraic expression of cubic polynomials and avoid the possible rising of spurious wiggles and other not realistic effects as Gibbs phenomenon. For an appropriate visualization of the flow, we use a computational method based on an appropriate distribution of the polynomials among the available processors, and a fine resolution for the polynomials variable values. The performances of the method are vey good, mainly reducing the number of floating-points computations by caching the numerical values of the polinomials parameter's powers, and reducing the necessity of communication among processes. The computation is performed using a customized parallel environment for the package Matlab on a multinode 8 processors Linux - Windows cluster. The work permits to deduce these conclusions:

1.

it's possible to obtain smooth and realistic rendering of a flow even in situations where the geometry of the interested region is not easily schematizable by standard mathematical methods such as finite elements;

2.

the parallel method used has a good level of computational efficiency (about 0.8 - 0.9 in our experiments, hence a quasi-linear speedup).

This work has been developed for the Research & Development Department of our company for planning advanced customized models of industrial burners; in particular the results are useful for simulating fluid flows in the combustion chambers.

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