Theme: Aerospace.- Theme: Electronic Industry.- Theme: Chemical Technology.- Theme: Life Sciences.- Theme: Materials.- Theme: Geophysics.- Theme: Financial Mathematics.- Theme: Water Flow.- Other Contributions.
Fast Numerical Computing for a Family of Smooth Trajectories in Fluids Flow ( p. 39)
G. Argentini Riello Group, via degli Alpini 1, 37045 Legnago (Verona), Italy gianluca.argentini@riellogroup.com
Summary.
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 for flows in a generic geometry. The principal result of the work was implementation and test of a method for interpolation of 3D points by regular parametric curves, and fast and efficient evaluation of these functions for a good resolution of rendering.
For this purpose I have used a parallel environment using a multiprocessor cluster architecture. The effciency of the used method is good, mainly reducing the number of floating-points computations by caching the numerical values of some line-parameter’s powers, and reducing the necessity of communication among processes. This work has been developed for the Research &, Development Department of my company for planning advanced customized models of industrial burners.
Key words:
computational fluid dynamics, cubic spline interpolation, parallel computing, parallel effciency.
1 Introduction
Industrial and power burners have some particular requirements, as a customized study of the geometry for combustion head and combustion chamber for an optimal shape of the flame. Rapid prototyping for an accurate design of the correct geometry involves a numerical simulation of the gas or oil flows in the burner’s components.
The necessity of an high graphic resolution requires a large amount of particles paths for tracing the streamlines of flow. Hence the numerical computation is memory and cpu very expensive for the used hardware environment. In a tipical simulation the number of paths to compute is some thousands, and the number of geometrical points to interpolate for each path is some thousands too. For the treatment of this large amount of data a parallel environment can be very useful.
2 Fitting trajectories with cubic polynomials
We suppose to have a dataset output from pre-processing and processing phases of a simulation, for example from numerical resolution of Navier-Stokes equations or from Cellular Automaton models [1].We would a fast and flexible method to obtain from those data an accurate paths tracking of fluid particles with a smooth 3D visualization of trajectories, possibly with continuous slope and curvature.
Our experience shows that Computational Fluid Dynamics packages have some limits in this post-processing phase, principally due to a rigid resolution of the initial mesh and to a small degree of parallelism. Let S the number of 3D points for each trajectory andMthe total number of trajectories from simulation dataset.
We have tested that usual interpolation methods have some disadvantages for our aims: for example Bezier-like is not realistic in case of twisting or diverging speed-fields, Chebychev or Least- Squares-like are too rigid for a customized application, polynomial flitting is simple but often shows spurious effects as Runge phenomenon [6]. We have elaborated a spline-based technique.
