Scientific Visualization of Multidimensional Geometric Data:a Study Inspired by the Simulator BedSim of LKAB
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This work began as a joint research project between Narvik University College and the R&D Department of LKAB (www.lkab.com), an international high-tech iron ore processing company seated in Kiruna in northern Sweden. The pelletizing process at the processing plants of LKAB has two main consecutive parts: ”the cold part” (not discussed here) and ”the warm part” (a thermo-chemical process of drying, preheating and heating up to about 1400◦C in the so-called Grate, induration process in the so-called Kiln, and cooling). One of our task in this research project was to develop methods and software for 3D and animated 4D space-time visualization of the ”warm part” of the iron-pellets production process, as modeled in LKAB’s own simulator developed in the course of the last 25-30 years, BedSim. The volume of the numerical results from BedSim increases very fast over time, especially if local or global refinement in space and/or time is needed to recover important details. Moreover, the BedSim model contains both non-smooth and very smooth components. We found that in the BedSim model wavelet methods offer an excellent trade-off between compression rates for smooth and non-smooth components (approximately, 95% and 80%, respectively, using an adaptive combination of threshold and non-threshold wavelet shrinkage algorithms of our in-house wavelet library GM-Waves). To compute intersections for level manifolds in 3D, 4D and higher dimensional data set, a ”marching simplex” algorithm were introduced in see , replacing the less efficient ”marching cube” algorithms (a comparison of these two methods is made in ). Rather than using parallel computing architecture to boost up the computational performance, we resorted to the cheaper alternative of GPU-programming. However, the use of GPUs for general computational purpose have a constraint on dimensionality. Therefore, an important and theoretical challenging sub-task became the design of a general computational geometric algorithm for reversible reduction of multidimensional data sets to 2D data sets. For this purpose, in ,  and  we developed an algorithm (based on orthonormal (multi)wavelets) for isometric immersion of smooth n-variate m-dimensional vector fields onto fractal curves and surfaces, m = 3, 4, . . . , n = 3, 4, . . . . During the ongoing development of the project another topic also proved to be of relevance: visualization of the solutions of boundary problems for PDEs of mixed type, due to the fact that the thermo-chemical phenomena in the pelletizing process are best described by PDEs, and, because of the sharp and spatially inhomogeneous changes that can take place, either within very short time (virtually, with a jump in time) or more gradually, due, e.g., to exothermic or endothermic chemical reactions, local change of aggregate phase, etc., the PDEs modelling the process may change type (as well as include various nonlinear and/or integral terms). In a follow-up sequence of 4 papers (, , , ) we studied the scientific visualization of Green’s functions and solutions of initial-value and boundary-value problems for PDEs of 2nd order, by making graphical comparisons for the elliptic, parabolic and hyperbolic case. Our software applications were based on Microsoft .NET2001 - .NET2005 (C++), OpenGL, together with our in-house libraries GM-Lib (for geometric modelling) and GM-Waves (for wavelet processing), and Trolltech Qt for the user interface. The research was a part of the research of the R&D Group for Mathematical Modelling, Numerical Simulation and Computer Visualization at Narvik University College within two consecutive Strategic Projects of the Norwegian Research Council – the now completed project ”GPGPU – Graphics Hardware as a High-end Computational Resource” (2004-2007), and the ongoing project ’Heterogeneous Computing’ (2008- 2010).