Master's Thesis - Studierstube Augmented Reality Project - Graz ...

2.1 Technical Visualization
Visualizing glyphs, arrows and color maps are based on rendering of simple geometrical
structures at each point of the grid. Therefore these techniques do not need
further explanations. The next step will be to make a definition of particle movements
in steady or unsteady flows.
Related trajectories can be calculated by different approaches.
This thesis concentrates on discrete particle injection methods and finite
differences approximations. Texture based advection techniques which are based on
Lagrangian approaches were not implemented.
∂x
∂t = v(x(t), τ), x(t 0) = x 0 , x : R → R n (2.1)
denotes the continuous movement of a massless particle under the influence of a even
varying vector field, [Kipfer2004; Krueger2005; Weiskopf2007]. v describes a sampled
vector field whose sampled values depend on the current position of an particle x(t).
In the case of a time varying vector field, which equals unsteady flow, τ serves as
parameterization of the time. Together with the initial condition x(t 0 ) this differential
equation is complete. Keeping in mind that the measurement method from chapter 3
produces a vector field defined on a regular spaced lattice and that general programming
approaches cannot solve differential equations in complete form, numerical integration
methods have to be used.
The simplest form to solve the initial value problem 2.1 numerically is the standard
explicit Euler-approach [Euler1768]. Choosing a discretization step size ∆t = h > 0 for
t k+1 = t k + h, (2.2)
leads to an approximation for the exact solution of
x k+1 = x k + hv(x k , t k , τ). (2.3)
The accuracy depends on the selected step size ∆t which is indirect proportional to the
computational costs for the same length of a certain trajectory.
To reduce the integration error or the computation effort of the former described numerical
solution, several correction mechanisms can be added. One example is the well
known Runge-Kutta method [Runge1895; Kutta1901] of a certain order [Albrecht1996;
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