**Next:** 2 Adding Time-Dependency to **Up:**
Visualizing Data from Time-Dependent **Previous:**
Visualizing Data from Time-Dependent

In PDE models of natural phenomena, most of the interesting effects correspond to certain nonlinearities. Many of them, such as shocks, cracks, flames, melting zones, vortex cores, are local in space and move in time. Numerical methods which are based on a fixed spatial discretization meet with serious difficulties in resolving all the important scales of the solution. Therefore, especially in the last decade, powerful adaptive methods on unstructured grids (e.g. triangular or tetrahedral meshes) have been developed [20][11][2], that, by use of local refinement and coarsening [27][5][4], capture local phenomena with a moderate total amount of unknowns.

When we try to understand visualization results, debug a numerical code
visually, or want to extract interesting features of the global geometry of the
calculated solution, advanced visualization comes into play. Here the flexible
and interactive handling of numerical data sets is an essential ingredient of an
efficient visualization environment [38][37][26][14][3]. A variety of
widespread visualization tools is currently in use to visualize simulation data [33][19][10][1]. Object oriented
concepts are taken into account to structure visualization environments [35][32][12]. Unfortunately, for numerical methods which
adapt the underlying grid and the time step width, the support from most of the
well-known visualization packages is quite limited. One has obtained faster and
more efficient simulations with better error control, but at the price of
complex adaptive meshes where discretization changes with time. At the first
glance it seems to be unclear how to set up an effective interface for the
visualization of the latter type of data. We have tried to fill this gap and
have developed a concept to handle arbitrary time-dependent processes in an
interactive graphical environment. Such a process is no longer taken for a more
or less dense list of keyframes, each containing the solution at a distinct
time. We deal with the entire process as one *time-object* [30]. In the sense of OOP, *time-object* is an
abstract expansion of any type of stationary *object*, i.e. a subclass of
some stationary class. It represents a continuous one parameter family of
objects of the initial type.

Adaptive simulation data we mainly consider here also form such a family of
objects and the result can be mapped to a specific *time-object*. In
general, the data base consists of a number of time steps. With an appropriate
interpolation scheme, the simulation process can be evaluated at any time in
between. This scheme is closely related to the adaptive numerical calculation
and takes into account changes in discretization. However, the concept of *time-objects*
is not restricted to handling the whole simulation process. Popular CFD
expedients like particle traces [28][24][17][7], stream surfaces [40][41][18] or moving test
sets can again be regarded as time-dependent processes which fit well into our
setting. It then depends on the display style for the specific *time-object*
whether we visualize the moving particles, curves and surfaces, or their traces.
In [25], local characteristics of the flow are
visualized by placing specific icons at points of interest. Icons are also used
by [16][13] to
visualize critical points together with some information on the underlying
tensor fields. A set of icons moving in time and undergoing modification will be
an additional example of extracted features forming dynamic processes.

Furthermore it should be possible, although it's still a lot of work, to handle topological information contained for instance in hyperstreamlines [9], vortex cores [6] or the boundaries of topological regions [16][15] in a similar manner.

Our concept has been implemented in GRAPE [42], an object oriented environment, developed at the SFB 256 at Bonn University and at the Institute for Applied Mathematics at Freiburg University [39] [31]. And yet, this presentation should underline that the methodology introduced here can easily be ported to other environments. Although the object oriented approach is the essential ingredient, we shall not go into details, but hope for an intuitive understanding of message passing, late binding and inheritance in the background.

The outline of the paper is as follows. First we will introduce the concept
of *time-objects* and describe how to work with them in an interactive
environment and with respect to animation purposes. The next section is
concerned with the structure of the dynamic process supervised by a *time-object*
when the underlying data is an adaptive time-dependent simulation. Next, it will
be pointed out that the integration techniques on CFD data fit into this
framework, and an appropriate integration scheme will be presented that takes
into account the properties of an adaptive solution on an unstructured grid.
Finally, we consider local probing at possibly moving positions in the
simulation domain.

**Next:** 2 Adding Time-Dependency to **Up:**
Visualizing Data from Time-Dependent **Previous:**
Visualizing Data from Time-Dependent

Universität Freiburg

Mon Jul 24 23:57:35 MDT 1995