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1 Introduction

  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.

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