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We computed separatrix persistence as described in Section 5.2.2 and applied it to two data sets from the fluid dynamics domain: In the first experiment, we extracted the most dominant extremal lines and surfaces from a scalar quantity, i.e., the Q-criterion, derived from fluid dynamics simulation of a Kármán vortex street. Similar as for the bonsai example in Section 5.3.3, we compared our result to the method of Peikert and Sadle [PS08]. The second experiment focuses on the most dominant parts of separa- tion surfaces in an FTLE field. The computation were done on an Intel Xeon E31225 (3.1GHz) CPU and 16 GB RAM. The running time of the complete pipeline as de- scribed in Section 5.2.3 was 3 minutes for the vortex street and 10 minutes for the FTLE field.

Figure 5.17 demonstrates the results of our method applied to a scalar quantity de- rived from a flow behind a cylinder. The data set was provided by Bernd R. Noack (TU Berlin) from a direct numerical Navier Stokes simulation by Gerd Mutschke (FZ Rossendorf). It resolves the so called “mode B” of the 3D cylinder wake at a Reynolds number of 300 and a spanwise wavelength of 1 diameter. The data is provided on a

method.

(c) Ridge lines (green) of Q extracted using [PS08]. (d) Close-up of a ridge line (green) extracted using [PS08].

(e) The separation surfaces (blue) of Q are a part of the strain skeletonfor Q< 0 [SWTH07]. They partition the domain into vortex regions. Inside these regions, maximal lines (red) of Q are shown, i.e., lines of maximal vortical behavior. Lines/surfaces extracted using our method.

(f) All extremal lines of V0.

(g) Most dominant extremal lines filtered and scaled by separatrix persistence.

Figure 5.17: 3D unsteady flow behind a cylinder. Shown are extremal features of the Q-criterion for t= π. The gray isosurface depicts the zero-level of Q while the level Q= 2.7 is illustrated as yellow isosurface.

265 × 337 × 65 curvilinear grid as a low-dimensional Galerkin model. The examined time range is[0, 2π]. The flow exhibits periodic vortex shedding leading to the well known von Kármán vortex street [ZFN+95]. This phenomenon plays an important role in many industrial applications such as mixing in heat exchangers or mass flow mea- surements with vortex counters. However, this vortex shedding can lead to undesirable periodic forces on obstacles such as chimneys, buildings, bridges and submarine tow- ers.

We analyze the Q-criterion [Hun87] of this flow, which is a derived scalar field that allows to distinguish between vortex (Q> 0) and strain (Q < 0) behavior. The latter measures the amount of stretching and folding which drives mixing to occur. As pointed out by Sahner et al. [SWTH07], the minimal points/lines/surfaces of Q repre- sent the strain skeleton, while the maximal features of Q denote the vortex skeleton.

Figures 5.17f-g provide a comparison between the unfiltered extremal lines and the lines filtered and scaled by separatrix persistence. Minimal lines are shown in blue, maximal lines in red. This exemplifies that separatrix persistence is able to reveal the most dominant features. We additionally applied a derived filter criterion here: the variance of separatrix persistence along a line. The idea is to favor lines that stay in the center of a vortex, i.e., that have a rather constant Q-value and therefore a rather constant separation strength.

(a) Volume rendering. (b) Unfiltered repelling separation surfaces.

(c) Surfaces filtered using separatrix persistence.

Figure 5.18: FTLE of the ABC Flow. Filtering the separation surfaces of the Morse- Smale complex using separatrix persistence reveals the most dominant surfaces of max- imal FTLE value.

The most dominant maximal lines and ridges of Q are shown in Figures 5.17a and 5.17c, respectively. The ridge lines are filtered by the F45filter [PS08]. We additionally

removed small isolated lines. Both extraction methods yield qualitatively very similar results. However, the close-ups in Figures 5.17b and 5.17d reveal an important differ- ence of the two approaches: The topological approach gives long, fully connected lines (Figure 5.17b). In contrast, ridge lines are often split into several smaller parts in this data set (Figure 5.17d). This is due to the fact that ridge lines are local features, i.e., it is locally decided whether or not a point is on a ridge or not. Due to numerical insta- bilities or noise, some of the local decisions along a ridge line may produce a “miss”, which then leads to disconnected results. This cannot happen for the topological ap- proach, since separatrices are global features. On the other hand, ridge lines do not suffer from deviations due to smoothing.

Figure 5.17e shows the separation surfaces emanating at the 2-saddles of Q re- stricted to Q< 0. Following Sahner et al. [SWTH07], this provides a partition of the domain into vortex regions, which is nicely confirmed by the shown vortex core lines in the center of each of these regions.

The Finite Time Lyapunov Exponent (FTLE) [Hal01] is a scalar field that describes the separation of particles in a flow: high FTLE values indicate a strong separation, i.e., neighboring particles diverge from each other during integration. Hence, one is interested in finding surfaces of maximal FTLE value. We computed FTLE for the well-known ABC flow from [Hal01]. This is an analytic flow given on a uniform 2563

grid and the numerically computed FTLE field exhibits almost no noise. We include this example to showcase that separatrix persistence is not only useful for filtering noise-induced structures, but also to find the most dominant parts of features, i.e., the parts with the highest feature strength. Figure 5.18a shows a volume rendering of this scalar field. The transfer function has been chosen to show only regions of very high FTLE values. Figure 5.18b shows all separation surfaces emanating at the 1-saddles. We find the most dominant parts of these surfaces by filtering them using separatrix persistence, as shown in Figure 5.18c.

Figure 5.19: Discrete (top) and smooth (bottom) representa- tion of a subregion of the cell membrane.

Figure 5.20: Extraction of a cell membrane in a cryo- electron tomogram using separatrix persistence (left col- umn) and ridge/valley definition (right column). The top row shows the original data. The bottom row shows the results after Gaussian smoothing.