Topology-Based Methods in Visualization 2015 Compute and Visualize Discontinuity Among Neighboring Integral Curves of 2D Vector Fields Lei Zhang and Guoning Chen, Department of Computer Science, University of Houston Robert S. Laramee, Swansea University David Thompson and Adrian Sescu, Mississippi State University 1 Outline Background and motivation Attribute Fields Definition and Computation Discontinuity in the Attribute Fields

Correlations Among Attributes Visual Exploration Based on Attribute Fields Conclusions and Discussions 2 Vector Fields and Vector Field Analysis Applications

Computational fluid dynamics Mechanical engineering Automobile/aircraft design Climate study Oceanography New energy Chemistry reaction Medical applications Computer graphics

[Chen and Xu 2015] [Edmunds et al.2013] [Edmunds et al.2012] [Bhatia et al. 2012] [Sanderson et al. 2011] 3 Integral Curve Attributes Sharp or rapid

change of attributes (highlighted by the arrows) at certain SPATIAL location Seeded Streamlines Plots of attributes along seeding segments 4 Integral Curve Attributes +

0 - Sharp or rapid change of attributes can be easily detected Texture-based method has employed this Eulerian representation! Seeded Streamlines Encode attributes to spatial locations

5 Attribute Fields and Their Computation Consider a vector field defined in a spatio-temporal domain , The attribute of a streamline passing through can be computed using 1 ( ) = ( ) ( ( ) , ) s is the arc-length parameter 0 while the attribute of a pathline passing through can be computed using 2

( , ) = ( ) ( , ( ) , ) 1 is the time parameter [Shi et al. 2008] Definition of attribute fields The attribute values at any given positions can be computed using the above formulas, which gives rise to an attribute field . 6 Attribute Fields and Their Computation Computation of an attribute field

We employ a uniform sampling strategy for all examples For steady flows, we compute streamlines until they reach the boundary of the domain, hit the fixed points, form loops, or longer than some user-specified threshold. For unsteady flows, we currently consider only forward integration of pathlines and compute them within a user-specified time window. The selected attributes are collected along the obtained integral curves using the following discrete version of the accumulation. ( , ) = ( ) =0 Assuming the integral curve is represented by integration points We use box-filter for all the examples, therefore,

7 Attribute Fields for 2D Steady and Unsteady Flows + 0 - Attribute fields of the 2D steady flows are defined in the original flow domain (i.e., 2D) Attribute fields of the 2D unsteady flows are defined in a spatio-temporal domain (i.e., 3D) 8

Choices of Attributes List of attributes to consider, based on the recently work of [Shi et al. 2009] and [Pobitzer et al. 2012] Rotation field (i.e., winding angle)i.e., winding angle) Length field

Average particle velocity field avgV Non straight velocity field nsV, Relative start end distance field seDist Average direction field avgDir Acceleration acce curl field, Hunts Q Although the framework of attribute field computation is much more general, in this paper, we concentrate on those scalar attributes 9 Discontinuity in Attribute Fields

Informal definition Places in the flow domain whose attribute values exhibit sharp/rapid change when compared with their neighboring values. avgDir acce The average direction field may have some artificial discontinuity due to choice of the color coding. Here we simply map the angle to white-red color.

10 Discontinuity in Attribute Fields Informal definition Places in the flow domain whose attribute values exhibit sharp/rapid change when compared with their neighboring values. How to detect? Using Canny edge detector

Using gradient magnitude field Parameters for Canny edge detector: 11 A Very Informal Discussion Topology and LCS FTLE Cusp-like seeding curves Topology field

field Start to end point direction Transportation structure of materials vs. LCS structure [Shi et al. 2008] 12 A Very Informal Discussion Topology and LCS Vortex regions

Cusp-like seeding curves Singularity paths curl field field 13 A Very Informal Discussion Topology and LCS Vortex regions

Cusp-like seeding curves Sampled pathlines Pathline-based field Singularity paths Streak line-based field Sampled streak lines 14 Correlations Among Attributes Correlation Study Via Pairwise Scatter Plots of different

attribute fields of the Double gyre flow 15 Correlations Among Attributes Length Field vs. Average Particle Velocity Field avgV These two fields are highly correlated. Because the arc-length of each integral curve is equal to the sum of the velocity magnitudes, scaled by the integration stepsize, measured along this curve 16

Correlations Among Attributes curl , curl, Q, fields Q curl All these four attribute fields are related to the rotational behavior of the flow. Except for the Q field which is always negative, the other three fields values can be both positive and negative. Among them, the plots related to curl field and field demonstrate some symmetry patterns.

17 Correlations Among Attributes Acceleration field acce vs. other attribute fields When the value of the acce field is small, the other attributes tend to be small. When the value of the acce field is increasing and becomes large enough, the other attribute values tend to be large as well. This matches the common knowledge that the accelerationa result of the external force based on Newton Second Law, is the source of many different flow behaviors, such as flow separation and rotation 18 Correlations Among Attributes FTLE vs. the gradient of the attribute fields

It shows quite some strong correlation between the gradient of the attribute fields and the FTLE field. This supports the discussion of the relationship between the transportation structure of certain materials and LCS. [Shi et al. 2008] 19 Correlations Among Attributes avgDir + Spatial Correlation via Combined Attribute Fields Black curves show stable edges under different combinations 20

Visual Exploration Based on Attribute Fields 21 System Screenshot User Interface Visualization Visualization Options Parameters Setup Some Results avgV field nsV

field field FTLE and LCS Attribute fields (left) of the flow behind cylinder and detected edges (right) The parameters of Canny edge detector are s =1.0, =0.3, = 0.8. 23 Some Results

Common edges Attribute fields and their detected edges of force duffing oscillator [Haller and Sapsis, 2011] 24 Some Results Curl-based attribute field of the Bernard data set The rotation field, , of the Lorenze strange attractor The length field of the

tornado data set 25 Conclusion and Discussions Potential benefits of the attribute fields Integral curve and surface seeding Flow visualization in higher-dimensional space Flow domain segmentation Flow pattern retrieval Multi-field analysis 26 Conclusion and Discussions

Potential benefits of the attribute fields Integral curve and surface seeding Flow visualization in higher-dimensional space Flow domain segmentation Flow pattern retrieval Multi-field analysis Problems of the attribute fields Still lack of a rigorous description of the discontinuity and its relation to standard flow features. Correlation analysis can be performed in a more general sense. The correlation and dependency among different attribute fields are not fully investigated The sampling strategy and integration time play an important role as FTLE computation. Memory consumption. 27

Acknowledgments People Jacqueline H. Chen, Sandia National Lab Mathew Maltude, Lawrence Livermore National Lab James Liburdy, Oregon State University Funding Angency National Science Foundation 28