Log

Streamline and trajectory are related topics in physical modeling, common seen in fluid and particle simulations.

First make it work, then make it better and fast.

One approach is called Pollock method.

Tracing in Unstructured Grid

Given an unstructured grid with node points and connectivity, how should you do the streamline tracing?

Brute force algorithm:

1. find the grid cell you are currently in;
2. move along the vector direction until you hit the boundary of that cell;
3. find the neighbour who shares the same edge and the intersection point;
4. use the vector direction in the next cell and move along that direction;
5. repeat 2-4 until you reach any of the stopping criteria: hit the boundary, exceed MaxIteration, or exceed MaxLength.

Some questions during the process:

• How to find the neighbouring cell?
• How to determine which boundary edge will you cross?
• How to improve the search speed?
• How to improve accuracy?

MATLAB

There is an implementation of streamline tracing in Matlab called tristream. It requires nodal data.

Inside the function, there is an intrinsic function called pointLocation, which returns the index of cell where the point locates.

YT

There is another implementation in yt library, which has many similarities to the one I borrowed from SpacePy.

Streamlining through a volume is useful for a variety of analysis tasks. By specifying a set of starting positions, the user is returned a set of 3D positions that can, in turn, be used to visualize the 3D path of the streamlines. Additionally, individual streamlines can be converted into YTStreamline objects, and queried for all the available fields along the streamline.

The implementation of streamlining in yt is described below.

1. Decompose the volume into a set of non-overlapping, fully domain tiling bricks, using the AMRKDTree homogenized volume.

2. For every streamline starting position:

• While the length of the streamline is less than the requested length:

  1. Find the brick that contains the current position.

  2. If not already present, generate vertex-centered data for the vector fields defining the streamline.

  3. While inside the brick:

  4. integrate the streamline path using a Runge-Kutta 4th order method and the vertex centered data.

  5. during the intermediate steps of each RK4 step, if the position is updated to outside the current brick, interrupt the integration and locate a new brick at the intermediate position.

1. The set of streamline positions are stored in the Streamlines object.

VTK

In the VTK library, there is a class called vtkPointLocator. It is a spatial search object to quickly locate points in 3D. vtkPointLocator works by dividing a specified region of space into a regular array of "rectangular" buckets, and then keeping a list of points that lie in each bucket. Typical operation involves giving a position in 3D and finding the closest point. It supports both nodal data and cell data.

vtkPointLocator has two distinct methods of interaction. In the first method, you supply it with a dataset, and it operates on the points in the dataset. In the second method, you supply it with an array of points, and the object operates on the array.