SOM Neural Network for Particle Tracking Velocimetry Aifeng Yao

What is Particle Tracking Velocimetry? PIV is a method for measuring a 2D velocity vector map of a flow field at an instant in time by matching the images of small suspended seeding particles in the flow taken in a small time interval. Particle tracking One of the first equations people learn, perhaps without realising it, is It is simply a mathematical way of expressing the practical point that, if you know how fast you are travelling and the time you use for the journey, you can work out the distance travelled. Any one of the three parameters can of course be calculated if you know the other two, simply by rearranging the equation. Hence, if you can measure the distance an element of a fluid travels in a known time interval, then you can calculate the speed of the flow field using the simple equation In cross-correlation PIV, we know the direction as well as the magnitude of displacements, so we use two vector quantities. That is, where v is the velocity vector (what we want to find) d is the displacement vector (what we measure) t is the time interval (what we know) Frame 1 Frame 2 displacement Velocity = How to match the particle images? time interval

SOM for Particle Tacking Neural Network Structure Position vector xi Weight vector wi Sub-network 1 i=1,2,…,N Sub-network 2 Weights vector Wj Position vector Xj j=1,2,…,M

SOM Neural Network Implementation (1) 1. Weight initialization, wi=xi, Wj=Xj (i=1,2,…,N, j=1,2,…,M) 2. Competition between neurons within radius Dmax wisub-network 2, for Wj-wiDmax (j=1,2,…,M) Euclidean distance dij=Wj-wi, min{dij}winner Wc Awarding winner and its neighbors Wk= Wk +c(wi-Wc), c=  for neurons Xc-XkR (neighbors) (accumulation) 0 otherwise Wjsub-network 1, for wi-WjDmax (i=1,2,…,N) Repeat competition and awarding

SOM Neural Network Implementation (2) 3. Updating weights for both sub-networks wi=wi+ wi (i=1,2,…,N) Wj=Wj+ Wj (j=1,2,…,M) 4. Decreasing neighborhood radius R R=R*Rcoef Increasing  =*coef 5. Repeat step 2 to step 4 nearest neighbor matching winner= Wj-wc  for j=1,2,…,M

Result (1) Shear flow constant gradient Ratio=254/254=1.0 Erroneous matching=7 Net ratio =247/254 =0.97

Results (2) rotational flows with out-of-plane particles 20/200 matching = 171/180=0.95 Mismatch= 20 Net ratio=0.84

Conclusions and Suggestion SOM alg. works well in Particle Tracking Velocimetry; Extension to real experimental scenarios, # of particles, unmatchable particles, Comparison to other alg. used in PTV, accuracy and speed.