Large Scale Parallel Lattice Boltzmann Model of Dendritic Growth
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1 Large Scale Parallel Lattice Boltzmann Model of Dendritic Growth Bohumir Jelinek Mohsen Eshraghi Sergio Felicelli CAVS, Mississippi State University March 3-7, 2013 San Antonio, Texas US Army Corps of Engineers
2 Why LBM-CA? When the fluid flow around solidifying dendrites is considered, lattice Boltzmann method is faster than alternatives
3 Lattice Boltzmann method Lattice-Boltzmann method (LBM) calculates time evolution of a quantity of interest governed by a partial differential equation subject to given initial and boundary conditions at regularly spaced nodes.
4 Lattice Boltzmann Method D2Q9 lattice Each node has 9 distribution functions f i representing portion of the mass density moving in the lattice direction e i
5 Lattice Boltzmann Method D2Q9 lattice Each node has 9 distribution functions f i representing portion of the mass density moving in the lattice direction e i
6 Evolution of the distribution functions
7 Evolution of the distribution functions For each lattice direction e i, i=0..8
8 Evolution of the distribution functions For each lattice direction e i, i=0..8 Collision: Adjusts the distribution function to approach equilibrium distribution
9 Evolution of the distribution functions For each lattice direction e i, i=0..8 Streaming: Shifts each distribution function to the neighboring node Collision: Adjusts the distribution function to approach equilibrium distribution
10 Evolution of the distribution functions For each lattice direction e i, i=0..8 ~kin. viscosity Streaming: Shifts each distribution function to the neighboring node Collision: Adjusts the distribution function to approach equilibrium distribution
11 Equilibrium distribution function
12 LBM parallelization CPU 7 CPU 8 CPU 9 Spatial domain decomposition CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3
13 LBM parallelization streaming For each lattice direction e i i=0..8 Streaming: Shifts each distribution function to the neighboring node
14 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
15 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
16 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
17 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
18 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
19 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
20 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
21 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
22 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
23 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
24 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
25 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
26 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
27 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
28 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
29 LBM parallelization streaming Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE)
30 LBM parallelization streaming CPU 7 CPU 8 CPU 4 CPU 5 CPU 9 CPU 6 Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE) CPU 1 CPU 2 CPU 3
31 LBM parallelization streaming CPU 7 CPU 8 CPU 4 CPU 5 CPU 9 CPU 6 Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE) CPU 1 CPU 2 CPU 3 buffer=send MPI_Sendrcv_replace( buffer, dst=6, src=4) recv=buffer
32 LBM parallelization streaming CPU 7 CPU 8 CPU 4 CPU 5 CPU 9 CPU 6 Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE) CPU 1 CPU 2 CPU 3 buffer=send MPI_Sendrcv_replace( buffer, dst=2, src=8) recv=buffer
33 LBM parallelization streaming CPU 7 CPU 8 CPU 4 CPU 5 CPU 9 CPU 6 Direction horizontal (W, E) vertical (N, S) diagonal (NW, NE, SW, SE) CPU 1 CPU 2 CPU 3 buffer=send MPI_Sendrcv_replace( buffer, dst=3, src=7) recv=buffer
34 LBM-CA solidification model C l, u, T Flow of solute between solidifying dendrites in a variable temperature field. Cooled at front and back boundaries, heated from left (inlet) and right (outlet) boundaries.
35 LBM-CA parallelization dendrite growth CPU 7 CPU 8 CPU 9 For dendrite growth, information from neighboring nodes is needed to update local node value CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3
36 LBM-CA parallelization ghost nodes CPU 7 CPU 8 CPU 9 Populate ghost nodes after each local update CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3
37 LBM-CA parallelization ghost nodes CPU 7 CPU 8 CPU 9 Populate ghost nodes after each local update east CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3 MPI_Sendrcv( send, recv, dst=6, src=4)
38 LBM-CA parallelization ghost nodes CPU 7 CPU 8 CPU 9 Populate ghost nodes after each local update west CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3 MPI_Sendrcv( send, recv, dst=4, src=6)
39 LBM-CA parallelization ghost nodes CPU 7 CPU 8 CPU 9 Populate ghost nodes after each local update north CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3 MPI_Sendrcv( send, recv, dst=8, src=2)
40 LBM-CA parallelization ghost nodes CPU 7 CPU 8 CPU 9 Populate ghost nodes after each local update south CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3 MPI_Sendrcv( send, recv, dst=2, src=8)
41 LBM-CA parallelization ghost nodes CPU 7 CPU 8 CPU 9 Populate ghost nodes after each local update south-east CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3 MPI_Sendrcv( send, recv, dst=3, src=7)
42 LBM-CA parallelization ghost nodes CPU 7 CPU 8 CPU 9 Populate ghost nodes after each local update north-west CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3 MPI_Sendrcv( send, recv, dst=7, src=3)
43 LBM-CA parallelization ghost nodes CPU 7 CPU 8 CPU 9 Populate ghost nodes after each local update south-west CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3 MPI_Sendrcv( send, recv, dst=1, src=9)
44 LBM-CA parallelization ghost nodes CPU 7 CPU 8 CPU 9 Populate ghost nodes after each local update north-east CPU 4 CPU 5 CPU 6 CPU 1 CPU 2 CPU 3 MPI_Sendrcv( send, recv, dst=9, src=1)
45 Computational resources Talon, MSU HPC 2 : 3072 cores, 12 cores/node (user limit 192 cores / job) Intel Xeon (Westmere) processors 24 GByte/node memory Voltaire quad data-rate InfiniBand (40Gb/s) peak performance of over 34.4 TeraFLOPS Kraken, NICS/ORNL: 112,896 cores, 12 cores/node (user limit cores / job) AMD Opteron (Istanbul) processors 16 GByte/node memory Cray SeaStar2+ router peak performance of 1.17 PetaFLOPS
46 Generating an initial configuration for parallel scaling tests Simulation domain: rectangular lattice, 8000x6000 grid points dimensions: 2.4 mm x 1.8 mm (0.3 µm/lattice distance) 3264 random dendrite nucleation sites constant cooling rate 100K/s across the whole domain forced melt flow through inlet (left) and outlet (right) boundaries almost 16 GB of memory = single node of Kraken 400k time steps took about 10 hours on 192 cores on MSU
47 Initial configuration
48 Magnified portion of initial configuration
49 Speed up Speed up (strong scaling) represents how much faster a task is solved utilizing multiple cores Speed up tests were performed by restarting simulation from the step when the dendrites were fairly grown in the incubation domain Incubation domain is split equally between varying number of cores, then executed for 587 time steps with a flow forced at the inlet (left) and outlet (right), and with a specified cooling flow rate at all boundaries
50 Speed up - constant task, 1 core Single core
51 Speed up - constant task, 2 cores Core 1 Core 2
52 Speed up - constant task, 4 cores Core 1 Core 2 Core 3 Core 4
53 Speed up - constant task, 12 cores Core 1 Core 2 Core 3 Core 4 Core 5 Core 6 Core 7 Core 8 Core 9 Core 10 Core 11 Core 12
54 Speed up - constant task, 12 cores Core 1 Core 2 Core 3 Core 4 Core 5 Core 6 Core 7 Core 8 Core 9 Core 10 Core 11 Core 12
55 Speed up - constant task, 24 cores
56 Speed up - results strong scaling (speed up) near perfect up to 3072 cores Algorithm is memory bandwidth limited on multi-core architecture (low FLOP/byte ratio)
57 Scale up Scale up (weak scaling) tests checks if the algorithm can solve larger task when more cores are utilized without a significant performance penalty Scale up tests were initialized from the stage when the dendrites were fairly grown in the incubation domain Incubated domain was duplicated equally onto varying number of nodes, then executed for 587 time steps with a flow forced at the inlet (left) and outlet (right), and with a specified cooling flow rate at all boundaries
58 Scale up - constant domain per node Base domain
59 Scale up - constant domain per node Duplication of the incubation domain onto 4 nodes
60 Scale up - constant domain per node Node 1 Node 2 Duplication of the incubation domain onto 4 nodes Node 3 Node 4
61 Scale up - constant domain per node Node 1 Node 2 Node3 Node 4 Duplication of the incubation domain onto 16 nodes Node 5 Node 6 Node7 Node 8 Node 9 Node 10 Node11 Node 12 Node 13 Node 14 Node15 Node 16
62 Scale up - results Goal: constant calculation time
63 Scale up - results Demonstrated nearly perfect scale up Largest domain: cores of Kraken over 165 billion grid nodes 11 millions of dendrites (only hundreds reported before) solute diffusion, melt convection, and heat transport dimensions cm x 8.64 cm 587 time steps 40 minutes of simulation time
64 3D LBM-CA parallelization C l, u 3D Dendrite growth in undercooled Al-3wt%Cu melt Equiaxial growth Effect of convection
65 3D columnar dendrites growing in undercooled melt of Al-3wt%Cu Domain size 180x180x144 (µm)³ By Mohsen Eshraghi
66 2D: Conclusions Parallelized 2D lattice Boltzmann / cellular automaton model of dendritic growth Tested the strong and weak parallel scaling of LBM/CA model with dendrites at advanced growth stage Demonstrated nearly ideal speed up and scale up 3D: Preliminary results exhibit similar speed up and scale up performance in 3D, measured tip growth velocity and solute concentration profiles Effects of convection, to be presented by Mohsen Eshraghi - Frontiers in Solidif. Science, Wed. 5:40 PM
67 Funding Acknowledgement U.S. Army Corps ERDC NSF Center for Advanced Vehicular MSU Computational resources MSU HPC ORNL XSEDE collaborative support by Reuben Budiardja Personal Mohsen Eshraghi for guiding me through the algorithm Sergio Felicelli and John Peters for opportunity to contribute to this interesting project
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