Graphics Card Computing for Materials Modelling

Transcription

1 Graphics Card Computing for Materials Modelling Case study: Analytic Bond Order Potentials B. Seiser, T. Hammerschmidt, R. Drautz, D. Pettifor Funded by EPSRC within the collaborative multi-scale project Alloys By Design: Nickel-base superalloys

2 Alloys by Design Materials for gas turbine blades: Challenge: CREEP RESISTANT STABLE 0.5 μm 2.5 μm Dislocation creep COATABLE Precipitation of detrimental phases CASTABLE Titanium Steel Nickel Aluminium 25 μm 25 cm Ni-based superalloys: Cr, Co, Mo, W, Al, Ti, Ta, Re, Ru, Hf, C, B (<10 wt%) alloy design still empirically rather than theoretically expensive, time-consuming, non-optimized alloys Reaction with coatings Freckling instabilities Need multi-scale modelling for alloy design

3 Materials Modelling with GPUs Molecular dynamics GPU codes Hierarchy in Materials Modelling AceMD (the biomolecular MD package used by GPUGRID) Ascalaph (molecular modelling suite) HOOMD (Highly Optimized Object Oriented Molecular Dynamics) VMD & NAMD (Visual Molecular Dynamics) Density functional theory codes TeraChem (GTO, J. Chem. Theory Comput., 2008, 4 (2), pp ) Single precision: x speed up BIGDFT (WL, see Journal of Chemical Physics 131, , 2009) Dwarfs are essential for most electronic structure calculation methods

4 Tight-binding method Total energy: Repulsive energy: E = E rep + E bond Summation of pair-wise interactions Bond energy: Bond integral: H kl k H = l H ik i H jl H ij j E F E bond = n(e) E de n(e) Density of states H ii H ij H ik 0 H ji H jj 0 H jl H ki 0 H kk H kl H ij = < i H j> = R T Hv = Ev x ppσ (r ij ) ppπ (r ij ) ppπ (r ij ) Matrices dimension depending on number of orbitals Lapack Scalapack Hv = Ev periodic crystal E x R E F 0 H lj H lk H ll Jacket n(e)

5 Analytic Bond Order potentials Moments of density of states: Moment theorem: Cyrot-Lackmann (1967) = 1 = centre of gravity = RMS width = skewness = bimodality Bond integral Interference path between atom i and j Bond order potential (BOP) bond energy: n = 3 Drautz and Pettifor (2006) n = 4 n = 5 where g n and is n th moment E f

6 BOPfox BOPfox tool (Fortran 90): Tight-binding, EAM, BOP -> Molecular dynamics, kmc Benchmark for fcc with 864 W atoms, 12 moments [s] [%] initialization neighbour lists bond matrix evaluate moments evaluate ainf,binf forces EAM Fermi level search self-consistency total % matrix multiplications rest is spent on path finding

7 Interference paths Calculation of interference paths: Length (n) = 2 l ( ) = ( ) ( ) li start and end on atom i lj ji + ( ) ( ) start and lk end ki 2 nd moment of atom i = sum of paths (n=2) that 4 nd moment of atom i = sum of paths (n=4) that on atom i j i k T ( ) ii = ( ) li ( ) li EP Set of end points

8 Interference paths Calculation of interference paths: Length = 3 ( ) = ( )( ) j k + ( )( ) +... i

9 Density of of states Number of matrix multiplications /atom Matrix multiplications EAM/PP TB 20 7x x10 4 5x10 4 4x10 4 3x10 4 2x10 4 1x Energy Number of moments Accuarcy Number of matrix multiplications scales linearly with number of atoms!

10 BOPfox goes GPU BOPfox tool (Fortran 90): Tight-binding, EAM, BOP -> Molecular dynamics, kmc Benchmark for fcc with 864 W atoms, 12 moments [s] [%] initialization neighbour lists bond matrix evaluate moments evaluate ainf,binf forces EAM Fermi level search self-consistency total hosttogpu_uploadatomicpositions(); hosttogpu_uploadneighbourlist(); gpu_gettodolist(); //Get list of matrix calculations gpu_calculatebondintegrals(); //r ik -> H ik for (i = 2; i <= ninterferencemax; i++){ gpu_matrixmultiplication(); gpu_matrixaddition(); gpu_momentcalculation(); gputohost_moments(); }

11 Graphics Card Computing for Materials Modelling BOPfox and BOPC BOPfox (CPU) Hardware Intel Core2 Dual CPU E GHz 4 GB memory Compiler options Gfortran Release modus (-03) BOPC (GPU) Hardware nvidia GeForce GTX multiprocessors 216 cores 1.5 Ghz Compiler options Nvcc release modus (-03), CUDA 2.0 Benchmark of BOPfox vs BOPC Task BOPfox (CPU) [ms] BOPC (GPU) [ms] Factor (Speed up) Calculation of matrices ~22 Path finding ~44 Matrix multiplication ~19 24 x overall speed up

12 Conclusions Materials modelling can benefit significantly from GPU parallelization Linear algebra and FFT are essential for most electronic structure calculation methods Models like analytic bond order potentials try to avoid expensive LA/FFT routines significant speed up possible