CEE 618 Scientific Parallel Computing (Lecture 12)

Transcription

1 1 / 26 CEE 618 Scientific Parallel Computing (Lecture 12) Dissipative Hydrodynamics (DHD) Albert S. Kim Department of Civil and Environmental Engineering University of Hawai i at Manoa 2540 Dole Street, Holmes 383, Honolulu, Hawaii 96822

2 2 / 26 Outline 1 Introduction Brownian Dynamics Stokesian Dynamics Lab work and Project 2 Raster3D Visualizing Spheres

3 3 / 26 What is? Introduction A study of motion of multiple particles, influenced by forces and torques

4 4 / 26 What is the force? Introduction FORCE ENERGY A push or pull that can cause an object with mass to accelerate Newton s second law: Acceleration: F = ma a = dv dt = d2 r dt 2 A scalar physical quantity that is a property of objects and systems which is conserved by nature The ability to do work: only if F = F(r). r2 E = F dr r 1

5 5 / 26 Introduction Statistical Mechanical Approaches 1 Nano-scale (10 9 m) MD (Molecular Dynamics) = Deterministic simulation of solving Newton s second law for ion species 2 Nano to Micro-scale (10 6 m) BD (Brownian Dynamics) = Updated simulation protocol of MD for ions in a fluid medium, but more applied to volumeless (point) colloidal/nano-particles: Random Forces/Torques DPD (Dissipative ) = Simulation method for Brownian motion of multiple particles using (approximate) pair-wise hydrodynamics. 3 Nano to Meso-scale (10 3 m) SD (Stokesian Dynamics) = Accurate simulation method for micro-hydrodynamics of spherical particles DHD = General simulation method for micro-hydrodynamics of Brownian and non-brownian particles

6 6 / 26 Brownian Dynamics Brownian Dynamics: Langevin s Equation The Langevin equations for the system of N Brownian particles: for particle i interacting with j s ṗ i = m i v i = F i (r) + j ( ) ξ ij v j + j α ij f j 1 Molecular Dynamics for conservative forces/torques 2 Stokesian Dynamics for hydrodynamic forces/torques 3 Dissipative for stochastic forces/torques * On the average hydrodynamic stochastic p i = m i v i is the momentum, ξ ij is the hydrodynamic friction tensor, F i is the sum of inter-particle and external forces, and j α ijf j represents the randomly fluctuating force exerted on a particle by the surrounding fluid: negligible if particles are much bigger than 1.0 µm.

7 Brownian Dynamics 7 / 26 Properties of Random Fluctuating Force, f i 1 Time average is zero: f i = 0 (1) 2 Independently exerted on i and j particle of different positions (i.e., r i and r j ) and at different times (i.e., t and t ) f i (t) f j ( t ) = 2δ ij δ ( t t ) (2) 3 δ is the Dirac-delta function: δ ij = 0 if i j; and δ ij = 1 if i = j; δ (t t ) = 0 if t t ; and δ (t t ) = 1 if t = t. 4 Related to the friction coefficient indicating α ξ. ξ ij = 1 k B T α ik α jk (3) k

8 Brownian Dynamics Brownian Dynamics Integration of the Langevin equation gives the time evolution equation: r i (t + t) = r i (t) + j D ij (t) k B T F j t + ( D) t + r G i (4) where the components of ri G are random displacements selected from 3N variate Gaussian distribution with zero means and covariance matrix r G i = 0 and r G i r G j = 2D ij t (5) The Oseen tensor (crude approximation) is given by D ij = k BT 1, for i = j (6a) 6πηa = k BT 8πηr ij ( 1 + r ijr ij r 2 ij ), for i j (6b) and one calculates D = 0. If F j 0, the random motion is dominant in multi-particle dynamics: ri G t. 8 / 26

9 Brownian Dynamics (BD) Brownian Dynamics Langevin equation 1 with inter-particle (conservative) forces f P, drag forces f H = ξv, and random Brownian forces f B m dv dt = f P + f H + f B (t) (7a) f H = ξv (7b) f B (t) = 0 (7c) f B (0) f B (t) = 6ξk B T δ (t) (7d) 1 Ermak and McCammon, J. Chem. Phys. 69 (1978) ; Langevin, C. R. Acad. Sci. (Paris) 146 (1908) / 26

10 10 / 26 Brownian Dynamics e.g., a falling body in liquid with x(0) = 0 & v(0) = 0 ma = mg βv + f B (t)

11 Stokesian Dynamics Stokesian Dynamics: Langevin s Equation The Langevin equations for the system of N force-free, non-brownian particles ṗ i = m i v i = ξ ij (v j U) F H j F H is the hydrodynamic forces/torques, p i = m i v i is the momentum, and ξ ij is the hydrodynamic friction tensor. If particles are at rest, U = M F H (8) F H = R U (9) R = (M ) 1 (10) where U is the translational/rotational velocity vector, and M and R are the grand mobility and grand resistance matrixes, respectively. R is dependent on particle positions and calculated as an inverse matrix of M. 11 / 26

12 12 / 26 Stokesian Dynamics (SD) Stokesian Dynamics Particles translate and rotate in a fluid field of V = U + r Ω + E : r where U is the uni-directional flow; and the vorticity Ω and rate of strain E are represented as Ω = 1 2 V (r) Eij = 1 ( Vi + V ) j = 1 2 x j x 2 ( jv i + i V j ) = E ji i respectively. If no shear, E = 0

13 13 / 26 Stokesian Dynamics Hydrodynamic Force Calculation with upflow U U = M F H and F H 6N p 1 =? Hydrodynamic Force where U 6Np 1 the relative velocities, F6N H p 1 is the hydrodynamic forces on particles, and Visualization: M 6N p 6N p the grand Two mobility Examples matrix. N = 4 3 = 64 =11, 303 p N p Hassonjee, Q., Ganatos, P., and Pfeffer, R., J. Fluid Mech., 197, 1-37 (1988) 70 minutes of running time using 25 processors

14 14 / 26 Stokesian Dynamics Parallel Computation of SD Simulation 6N p 6N p Parallel computing of large matrixes using 1,600 processors (out of 5400) of Jaws at Maui High Performance Computing Center (MHPCC) 1 The grand mobility matrix M has a dimension 6N p 6N p 2 N p = 40 3 = 64, 000 (6N p ) 2 = 147 billion elements 3 Memory = 1.18 TB 738 MB per processor 4 Time to calculate F H = 47 min. Using Tachyon at KISTI 1 N p = = 147, 456 (6N p ) 2 = 783 billion elements cores, 6.3 TB, and 16.5 hours

15 15 / 26 SD Simulation Stokesian Dynamics When particles are at rest, and a uniform upflow approaches N p particles with a constant velocity U 0 = 1 (in dimensionless unit), U 0 = M F H then the hydrodynamic forces acting on the particles are calculated as F 6Np 1. Each particle has six component of in F 6Np 1. 1 F 1 F 3 are forces on particle 1 in x, y, and z-directions, and F 4 F 6 are torques on particle 1 in x, y, and z-directions. 2 F 7 F 9 are forces on particle 2 in x, y, and z-directions, and F 10 F 12 are torques on particle 2 in x, y, and z-directions. 3 And so forth... For upward velocity, U j = 1 if j = 3 + 6(i 1) otherwise U j = 0: non-zero U j for j = 3, 9, 15,. An example calculation was included in Hassonjee, Q., Ganatos, P., & Pfeffer, R. (1988). J. Fluid Mech., 197, 1 37.

16 16 / 26 Stokesian Dynamics Cubic configuration of 64 particles with D/a = Figure: array

17 17 / 26 Stokesian Dynamics Force calculation: D/a = 16.12: degenerated z-forces

18 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E / 26 Stokesian Dynamics Results: F x, F y, F z, T x, T y, T z for 64 particles 1 Which column is always positive and why? 2 Compare the fourth column-values with F z s in the previous page. 3 How to get unique values of F z in the fourth column? Use cat, cut, and sort.

19 Stokesian Dynamics 19 / 26 Directions of force/torque: F x, F y, F z, T x, T y, T z Exerted on each particle with upflow, U = +1 ( ) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-04

20 20 / 26 Lab work Lab work and Project SD code code for hydrodynamic force/torque calculation is in /opt/cee618s13/class12/hasonjee/

21 21 / 26 Outline Raster3D 1 Introduction Brownian Dynamics Stokesian Dynamics Lab work and Project 2 Raster3D Visualizing Spheres

22 1 Raster3D is a set of tools for generating high quality raster images of proteins or other molecules. 2 The core program renders spheres, triangles, cylinders, and quadric surfaces with specular highlighting, Phong shading, and shadowing. 22 / 26 Raster3D Raster3D Visualizing Spheres

23 23 / 26 Example 1 Raster3D Visualizing Spheres 1 Copy all the files from /opt/cee618s13/class12/raster3d/example1/ to your own directory. 2 Type and enter: qsub raster_ex1.pbs 3 This pbs script will execute example1h.script and generate an image file, example1h.tff

24 24 / 26 Raster3D Visualizing Spheres Sphere configuration: array Under /mnt/home/albertsk/uhtraining/cee618-sp2012/class09/dhd 1 In shsnj_obsd_fts_64.f To rotate image change Euler angles of alpha0, beta0, and gamma0. To change the distance between the center and your eyes, control distance shsnj_obsd_fts_64.f. 2 Raster3Dspheres.f is included in the main code shsnj_obsd_fts_64.f. 3 There will be three output files from this serial run: 1 sforcefts.dat stores force/torque calculation data. 2 scoordxyz.dat includes (x, y, z) coordinates of N p particles. 3 scoordxyz.r3d contains Raster3D format coordinate data, translated to the center of mass.

25 25 / 26 Raster3D How to generate an image Visualizing Spheres 1 Copy all the files in /opt/cee618s13/class12/dhd-raster3d/ to your own directory. 2 Execute $ make $ make run 3 Then, a file like scoordxyz.tff will be generated. 4 Download the.tff file and view it.

26 1 Example of material properties and file indirection tiles in x,y pixels (x,y) per tile 4 4 3x3 virtual pixels -> 2x2 pixels background colour 6 T cast shadows 7 25 Phong power secondary light contribution ambient light contribution specular reflection component eye position main light source position E E E E E E E E E E E E E E E E mixed objects 18 * 19 * 20 * 21 # Draw a bunch of spheres 22 # 23 # 24 # E E E E E E E E E E E E E E / 26 Raster3D Visualizing Spheres Raster file: scoordxyz.r3d, x, y, z, a, and 3 more