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Overview of numerical methods

• Many CFD techniques exist.
• The most common in commercially available CFD
  programs are
• The finite volume method has the broadest
  applicability (80).
• Finite element (15).
• Here we will focus on the finite volume method.
• There are certainly many other approaches (5),
  including
• Finite difference.
• Finite element.
• Spectral methods.
• Boundary element.
• Vorticity based methods.
• Lattice gas/lattice Boltzmann.
• And more!

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LBM Basics I
Lattice Unit, lu

• Notes
• Based on statistical mechanics and kinetic
  theory.
• Solute and buoyancy will be covered in later
  slides.

Unfold
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Lattice Boltzmann Model
Unit Vectors ea
Direction-specific particle densities fa
f7 (rest)
Macroscopic flows
Density
Velocity
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Single Relaxation Time BGK (Bhatnagar-Gross-Krook)
Approximation
Streaming
Collision (i.e., relaxation towards local
equilibrium)
Collision and streaming steps must be separated
if solid boundaries present (bounce back boundary
is a separate collision)

• t relaxation time
• d0 fraction of rest particles
• b number of unit velocity directions

• D dimension of space
• c maximum speed on lattice (1 lu /time step)

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D2Q9 feq
wa are 4/9 for the rest particles (a 0), 1/9
for a 1, 2, 3, 4, and 1/36 for a 5, 6, 7, 8.
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Goal of NGrid
Enhancing Koreas Competitiveness in Science and
Technology
Grid Applications
Computational, Data Access Grid
Supercomputers, Storages Instruments
Grid Middleware
Internet Infrastructure
High Speed Network Infrastructure
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LBE3D - Lattice Boltzmann simulation of miscible
fluids

• Parallel (MPI), F90 code
• Large number of steerable/monitored parameters
  (configured in input file)
• Control of output of colour data field allows
  progress of e.g. phase separation to be monitored
  via on-line visualization
• visualization is bespoke vtk application using
  steering library to read data
• Uses Tcl for GUI and event loop to allow
  automatic updates when new data available
• Have integrated with Chromium for
  higher-resolution AG displays
• Pause facility (useful for demonstrations/delibera
  tions)

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LBE3D functionality

• Checkpoint, restart and rewind
• LB3D registers all checkpoints it creates with
  the steering library
• Checkpoint tree built up over time
• Persistent record of work done (including all
  steering activity)
• A running LBE3D can be requested to rewind
• Steering used to investigate phase space of
  oil/water/surfactant systems by adjusting
  strength of oil-water interaction
• used with small models to inform what jobs to run
  with large models

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Simulation portal
A general overview / possible scenario
Medical simulation service Portal networked
compute resources
User-site (SW installed)
Simulation Service System
GRID SW (interface)
GRID SW
Pre- Post- processing
Applications SW
GRID SW (service use)
Internet or Intranet
Could also be moved to the services portal
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Grid Architecture
Application Layer Collective Layer Resource
Layer Fabric Layer
Applications, Portals, Environments
Higher Services, APIs, Protocols
APIs, Protocols, SDKs
Computers, Networks, Devices
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Lattice Boltzmann Model (LBM)

• CFD method on the lattice
• Numerical calculations
• Stream
• Collision
• Yields Navier-Stokes for incompressible fluid
• Greatly flexible in specifying complex boundaries

A cell of the D3Q19 lattice
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LBM on a Single GPU

• Li et al., Visual Computer 03
• Program fragment processing stage

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Store LBM Data in Textures
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Scale-up LBM

• Domain decomposing
• Communication
• Read out from GPU
• Network transfer
• Write into GPUs

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GPU lt-gt PC data transfer

• Read data from GPU
• Aggregate necessary boundary data together into a
  texture
• Read them out in a single operation
• Write data into GPU
• Reverse the above procedure

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Network Transfer

• MPI
• To minimize communication cost
• Overlap network transfer time with computation
  time
• Simplify communication pattern

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Navier-Stokes Equations

• Describe flow of an incompressible fluid

Advection
Pressure Gradient
External Force
Diffusion (viscosity)
Velocity is divergence-free
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Supercomputer
Nodes
Host
Image from http//www.olympusmicro.com/micd/gall
eries/oblique/glasswool.html
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Advantages of LBM

• Method is fully parallel (collision step requires
  only data from its own lattice site) and local
  (only nearest neighbors interact)
• Complex boundary conditions easily handled
• Simple to program and tune for performance

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Low Reynolds Number Laminar Flow
V velocity R characteristic distance r
density h viscosity For bacteria, Re .00003
www.zoology.ubc.ca/courses/bio332
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Microfluidics

• Microfluidics is the science of designing,
  manufacturing, and formulating devices and
  processes that deal with volumes of fluid on the
  order of nanoliters (symbolized nl and
  representing units of 10-9 liter) or picoliters
  (symbolized pl and representing units of 1 E -12
  liter)

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• Various molecules were developed as
• nanodevices for nanotherapeutic
• applications as shown below

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CACTUS Parallelism

• Cactus is designed around a distributed memory
  model. Each thorn is passed a section of the
  global grid.
• The actual parallel driver (implemented in a
  thorn) can use multiple methods to decompose the
  grid across processors and exchange ghost zone
  information

• Standard driver distributed with Cactus (PUGH) is
  for a parallel unigrid and uses MPI for the
  communication layer
• PUGH can do custom processor decomposition and/or
  static load balancing
• Expect high parallel efficiency

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Development of discrete models of medium
Molecular dynamics (Alder, 1960)
Kinetic Boltzmann equation (1872)
1964
Boltzmann equations with discrete
set of velocities
Lattice Gas Automata
1988
1997
Lattice Boltzmann Equation

Chapman Enskog expansion
Macroscopic equations of
hydrodynamics (Navier Stokes equations)
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Boltzmann equations with discrete velocities
The discrete finite set of vectors ck of
particle velocities could be used for Boltzmann
equation at hydrodynamic stage
For 1D
Usually the populations Nk are used for
each group of particles
Hydrodynamic variables
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Lattice Boltzmann equation method (LBE)
The main idea is that time step must be so that
One-dimensional isothermal variant (D1Q3)
Two-dimensional variants
(D2Q9)
(D2Q13)
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Lattice Boltzmann equation method (LBE)
The discrete single-particle distribution
functions Nk are used as variables
Hydrodynamic variables
Evolution equations of LBE method
is the collision operator
in BGK form (relaxation to the equilibrium state
with relaxation time ?).
Viscosity
Expansion in u
is the body force term.
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New general method of incorporatinga body force
term into LBE
Kinetic Boltzmann equation for single particle
distribution function f(r,?,t)
Perturbation method
For any equilibrium distribution function
Hence
From the other hand, the full derivative along
the Lagrange coordinate at a constant density
is equal to
Thus, we obtained the Boltzmann equation in form
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Exact difference method for lattice Boltzmann
equation
After discretization of Boltzmann equation in
velocity space we have
Here the changes of the distribution functions Nk
due to the force F are equal to the exact
differences of equilibrium distribution functions
at constant density
The commutative property of body force term and
the collision operator indicates the second order
accuracy in time. The distribution function that
is equilibrium in local region of space, is
simply shifting under the action of body force by
the value
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Credits Mathies Lab, UC-Berkeley Quake Lab,
Stanford Agilent, Inc.
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Basics of Microfluidics
Small devices with micron-scale channels All
flow is laminar no turbulence all mixing by
diffusion
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MATHEMATICAL MODELING OF ENGINEERING PROBLEMS
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What is a protein?

• Large molecule chain of amino acids
• Several tens to thousands residues
• Folds to specific shape
• Biological machines

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Proteins as interdisciplinary molecules
Proteins are evolving molecular machines, at the
border between Physics and Biology. PHYSICS They
are molecular machines that obey the laws of
statistical mechanics. BIOLOGY They are evolving
machines, produced through the action of mutation
and natural selection. BIOINFORMATICS
integrates both sources of information to predict
biological properties. Thermodynamics sheds
light on protein evolution, and evolutionary
considerations sheds light on protein folding.