Numerical Simulation of Incompressible Viscous Fluids using Finite Differences

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Numerical Simulation of Incompressible Viscous
Fluids using Finite Differences
Ferienakademie 2005 - Sarntal (Italy)
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PART I

• Description of the Sequential Code

Presented by Niko Manopulo
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Agenda

• Introduction and Governing Equations
• Navier Stokes Equations
• Finite Difference Discretization
• The Algorithm
• Derivation of the Pressure Equation
• Overview of the Algorithm
• 3. Obstacles and Free Surface Flows
• Treatment of Obstacles
• Treatment of Free Surface Flows

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Navier Stokes Equations

• Conservation of Mass
• Conservation of Momentum
• Conservation of Energy

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Assumptions

• Incompressible Flow
• Newtonian Fluid
• Constant Temperature
• 2D Cartesian Coordinates

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Simplified Equations

• Conservation of Mass
• Conservation of Momentum

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Boundary Conditions

• No Slip Condition
• Free Slip Condition

• Inflow Condition
• Outflow Condition

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Finite Difference Discretization

• 2nd Order Difference Scheme
• Donor Cell Scheme
• Explicit Time Discretization

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The Staggered Grid
j 1
vi,,j
vi1,,j
ui,,j
ui-1,,j
ui1,,j
j
pi,,j
pi1,,j
vi,,j-1
vi1,,j-1
j - 1
i - 1
i
i 1
i 2
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Solution Domain
Domain Boundary
j jmax 1
j jmax
j 2
j 1
j 0
i 0
i 1
i 2
i imax
i imax1
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Computation of the Pressure

• Consider again the Momentum Equation

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Computation of the Pressure

• Please follow the blackboard

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Calculation of the Time Step

• Adaptive Stepsize Control
• Courant Friedrichs Levi (CFL) Conditions
• Safety factor (t)

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Overview of the Algorithm
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Overview of the Algorithm
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Treatment of Obstacles

• The Flag Matrix
• Cell Marking Loops

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Treatment of Free Boundaries

• Falling Drop Breaking Dam

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Marker and Cell

• Domain Covered with a set of particles
• Loop moving the particles
• Loop marking the cell flags
• Additional set of Flags
• Additional conditions on
• the free boundary

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Particle Data Structure

• struct particle
• double x,y
• struct particle next
• struct particle_line
• int length
• struct particle Particle
• struct particle_line Particle_Lines

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Additional Flags
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Free Boundary Conditions

• Vanishing surface tension

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Modified Algorithm
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Modified Algorithm
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PART II

• Parallel programming applied to free boundary
  value problems

Presented by Darya Popiv
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Problem to Solve

• We have sequential code of free boundary problem
• Specifically braking dam and falling drop into
  the basin
• Goal parallelize the code to achieve the
  reasonable speed up of execution time

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Investigating Architectures

• SISD single instruction single data (PC)
• MISD multiple instruction single data (special
  construction implementing conveyor mode, if
  exists in pure form)
• SIMD single instruction multiple data (vector
  computers)
• MIMD multiple instruction multiple data (true
  parallel computers, clusters)

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MIMD

• Why MIMD? Promising architecture Possibility to
  use
• Where MIMD is realized? Distributed address space
  machines Global address space machines

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Distributed vs. Global Address Space Machines

• Distributed address space machines message
  passing machines. Hardware level - computer
  clusters. Software libraries MPI (1.2, 2.0),
  PVM, Parmacs, Express.
• Global address space machines shared memory.
  Software libraries OpenMP.

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Resources for our solution

• MIMD
• Computer cluster
• MPI as software library

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Now back to the roots ?

• Input domain dimensions, domain resolution,
  initial boundary conditions, particle
  distribution over domain, initial particle
  velocities
• Processing using SOR we find pressure,
  velocities and therefore particle position at
  each time step
• Output velocities, pressure, position of
  particles

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Domain Decomposition
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Domain Decomposition 2

• Calculating pressure and velocities

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Domain Decomposition 3

• Exchanging pressure and velocities between
  subdomains

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Domain Decomposition 4

• Exchanging of particles between subdomains

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Parallel Algorithm
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Parallel Algorithm 2
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MPI Functions

• MPI_Init
• MPI_Finalize
• MPI_Comm_rank
• MPI_Comm_size
• MPI_Barrier
• MPI_Send, MPI_Recv
• MPI_Sendrecv
• MPI_Reduce
• MPI_Bcast

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Measuring efficiency

• Speedup
• Parallel Efficiency

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Falling Drop
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Breaking Dam
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Possible improvements

• We do not use dynamic load balancing gt speedup
  is not that big
• You wanna try? ?