Louis Howell Center for Applied Scientific Computing AX Division Lawrence Livermore National Laborat

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Louis HowellCenter for Applied Scientific
Computing/AX DivisionLawrence Livermore
National Laboratory
Parallel Adaptive Mesh Refinement for Radiation
Transport and Diffusion
May 18, 2005
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Raptor Code Overview

• Block-structured Adaptive Mesh Refinement (AMR)
• Multifluid Eulerian representation
• Explicit Godunov hydrodynamics
• Timestep varies with refinement level
• Single-group radiation diffusion (implicit,
  multigrid)
• Multi-group radiation diffusion under development
• Heat conduction, also implicit
• Now adding discrete ordinate (Sn) transport
  solvers
• AMR timestep requires both single and multilevel
  Sn
• Parallel implementation and scaling issues

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Raptor Code Core Algorithm Developers

• Rick Pember
• Jeff Greenough
• Sisira Weeratunga
• Alex Shestakov
• Louis Howell

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Radiation Diffusion Capability

Single-group radiation diffusion is coupled with
multi-fluid Eulerian hydrodynamics on a regular
grid using block-structured adaptive mesh
refinement (AMR).
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Radiation Diffusion Contrasted with Discrete
Ordinates

• All three calculations conserve energy by using
  multilevel coarse-fine synchronization at the end
  of each coarse timestep. Fluid energy is shown
  (overexposed to bring out detail). Transport
  uses step characteristic discretization.

Flux-limited Diffusion
S16 (144 ordinates)
144 equally-spaced ordinates
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Coupling of Radiation with Fluid Energy

• Advection and Conduction
• Implicit Radiation Diffusion (gray, flux-limited)

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Coupling of Radiation with Fluid Energy

• Advection and Conduction
• Implicit Radiation Transport (gray, isotropic
  scattering)

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Implicit Radiation Update

• Extrapolate Emission to New Temperature

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Implicit Radiation Update

• Iterative Form of Diffusion Update

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Implicit Radiation Update

• Iterative Form of Transport Update

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Simplified Transport Equation

• Gather Similar Terms
• Simplified Gray Semi-discrete Form

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Discrete Ordinate Discretization

• Angular Discretization
• Spatial Discretization in 2D Cartesian
  Coordinates
• Other Coordinate Systems 1D 3D Cartesian,
  1D Spherical, 2D Axisymmetric (RZ)

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Spatial Transport Discretizations

• Step
• First order upwind, positive, inaccurate in both
  thick and thin limits
• Diamond Difference
• Second order but very vulnerable to oscillations
• Simple Corner Balance (SCB)
• More accurate in thick limit, groups cells in 2x2
  blocks, each block requires 4x4 matrix inversion
  (8x8 in 3D).
• Upstream Corner Balance
• Attempts to improve on SCB in streaming limit,
  breaks conjugate gradient acceleration
  (implemented in 2D Cartesian only)
• Step Characteristic
• Gives sharp rays in thin streaming limit,
  positive, inaccurate in thick diffusion limit
  (implemented in 2D Cartesian only)

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Axisymmetric Crooked Pipe Problem

• Diffusion S2 Step S8 Step
  S2 SCB S8 SCB

Radiation Energy Density
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Axisymmetric Crooked Pipe Problem

• Diffusion S2 Step S8 Step
  S2 SCB S8 SCB

Fluid Temperature
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AMR Timestep

• Advance Coarse (L0)
• Advance Finer (L1)
• Advance Finest (L2)

?t0
?t1
?t2
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AMR Timestep

• Synchronize L1 and L2
• (Multilevel solve)
• Repeat (L1 and L2)
• Synchronize L0 and L1
• (Multilevel solve)

?t1
?t0
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Requirements for Radiation Package

• Features controled by the package
• Nonlinear implicit update with fluid energy
  coupling
• Single level transport solver (for advancing each
  level)
• Multilevel transport solver (for synchronization)
• Features not directly controled by the package
• Refinement criteria
• Grid layout
• Load balancing
• Timestep size
• Parallel support provided by BoxLib
• Each refinement level distributed grid-by-grid
  over all processors
• Coarse and fine grids in same region may be on
  different processors

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Multilevel Transport Sweeps

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Sources Updated Iteratively

• Three sources must be recomputed after each
  sweep, and iterated to convergence
• Scattering source
• Reflecting boundaries
• AMR refluxing source
• The AMR source converges most quickly, while the
  scattering source is often so slow that
  convergence acceleration is required.

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Parallel Communication

• Four different communication operations are
  required
• From grid to grid on the same level
• From coarse level to upstream edges of fine level
• From coarse level to downstream edges of fine
  level (to initialize flux registers)
• From fine level back to coarse as a refluxing
  source
• Operations 2 and 3 only needed when preparing to
  transfer control from coarse to fine level
• Operation 3 could be eliminated and 4 reduced if
  a data structure existed on the coarse processor
  to hold the information

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Parallel Grid Sequencing

• To sweep a single ordinate, a grid needs
  information from the grids on its upstream faces
• Different grids sweep different ordinates at the
  same time

2D Cartesian, first quadrant only of S4 ordinate
set 13 stages for 3 ordinates
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Parallel Grid Sequencing

• In practice, ordinates from all four quadrants
  are interleaved as much as possible
• Execution begins at the four corners of the
  domain and moves toward the center

2D Cartesian, all quadrants of S4 ordinate
set 22 stages for 12 ordinates
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Parallel Grid Sequencing RZ

• In axisymmetric (RZ) coordinates, angular
  differencing transfers energy from ordinates
  directed inward towards the axis into more
  outward ordinates. The inward ordinates must
  therefore be swept first.

2D RZ, S4 ordinate set requires 26 stages for 12
ordinates, up from 22 for Cartesian
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Parallel Grid Sequencing AMR

• 43 level 1 grids, 66 stages for 40 ordinates (S8)
  (20 waves in each direction)

Stage 4
Stage 15
Stage 34
Stage 62
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Parallel Grid Sequencing 3D AMR

• In 2D, grids are sorted for each ordinate
  direction
• In 3D, sorting isnt always possibleloops can
  form
• The solution is to split grids to break the loops
• Communication with split grids is implemented
• So is a heuristic for determining which grids to
  split
• It is possible to always choose splits in the z
  direction only

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Acceleration by Conjugate Gradient

• A strong scattering term may make iterated
  transport sweeps slow to converge
• Conjugate gradient acceleration speeds up
  convergence dramatically
• The parallel operations required are then
• Transport sweeps
• Inner products
• A diagonal preconditioner may be used, or for
  larger ordinate sets, approximate solution of a
  related problem using a minimal S2 ordinate set
• No new parallel building blocks are required

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2D Scaling (MCR Linux Cluster)Single Level, Not
AMR
40,47,52, 58, 64, 68, 74
Stages
Grids arranged in square array, one grid per
processor, each grid is 400x400 cells. Sn
tranport sweeps (Step and SCB) are for all 40
ordinates of an S8 ordinate set. Uses icc, ifc,
hypre version 1.8.2b on MCR (2.4 GHz Xeon,
Quadrics QsNet Elan3)
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3D Scaling (MCR Linux Cluster)Single Level, Not
AMR
85,99, 113, 129
Stages
Grids arranged in cubical array, one grid per
processor, each grid is 40x40x40 cells. Sn
tranport sweeps (Step and SCB) are for all 80
ordinates of an S8 ordinate set.
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AMR Scaling 2D Grid LayoutCase 1 Separate
Clusters of Fine Grids

• To investigate scaling in AMR problems, I need to
  be able to generate similar problems of
  different sizes.
• I use repetitions of a unit cell of 4 coarse and
  18 fine grids.
• Each processor gets 1 coarse grid. Due to load
  balancing, different processors get different
  numbers of fine grids.

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AMR Scaling 2D Grid Layout Case
2 Coupled Fine Grids

• The decoupled groups of fine grids in the
  previous AMR problem give the transport
  algorithms an advantage, since groups do not
  depend on each other.
• This new problem couples fine grids across the
  entire width of the domain.
• Note the minor variations in grid layout from one
  tile to the next, due to the sequential nature of
  the regridding algorithm.

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2D Fine Scaling (MCR Linux Cluster) Case 1
Separate Clusters of Fine Grids
Grids arranged in square array, 4 coarse grids
and 18 fine grids for every four processors, each
coarse grid is 256x256 cells, 41984 fine cells
per processor. Sn tranport sweeps are for all 40
ordinates of an S8 ordinate set.
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2D Fine Scaling (MCR Linux Cluster) Case 2
Coupled Fine Grids
Grids arranged in square array, one coarse grid
and 5-6 fine grids for every processor, each
coarse grid is 256x256 cells, 51000 fine cells
per processor. Sn tranport sweeps are for all 40
ordinates of an S8 ordinate set.
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3D Fine Scaling (MCR Linux Cluster)Case 1
Separate Clusters of Fine Grids
Grids arranged in cubical array, 8 coarse grids
and 58 fine grids for every eight processors,
each coarse grid is 32x32x32 cells, 28800 fine
cells per processor. Sn tranport sweeps are for
all 80 ordinates of an S8 ordinate set.
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3D Fine Scaling (MCR Linux Cluster) Case 2
Coupled Fine Grids
Grids arranged in cubical array, one coarse grid
and 33 fine grids for every processor, each
coarse grid is 32x32x32 cells, 47600 fine cells
per processor. Sn tranport sweeps are for all 80
ordinates of an S8 ordinate set.
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2D AMR Scaling (MCR Linux Cluster)Case 1
Separate Clusters of Fine Grids
Grids arranged in square array, 4 coarse grids
and 18 fine grids for every four processors, each
coarse grid is 256x256 cells, 41984 fine cells
per processor. Sn tranport sweeps are for all 40
ordinates of an S8 ordinate set.
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2D AMR Scaling (MCR Linux Cluster) Case 2
Coupled Fine Grids
Grids arranged in square array, one coarse grid
and 5-6 fine grids for every processor, each
coarse grid is 256x256 cells, 51000 fine cells
per processor. Sn tranport sweeps are for all 40
ordinates of an S8 ordinate set.
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2D AMR Scaling (MCR Linux Cluster) Case 2
Coupled Fine (Optimized Setup)
This version has neighbor calculation in wave
setup implemented using an O(n) bin sort,
depth-first traversal for building waves (makes
little difference). In stage setup wave
intersections optimized and stored. All
optimizations serial.
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3D AMR Scaling (MCR Linux Cluster)Case 1
Separate Clusters of Fine Grids
Grids arranged in cubical array, 8 coarse grids
and 58 fine grids for every eight processors,
each coarse grid is 32x32x32 cells, 28800 fine
cells per processor. Sn tranport sweeps are for
all 80 ordinates of an S8 ordinate set.
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3D AMR Scaling (MCR Linux Cluster)Case 1
Separate Clusters (Optimized)
This version has neighbor calculation in wave
setup implemented using an O(n) bin sort. In
stage setup wave intersections optimized and
stored. All optimizations serial.
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3D AMR Scaling (MCR Linux Cluster) Case 2
Coupled Fine Grids (Optimized)
Grids arranged in cubical array, one coarse grid
and 33 fine grids for every processor, each
coarse grid is 32x32x32 cells, 47600 fine cells
per processor. Sn tranport sweeps are for all 80
ordinates of an S8 ordinate set.
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Transport Scaling Conclusions

• A sweep through an S8 ordinate set and a
  multigrid V-cycle take similar amounts of time,
  and scale in similar ways on up to 500
  processors.
• Setup expenses for transport are amortized over
  several sweeps. This is code for determining the
  communication patterns between grids, including
  such things as the grid splitting algorithm in
  3D.
• So far, optimized scalar setup code has given
  acceptable performance, even in 3D.

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Acceleration by Conjugate Gradient

• Solve by sweeps, holding right hand side fixed
• Solve homogeneous problem by conjugate gradient
• Matrix form

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Acceleration by Conjugate Gradient

• Inner product
• Preconditioners
• Diagonal
• Solution of smaller (S2) system by DPCG
• This system can be solved to a weak (inaccurate)
  tolerance without spoiling the accuracy of the
  overall iteration

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Clouds Test Problem Acceleration

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Clouds Test Problem Acceleration

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Clouds Test Problem

• 1 km square domain
• No absorption or emission
• 400000 erg/cm2/s isotropic flux incoming at top
• Specular reflection at sides
• Absorbing bottom
• ?s10-2 cm-1 inside clouds
• ?s10-6 cm-1 elsewhere
• S2 uses DPCG
• S8 uses S2PCG
• Serial timings on GPS (1GHz Alpha EV6.8)

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Clouds Test Problem SCB Fluxes

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UCRL-PRES-212183
This work was performed under the auspices of the
U. S. Department of Energy by the
University of California Lawrence Livermore
National Laboratory under Contract
W-7405-Eng-48.