Recent Developments in Overture Bill Henshaw Center for Applied Scientific Computing Lawrence Livermore National Laboratory Livermore, CA 10th Symposium on Overset Composite Grids and Solution Technology, Nasa Ames Research Center, Moffett Field, CA October 2010. Henshaw (LLNL) Recent Developments in Overture OGS2010 1 / 25
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Recent Developments in Overture
Bill Henshaw
Center for Applied Scientific ComputingLawrence Livermore National Laboratory
Livermore, CA
10th Symposium on Overset Composite Grids and Solution Technology,Nasa Ames Research Center, Moffett Field, CA
October 2010.
Henshaw (LLNL) Recent Developments in Overture OGS2010 1 / 25
Acknowledgments.
Supported byDepartment of Energy, Office of Science
ASCR Applied Math ProgramLLNL: Laboratory Directed Research and Development (LDRD) program
Current Overture developersKyle ChandBill Henshaw
Major ContributorsDon Schwendeman (RPI),Jeff Banks (LLNL).
Henshaw (LLNL) Recent Developments in Overture OGS2010 2 / 25
Overview of Recent Work
Solid mechanics on overlapping grids (talk by Jeff Banks).
Efficient high-order scheme for incompressible flows (talk by KyleChand).
Grid generation: large scale (billions of grid points), parallel, andmoving.
Multigrid solvers for overlapping grids (new high-order accurateand parallel algorithms),
grids ordered by priority; interpolation preferred from higher priority grids.
robust algorithm with backup rules and interactive error diagnostics.
Brief description of capabilities:
arbitrary stencil widths (1,2,3... fringes),
arbitrary order of interpolation (linear, quadratic, cubic,...),
fast searching algorithms and fast “inverse” map.
inverse optimized for common mappings (spheres, cylinders, ...),
optimized for Cartesian grids,
script files with embedded perl commands for "automatic" parameterizedgrid generation.
Henshaw (LLNL) Recent Developments in Overture OGS2010 9 / 25
Ogen: two examples
Grid order of grid points processors cpu (s)accuracy (nodes× p/n)
Sphere in a box 2 2.1 billion 16 (8 × 2) 136Re-entry vehicle 4 215 million 128 (16 × 8) 1990
Significant performance improvements can still be made.
Henshaw (LLNL) Recent Developments in Overture OGS2010 10 / 25
Parallel Flow Solution on Moving Grids.
Flow past a pitching andplunging airfoil.
Demonstrates new parallelmoving grid capabilities.
Henshaw (LLNL) Recent Developments in Overture OGS2010 11 / 25
Multigrid: fast in theory and practice (if careful)
For elliptic problems, multigrid algorithms can have near optimalcomplexity requiring O(N) work to solve for N unknowns (c.f.conjugate gradient: O(N3/2)).
multigrid uses a sequence of coarser and coarser meshes toaccelerate the convergence rate on the finest grid.
For overlapping grids, coarse grid generation is a difficulty.
Henshaw (LLNL) Recent Developments in Overture OGS2010 12 / 25
Ogmg: Overlapping Grid Multigrid Solver
Ogmg: solves scalar elliptic boundary value problems.
automatic coarse grid generation of “any” number of levels.
adaptive smoothing
variable sub-smooths per component gridinterpolation-boundary smoothing (IBS)over-relaxed Red-Black smoothers
Henshaw (LLNL) Recent Developments in Overture OGS2010 15 / 25
Ogmg: Results: fourth order accuracy.
10−2
10−10
10−9
10−8
10−7
10−6
10−5
10−4
h
Max
imum
err
orCircle in a Channel, Errors versus h
Slope 4
DirichletNeumann
10−2
10−1
10−7
10−6
10−5
10−4
10−3
10−2
h
Max
imum
err
or
Sphere in a Box, Errors versus h
Slope 4
DirichletNeumann
Henshaw (LLNL) Recent Developments in Overture OGS2010 16 / 25
Ogmg: Convergence rates - Cartesian grids.
0 1 2 3 4 5 6 7 8
10−5
100
105
Square 10242, Order 4, V[2,1]
max
imum
res
idua
l
multigrid cycle
← CR=0.117, ECR=0.65
← CR=0.061, ECR=0.57
CR=0.018, ECR=0.46 →
ω=1ω=1.15Op Ave, ω=1.15
0 1 2 3 4 5 6 7 810
−8
10−6
10−4
10−2
100
102
104
106
Box 2563, Order 4, V[2,1]
max
imum
res
idua
l
multigrid cycle
← CR=0.188, ECR=0.70
← CR=0.118, ECR=0.63
CR=0.042, ECR=0.50 →
ω=1ω=1.15Op Ave, ω=1.15
Figure: Multigrid convergence rates for Poisson’s equation with Dirichletboundary conditions on a 10242 square and a 2563 box with a V[2,1] cycle.The rates are significantly improved using operator averaging andover-relaxed Red-Black smoothers with a relaxation parameter ω.
Henshaw (LLNL) Recent Developments in Overture OGS2010 17 / 25