Computational Analysis Tools for Large-Scale Computational Fluid Dynamics Codes

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Computational Analysis Tools for Large-Scale
Computational Fluid Dynamics Codes
Boris Diskin, National Institute of Aerospace
(NIA)
in collaboration with FUN3D development group at
NASA Langley Research Center

• Southeastern Atlantic Mathematical Science
  (SEAMS) Workshop
• University of North Carolina, Chapel Hill, NC,
• November 2, 2008

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Outline

• Accuracy analysis of finite volume discretization
  (FVD) schemes on irregular grids
• Convergence orders of discretization and
  truncation errors
• Consistent refinement
• Regular and irregular grids
• Windowing and downscaling
• Examples of accuracy analysis
• New findings and applications

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Outline (cont.)

• Analysis of efficiency of multigrid solutions
• Components of a multigrid cycle
• Textbook Multigrid Efficiency (TME)
• Idealized Relaxation (IR) and Idealized Coarse
  Grid (ICG) iterations
• Examples of analysis
• Summary and new findings

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Accuracy analysis of FVD schemes on irregular
grids
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DPW2/3 Configurations (Mavriplis, AIAA CFD
Conference Miami, 2007)

• Up to 72M point meshes

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3rd CFD Drag Prediction Workshop San Francisco,
California June 2006
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Finite volume discretization (FVD) schemes

• General mixed-element meshes
• Node-centered edge-based FVD scheme, median-dual
  partitioning
• Cell-centered FVD scheme

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Accuracy measures
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Relations between convergence of truncation and
discretization errors
The convergence orders do not have to be the same
Lax theorem stability consistency (convergence
of the truncation errors) imply convergence of
the discretization errors Consistency is
sufficient, not a necessary condition Observations
and some rigorous proofs that some FVD schemes
formally non-consistent on unstructured grids can
converge
T. Manteuffuel and A. White, Math.
Of Comp., vol. 47, 1986
B. Despres, Math. of Comp.,
vol. 73, 2003
M. Giles,
Lectrure Notes in Physics, vol. 323, 1989
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Convergence of truncation and discretization
errors
Truncation error for finite-volumes (Turkel,
1986)
The difference in orders occurs for solution
components that are non-smooth on the grid scale
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Consistently Refined Grids

• Constraint to enable meaningful assessment of
  convergence orders with unstructured grids
• Define equivalent mesh sizes (3-D)
• Consistent refinement

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Unstructured Sphere GridsTetrahedral Elements
Far-Field Finest Grid
Far-Field Coarsest Grid
Near-Field Coarse Grid
Near-Field Finest Grid
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Consistent Refinement Check
X Fails Consistent Refinement Check.
Ideal Consistently Refined Mesh
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Computational Grids Regular Grids
Structured smoothly mapped grids

• Periodic topological structures
• Smooth metrics
• Smooth mapping

Examples Cartesian, regular triangular,
curvilinear, stretched, etc.
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Computational Grids Irregular Grids
Unstructured grids
Randomly perturbed grids
Both and more
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Local Elementary Transformations
Bold assumption practical irregular grids can be
derived from regular grids by local elementary
transformations
List of local elementary transformations
(incomplete?)
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Properties of Local Elementary Transformations

• Change fluxes at small number of faces
• Flux contributes to the truncation errors on
  both sides of its face with amounts equal in size
  and opposite in sign.
• Recall
• Elementary transformations do not change the
  total truncation error over a local neighborhood
• May strongly affect discretization errors
  locally have only higher-order effects on
  distant discretization errors. Downscaling test
  is an efficient tool to asses local errors

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Downscaling Test Shrinking Domains
Downscaling with identical meshes
Independent mesh generation at each scale
Downscaling at curved boundary
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Downscaling Test Overview

• General technique that can be applied to
  arbitrary unstructured grids and geometries
• Performs inexpensive computational tests with a
  manufactured solution on several scales.
• Provides sharp estimates for the convergence
  orders of the local discretization and truncation
  errors by comparing errors obtained on different
  scales
• Analyzes accuracy of the interior discretization
  schemes as well as accuracy at the boundary
  and/or in vicinity of singularities

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One-dimensional examples node-based grids
Primal mesh - uniform, Dual mesh - unbiased
unperturbed
Primal mesh - uniform, Dual mesh - unbiased
randomly perturbed
Primal mesh - random, Dual mesh - unbiased
unperturbed
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One-dimensional convection equation Random
primal mesh unbiased, unperturbed dual mesh
Fluxes are linearly interpolated from the
neighboring nodes
Downscaling test
Grid refinement test
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One-dimensional diffusion equation Random primal
mesh randomly perturbed dual mesh
Downscaling test
Grid refinement test
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Accuracy verification methodology
Windows defined by features Problem equations,
boundary conditions, curvature, corners,
etc. Solution attached, separated, shock,
stagnation, vortical, etc. Discretization/Grid
element-types, interfaces, boundary
approximation,etc.
Each combination of features requires a
designated test
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Elliptic equation, node-centered discretization
Downscaling (DS)
Grid Refinement
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Incompressible Inviscid flow, inflow/outflow
boundaries
Cylinder
Triangular Grid
Mixed-Element Grid
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Incompressible inviscid flows
In both downscaling and grid-refinement

• Reduced order of mixed-element node-centered
  scheme results from edge-based flux integration

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Incompressible Inviscid Flows, Tangency Boundary
Far-Field Boundary
Inflow Boundary
Outflow Boundary
Tangency Boundary
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Interior Tangency
2nd order
1st order
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Inflow/tangency corner

• X Fails Design Order Tests
• Local accuracy loss
• Can be repaired

2nd order
1st order
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New findings regarding accuracy of FVD schemes

• Established discretization accuracy for a common
  node-based median-dual FVD scheme
• For inviscid equations
• 2nd order on triangular grids
• 1st order on general mixed-element grids
• For viscous equations
• 2nd order on all grids
• Established accuracy for a general mixed-element
  cell-centered FVD scheme with least-square
  gradient reconstruction
• 2nd order on all grids
• Established local accuracy deterioration at the
  tangency/inflow corner

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New findings regarding accuracy of FVD schemes
(not shown in the talk)

• Demonstrated FVD schemes providing 3rd order
  accuracy on general irregular grids.
• Established local discretization accuracy
  deterioration for inviscid equations at
  stagnation
• Demonstrated 2nd order accuracy on general 3D
  agglomerated grids for viscous fluxes
• Analyzed flat-panel boundary approximations in 3D
• Derived efficient discretization scheme for
  cell-centered and node-centered formulations and
  for agglomerated grids.
• Analyzed accuracy of many schemes used in
  practical codes.

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Computational analysis of multigrid
efficiency Idealized Relaxation (IR) Idealized
Coarse-Grid (ICG) Iterations
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Influence of Gauss-Seidel Relaxation on the Error
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Multigrid V-cycle
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Full MultiGrid (FMG) algorithm
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Textbook Multigrid Efficiency
Solution attained in a total computational work
that is a small (lt 10) multiple of the work
required for one residual evaluation
An order of magnitude error reduction in one
inexpensive multigrid cycle
Computational framework FMG- algorithm. Convergen
ce to discretization level independent of
parameters (Reynolds, Mach, etc.) and mesh sizes.
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Demonstrations of TME for CFD

• In 1984, Brandt demonstrated first TME
  computations for incompressible Navier-Stokes and
  compressible Euler equations in simple
  vortex-free geometries without viscous boundary
  effects.
• In 1990s, Brandt, Yavneh, Oosterlee, and Verner
  showed TME for vortical flows.
• In 2000s, a group at NASA LaRC extended TME to
  more practical CFD computations.
  (Thomas, Diskin,
  Brandt, Textbook Multigrid Efficiency for Fluid
  Simulations, Annu. Rev. Fluid Mech. 2003.
  3517-40)
• The latest advancement demonstrated TME for 3D
  unsteady compressible Navier-Stokes equations.
  (Liao,
  Diskin, Peng, Luo, JCP 227(15),  2008)

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Multigrid convergence observed in practical
computations (Mavriplis, AIAA CFD Conference
Miami, 2007)
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General quantitative multigrid analysis tools

• Qualitative methods do not provide sharp
  estimates for convergence rates and rarely can
  guide practical developments
• Linear algebra analysis of iterations matrix are
  accurate and useful, but prohibitively expensive
  for large-scale applications (unless applied in
  computational windows)
• The main practical quantitative analysis tool,
  Local Mode Fourier analysis, does not predict the
  effects of geometry, boundary conditions, and
  incapable to analyze unstructured-grid
  applications

• Alternative computational-analysis approach
• Employ an available, non-perfect multigrid solver
• Identify, isolate, and improve the parts of the
  solver responsible for the less-than-optimal
  performance.

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Two-grid (?1, ?2) cycle
Fine-grid relaxation sweeps
Standard assignments Relaxation smoothes the
fine-grid error Coarse-grid correction reduces
smooth error components.
Coarse-grid correction
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Idealized relaxation (IR) iterations
Idealized relaxation is an efficient smoothing
procedure directly acting on known algebraic
error Examples explicit averaging of the
algebraic error, Gauss-Seidel or another
efficient relaxation for a nice elliptic operator
(Laplace), etc.

• Perform IR iterations that employ the actual
  coarse-grid correction and an idealized
  relaxation
• IR iterations converge fast the actual
  relaxation is lacking
• IR iterations converge slow the actual
  coarse-grid correction is not efficient

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Idealized coarse-grid (ICG) iterations
Idealized coarse-grid is an explicit procedure
efficiently removing smooth components of the
algebraic error
Operators and should approximate each
other well. Ideal choice two identity operators

• Perform ICG iterations that employ idealized
  coarse-grid and actual relaxation
• ICG iterations converge fast
• the actual coarse-grid correction is lacking
• ICG iterations converge slow
• the actual relaxation is not efficient

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Reference cycle

• Reference cycle employs both idealized relaxation
  and idealized coarse-grid correction
• Efficiency of the reference cycle should be
  better than efficiency goal for the two-grid
  cycle
• For meaningful conclusions, efficiency of the
  reference cycle should be much better than
  efficiency of the actual cycle

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Example 1 Skew Laplacian
Fast convergence of IR iterations and slow
convergence of ICG iterations indicate

• coarse-grid correction is efficient
• relaxation is not efficient

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Example 2 Convection Equation
Slow convergence of IR iterations and fast
convergence of ICG iterations indicate

• coarse-grid correction is not efficient
• relaxation is efficient

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Agglomeration Multigrid
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Summary and new findings

• The IR and ICG iterations are very general and
  practical analysis tool because averaging
  operators required for both IR and ICG iterations
  are readily available in all solvers on any grid.
• ICG and IR iterations have been implemented in
  FUN3D and are considered the main analysis tools
  for multigrid developments.
• Optimally efficient multigrid has been developed
  for solutions on general irregular grids

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QUESTIONS?