NASA Technical Reports Server (NTRS) 20100024103: Computations of Aerodynamic Performance Databases Using Output-Based Refinement

9.6 MB·English

by  NASA Technical Reports Server (NTRS)

Computations of Aerodynamic Performance
Databases using Output-Based Refinement
Marian Nemec and Michael J. Aftosmis
ELORET Corp. NASA Ames
Advanced Supercomputing Division
NASA Ames Research Center
47th AIAA Aerospace Sciences Meeting, Orlando, FL
January 11, 2009
Oral Presentation Only
Motivation
Langley Glide-Back Booster
Database: ~2900 Cases
• How well is the vehicle’s aerodynamic performance estimated?
• Is the mesh appropriate for every flow condition and vehicle configuration?
Objectives
Toward automation of CFD analysis
• Handle complex geometry problems
• Control discretization errors via solution-adaptive
mesh refinement
• Focus on aerodynamic databases of parametric
and optimization studies
1. Accuracy: satisfy prescribed error bounds
2. Robustness and speed: may require over 105
mesh generations
3. Automation: avoid user supervision
• Obtain “expert meshes” independent of user skill
• Run every case adaptively in production settings
Approach
1. Embedded-boundary Cartesian mesh method (1990’s)
•
Arbitrarily complex domains, efficient and accurate
•
Irregularity confined to body intersecting cells
2. Incremental strategy for h-refinement of
nested Cartesian meshes (2002)
•
Fast local re-meshing of flagged cells
•
Guaranteed reliability
•
Early work used feature detection and τ-
extrapolation
3. Adjoint-weighted residual error estimates (2007)
•
Mesh enrichment targets output functionals
•
Functional error-bound estimates
•
Implementation exploits nesting of Cartesian
meshes for fast interpolation
Numerical Error
Uniform Mesh Refinement
Exact Functional:
J

) • Numerical solution on a mesh
.
. E
.

, with cell-size H gives
D
C
approximate functional:

,
L
C J (U )
H
(
J (U )

l H
a
n
o Exact Solution • Goal is to estimate functional
i
t
c
n Approximate Functional error:
u
F
(U) = J (U ) + E
H
J
4 5 6 7 8 9 • Express the error as a function
10 10 10 10 10 10
Number of Cells of the flow solution
E = f (U )
H
Discrete Estimate of Numerical Error
• Consider a simpler problem of
computing relative error:
J (U ) = J (U ) + e
h H

)
.
.
. J (U ) • For second-order accurate
, h
D
e
C
spatial discretization and cell-

,
L
C J (U ) size in the asymptotic range, the
H
(

l functional error is:
a
n
o Exact Solution 1 1
i
t E = e + e + e +
c
Approximate Functional
n 4 42 · · ·
u
F
4
= e
3
4 5 6 7 8 9
10 10 10 10 10 10
• We will use an adjoint solution
Number of Cells
on mesh H to estimate
e = f (U , ψ )
H H
Adjoint Error Estimates
• Consider a functional J ( U ) computed from the solution of Euler
H
equations discretized on an affordable mesh with cell-size H:
R(U ) = 0
H
H h
• In addition, consider an embedded mesh with cell-size h obtained via
uniform refinement of the baseline mesh
• We seek to compute the error relative to the embedded mesh without
solving the problem on the fine mesh
H
e = J (U ) J (U )
h
h
| − |
Venditti & Darmofal, 2002
• Estimate of functional on embedded mesh is obtained from Taylor
series expansions about the coarse mesh solution
H
∂J (U )
H h H
J (U ) J (U ) + (U U )
h h
h h
≈ ∂U −
h
H
∂R(U )
H h H
R(U ) = 0 R(U ) + (U U )
h h
h h
≈ ∂U −
h
• These equations are combined to give
H T H
J (U ) J (U ) ψ R(U )
h
h h h
≈ −
ψ
where satisfies the adjoint equation
T T
H H
∂R(U ) ∂J (U )
h h
ψ =
h
∂U ∂U
h h
! "
• Estimate of functional on embedded mesh is obtained from Taylor
series expansions about the coarse mesh solution
H
∂J (U )
H h H
J (U ) J (U ) + (U U )
h h
h h
≈ ∂U −
h
H
∂R(U )
H h H
R(U ) = 0 R(U ) + (U U )
h h
h h
≈ ∂U −
h
• These equations are combined to give
Adjoints provide
H T H
J (U ) J (U ) ψ R(U ) a weighting on
h
h h h
≈ −
residual errors
ψ
where satisfies the adjoint equation
T T
H H
∂R(U ) ∂J (U )
h h
ψ =
h
∂U ∂U
h h
! "
Adjoint Correction and Error Bound
• Since the adjoint solution is not known on the embedded mesh,
we use an approximate solution from the coarse mesh to obtain
H H T H H T H
J (U ) J (U ) (ψ ) R(U ) (ψ ψ ) R(U )
h h
h h h h h
≈ − − −
Adjoint Correction Remaining Error
H H
• U , ψ denote reconstructed solutions lifted from coarse mesh
h h
to embedded mesh. We use linear interpolation
• ψ is unknown. We approximate it with a U
h h
U
H
quadratic interpolant