application of the discrete adjoint method to coupled multidisciplinary unsteady flow problems for...
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Application of the Discrete Adjoint Method to Coupled Multidisciplinary Unsteady Flow
Problems for Error Estimation and Optimization
Karthik ManiDepartment of Mechanical Engineering
University of Wyoming
Ph.D. Defense
March 27, 2009
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Outline
• Motivation
• Background
• Goals
• Analysis problem
• Adjoint gradient and optimization results
• Adjoint error estimates and adaptation results
• Concluding remarks
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Motivation – Part 1• Began with shape optimization in unsteady flow problems• Evolved to address multiple coupled unsteady disciplines• Airfoil design for unsteady applications such as dynamic stall , flutter control, highly
unsteady flows such as rotorcraft forward flight, rotor-stator interactions in turbomachines etc
• Primary mechanism – use adjoint equations for computing gradient vector
ONERA NLR
ONERA
CFX
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Motivation – Part 2
• Adjoint variables can also be used for goal-based error estimation
• Can be used to drive adaptive solutions
• Spatial goal-based adaptation for steady-state problems has been done
• Time-dependent adjoint variables now available. Can we adapt time-domain in unsteady problems?
Vendetti & DarmofalMIT
FUN3D - NASA Langley
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Background: Gradient-based Optimization
)(DFL Output functional Input parametersFunction
Minimization of a functional by modifying inputs
D
L
dL/dD
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Background:Computation of the Gradient
• F = Flow equations or discrete flow solution (CFD-code)
• D = parametric representation of geometry
• L = functional of interest (ex: lift,drag etc)
Input D CFD Code Output L
Input D+ CFD Code Output Lnew
Change D (i.e shape) to change L (Ex: lift) => need gradient dL/dD => how?
Finite difference
dL/dD ~ L/D = (Lnew – L)/
• is heuristically determined• Large -> wrong slope due to nonlinear effects• Small -> machine error
Multiple D => multiple evaluations of F
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Background:Differentiating F for Gradient
L = F(D)dL/dD = dF(D)/dD
Possible to directly differentiate analytic F to obtain expression for analytic dL/dD
Can also sequentially differentiate operations in CFD code which computes L=F(D)
Input D CFD Code Output L
CFD Coded
dDdL/dD
Continuous Linearization
Discrete Linearization
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Background:Continuous vs. Discrete Linearization
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Background:Adaptive Solutions - Spatial Domain
• Goal is to efficiently use of computational resources
• Add points to the mesh on-the-fly only where it is important
• But which areas are important? High gradients? Local error?
Stagnation points? Wakes?
Shock waves?Boundary layers?
VendittiMIT
• Whatever is relevant to our functional of interest (Ex: lift, drag) is what is important !• Same argument in time domain.• Adjoint equations establish a relationship between the functional and the regions of the mesh that are relevant to it
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Goals:
• Develop a unified framework for computing adjoint variables for coupled unsteady equations
• Compute functional gradients using adjoint variables and demonstrate shape optimization for aeroelastic problems
• Extend utility of adjoint variables for estimating functional relevant temporal discretization error and algebraic error
• Demonstrate adaptation of the time-domain and adaptation of convergence tolerances
• Do this in an unstructured framework with finite-volume formulation applied to dynamic meshes.
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Analysis Problem:Flow Equations
Conservative form of Euler equations
Integrate over a moving control volume
Define flowresidual as:
Newton solver
• Spatial discretization uses second-order accurate matrix dissipation scheme• Temporal discretization uses second-order accurate BDF2 scheme
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Analysis Problem Mesh Motion Equations
Approximate edges as springs
2 Spring equations (x and y) for each node that can be solved as
Overall linear system: relates interior displacements to boundary displacements
Solve using Gauss-Seidel or linear multigrid
0][ surfxxKG
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Analysis Problem:Structural Equations
Airfoil with 2 pitch and plunge degrees-of-freedom
Equations of motion
Matrix form:
Transform as:
Structure residual:
Newton solver:
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Analysis Problem:Geometry Parameterization
Hicks-Henne Sine bump function:
bump location
node location
node displacement
bump magnitude
10 xValid for:
Bump magnitudes amax are the design variables D
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Analysis Problem:Coupled Aeroelastic Solution
Coupled residual equations:
Coupled iterative solution
Coupled iterations convergence
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Analysis Problem:Overall Solution Procedure
Steady-State solution from freestream
conditions
Initiate unsteady solution using steady-state solution
Unsteady aeroelastic solution initiated by prescribed unsteady
solution
Read flow parameters and
mesh
Deform mesh to correct AOA
Deform mesh to prescribed orientation at current time-step
Flow solution at current time-step
Prescribed unsteadytime-integration Time-integration with coupled
iterative aeroelastic solution at each time-step
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Adjoint-based Gradient:A Simple Steady-State Example
))(,( DUDLL
D
U
UDD
LL
d
dL
Consider a functional L evaluated using solution state U:
Linearize for gradient:
Linearization of scalar – easily computable
state sensitivity to D is a matrix
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Adjoint-based Gradient:A Simple Steady-State Example
Residual equation used to obtain solution U 0))(,( DUDR
Residual has to evaluate to zero at all points, therefore derivative is also zero
D
R
D
U
U
R
0
D
U
U
R
D
R
• Limiting case of a single D, this is a linear system that can be solved to be obtain state sensitivity (vector)• For multiple D, multiple linear solutions required to construct state sensitivity one column at a time• Better than finite-difference in terms of accuracy, but no better in cost
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Adjoint-based Gradient:A Simple Steady-State Example
TTTT LL
d
dL
UD
U
DD
TTT
U
R
D
R
D
U
U
UU
R
D
R
DD
TTTTT LL
d
dL
Transpose to obtain adjoint linearization:
TTL
UU
RU
Rearrange state sensitivity expression:
Substitute into adjoint linearization and introduce adjoint variable:
Linear adjoint equation D
R
D
U
U
R
• No dependence on D during linear solution• Effect of D confined to final matrix-vector product UD
R
DD
TTT L
d
dL
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Generalized Form for Multidisciplinary Unsteady Coupled Equations
m coupled disciplines=> L is a functional computed using multidisciplinary solution set:
Linearize using chain rule with respect to D:
Inner-product form: Tranpose for adjoint total sensitivity:
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Generalized Form for Multidisciplinary Unsteady Coupled Equations
Residual equations for m disciplines:Linearize with respect to D:
Write in combined matrix form:
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Generalized Form for Multidisciplinary Unsteady Coupled Equations
Transpose and rearrange for vector of state sensitivity matrices:
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Generalized Form for Multidisciplinary Unsteady Coupled Equations
Substitute into adjoint total sensitivity equation:
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Generalized Form for Multidisciplinary Unsteady Coupled Equations
Define vector of disciplinary adjoint variables:
Rearrange to recover linear adjoint system:
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Generalized Form for Multidisciplinary Unsteady Coupled Equations
General Jacobian matrix when expanded discretely in time is lower triangular due to hyperbolic nature of time:
Swap rows and columns to obtain an upper triangular linear system that can be solved by back substitution
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Generalized Form for Multidisciplinary Unsteady Coupled Equations
Finally, substitute vector of disciplinary adjoint variables into total sensitivity equation to obtain
gradient
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Unsteady Aeroelastic ProblemFunctional based on solution to unsteady flow, structure and mesh equations
Linearize, tranpose, and write in inner-product form for adjoint total sensitivity:
Tap flow, structure and mesh residuals for state variable sensitivity matrix expression
Mesh residual is function of structural state
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Unsteady Aeroelastic Problem
Linearize residuals:
Combined matrix form:
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Unsteady Aeroelastic ProblemSubstitute expression for vector of state variables sensitivity matrices into
adjoint total sensitivity equation and define adjoint variables:
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Unsteady Aeroelastic ProblemNow determine form of discrete temporal expansion of Jacobians
Flow residual at time-step n:
Tri-diagonal for both residual to flow state and residual to mesh coordinate sensitivity
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Unsteady Aeroelastic ProblemMesh residual at time-step n:
All Jacobians are diagonal since there is no dependence beyond time-step n:
Structure residual:
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Unsteady Aeroelastic Problem
Substitute expanded Jacobians back into adjoint system:
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Unsteady Aeroelastic ProblemSwap rows and columns to form an upper triangular system:
•Each diagonal block represents one time-step
•Diagonal block is coupled – requires solution procedure similar to analysis problem
•Backward sweep in time with coupled iterative solution at each time step
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Unsteady Aeroelastic Problem
Segregated form at an arbitrary time-step:
When adjoint variables are available, substitute back into total sensitivity equation:
Complete gradient then:
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Uncoupled Unsteady ProblemFunctional based on only time-dependent flow and mesh coordinates:
Linearize, tranpose and convert to inner-product form:
Residual equations:
Linearize, combine into matrix form, tranpose and rearrange for vector of state variable sensitivity matrices
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Uncoupled Unsteady Problem
Substitute into total sensitivity and define vector of adjoint variables:
Discretely expand in time using definitions for Jacobians from before:
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Uncoupled Unsteady Problem
Rearrange to form upper triangular adjoint system
Note: Diagonal blocks are also upper triangular => uncoupled equations
Segregated form:
First solve flow adjoint then solve mesh adjoint
Substitute adjoint variables into total sensitivity.
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Steady-State ProblemAdjoint system is same as the uncoupled unsteady problem. No
discrete temporal expansion since solutions span only space.
Solve directly by forward substitution
First solve flow adjoint as: Then mesh adjoint as:
Mesh residual for steady-state:
Final gradient then becomes:
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Validation of Adjoint-based Gradient
• Forward linearization is first verified against finite-difference• Dual consistency between transpose operations is used to verify adjoint gradient
Primal operationDual operation
Comparison of adjoint gradient with finite-difference
Dual consistency requires:
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Steady-State Shape Optimization
Lift constrained drag minimization of a NACA0012 in transonic flow
Mach number = 0.8, AOA = 1.25 degrees
Functional:
Computational mesh ~ 20,000 elements
•L-BFGS-B algorithm for optimization
•Uses gradient and builds approximate Hessian on-the-fly using history
•32 bump functions – 16 on upper surface and 16 lower surface
•Magnitudes of bumps form vector design variables
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Steady-State Shape Optimization
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Steady-State Shape Optimization
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Steady-State Shape Optimization
Entropy Contours
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Uncoupled Unsteady ProblemTime-Dependent Load Matching
AGARD test case No.5: Mach number = 0.755, =0.016o, kc=0.0814, m=2.51o
• L-BFGS-B Optimization Algorithm
• 290 design variables
• Bump at each surface node
• Same mesh as steady-state problem
• Target load is that of NACA64210
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Uncoupled Unsteady ProblemTime-Dependent Load Matching
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Uncoupled Unsteady ProblemTime-Dependent Load Matching
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Uncoupled Unsteady ProblemTime-Dependent Load Matching
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Uncoupled Unsteady ProblemTime-Dependent Pressure Distribution Matching
Same unsteady problem and global functional formulation.Local functional at each time-step is the difference between pressures
• Inverse design in unsteady flow from NACA0012 to NACA64210• Target airfoil is NACA64210 “type” airfoil • Target is closest match possible using the parameterization of the geometry• 290 design variables (Convergence stalls)• Reduced to 14 design variables and true optimum achieved in 28 iterations
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Uncoupled Unsteady ProblemTime-Dependent Pressure Distribution Matching
14 design variables290 design variables
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Uncoupled Unsteady ProblemTime-Dependent Pressure Distribution Matching
290 design variables case
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Uncoupled Unsteady ProblemTime-Dependent Pressure Distribution Matching
14 design variables case
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Uncoupled Unsteady ProblemTime-Dependent Lift Constrained Drag
Minimization
Same unsteady problem functional formulation as time-dependent load matching.Target drag now set to zero at all time-steps. Target lift is that of NACA0012
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Uncoupled Unsteady ProblemTime-Dependent Lift Constrained Drag
Minimization
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Unsteady Aeroelastic ProblemSingle Element Airfoil – Flutter Control
Functional formulation for zero velocities and displacements at final time-step, constrained by steady-state lift at max forced pitch:
NACA64A010 airfoil at Mach number = 0.825, Flutter velocity = 0.7532 design variables, 16 upper, 16 lower. L-BFGS-B algorithm
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Unsteady Aeroelastic ProblemSingle Element Airfoil – Flutter Control
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Unsteady Aeroelastic ProblemSingle Element Airfoil – Flutter Control
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Unsteady Aeroelastic ProblemTwo-Element Slotted Airfoil – Flutter Control
Functional formulation same as previous case but converted to sum over final 10 time-steps:
Slotted airfoil with identical structural parameters at Mach number = 0.6, Flutter velocity = 1.8532 design variables, 8x4 upper and lower on both airfoils. L-BFGS-B algorithm
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Unsteady Aeroelastic ProblemTwo-Element Slotted Airfoil – Flutter Control
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Unsteady Aeroelastic ProblemTwo-Element Slotted Airfoil – Flutter Control
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Functional Relevant ErrorA Simple Spatial Example
Arbitrary fine resolution - hArbitrary coarse resolution - H
Lh(Uh)LH(UH)Functional using real solution on h:
Functional using real solution on H:
Can we estimate true fine level functional using only coarse level solution?
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Functional Relevant ErrorA Simple Spatial Example
Project coarse solution onto fine solution
UH UHh
Evaluate functional on fine level using projected solution, then expand using Taylor series:
Hhh
Hhhhh
Hh
LLL UU
UUU
U
)()(
Up to first order
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Functional Relevant ErrorA Simple Spatial Example
Hhh
Hhhhh
Hh
LLL UU
UUU
U
)()(
Rearrange and approximate the error in the functional as:
Easily computableWe don’t know this because we don’t want to solve on h to get Uh
0)()(
Hhh
Hhhhh
Hh
UUU
RURUR
U
)(1
Hhh
Hhh
Hh
URU
RUU
U
Expand residual on fine level the same way:
Rearrange for unknown:
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Functional Relevant ErrorA Simple Spatial Example
)()()(1
Hhh
Hhhhh
Hh
Hh
LLL UR
U
R
UUU
UU
1
Hh
Hh
h
LT
UUU U
R
U
TT
Hh
hHh
L
UU
U UU
R
Substitute into error equation:
Define adjoint variable on fine level: Rearrange for fine adjoint system:
Approximate on coarse (to avoid fine level solution):
TT
H
H
H
L
UU
U UU
R
H
Hhh I UU
Project to fine level:
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Functional Relevant ErrorA Simple Spatial Example
)()()( Hhh
THhhhh h
LL URUU USubstitute adjoint and then error estimate becomes:
ncells
iii
1
R
•This is a distribution in space of the error relevant to functional L.•Sum of distribution (i.e. ) is also a correction to the functional.•Use error distribution to adapt relevant regions of mesh.
Inner-product between Adjoint and Nonzero residual.Residual evaluated of fine level using projection of solution from coarse level.
Interpret inner-product as sum of adjoint-weighted nonzero residual over all cells.Non-zero residual is an estimate of local error in each cell.
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Sources of Error
Multidisciplinary solution
Temporal discretization
error
Spatial discretization
error
Algebraic error
Within each type of error, contributions from each discipline
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Temporal Discretization Error:Convention is now: h-fine temporal mesh, H-coarse temporal mesh
Functional on some arbitrary fine temporal level using multidisciplinary solution set:
Assuming full convergence, expand using Taylor series about functional computed using projected solution from coarse temporal mesh:
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Truncate to remove higher-order terms and write in inner-product form:
Easily computable
Unknown at this point
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Residual equation of arbitrary discipline j on fine temporal mesh expanded about residual constructed using projected solution from coarse temporal mesh:
Do this for all disciplines and write in combined matrix form:
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Rearrange to get an expression for vector state variable differences due to projection:
Substitute back into error equation and define vector of disciplinary adjoint variables:
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Recover adjoint system (this is on the fine temporal mesh):
Recast on coarse temporal mesh – Now this system is identical to that presented in the gradient derivation:
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Project adjoint variables onto fine level and now error is the inner-product between disciplinary adjoint and corresponding disciplinary non-zero residual:
Total temporal discretization error = sum of disciplinary adjoint weighted disciplinary residuals.
Tells us how much discretization error arises from each discipline.
But each disciplinary inner product is again a sum in time.Tells us how much temporal discretization error arises from each time-step within each discipline
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Algebraic Error:Convention is now: overbar indicates partial convergence
All operations on coarse temporal level
Expand functional on coarse level computed using fully converged solution about functional based on partially converged solution:
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Truncate and write in inner-product form:
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Expand fully converged disciplinary residual about partially converged residual:
Do for all disciplines and write in combined matrix form. Rearrange to extract unknown:
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Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Substitute into algebraic error equation and define adjoint variables to recover adjoint system as:
Nearly identical to adjoint system for temporal discretization error. Except now use partially converged solution on coarse level
Disciplinary algebraic error is now again adjoint-weighted nonzero residual (due to partial convergence)
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Separation of Algebraic and Temporal Discretization Errors
For temporal discretization we assumed full convergence on coarse level before projecting to fine level to get:
If we partially converge on coarse level and then project to fine level then:
This includes effects due to partial convergence and projection = Total error.Separate temporal discretization error by subtracting algebraic error.
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Validation of Error EstimatesTemporal discretization error only:
Flow algebraic error:
Mesh algebraic error:
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Validation of Error Estimates
Combined total error:
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Adaptation ResultsGeneral Notes:
Targeted temporal adaptation compared against local error-based adaptation:
Local error estimated as:
Adaptation strategy:Sort by time-steps by error contribution - decreasing order
Parse down list and flag time-steps for refinement until 99% error is coveredSame for all error components
Temporal resolution adaptation = divide time-step by twoConvergence tolerance adaptation = tighten by factor of 3
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Adaptation ResultsNon-Time-Integrated Functional
NACA64A010 Airfoil with prescribed pitching motion operating at Mach number of 0.3
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Adaptation ResultsNon-Time-Integrated Functional
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Adaptation ResultsNon-Time-Integrated Functional
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Adaptation ResultsNon-Time-Integrated Functional
At first adaptation cycle
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Adaptation ResultsNon-Time-Integrated Functional
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Adaptation ResultsNon-Time-Integrated Functional
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Adaptation ResultsNon-Time-Integrated Functional
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Adaptation ResultsNon-Time-Integrated Functional
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Adaptation ResultsTime-Integrated Functional
Interaction of a convecting vortex with a slowly pitching airfoil.NACA0012 airfoil pitching at kc=0.001. Mach number is 0.4225. Starting at 50 steps uniform time-steps.
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Adaptation ResultsTime-Integrated Functional
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Adaptation ResultsTime-Integrated Functional
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Adaptation ResultsTime-Integrated Functional
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Adaptation ResultsTime-Integrated Functional
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Adaptation ResultsTime-Integrated Functional
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Adaptation ResultsTime-Integrated Functional
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Adaptation ResultsTime-Integrated Functional
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Adaptation ResultsTime-Integrated Functional
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Concluding RemarksSummary:
• Developed a unified framework for computing adjoint variables for multidisciplinary coupled unsteady equations
• Adjoint variables were used to construct gradient for use in shape optimization
• Goal-based error estimates for temporal discretization and algebraic errors formulated using adjoint variables
• Adaptation of time domain and convergence tolerances were demonstrated
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Concluding RemarksPotential Applications:
• Rapid determination of gradient vector or sensitivity derivatives for coupled unsteady equations for design optimization
• Model parameter sensitivity
• Improved unsteady simulation capability
• Quantification and compensation for algebraic errors
• Error due to coupling
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Concluding RemarksFuture work:
• Demonstrate algorithms in realistic 3D problems
• Target all error components (spatial, temporal, algebraic, coupling etc) to develop efficient solvers
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Thank you.
Questions?