Error Estimation and Control for Unsteady Flow Problems with Dynamic Meshes using a Discrete Adjoint

1
Error Estimation and Control for Unsteady Flow
Problems with Dynamic Meshes using a Discrete
Adjoint Approach

• Dimitri Mavriplis
• Department of Mechanical Engineering
• University of Wyoming

2
Motivation

• Complex simulations have multiple error sources
• Engineering simulations concerned with specific
  output objectives
• Adjoint methods / Goal Oriented Approach
• Methodical approach for constructing discrete
  adjoint
• Use for a posteriori error estimation
• Spatial error
• Temporal error
• Other error sources
• Use to drive adaptive process

3
Overview

• Discrete Adjoint formulation for steady
  aerodynamics
• Design Optimization problem (3D)
• Discrete Adjoint construction for unsteady
  multiphysics problem
• Dynamically deforming meshes
• Unsteady flow solution
• Unsteady design optimization
• Unsteady error estimation

4
NSU3DG CFL3D1,2 OverflowH,K,..
Grid Convergence All Solutions
5
Drag Prediction Workshop Test Case

• Wing-Body Configuration
• 72 million grid points
• Transonic Flow
• Mach0.75, Incidence 0 degrees, Reynolds
  number3,000,000

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Wing1-Wing2 Grid Convergence Study (DPW3)

• NSU3D delivers consistent grid convergence
  provided
• Flow is mostly attached
• Sequence of refined grids originate from same
  family with same relative resolution distributions

7
Discrete Adjoint Approach

• Solution of adjoint problem enables
• Rapid calculation of sensitivities for
  design-optimization
• Error estimation based on functional outputs
  (lift, drag)
• Sensitivity of local errors to output of interest
  adaptive meshing/time stepping
• Sensitivity can be obtained by
• Finite Difference (perturb input, rerun)
• Tangent Problem Solve linearization of analysis
  problem
• Suitable for single design variable (input), many
  objectives (outputs)
• Adjoint Problem Transpose of Tangent problem or
  linearization
• Suitable for many design variables, few outputs
  (objective)
• Discrete adjoint constructed using modular
  linearization of individual solver subroutines
• Flow Adjoint, Mesh Motion Adjoint
• Adjoint Problems Solved using line-implicit
  multigrid scheme
• Similar convergence as analysis problem (same
  eigenvalues)
• Formulation enables extension to unsteady and
  multidisciplinary problems

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Drag Minimization Problem

• DLR-F6 Wing body configuration
• Mach0.75, Incidence1o , Re3M
• 1.12M grid points, 4.2M cells

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Drag Minimization Problem

• Mach0.75, Incidence1o , CL0.673
• Convergence lt 500 MG cycles, 40 minutes on 16
  cpus
• Change Wing Shape to Reduce Drag
• Objective Drag Design Variables Surface
  Grid Point Positions

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Drag Minimization Problem

• Total Optimization Time for 15 Design Cycles
• 6 hours on 16 cpus of PC cluster
• Flow Solver 150 MG cycles
• Flow Adjoint 50 Defect-Correction cycles (x 4
  MG)
• Mesh Adjoint 25 MG cycles
• Mesh Motion 25 MG cycles

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Goal-Oriented Spatial Adaptivity
DG Discretization
p-adaptivity (Drag)

• h-adaptivity
• (Drag)

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Extension to Unsteady Problems

• ALE finite-volume formulation (cell-centered)
• Satisfies Geometric Conservation Law (GCL)
• Unstructured triangular meshes
• 2nd-order spatial accuracy
• Gradient based reconstruction for 2nd-order
  accuracy
• Temporally 2nd-order accurate (BDF2)
• Mesh deformation via linear tension spring
  analogy
• Roes flux model
• Linear multigrid for convergence acceleration

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Unsteady Discrete Adjoint

• Using chain rule linearization
• New time-step values depend on previous time step
  values
• Integrate linearized equation in time (tangent
  problem)
• Transpose all to get adjoint (and reverse order
  of matrix multiplication)
• Integrate backwards in time (Flow and Mesh
  Motion)
• Gives sensitivity of solution at time ttfinal to
  design variables
• Do not use variational formulation
• Finite volume discretization

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Flow equations
Conservative form of Euler equations
Integrate over moving control volume to get
Arbitrary-Lagrangian-Eulerian (ALE) finite-volume
form
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Mesh Deformation

• Linear Tension Spring Analogy
• Mesh is a series of interconnected springs
• Kdxint dxsurf
• 2 independent force balance equations at each
  node
• Computational Mesh
• Unstructured triangular elements for inviscid
  flows
• 20,000 elements
• 289 surface nodes

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Basic Sensitivity Formulation
Objective function
Forward linearization
Need expressions for these 2 terms
Transpose reverse linearization
Easy to compute
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Details of Unsteady Adjoint
Flow constraint equation for BDF1 scheme
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Continued..
Flow Adjoint at n
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Continued
Mesh Motion constraint equation
Mesh adjoint at n
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Continued.
Will need expression from mesh constraint at n-1
Derivative of equation that prescribes geometry
motion
Will need expression from flow constraint at n-1

• Results in backward time-integration
• Solve one flow adjoint and one mesh adjoint at
  each time-step
• Recurrence stops at n0 via steady state
  adjoint solution for problems initiated from a
  steady state solution

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Typical Multigrid Convergence
Mesh motion
Flow equations
Mesh adjoint
Flow adjoint

• All systems converge within 100 MG cycles
• Flow equations adjoint 45 seconds each
• Mesh equations adjoint 10 seconds each

Formulated as duality preserving iterative scheme
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Optimization Procedure
Run flow solver to obtain unsteady solution
forward time-integration
Deform airfoil surface as a function of D
Compute objective function Lg
Perturb D by dD
Set dD 0
Check Lg
start
stop
Compute design variable perturbation as
Run adjoint solver to compute sensitivities
backward time-integration
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Case 1 Objective Formulation
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Time-Dependent Load Convergence/Comparison
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Unsteady Flow Solution
Pressure Contours for Pitching Airfoils Minf
0.755, a0 0.016o, amax 2.51o, w 0.1628, t0
to 54 27 time-steps with dt2.0
Optimized Airfoil
NACA0012 Baseline Airfoil
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Error Estimation
Test Case Description
Sinusoidally pitching airfoil Functional scalar
is Lift after 1 period
Easily extended to estimate error in
time-integrated Lift history (future work)
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Sources of Error
Error in Time domain
Partial convergence error
Temporal resolution error
Flow equations
Mesh motion equations
Flow equations
Mesh motion equations
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Flow equations
Conservative form of Euler equations
Integrate over moving control volume to get
Arbitrary-Lagrangian-Eulerian (ALE) finite-volume
form
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Mesh Deformation
Linear Tension Spring Analogy Mesh is a series
of interconnected springs Kdxintdxsurf 2
independent force balance equations at each node
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Temporal Resolution Error
Fine level Taylor expansion of functional
objective L
h fine time domain Dt/2 H coarse time
domain Dt
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Evaluation of Flow Contribution to Temporal
Resolution Error
Fine level flow residual Taylor expansion
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Continued
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Continued
Adjoint equation over entire fine time domain
Recast on coarse time domain
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At each time level n for BDF1
Discrete form for BDF1
Can be solved by block backsubstitution
backward recurrence relation in time
(backward integral in time)
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Continued
Contribution to temporal resolution error from
flow equations
Remaining temporal resolution error (due to mesh
motion equations, but also feeds into flow state )
Intepret as sum of dot products of adjoint and
residual at each time step
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Evaluation of Mesh contribution to Temporal
Resolution Error
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Continued
Contribution temporal resolution error from mesh
motion equations
Contribution temporal resolution error from flow
equations
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Summary of Temporal Resolution Error Evaluation

• Compute unsteady flow solution on coarse time
  domain
• Compute adjoint variables on coarse time domain
• Integrating backward in time
• Project adjoint variables, flow solution and mesh
  solution onto fine time domain
• Temporal resolution error is then dot product of
  adjoint with corresponding non-zero residual on
  fine time domain
• Distribution in time is used to drive adaptation

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Validation

• Adjoint is linearization about current state (16
  time steps) to predict objective value on
  modified state (32 time steps)
  
  

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Partial Convergence Error
Coarse level Taylor expansion about partial
solution functional
Partially converged flow and mesh solution
Fully converged flow and mesh solution
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Partial Convergence Error
Fine level flow residual Taylor expansion
non-zero due to partial convergence
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Continued
Same adjoint equations as those for temporal
resolution error
Partial convergence error due to flow equations
Non-zero residuals due to partial convergence
Partial convergence error due to mesh equations
43
Summary of Total Error Evaluation and
Decomposition

• Compute partially converged flow and mesh
  solution on coarse time domain
• Compute adjoint variables on coarse domain using
  partially converged solution
• Compute partial convergence error on coarse level
  time domain
• Inner product of adjoint with partially converged
  (non-zero) residual
• Project partially converged solution and adjoint
  variables onto fine time domain
• Evaluate fine level error estimate as previously
• Combined temporal resolution and partial
  convergence error
• Determine temporal resolution error by
  subtracting partial convergence error from total
  error estimate on fine time domain

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Validation
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Adaptation

• Compute time-integrated averages of error
  component distributions
• Adapt where error is greater than time-integrated
  average
• Time resolution error divide time step by 2
• Convergence error tighten tolerance by
  predetermined factor

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Adaptive Time Step and Convergence Criteria
Example
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Distribution of time steps and convergence limits
in the time domain after 6 adaptation cycles
Time steps
Flow limits
Mesh limits
48
Distribution of Error Components
Flow convergence error
Mesh convergence error
Resolution error
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Conclusions

• Rigorous procedure for determining global error
• Identifies individual error contributions from
  each set of governing equations
• Distribution of each component available and can
  be used for adaptation
• Can be extended to problems involving multiple
  sets of governing equations such as conjugate
  heat transfers, structural equations etc.
• Address other errors such as coupling errors
• Aero-structural coupling
• Further extensions to incorporate space
  adaptivity non-trivial
• Space-time formulations most obvious path forward