1. Department of Mechanical Engineering
Application of the Discrete Adjoint Method to Coupled Multidisciplinary Unsteady Flow Problems for Error Estimation and Optimization
Karthik Mani
Department of Mechanical Engineering
University of Wyoming
Ph.D. Defense
March 27, 2009

2. Outline Motivation Background Goals Analysis problem
Adjoint gradient and optimization results
Adjoint error estimates and adaptation results
Concluding remarks

3. 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
CFX
ONERA
NLR
ONERA

4. 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 & Darmofal
MIT
FUN3D - NASA Langley

5. Background: Gradient-based Optimization
Minimization of a functional by modifying inputs
Output functional
Function
Input parameters
dL/dD L D

6. 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)
Change D (i.e shape) to change L (Ex: lift) => need gradient dL/dD => how?
Finite difference
Input D
CFD Code
Output L
Input D+e
CFD Code
Output Lnew
dL/dD ~ DL/DD = (Lnew – L)/e
e is heuristically determined
Large -> wrong slope due to nonlinear effects
Small -> machine error
Multiple D => multiple evaluations of F

7. Background: Differentiating F for Gradient
Possible to directly differentiate analytic F to obtain expression for analytic dL/dD
L = F(D)
dL/dD = dF(D)/dD
Continuous Linearization
Can also sequentially differentiate operations in CFD code which computes L=F(D)
Input D
CFD Code
Output L d
CFD Code
dL/dD dD
Discrete Linearization

8. Background: Continuous vs. Discrete Linearization

9. 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?
Boundary layers?
Shock waves?
Venditti
MIT
Wakes?
Stagnation points?
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

10. 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.

11. Analysis Problem: Flow Equations
Conservative form of Euler equations
Integrate over a moving control volume
Spatial discretization uses second-order accurate matrix dissipation scheme
Temporal discretization uses second-order accurate BDF2 scheme
Define flow
residual as:
Newton solver

12. Analysis Problem Mesh Motion Equations
Approximate
edges as
springs
Overall linear system: relates interior displacements to boundary displacements
2 Spring equations (x and y) for each node that can be solved as
Solve using Gauss-Seidel or linear multigrid

13. Analysis Problem: Structural Equations
Equations of motion
Airfoil with 2 pitch and plunge degrees-of-freedom
Matrix form:
Transform as:
Structure residual:
Newton solver:

14. Analysis Problem: Geometry Parameterization
Hicks-Henne Sine bump function:
node displacement
bump magnitude
node location
bump location
Valid for:
Bump magnitudes amax are the design variables D

15. Analysis Problem: Coupled Aeroelastic Solution
Coupled residual equations:
Coupled iterative solution
Coupled iterations convergence

16. Analysis Problem: Overall Solution Procedure
Read flow parameters and mesh
Steady-State solution from freestream conditions
Deform mesh to correct AOA
Deform mesh to prescribed orientation at current time-step
Initiate unsteady solution using steady-state solution
Unsteady aeroelastic solution initiated by prescribed unsteady solution
Flow solution at current time-step
Prescribed unsteady
time-integration
Time-integration with coupled iterative aeroelastic solution at each time-step

17. Adjoint-based Gradient: A Simple Steady-State Example
Consider a functional L evaluated using solution state U:
state sensitivity to D is a matrix
Linearize for gradient:
Linearization of scalar – easily computable

18. Adjoint-based Gradient: A Simple Steady-State Example
Residual equation used to obtain solution U
Residual has to evaluate to zero at all points, therefore derivative is also zero
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

19. Adjoint-based Gradient: A Simple Steady-State Example
Transpose to obtain adjoint linearization:
Rearrange state sensitivity expression:
Substitute into adjoint linearization and introduce adjoint variable:
Linear adjoint equation
No dependence on D during linear solution
Effect of D confined to final matrix-vector product

20. 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:

21. Generalized Form for Multidisciplinary Unsteady Coupled Equations
Linearize with respect to D:
Residual equations for m disciplines:
Write in combined matrix form:

22. Generalized Form for Multidisciplinary Unsteady Coupled Equations
Transpose and rearrange for vector of state sensitivity matrices:

23. Generalized Form for Multidisciplinary Unsteady Coupled Equations
Substitute into adjoint total sensitivity equation:

24. Generalized Form for Multidisciplinary Unsteady Coupled Equations
Define vector of disciplinary adjoint variables:
Rearrange to recover linear adjoint system:

25. 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

26. Generalized Form for Multidisciplinary Unsteady Coupled Equations
Finally, substitute vector of disciplinary adjoint variables into total sensitivity equation to obtain gradient

27. Unsteady Aeroelastic Problem
Functional 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

28. Unsteady Aeroelastic Problem
Linearize residuals:
Combined matrix form:

29. Unsteady Aeroelastic Problem
Substitute expression for vector of state variables sensitivity matrices into adjoint total sensitivity equation and define adjoint variables:

30. Unsteady Aeroelastic Problem
Now 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

31. Unsteady Aeroelastic Problem
Mesh residual at time-step n:
All Jacobians are diagonal since there is no dependence beyond time-step n:
Structure residual:

32. Unsteady Aeroelastic Problem
Substitute expanded Jacobians back into adjoint system:

33. Unsteady Aeroelastic Problem
Swap 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

34. Unsteady Aeroelastic Problem
Segregated form at an arbitrary time-step:
When adjoint variables are available, substitute back into total sensitivity equation:
Complete gradient then:

35. Uncoupled Unsteady Problem
Functional 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

36. Uncoupled Unsteady Problem
Substitute into total sensitivity and define vector of adjoint variables:
Discretely expand in time using definitions for Jacobians from before:

37. 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.

38. Steady-State Problem
Adjoint 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:

39. 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 operation
Dual operation
Dual consistency requires:
Comparison of adjoint gradient with finite-difference

40. 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

41. Steady-State Shape Optimization

42. Steady-State Shape Optimization

43. Steady-State Shape Optimization
Entropy Contours

44. Uncoupled Unsteady Problem Time-Dependent Load Matching
AGARD test case No.5:
Mach number = 0.755, a=0.016o, kc=0.0814, am=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

45. Uncoupled Unsteady Problem Time-Dependent Load Matching

46. Uncoupled Unsteady Problem Time-Dependent Load Matching

47. Uncoupled Unsteady Problem Time-Dependent Load Matching

48. Uncoupled Unsteady Problem Time-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

49. Uncoupled Unsteady Problem Time-Dependent Pressure Distribution Matching
290 design variables
14 design variables

50. Uncoupled Unsteady Problem Time-Dependent Pressure Distribution Matching
290 design variables case

51. Uncoupled Unsteady Problem Time-Dependent Pressure Distribution Matching
14 design variables case

52. Uncoupled Unsteady Problem Time-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

53. Uncoupled Unsteady Problem Time-Dependent Lift Constrained Drag Minimization

54. Unsteady Aeroelastic Problem Single 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 = 0.75
32 design variables, 16 upper, 16 lower. L-BFGS-B algorithm

55. Unsteady Aeroelastic Problem Single Element Airfoil – Flutter Control

56. Unsteady Aeroelastic Problem Single Element Airfoil – Flutter Control

57. Unsteady Aeroelastic Problem Two-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 = 1.85
32 design variables, 8x4 upper and lower on both airfoils. L-BFGS-B algorithm

58. Unsteady Aeroelastic Problem Two-Element Slotted Airfoil – Flutter Control

59. Unsteady Aeroelastic Problem Two-Element Slotted Airfoil – Flutter Control

60. Functional Relevant Error A Simple Spatial Example
Arbitrary coarse resolution - H
Arbitrary fine resolution - h
Functional using real solution on h:
Functional using real solution on H:
Lh(Uh)
LH(UH)
Can we estimate true fine level functional using only coarse level solution?

61. Functional Relevant Error A Simple Spatial Example
Project coarse solution onto fine solution UH
UHh
Evaluate functional on fine level using projected solution, then expand using Taylor series:
Up to first order

62. Functional Relevant Error A Simple Spatial Example
Rearrange and approximate the error in the functional as:
We don’t know this because we don’t want to solve on h to get Uh
Easily computable
Expand residual on fine level the same way:
Rearrange for unknown:

63. Functional Relevant Error A Simple Spatial Example
Substitute into error equation:
Define adjoint variable on fine level:
Rearrange for fine adjoint system:
Approximate on coarse (to avoid fine level solution):
Project to fine level:

64. Functional Relevant Error A Simple Spatial Example
Substitute adjoint and then error estimate becomes:
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.
This is a distribution in space of the error relevant to functional L.
Sum of distribution (i.e. e) is also a correction to the functional.
Use error distribution to adapt relevant regions of mesh.

65. Sources of Error Multidisciplinary solution
Temporal discretization error
Spatial discretization error
Algebraic error
Within each type of error, contributions from each discipline

66. Temporal Discretization Error:
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:

67. 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

68. 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:

69. 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:

70. 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:

71. 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

72. 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:

73. Generalized Error Form for Multidisciplinary Unsteady Coupled Equations
Truncate and write in inner-product form:

74. 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:

75. 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)

76. 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.

77. Validation of Error Estimates
Temporal discretization error only:
Flow algebraic error:
Mesh algebraic error:

78. Validation of Error Estimates
Combined total error:

79. Adaptation Results General 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 covered
Same for all error components
Temporal resolution adaptation = divide time-step by two
Convergence tolerance adaptation = tighten by factor of 3

80. Adaptation Results Non-Time-Integrated Functional
NACA64A010 Airfoil with prescribed pitching motion operating at Mach number of 0.3

81. Adaptation Results Non-Time-Integrated Functional

82. Adaptation Results Non-Time-Integrated Functional

83. Adaptation Results Non-Time-Integrated Functional
At first adaptation cycle

84. Adaptation Results Non-Time-Integrated Functional

85. Adaptation Results Non-Time-Integrated Functional

86. Adaptation Results Non-Time-Integrated Functional

87. Adaptation Results Non-Time-Integrated Functional

88. Adaptation Results Time-Integrated Functional
Interaction of a convecting vortex with a slowly pitching airfoil.
NACA0012 airfoil pitching at kc= Mach number is Starting at 50 steps uniform time-steps.

89. Adaptation Results Time-Integrated Functional

90. Adaptation Results Time-Integrated Functional

91. Adaptation Results Time-Integrated Functional

92. Adaptation Results Time-Integrated Functional

93. Adaptation Results Time-Integrated Functional

94. Adaptation Results Time-Integrated Functional

95. Adaptation Results Time-Integrated Functional

96. Adaptation Results Time-Integrated Functional

97. Concluding Remarks Summary:
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

98. Concluding Remarks Potential 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

99. Concluding Remarks Future work:
Demonstrate algorithms in realistic 3D problems
Target all error components (spatial, temporal, algebraic, coupling etc) to develop efficient solvers

100. Thank you.
Questions?