1. Multi-Physics Adjoints and Solution Verification
Stanford PSAAP Center
Multi-Physics Adjoints and Solution Verification
Karthik Duraisamy
Francisco Palacios
Juan Alonso
Thomas Taylor

2. Predictive Science: Verification and Error Budgets
Real world problem
Mathematical Model
Assumptions + Modeling
Uncertainties
Numerical solution
Discretization
Numerical Errors
Certification,QMU
Use
Quantifying numerical / discretization errors is a necessary first step to quantify sources of uncertainty. Controlling numerical errors is necessary to achieve certification. Computational budget must be balanced between addressing numerical and UQ errors.

3. Key Accomplishments
Full Discrete Adjoint Solver for Compressible RANS Equations with turbulent combustion
 Fully integrated with flow solver
 Massively parallel
 Robust Convergence
Application to variety of PSAAP center problems including full Scramjet combustor
New developments
 Stochastic adjoints
 Hybrid adjoints
 Robust grids for UQ
Focus on three key contributions:

4. Use of Adjoints in V & V Capability Case Typ. Verific. Validation
Mesh conv
Error Est
Goal
Adapt
UQ Loop
Inviscid Disc / BC
Ringleb 2D
Analytic
Inviscid
Disc/Shocks
hyshot/1D comb model
DLR
Table lookup
Shock-ind Comb
Numeric
Viscous Disc
Lam SBLI
6th Order
Hakkinen
Turbulence
STBLI
LES-Morgan
Shock train
UQ Expt
2D/3D
Eaton/LES
Cold Hyshot
Turb+Comb
React Mix Layer
Hyshot 3D

5. RANS + Combustion: Governing equations
5 Flow equations +
2 Turbulence model equations +
3 Combustion model equations (FPVA), Peters 2000; Terrapon 2010
Table lookup
(Functions of transported variables and pressure) +
Equations of state +
Material properties

6. The Discrete Adjoint Equations
Conserved Variables
Flow Equations
Adjoint Equations
Computed using Automatic Differentiation, so can be arbitrarily complex
Note: Interpolation operators can also be differentiated
Non-zero elements in Jacobian:
33x10x10xN
[For 3D structured mesh]

7. Sample QoI : Shock crossing point in UQ Experiment
Contours of
n=2: QoI = e-01
n=4: QoI = e-01
n=8: QoI = e-01

8. Adjoint Equations : Solution
Truly unstructured grids with shocks and thin features result in very poorly conditioned systems
Original system : Preconditioned
GMRES not effective
Iterative solution: More robust
Laminar Rex = 3x105
Exact or approximate Jacobians

9. Supersonic Combustion model problem
OH Mass
Fraction
Air: V=1800 m/s,
T= 1550 K
Splitter plate
H2: V=1500 m/s,
T= 300 K
Pressure
K-w SST with
FPVA model on a mesh of 5000 CVs
QoI

10. Supersonic Combustion model problem:
Full Adjoint
Frozen turbulence
Exact Jacobians : CFL ~ 1000+
Approx Jacobians : CFL ~ 0.1

11. Goal oriented Error estimation
Governing equation and functional on
Error estimate on
(Venditti & Darmofal)
Have also extended it to estimate and control stochastic errors

12. Test 1: Shock-Turbulent Boundary Layer Interaction
Incoming BL: Mach number = 2.28, Rϑ = 1500, Shock deflection angle = 8o
LES
RANS
Reference Error: 3.1 e-04

13. Adapive Mesh refinement QoI: Integrated pressure on lower wall
2.5 % flagged % flagged % flagged
Gradient based
Adjoint based

14. Application to Scramjet Combustion
Forebody Ramp
Inlet/Isolator
Combustor
Nozzle/Afterbody
Fuel
Injection
Flow
Mach ~8
Air
1800 m/s, 1300 K, 1.2 bar H2
300K, 5 bar (total)

15. Wall pressures
Upper wall Lower wall

16. Adjoint Solution QoI : avg pressure at Comb exit (lower wall)
GMRES
24 hrs, 840 procs:
Local LU preconditioning + GMRES

17. Adjoint Error estimates QoI : avg pressure at Comb exit (lower wall)
Top : Estimated error contribution to QoI
Middle: Adjoint solution (adjoint of energy variable)
Bottom: Truncation error estimate (in energy equation)
QoI : kPa ; Error estimate: kPa (0.98%)

18. Goal oriented refinement QoI : Stagnation pressure at Nozzle exit

19. Goal oriented mesh refinement : Results
Baseline mesh
Adapted mesh

20. Towards a hybrid adjoint
Governing
Equations
Discrete
Linearized
Hybrid
Adjoint
Discretized Adjoint
Discretize
Linearize
Continuous
Equations that are difficult/impossible analytically
Equations with existing analytical formulations/code

21. Towards a hybrid adjoint
Discrete
Continuous
Hybrid
Development + –
Compatibility with discretized PDE ?
Compatibility with continuous PDE
Surface formulation for gradients
Arbitrary functionals
Non-differentiability
Computational cost
Flexibility in solution
See Tom Taylor Poster

22. Adjoint Solver Status & Applications
A full discrete adjoint implementation (using automatic differentiation) has been developed & verfied in Joe for the compressible RANS equations with the following features
 Turbulence (k-w, SST and SA models)
 Multi-species mixing
 Combustion with FPVA
Capabilities are used in different applications in PSAAP
 Estimation of numerical errors
 Mesh adaptation
 Robust grids for UQ
 Estimation and control of uncertainty propagation errors
 Sensitivity and risk analysis (acceleration of MC sampling) (Q. Wang)
 Balance of Errors and uncertainties (J. Witteveen)
Continuous adjoint also available in Joe for the compressible laminar NS equations
A new hybrid adjoint formulation developed and applied to idealized problems
Massively parallel implementation available using MUM and PETSC