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Stanford PSAAP Center

Multi-Physics Adjoints and Solution Verification Karthik Duraisamy Francisco Palacios Juan Alonso Thomas Taylor. Stanford PSAAP Center. Predictive Science: Verification and Error Budgets. Real world problem. Assumptions + Modeling . Mathematical Model. Discretization.

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Stanford PSAAP Center

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

  2. Predictive Science: Verification and Error Budgets Real world problem Assumptions + Modeling Mathematical Model Discretization Numerical solution Numerical Errors Uncertainties 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

  4. Use of Adjoints in V & V

  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 = 2.1362e-01 n=4: QoI = 2.1161e-01 n=8: QoI = 2.1146e-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 SBLI @ 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 5 % flagged 25 % flagged Gradient based Adjoint based

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

  15. Wall pressures Upper wall Lower wall

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

  17. Adjoint Error estimatesQoI : avg pressure at Comb exit (lower wall) QoI : 282.58 kPa ; Error estimate: 2.76 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 Linearized Governing Equations Continuous Adjoint Equations Linearize Discretize Discretized Adjoint Equations Equations with existing analytical formulations/code Governing Equations Hybrid Adjoint Equations Discretize Equations that are difficult/impossible analytically Discretize Linearize Discrete Governing Equations Linearize

  21. Towards a hybrid adjoint See Tom Taylor Poster

  22. Adjoint Solver Status & Applications • A full discrete adjointimplementation (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

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