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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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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 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.
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
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
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]
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
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
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
Supersonic Combustion model problem: Full Adjoint Frozen turbulence Exact Jacobians : CFL ~ 1000+ Approx Jacobians : CFL ~ 0.1
Goal oriented Error estimation Governing equation and functional on Error estimate on (Venditti & Darmofal) Have also extended it to estimate and control stochastic errors
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
Adapive Mesh refinement QoI: Integrated pressure on lower wall 2.5 % flagged 5 % flagged 25 % flagged Gradient based Adjoint based
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
Wall pressures Upper wall Lower wall
Adjoint SolutionQoI : avg pressure at Comb exit (lower wall) 24 hrs, 840 procs: Local LU preconditioning + GMRES
Adjoint Error estimatesQoI : avg pressure at Comb exit (lower wall) QoI : 282.58 kPa ; Error estimate: 2.76 kPa (0.98%)
Goal oriented refinement QoI : Stagnation pressure at Nozzle exit
Goal oriented mesh refinement : Results Baseline mesh Adapted mesh
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
Towards a hybrid adjoint See Tom Taylor Poster
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