Stanford PSAAP CenterMulti-Physics Adjoints and Solution Verification

Karthik DuraisamyFrancisco Palacios

Juan AlonsoThomas Taylor

Predictive Science: Verification and Error Budgets

Certification,QMU

Real world problem

Numerical Errors

Uncertainties

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.

Mathematical Model

Assumptions + Modeling

Numerical solution

Discretization

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

Capability Case Typ. Verific. Validation Mesh conv

Error Est

GoalAdapt

UQ Loop

Inviscid Disc / BC

Ringleb 2D Analytic

InviscidDisc/Shocks

hyshot/1D comb model

2D DLR

Table lookup Shock-ind Comb

2D Numeric

Viscous Disc Lam SBLI 2D 6th Order Hakkinen

Turbulence STBLI 2D LES-Morgan

Turbulence Shock train 2D LES-Morgan

Turbulence UQ Expt 2D/3D Eaton/LES

Turbulence Cold Hyshot 2D DLR

Turb+Comb React Mix Layer

2D

Turb+Comb Hyshot 3D DLR

Use of Adjoints in V & V

RANS + Combustion: Governing equations 5 Flow equations

2 Turbulence model equations

+

+

3 Combustion model equations (FPVA), Peters 2000; Terrapon 2010

+

Equations of state

+

Material properties

Table lookup(Functions of transported variables and pressure)

The Discrete Adjoint Equations

Conserved Variables

Flow Equations

Adjoint Equations

Computed using Automatic Differentiation, so can be arbitrarily complexNote: Interpolation operators can also be differentiated

Non-zero elements in Jacobian:33x10x10xN[For 3D structured mesh]

Contours of

Sample QoI : Shock crossing point in UQ Experiment

n=2: QoI = 2.1362e-01n=4: QoI = 2.1161e-01n=8: QoI = 2.1146e-01

Truly unstructured grids with shocks and thin features result in very poorly conditioned systems

Original system : Preconditioned GMRES not effective

Iterative solution: More robust

Adjoint Equations : Solution

Exact or approximate Jacobians

Laminar SBLI @ Rex = 3x105

Air: V=1800 m/s,T= 1550 K

H2: V=1500 m/s,T= 300 K

Splitter plate

QoI

OH MassFraction

Pressure

K-w SST withFPVA model on a mesh of 5000 CVs

Supersonic Combustion model problem

Supersonic Combustion model problem:

Exact Jacobians : CFL ~ 1000+Approx Jacobians : CFL ~ 0.1

Full Adjoint

Frozen turbulence

Governing equation and functional on Error estimate on

Goal oriented Error estimation

Have also extended it to estimate and control stochastic errors

(Venditti & Darmofal)

Test 1: Shock-Turbulent Boundary Layer Interaction

Reference Error: 3.1 e-04

LES RANS

Incoming BL: Mach number = 2.28, Rϑ = 1500, Shock deflection angle = 8o

Adapive Mesh refinement QoI: Integrated pressure on lower wall

2.5 % flagged 5 % flagged 25 % flagged

Gradient based

Adjoint based

Forebody Ramp Inlet/Isolator Combustor Nozzle/Afterbody

Fuel Injection

FlowMach ~8

Application to Scramjet Combustion

Air 1800 m/s, 1300 K, 1.2 bar

H2

300K, 5 bar (total)

Wall pressures

Upper wall Lower wall

Adjoint Solution QoI : avg pressure at Comb exit (lower wall)

24 hrs, 840 procs:Local LU preconditioning + GMRES

Adjoint Error estimates QoI : 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 meshAdapted mesh

GoverningEquations

DiscreteGoverningEquations

LinearizedGoverningEquations

HybridAdjoint

Equations

Discretized Adjoint

Equations

DiscretizeLinearize

Discretize

Discretize

Linearize

Linearize

ContinuousAdjoint

Equations

Equations that are difficult/impossible

analytically

Equations with existing analytical

formulations/code

Towards a hybrid adjoint

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

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