Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver

1. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Error Estimation using Overflow Discrete Adjoint and its Dual, a Linearized Error Solver J. Elliott Boeing Airplane Company 14th Overset Composite Grid and Solution Technology Symposium

2. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Outline • Introduction • Baseline adjointerror estimation method • Single block results • NACA0012 inviscid half-plane, subsonic, a=0 • NACA0012 inviscid full-plane, subsonic, a=0 • NACA0012 inviscid full-plane, subsonic, a=1 • NACA0012 inviscid full-plane, transonic • Dual (forward) linearized error estimation • Conclusion

3. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Discrete Method Sensitivities (review of theory for design) • Finite Difference (from Taylor series expansion):- • Direct Method (one solve per design variable) • Adjoint Method: Introduce  such that (one solve per cost function)

4. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver

5. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver • Theory for error estimation is similar • Based on approximation, , of solution, , on fine (truth) grid by bilinear interpolation (prolongation) from coarse grid, • We have, by Taylor series expansion (Darmofal et al, 1999):- • As for design, we introduce variable, , such that • And as for design, we can also solve for error, directly:-

6. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Discrete adjoint algorithms for error estimation As shown, adjointscan be used to calculate error associated with estimate of solution on finer (truth) grid, QHh , and the actual solution, Qh We may also wish to approximate the adjoint, h, on the grid h, with Hh its interpolation from coarser grid H. The following slides show the mechanics of this baseline process. In the following charts:- • W = adjoint corresponding to interior residuals • V = adjoint corresponding to boundary condition residuals

7. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Baseline flow solution (Cp) on 513x513 grid

8. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Baseline adjoint solution, W5, on 257x257 grid

9. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Adjoint and flow convergence on the 65x65 grid

10. Qh QHh QH Rh(QHh) WhRh(QHh) Wh Steps involved in building up drag correction estimate

11. Wh WHh WH Vh VHh VH Alternative interior and boundary adjoints that can be used

12. Wh WHh WH Vh VHh VH /project/brt_aero/tech_dev/overadj/overflow2.1x_nasa/runs/2d/opt-ws/naca0012sym-c5/aaa/fieldgrd/flow_65_ibtyp=1/

13. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Pointwise value of error estimate corresponding to 5th conservative variable component using the 513x513 grid as the “truth grid” and the 257x257 grid as the “affordable grid”.

14. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Inclusion of error estimate in drag prediction gives increasingly good prediction of drag on next finer grid

15. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Inclusion of error estimate in drag prediction gives increasingly good prediction of drag on next finer grid. Error in prediction is 2nd order

16. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Extension to NACA0012 with full domain O-grid WhRh(QHh) /project/brt_aero/tech_dev/overadj/overflow2.1x_nasa/runs/2d/opt-ws/naca0012sym-c5/aaa/fieldgrd/flow_513 _ibtyp=1/errest_bciwpx_bcsymi_bcchar_fineadjb aab/fieldgrd/flow_513_ibtyp=1/

17. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Inclusion of error estimate in drag prediction gives increasingly good prediction of drag on next finer grid

18. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Inclusion of error estimate in drag prediction gives increasingly good prediction of drag on next finer grid. Error in prediction is 2nd order

19. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Extension to full domain NACA0012 using finer O-grid dz=1.5e-4 Cp on 1025x513 grid

20. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Extension to full domain NACA0012 using finer O-grid dz=1.5e-4

21. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Extension to full domain NACA0012 using finer O-grid dz=1.5e-4. Error in prediction is 2nd order

22. Error estimation using Overflow Discrete Adjoint and a Linearized Error Solver Extension to full domain NACA0012 using C-grid Cp on 1281x513 grid /project/brt_aero/tech_dev/overadj/overflow2.1x_nasa/runs/2d/opt-ws/naca0012sym-c5/aag/fieldgrd/flow_513_ibtyp=51/ aab/fieldgrd/flow_65_ibtyp=1/