Overview grid generation sensitivity analysis adjoint approach applications Aerodynamic optimisation for complex geometries 2

Transcription

1 optimisation Aerodynamic complex geometries for Giles Michael University Computing Laboratory Oxford April 25th, 1997 Aerodynamic optimisation for complex geometries 1

2 Overview grid generation sensitivity analysis adjoint approach applications Aerodynamic optimisation for complex geometries 2

3 design purposes we need to be able to For create parameters perturbed grids for perturbed values linear/nonlinear perturbed grids have the The topology as the base grid, so any same Grid Generation a base grid for given values of design objective function varies smoothly. Aerodynamic optimisation for complex geometries 3

4 grid generation: Base parametric solids-based EPD system solid surface as a collection of denes patches separated by lines surface by points terminated surface is gridded in order of dimensionality (point, line, increasing patch) surface Grid Generation interior grid nodes are then created by front or Delauney advancing algorithms Aerodynamic optimisation for complex geometries 4

5 grid generation: Perturbed parametric perturbation denes to solid surfaces and perturbation lines separating (point, line, surface dimensionality patch) Grid Generation surface grid node perturbations are in order of increasing dened interior grid nodes are perturbed using of springs' or elliptic p.d.e. `method Aerodynamic optimisation for complex geometries 5

6 of springs: Method edges of grid are modelled as springs equilibrium perturbation to surface points disturbs leading to perturbation to equilibrium, nodes to re-establish it interior Grid Generation base grid is dened to be in strength of springs is dened to ensure cross-over in the boundary layer no (similar idea can be used to dene perturbations in the rst place) surface Aerodynamic optimisation for complex geometries 6

7 the elliptic p.d.e. approach, grid node In ex(x) are dened by perturbations is dened to ensure no cross-over in k(x) layers. boundary Grid Generation r (k(x)rex) = 0; subject to specied boundary conditions. Aerodynamic optimisation for complex geometries 7

8 a single design variable, discrete For equations ow of objective function I(U ; ) can Gradient approximated by be I(U(+); +) I(U(); ) : generalised to multiple design Easily at cost of extra calculations. variables, Nonlinear Sensitivity F (U ; ) = 0; dene ow eld U as a function of. di d Aerodynamic optimisation for complex geometries 8

9 + F fu = 0; F F X : change in perturbs grid coordinates i.e. perturb ux residuals. which fu represents ow perturbation as seen by Linear Sensitivity Linearising discrete ow equations gives F U where X perturbed grid point Aerodynamic optimisation for complex geometries 9

10 to multiple design Generalisation requires separate calculation parameters each, so no particular benet for to nonlinear sensitivities. compared Linear Sensitivity Gradient of objective function is given by di = I d + I fu : U Aerodynamic optimisation for complex geometries 10

11 ! T V + I T = 0: Discrete Adjoint Substituting for f U gives di = I d 1! F + I ; F U U which can be written as di = V T F + I ; d where V satises the adjoint equation F U U Aerodynamic optimisation for complex geometries 11

12 advantage of the adjoint approach is The the same adjoint solution V can be that for each design variable, since V used on I but not. depends drawback is that because V depends The I a separate calculation must be on in real engineering applications, Question: many design variables and constraints how Discrete Adjoint performed for each constraint function. are there? Aerodynamic optimisation for complex geometries 12

13 analytic adjoint is more complicated. The rst step is formulation of linear Critical x F x(u) + y F y(u) = 0; x (A x e U) + y (A y e U) = 0; Analytic Adjoint perturbation equations. Simple linearisation of 2D Euler equations yields where e U is perturbation at a xed point. Aerodynamic optimisation for complex geometries 13

14 is similar to discrete adjoint This with no perturbation to interior treatment Analytic Adjoint However, linearising the b.c. un = 0; gives eun + (exru) n + u en = 0; which is hard to discretise accurately. grid points. Aerodynamic optimisation for complex geometries 14

15 X(; ) and Y (; ) are smooth where which match the surface functions = 0: Analytic Adjoint Start instead with generalised coordinates,!! y y F F x y x F x y + F x Now dene perturbed coordinates as x = + X(; ); y = + Y (; ); perturbations due to the design variable. Aerodynamic optimisation for complex geometries 15

16 e U is now the perturbation in the where variables for xed (; ) rather than ow linearisation of the b.c.'s is simple, The the overall accuracy is much better. and Analytic Adjoint Linearising with respect to yields! (A x e U) + (A y e U) = Y F F X y x! X y F Y x F ; xed (x; y). Aerodynamic optimisation for complex geometries 16

17 Shrinivas G.N. Crumpton P.I. Giles M.B Oxford, OGV Design Aerodynamic optimisation for complex geometries 17

18 OGV Design Aerodynamic optimisation for complex geometries 18

19 OGV Design Aerodynamic optimisation for complex geometries 19

20 OGV Design Outlet guide vanes Fan Pylon Aerodynamic optimisation for complex geometries 20

21 is to minimise circumferential Objective variation upstream of the OGV's pressure uses Optimisation unstructured grid with 560k tetrahedra OGV Design by changing their camber. Euler equations multigrid and parallel computing elliptic p.d.e. for grid perturbation nonlinear sensitivities and quasi-newton optimisation Aerodynamic optimisation for complex geometries 21

22 edge of each OGV is left Leading due to uniform ow incidence; unchanged change varies linearly with camber from leading edge to change the distance design exercise uses a camber First which varies sinusoidally with change 2 design variables: maximum change Only hub and tip at OGV Design outow angle. circumferential angle. Aerodynamic optimisation for complex geometries 22

23 OGV Design *10-2 Optimisation using sinusoidal variation (p - p_mean) Y Aerodynamic optimisation for complex geometries 23

24 OGV Design Optimisation using sinusoidal variation 1.0 Index of Pressure Variation Itn Number Aerodynamic optimisation for complex geometries 24

25 of this design is that all OGV's Drawback dierent. are design exercise uses just 3 blade Second the original, one with overturning types, one with an equal amount of and underturning. only 2 design variables: maximum Still at hub and tip change OGV Design Aerodynamic optimisation for complex geometries 25

26 OGV Design Datum Overturned Datum Underturned Datum Aerodynamic optimisation for complex geometries 26

27 OGV Design using 3 blade types Optimisation * (p - p_mean) Y Aerodynamic optimisation for complex geometries 27

28 OGV Design Optimisation using 3 blade types 1.0 Index of Pressure Variation Iteration Number Aerodynamic optimisation for complex geometries 28

29 Business Jet J. Elliott and J. Peraire, MIT, 1996 Aerodynamic optimisation for complex geometries 29

30 function is mean-square Objective from a target pressure deviation for a `clean' wing in the distribution of the rear nacelle. absence design variables are used to dene 6 perturbations to the wing. smooth Business Jet Aerodynamic optimisation for complex geometries 30

31 uses Optimisation unstructured grid with 860k tetrahedra perturbation BFGS optimisation method with adjoint formulation for discrete gradients Business Jet Euler equations multigrid and parallel computing method of springs for grid Aerodynamic optimisation for complex geometries 31

32 Business Jet Aerodynamic optimisation for complex geometries 32

33 Business Jet Aerodynamic optimisation for complex geometries 33

34 viscous modelling is under modelling; but will cost up to 5 development as much times grid generation for base grids and grids is a critical component perturbed pros and cons of dierent optimisation has yet to be properly methods investigated Conclusions/Future aerodynamic optimisation for complex is becoming a reality geometries with multigrid and parallel computing, are now acceptable for inviscid costs Aerodynamic optimisation for complex geometries 34