TOWARDS GRADIENT BASED AERODYNAMIC OPTIMIZATION OF WIND TURBINE BLADES USING OVERSET GRIDS
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1 TOWARDS GRADIENT BASED AERODYNAMIC OPTIMIZATION OF WIND TURBINE BLADES USING OVERSET GRIDS S. H. Jongsm E. T. A. vn de Weide H. W. M. Hoeijmkers Overset symposium Deprtment of mechnicl engineering 1
2 OUTLINE Motivtion Optimiztion procedure Prmeteriztion method Flow model Hole cutting Grdient computtion Test cse nd results 2
3 MOTIVATION (1 Conventionl pproch in design of lde: Design 2D erofoil sections Comine sections to otin lde geometry Incorporte effects 3D flow fetures very difficult even for experienced designers Current pproch: Simultion sed design of lde shpe in 3D 3D flow fetures tken into ccount Design expertise required for defining ojective function nd constrints for optimiztion method 3
4 GRADIENT BASED OPTIMIZATION PROCEDURE 4
5 FLOW MODEL (1 Consider rigid isolted rotor: i.e. effects of presence of tower nd ground plne re neglected effect of lde deformtion is neglected Assume rotor plne perpendiculr to stedy wind: flow is periodic Stedy flow round single lde in rotting frme i.e. effect of unstedy inflow nd yw is not considered Currently solve Euler equtions Lter: RANS 5
6 FLOW MODEL (2 Cell centered finite volume method 2 nd Order sptil ccurcy Integrtion to stedy stte y: Newton s method Runge-Kutt time stepping Use PETSc to solve the resulting system of liner equtions Re c = M = 0.2 = 2.3 o O-grid nd Crtesin ckground grid 6
7 PARAMETERIZATION METHOD Requirements: Spn complete design spce Limit numer of design vriles Choice: Non-uniform rtionl sis spline (NURBS surfce Flexile Comptile with CAD Design vriles: (Selection of coordintes of control points Weights (if desired 7
8 FLOW DOMAIN DISCRETIZATION Requirements: Account for chnge of shpe fter ech design itertion Not computtionlly intensive Fully utomtic Choice: Multi-lock overset grids Use hyperolic grid genertion for field grids 8
9 GRID PREPROCESSING (1 Approch Evlution of surfce integrls Zipper grid (Chn 2009 Elimintion cells inside geometricl entities Ry csting Block connectivity Implicit hole cutting (Lee 2008 No sequentil ottleneck Resonly efficient prllel implementtion Chn (2009, Enhncements to the Hyrid Mesh Approch to Surfce Lods Integrtion on Overset Structured Grids Lee (2008, On Overset Grids Connectivity nd Vortex Trcking in Rotorcrft CFD (PhD thesis 9
10 GRID PREPROCESSING (2 ZIPPER GRID (1 Closed non-overlpping surfce grid required for: Evlution of surfce integrls Ry-csting procedure Achieved y ppliction of zipper grids: remove overlpping prt of surfce grids connect neighoring grids y creting tringles in etween Works for prolems considered so fr 10
11 GRID PREPROCESSING (3 ZIPPER GRID (2 11
12 GRID PREPROCESSING (4 RAY-CASTING Approch: Choose p nd csting direction Find possile mtches using tree serch Check possile mtches with ccurte check Accurte check: Split qudrilterl in 2 tringles Trnslte tringle to origin: Project p' nd tringle on fce perpendiculr to csting direction Use trnsformtion to rycentric coordintes: Solve: Ry crosses tringle if: 12
13 GRID PREPROCESSING (5 IMPLICIT HOLE CUTTING (1 Concept Use dul grid for hole cutting Identify cells tht reside in region covered y multiple locks In region of overlp: Consider user defined priority of locks Consider index distnce to wll of cells Consider cell volume Use cells with the smllest volume to solve governing equtions (field cell Cell with lrger volume gets sttus: hole cell Specify fringe cells t interfce etween hole nd field cells (stencil dependent 13
14 GRID PREPROCESSING (6 IMPLICIT HOLE CUTTING (2 Approch Construct lternting digitl tree for ll cells in ech lock Perform tree serch to look for cells for which the ounding ox overlps prticulr cell Use tri-liner trnsformtion to identify overlp for potentil cndidtes from tree serch Use criteri on lock priority, index distnce nd cell volume to identify field nd hole cells Identify fringe cells 14
15 GRADIENT COMPUTATION (1 INTRODUCTION (1 Requirements: Accurte Otined efficiently Methods: Finite difference Complex step finite difference Discrete djoint eqution method 15
16 GRADIENT COMPUTATION (2 DISCRETE ADJOINT EQUATION METHOD (1 Design vrile α Computtionl mesh X = X (α Flow vriles u = u(α,x Ojective function I = I(α,u, X Residul of flow solution R = R(α,u, X Totl derivtive of ojective function: 16
17 GRADIENT COMPUTATION (3 DISCRETE ADJOINT EQUATION METHOD (2 Totl derivtive of residul: Rewrite to: Sustitute: 17
18 GRADIENT COMPUTATION (4 DISCRETE ADJOINT EQUATION METHOD (3 Define djoint vector: Sustitute to otin new expression for totl derivtive: Advntge: Only one flow solution needed for ll derivtives, once djoint vector is found y solving: 18
19 GRADIENT COMPUTATION (5 DUAL NUMBERS (1 (Fike et l Choose h = 1 Fike & Alonso (2011, The Development of Hyper-Dul Numers for Exct Second- Derivtive Clcultions, AIAA pper
20 20 GRADIENT COMPUTATION (6 DUAL NUMBERS (2 / / ( / 1/( ( /( ( / 1/ d 1/(c Division : ( ( ( Multipliction : ( ( ( ( Addition : 2 2 c d c c d c d c c d c c d c d c d c d c = = = = = ] ln( exp[( ( / ln( ln( exp( exp( exp( 1] ( [tn tn( tn( cos( sin( sin( sin( cos( Mth functions : cos( d (c 2 d c = = = = = = Computtionl rules
21 GRADIENT COMPUTATION (7 CONSIDERATIONS Limit implementtion time y reusing existing functions Use of C templte functions Use grph coloring to limit numer of function evlutions Existence of fringe cells nd hlo cells requires specil ttention Tke effect of oundry conditions into ccount explicitly Determine corresponding mtrix index of donors Computtion djoint vector Use first order Jcoin for construction ILU(k preconditioning mtrix Use PETSc to solve system of liner equtions y mens of GMRES itertions 21
22 GRADIENT COMPUTATION (8 HANDLING FRINGE CELLS Construction Jcoin mtrix: Fringe cells re represented y row in the mtrix Mke sure ech fringe cell hs explicit representtion y field cells Required to fcilitte implicit solution method Updte solution fter prtil convergence GMRES itertions For consistency, fringe cells re updted using solution of field cells 22
23 OPTIMIZATION (1 APPROACH Choose design vriles Specify ojective function nd constrint functions Specify ounds on design vriles Evlute ojective function nd constrint functions Compute grdients Provide vlue of ojective function, constrint functions nd grdients to grdient sed optimiztion lgorithm: SNOPT (sprse non-liner optimizer (Stnford University (Gill et l BFGS qusi Newton method Provides new design vriles o Repet procedure Gill, Murry & Sunders (2005, SNOPT: An SQP Algorithm for Lrge-Scle Constrined Optimiztion, SIAM Review, Vol. 47, No
24 OPTIMIZATION (2 PROBLEM DEFINITION (1 Inverse(/reverse design: Ojective: Find irfoil geometry which minimizes (c p c p_trget 2 Approch: Red input file with c p dt nd corresponding x-coordinte Interpolte input dt for comprison of c p current design for different x-coordinte center of fce Sum squre of difference Minimize resulting sum x 24
25 OPTIMIZATION (3 PROBLEM DEFINITION (2 Constrint functions: 25
26 OPTIMIZATION (4 SETUP Initil design: NACA 0012 t 0 ngle of ttck 13 control points 36 design vriles Flow conditions: M = 0.3 Grid dimensions: Grid n i n j O-grid Bckground Fr field: At 50 chord lengths 26
27 OPTIMIZATION (5 RESULTS Trget nd initil c p -distriution Finl nd initil geometry Fit Originl dt point 27
28 CONCLUDING REMARKS Prmeteriztion using NURBS surfce Discretiztion using hyperolic grid genertion method Fully utomtic hole cutting procedure Derivtives computed using: Discrete djoint eqution method Dul numer rithmetic Method successfully pplied for solving 2D optimiztion prolems on overset grids Fit of c p distriution Drg minimiztion in trnsonic flow 28
29 FURTURE WORK Testing the method for simple 3D optimiztion prolem Applying the method for full 3D wind turine lde in co-rotting frme of reference Use RANS equtions Extension to time-periodic optimiztion vi time-spectrl method 29
30
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