Adjoint-Based Sensitivity Analysis for Computational Fluid Dynamics

Transcription

1 Adjoint-Based Sensitivity Analysis for Comptational Flid Dynamics Dimitri J. Mavriplis Department of Mecanical Engineering niversity of Wyoming Laramie, WY

2 Motivation Comptational flid dynamics analysis capabilities commonplace today In addition to analysis capability, sensitivity capability is igly desirable Design optimization Error estimation Parameter sensitivity Sensitivities may be obtained by: Pertrb inpt, rern analysis code Finite difference Linearizing analysis code tangent metod Good for 1 inpt, many otpts Adjoint metod Good for many inpts, one otpt

3 Motivation Continos vs. Discrete Adjoint Approaces Continos: Linearize ten discretize Discrete: Discretize ten Linearize Continos Approac: More fleible adjoint discretizations Framework for non-differentiable tasks limiters Often invoked sing flow soltion as constraint sing Lagrange mltipliers Discrete Approac: eprodces eact sensitivities of code Verifiable trog finite differences elatively simple implementation Cain rle differentiation of analysis code ranspose tese derivates transpose and reverse order Incldes bondary conditions

4 Generalized Discrete Sensitivities Consider a mlti-pase analysis code: L = Objectives D = Design variables Sensitivity Analysis sing cain rle:

5 angent Model Special Case: 1 Design variable D, many objectives L Precompte all stff depending on single D Constrct dl/dd elements as:

6 Adjoint Model Special Case: 1 Objective L, Many Design Variables D Wold like to precompte all left terms ranspose entire eqation:

7 Adjoint Model Special Case: 1 Objective L, Many Design Variables D Wold like to precompte all left terms ranspose entire eqation: precompte as:

8 Sape Optimization Problem Mlti-pase process:

9 angent Problem forward linearization Eamine Individal erms: : Design variable definition CAD : Objective fnction definition

10 angent Problem forward linearization Eamine Individal erms:

11 Sensitivity Analysis angent Problem: Adjoint Problem

12 angent Problem 1: Srface mes sensitivity: 2: Interior mes sensitivity: 3: esidal sensitivity: 4: Flow variable sensitivity: 5: Final sensitivity

13 Adjoint Problem 1: Objective flow sensitivity: 2: Flow adjoint: 3:Objective sens. wrt mes: 4: Mes adjoint: 5: Final sensitivity:

14 General Approac Linearize eac sbrotine/process individally in analysis code Ceck linearization by finite difference ranspose, and ceck dality relation Bild p larger components Ceck linearization, dality relation Ceck entire process for FD and dality se single modlar AMG solver for all pases

15 General Dality elation Analysis otine: f = f angent Model: Adjoint Model: Dality elation: f δf = δ 1 1 δ f f = f 2 2 = f 1 Necessary bt not sfficient test Ceck sing series of arbitrary inpt vectors f 1 2

16 Drag Minimization Problem DL-F6 Wing body configration 1.12M vertices, 4.2M cells

17 Drag Minimization Problem DL-F6 Wing body configration Mac=0.75, Incidence=1 o, e=3m

18 Drag Minimization Problem Sbstantial redction in sock strengt after 15 design cycles CD: 302 conts 288 conts : -14 conts Wave drag

19 Drag Minimization Problem otal Optimization ime for 15 Design Cycles: 6 ors on 16 cps of PC clster Flow Solver: 150 MG cycles Flow Adjoint: 50 Defect-Correction cycles 4 MG Mes Adjoint: 25 MG cycles Mes Motion: 25 MG cycles

20 Etension to nsteady Problems Pressre Contors for Pitcing Airfoils M inf = 0.755, α 0 = o, α ma = 2.51 o, ω = , t=0 to time-steps wit dt=2.0 NACA0012 Baseline Airfoil Optimized Airfoil

21 Governing Eqations ALE In ALE Form: Vt = control volme. Face integrated mes velocity. ndb = Formlated to obey GCL = f n, n-1, n-2, Mavriplis and Yang AIAA for ig order IK Mavriplis and Nastase AIAA/JCP to appear for DG metods

22 BDF1: nsteady esidal Form De niqely to ALE grid speed terms BDF2: Similar epression depending on w n,w n-1,w n-2, n, n-1, n-2

23 Sape Optimization Per design cycle Steady Case One mes deformation problem One flow analysis One flow adjoint soltion One mes adjoint soltion nsteady Sape Optimization General fnctional dependence involves previos time step vales Cain rle reslts in forward time recrrence relation Wen transposed adjoint reslts in backwards integration in time ime istory of soltion mst be stored for se by adjoint in reverse time integration Write ot to local clster disks, read back in dring adjoint pase

24 Validation of Sensitivities Steady/nsteady sensitivities compare well wit finite difference vales angent/adjoint vales eqivalent to macine precision Dality principle

25 ime-dependent Load Convergence/Comparison Mani and Mavriplis AIAA

26 Adjoint-Based Error Estimation Comple simlations ave mltiple error sorces Engineering simlations concerned wit specific otpt objectives Adjoint metods / Goal Oriented Approac Metodical approac for constrcting discrete adjoint se for a posteriori error estimation Spatial error emporal error Oter error sorces se to drive adaptive process

27 Goal-Oriented Spatial Adaptivity steady-state -adaptivity Drag DG Discretization p-adaptivity Drag

28 , adjoint soltion Λ I = J J J -1 0 = + + = L L + + = J J J - Formlation aylor series ADJOIN-BASED EO ESIMAION Li Wang, Dimitri Mavriplis W 28

29 J J Λ Approimated objective becomes 29 Formlation Discrete adjoint problem Linear system of eqations Delivers similar convergence rate as te flow solver ADJOIN-BASED EO ESIMAION Li Wang, Dimitri Mavriplis W -1 = Λ J J = Λ or ranspose of Jacobian matri

30 Approimated objective becomes or ADJOIN-BASED EO ESIMAION Li Wang, Dimitri Mavriplis W 30 Formlation Avoid solving te adjoint variables on te fine mes, instead, Solve on te coarse mes econstrct onto te fine mes by sing least sqares metod Λ Λ Λ = Λ J ε 1 ε 2 correction J J Λ J J J J Λ J = Λ

31 ADAPIVE MES SAEGIES -refinement Local mes sbdivision P P P P p p+1 p-enricment Local variation of discretization orders p-refinement Local implementation of te - or p-refinement individally 31 Li Wang, Dimitri Mavriplis W

32 ADAPIVE MES SAEGIES p-refinement contd. For EAC flagged element, ow to make a decision between - and p-refinement?? Local smootness / sock indicator Li Wang, Dimitri Mavriplis W 32

33 ADAPIVE MES SAEGIES Local smootness / sock indicator Element-wise smootness [Persson, Peraire], S ek p p 1, p p 1 p, p k N p k = were p = i i i= 1 φ N p p 1 = 1 i i= 1 φ i s = ek log 10 S ek s ek > 1 4 κ p -refinement; oterwise, p-enricment Inter-element jmps[krivodonova,xin,cevageon,flaerty], S ek = 1 l l 1 2 q + q + + q q 1 s ek > -refinement; oterwise, p-enricment K ds Li Wang, Dimitri Mavriplis W 33

34 NMEICAL ESLS Sbsonic flow over a for-element airfoil M =0.2 Zero angle of attack, α=0 arget fnction: lift or drag J = pn ds J = pn ds Starting interpolation order p = 1 LLC iemann solver & special wall bondary treatment p-mltigrid accelerator Varios adaptation algoritms Ω w -refinement p-refinement p-refinement Ω w y initial mes 1508 elements 34 Li Wang, Dimitri Mavriplis W

35 NMEICAL ESLS Sbsonic flow over a for-element airfoil M =0.2 Primal and dal problems target fnction : lift Comparisons on p-mltigrid convergence for te flow and adjoint soltions Li Wang, Dimitri Mavriplis W 35

36 NMEICAL ESLS Sbsonic flow over a for-element airfoil arget fnction of lift p = 1 M =0.2 -refinement, adapted mes 8387 elements spatial error distribtion in te objective fnctional of lift 36 Li Wang, Dimitri Mavriplis W

37 NMEICAL ESLS Sbsonic flow over a for-element airfoil M =0.2 arget fnction of lift p = 1 -refinement, error convergence istory vs. degrees of freedom error convergence istory vs. CP time sec Li Wang, Dimitri Mavriplis W 37

38 NMEICAL ESLS Sbsonic flow over a for-element airfoil M =0.2 p-enricment, arget fnction of drag adapted mes 1508 elements discretization orders: p=14 spatial error distribtion for te objective fnctional of drag Li Wang, Dimitri Mavriplis W 38

39 NMEICAL ESLS Sbsonic flow over a for-element airfoil M =0.2 p-enricment, arget fnction of drag error convergence istory vs. degrees of freedom error convergence istory vs. CP time sec Li Wang, Dimitri Mavriplis W 39

40 NMEICAL ESLS Sbsonic flow over a for-element airfoil M =0.2 p-refinement, arget fnction of drag adapted mes 7,105 elements discretization orders: p=14 smootness indicator Li Wang, Dimitri Mavriplis W 40

41 NMEICAL ESLS Sbsonic flow over a for-element airfoil M =0.2 p-refinement, arget fnction of drag error convergence istory vs. degrees of freedom error convergence istory vs. CP time sec Li Wang, Dimitri Mavriplis W 41

42 NMEICAL ESLS Comparisons on convergence of objective fnctionals -refinement 42 Li Wang, Dimitri Mavriplis W

43 NMEICAL ESLS ig-speed flow over a alf circlar-cylinder Marget =6 fnction of integrated temperatre J = ds p-refinement starting discretization order p = 0 first-order accrate Ω w initial mes: 17,072 elements Li Wang, Dimitri Mavriplis W 43

44 NMEICAL ESLS ig-speed flow over a alf circlar-cylinder M =6 adapted mes: 42,234 elements, discretization orders p=03 Li Wang, Dimitri Mavriplis W 44

45 Error Estimation for ime Dependent Problems est Case Description Sinsoidally pitcing airfoil Fnctional scalar is Lift after 1 period Easily etended to estimate error in time-integrated Lift istory

46 Sorces of Error Error in ime domain emporal resoltion error Partial convergence error Flow eqations Mes motion eqations Flow eqations Mes motion eqations

47 Flow eqations Conservative form of Eler eqations: +. F = 0 t Integrate over moving control volme to get Arbitrary-Lagrangian-Elerian ALE finite-volme form: A t + db t [ F & v ] ndb = 0

48 Mes Deformation Linear ension Spring Analogy: Mes is a series of interconnected springs [K]δ int =δ srf 2 independent force balance eqations at eac node

49 emporal esoltion Error =, L, L L L + + Fine level aylor epansion of fnctional objective L: = fine time domain : Δt/2 = coarse time domain : Δt

50 Evalation of Flow Contribtion to emporal esoltion Error Fine level flow residal aylor epansion =, 0, = + +

51 Contined + =, 1 = L +, 1 L = Λ

52 Contined Adjoint eqation over entire fine time domain: ecast on coarse time domain: L = Λ L = Λ I Λ = Λ

53 Contined, L + Λ + Λ Contribtion to temporal resoltion error from flow eqations E emaining temporal resoltion error de to mes motion eqations, bt also feeds into flow state +K + Λ + Λ = Λ Λ , Intepret as sm of dot prodcts of adjoint and residal at eac time step

54 Evalation of Mes contribtion to emporal esoltion Error 0 ] [ = = srf K G δ δ 0 = + = G G G ] [ 1 G K = ] [ 1 G K E E = = Λ

55 Contined [ K] Λ = E [ K] Λ = E Λ = I Λ Λ G Λ, Contribtion temporal resoltion error from mes motion eqations Contribtion temporal resoltion error from flow eqations

56 Smmary of emporal esoltion Error Evalation Compte nsteady flow soltion on coarse time domain Compte adjoint variables on coarse time domain Integrating backward in time Project adjoint variables, flow soltion and mes soltion onto fine time domain emporal resoltion error is ten inner prodct of adjoint wit corresponding non-zero residal on fine time domain Distribtion in time is sed to drive adaptation

57 Validation Adjoint is linearization abot crrent state 16 time steps to predict objective vale on modified state 32 time steps

58 Partial Convergence Error Coarse level aylor epansion abot partial soltion fnctional: =, L, L L L + + =, =, Partially converged flow and mes soltion Flly converged flow and mes soltion

59 Partial Convergence Error =,, + + Fine level flow residal aylor epansion non-zero de to partial convergence

60 Contined Λ = [ K ] Λ = E L Same adjoint eqations as tose for temporal resoltion error Partial convergence error de to flow eqations: Partial convergence error de to mes eqations: Λ Λ, G Non-zero residals de to partial convergence

61 Smmary of otal Error Evalation and Decomposition Compte partially converged flow and mes soltion on coarse time domain Compte adjoint variables on coarse domain sing partially converged soltion Compte partial convergence error on coarse level time domain Inner prodct of adjoint wit partially converged non-zero residal Project partially converged soltion and adjoint variables onto fine time domain Evalate fine level error estimate as previosly Combined temporal resoltion and partial convergence error Determine temporal resoltion error by sbtracting partial convergence error from total error estimate on fine time domain

62 Adaptation Compte time-integrated averages of error component distribtions Adapt were error is greater tan timeintegrated average ime resoltion error: divide time step by 2 Convergence error: tigten tolerance by predetermined factor

63 Adaptive ime Step and Convergence Criteria Eample

64 Distribtion of time steps and convergence limits in te time domain after 6 adaptation cycles ime steps Flow limits Mes limits

65 Distribtion of Error Components esoltion error Flow convergence error Mes convergence error

66 Conclsions igoros procedre for determining global error Identifies individal error contribtions from eac set of governing eqations Distribtion of eac component available and can be sed for adaptation Can be etended to problems involving mltiple sets of governing eqations sc as conjgate eat transfers, strctral eqations etc. Ftre Work emporal adaptivity work etended to BDF2 emporal adaptivity for DG metods Combined Spatial emporal Adaptivity Mlti-pysics and copling error sensitivity Parameter sensitivities