Discrete and continuous adjoint method for compressible CFD J. Peter ONERA

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1 Discrete and continuous adjoint method for compressible CFD J. Peter ONERA J. Peter 1 1 ONERA DMFN October 2, 2014 J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

2 Outline Introduction 1 Introduction 2 Discrete adjoint method 3 Continuous adjoint method 4 Discrete vs Continuous adjoint 5 Conclusions J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

3 Introduction Introduction (1/4) Well-known aerodynamic optimization problems of the utmost importance Aircraft drag reduction Reduction of total pressure losses of a blade row. Strongly constrained problems (from aerodynamics, structure...) Several approaches for researches and studies in external aerodynamics Flight tests Wind tunnel experiments (with flight Re/lower than flight Re) Numerical simulation Numerical simulation most adapted J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

4 Introduction Introduction (2/4) (Non solvable) pde numerical simulation. Finite-volume simulation in this talk. Infinite dimension possible deformation parametrization Finite dimensional maths Which type of optimization method? Local or global optimization? J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

5 Introduction (3/4) Introduction Global optimization genetic/evolutionnary algorithms, particle swarm, aunt colony, CMA-ES... large number of function evaluations required combined with surrogate models in particular used for design space exploration with low fidelity models Local optimization very valuable when starting from pre-optimized shapes pattern methods. e.g. simplex method gradient-based methods. e.g. steepest descent, conjugate gradient Popular and efficient descent methods require objective and constraint sensitivities w.r.t. design parameters J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

6 Introduction Introduction (4/4) Needed sensitivities w.r.t. design parameters...not a trivial task in numerical simulation as state variables change with shape via the equations of the mechanical problem Sensitivty calculation 70 s 80 s finite differences. Scaling with number of shape parameters Control theory [Lions 71, Pironneau 73,74] aerodynamics shape optimization [Jameson 88] adjoint method. Scaling with the number of functions to be differentiated Other applications of adjoint method: understanding zones of influence for function value, goal-oriented mesh refinement J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

7 Outline Discrete adjoint method 1 Introduction 2 Discrete adjoint method 3 Continuous adjoint method 4 Discrete vs Continuous adjoint 5 Conclusions J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

8 Discrete adjoint method Discrete adjoint method Framework: compressible flow simulation using finite volume method. Discrete approach for sensitivity analysis Notations Volume mesh X, flowfield W (size n a) Wall surface mesh S Residual R, C 1 regular w.r.t. X and W steady state: R(W, X ) = 0 Vector of design parameters α (size n d ), X (α) S(α) C 1 regular Assumption of implicit function theorem (W i, X i ) / R(W i, X i ) = 0 ( R/ W )(W i, X i ) 0 Unique steady flow corresponding to a mesh J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

9 Discrete adjoint method Introduction Discrete gradient calculation methods Functions of interest J k (α) = J k (W (α), X (α)) k [1, n f ] Flowfield and volume mesh linked by flow equations R(W (α), X (α)) = 0 Sensitivities dj k /dα i k [1, n f ] i [1, n d ] to be computed Discrete gradient computation methods Finite differences 2n d flow computations (non linear problems, size n a) Direct differentiation method n d linear systems (size n a) Adjoint vector method n f linear systems (size n a) J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

10 Discrete adjoint method Finite difference method Choose steps δα i. Get shifted meshes X (α + δα i ), X (α δα i ) Solve flows R(W (α + δα i ), X (α + δα i )) = 0 R(W (α δα i ), X (α δα i )) = 0 Compute outputs sensitivities dw = W (α + δα i) W (α δα i ) dα i (FD) 2δα i dj k = J k(w (α + δα i ), X (α + δα i )) J k (W (α δα i ), X (α + δα i )) dα i (FD) 2δα i Two issues: definition of δα i, cost of shifted flow solves J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

11 Discrete adjoint method Direct differentiation method (1/2) Discrete equations for mechanics (set of n a non-linear equations ) R(W (α), X (α)) = 0 Differentiation with respect to α i i [1, n d ]. Derivation of n d linear system of size n a R dw = ( R dx ) W dα i X dα i Calculation of derivatives dj k dα i = J k X dx + J k dw dα i W dα i J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

12 Discrete adjoint method Direct differentiation method (2/2) Gradient vectors α J k (α) = J k dx X dα + J k W Check the flow sensitivities using finite differences dw dα R(W (α + δα i ), X (α + δα i )) = 0 R(W (α δα i ), X (α δα i )) = 0 Check the outputs sensitivities dw dα i? W (α + δα i) W (α δα i ) 2δα i dj k? J k(w (α + δα i ), X (α + δα i )) J k (W (α δα i ), X (α + δα i )) dα i 2δα i J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

13 Discrete adjoint method Mathematical game (1/4) Mathematical game in R n to understand adjoint method given (f, b i ) R n (i {1, n d }), given A M(R n ) Calculate the values of x i.f A x i = b i i {1, n d } Solution solving one linear system instead of n d linear systems??? J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

14 Discrete adjoint method Mathematical game (2/4) Linear algebra reminder: the inverse of the transpose is the transpose of the inverse M T (M 1 ) T = (M 1 M) T = I T = I (M 1 ) T M T = (MM 1 ) T = I T = I The notation M T is suitable for (M T ) 1 / (M 1 ) T J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

15 Discrete adjoint method Mathematical game (3/4) Mathematical game in R n to understand adjoint method given (f, b i ) R n (i {1, n d }), given A M(R n ) Calculate the values of f.x i A x i = b i i {1, n d } f.x i = f.(a 1 b i ) = ((A 1 ) T f ).b i = (A T f ).b i efficient solution Solve A T λ = f Calculate λ.b i i {1, n d } J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

16 Discrete adjoint method Mathematical game (4/4) Mathematical game in R n to understand adjoint method given (f j, b i ) R n (i {1, n d } j {1, n f }), given A M(R n ) Calculate the values of x i.f j A x i = b i i {1, n d } Solution solving n d linear systems Solution solving n f linear systems J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

17 Discrete adjoint method Discrete adjoint parameter method (1/5) Several ways of deriving the equations of discrete adjoint method. The following also helps understanding continuous adjoint Following equalities hold λ k R na dj k (α) dα i dj k (α) dα i = J k X λ T k R dw + λ T k ( R dx ) = 0 W dα i X dα i dx + J k dw + λ T k dα i W dα i = ( J k W + λt k R W )dw + J k dα i X R dw + λ T k ( R W dα i X dx + λ T k ( R dα i X dx dα i ) dx dα i ) J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

18 Discrete adjoint method Discrete adjoint parameter method (2/5) Vector λ k defined in order to cancel the factor of the flow sensitivity dw dα i... the adjoint equation. λ k actually appears to be linked to functions J k Calculation of derivatives i [1, n d ] J k W + λt k dj k (α) dα i R W = 0 = J k X dx + λ T k ( R dα i X dx dα i ) α J k (α) = J k dx X dα + λt k ( R dx X dα ) Method with n f and not n d linear systems to solve J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

19 Discrete adjoint method Discrete adjoint parameter method (3/5) Other ways to derive the discrete adjoint equation Introduce a Lagrangian Manipulate direct differentiation gradient expression (like in the mathematical game) From direct method gradient expression Define λ k column vector α J k (α) = J k dx X dα + J k W α J k (α) = J k dx X α J k (α) = J k dx X dα dw dα dα J ( ) 1 k dr dr W dw dx ( ( ) ) 1 J k dr W dw dx dα dr dx dx dα J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

20 Discrete adjoint method Discrete adjoint parameter method (4/5) From direct method gradient expression α J k (α) = J k dx X dα ( ( ) ) 1 J k dr dr dx W dw dx dα Define λ k λ T k = J k W Expresion of sensitivity ( ) dr 1 or λ T k dw ( dr dw ) = J k W or ( ) dr T λ k = J T k dw W α J k (α) = J k dx X dα + dr dx λt k dx dα J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

21 Discrete adjoint method Iterative solution of direct and adjoint equation (1/3) CFD teams tend to mimic the solution of steady state flow altough flow equations are non-linear whereas direct/adjoint equation are linear Storing the jacobian of the scheme and sending to direct solver has been done but is rare and is not tractable for large cases Iterative resolution is much more common. Newton/relaxation algorithm ( ) (APP) T R ( λ (l+1) k W ) ( λ (l) k = ( R W )T λ (l) k + ( J ) k W )T J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

22 Discrete adjoint method Iterative solution of direct and adjoint equation (2/3) Common Newton/relaxation algorithm for adjoint ( ) R (APP) T ( λ (l+1) k W ) ( λ (l) k = ( R Common Newton/relaxation algorithm for direct ( ) R (APP) ( ( dw ) (l+1) ( dw ) ( ) (l) = W dα i dα i W )T λ (l) k ( R W ) dw dα i + J ) k W )T ) (l) + R dx X dα i Defining an approximate Jacobians ( R W )(APP) is an old subject in compressible CFD (definition of implicit stages for backward-euler schemes...) upwind approximate linearization of convective flux neglecting cross derivatives in linearization of viscous fluxes... Possibly adapting implicit stages and mutigrid algorithm (flow solver to adjoint solver) J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

23 Discrete adjoint method Iterative solution of direct and adjoint equation (3/3) Common Newton/relaxation algorithm for adjoint ( ) R (APP) T ( λ (l+1) k W ) ( λ (l) k = ( R W )T λ (l) k + J ) k W )T Accuracy of adjoint vector only depends on ( R W ). Only minor simplifcations are allowed at this stage to perserve an acceptable accuracy Convergence towards solution of the linear system depends on ( R ( R W ), W )(APP), multigrid (if active), other operations like smoothing (if active) J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

24 Discrete adjoint method Discrete adjoint parameter method (5/5) Checking adjoint method... much more difficult than checking direct differentiation method. If dj k <> J k(w (α + δα i ), X (α + δα i )) J k (W (α δα i ), X (α + δα i )) dα i 2δα i no easy checking procedure In the iterative resolution method, the gradient accuracy depends on the ( R W )T λ (l) k operation If direct mode is coded, duality checks between direct and adjoint code are useful. (U, V ) two column vectors of R na U T ( R ( W )V = U T ( R ) ( W ).V = U T. ( R ) adj code W )V lin code Valid for individual fluxes routine. Valid for part of the interfaces (border, joins...) J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

25 Discrete adjoint method Discrete adjoint mesh method (1/3) Vector λ k defined by Calculation of derivatives i [1, n f ] J k W + λt k dj k (α) dα i R W = 0 = J k X dx + λ T k ( R dα i X dx dα i ) i [1, n f ] dj k (α) dα i = ( J k X + R λt k X ) dx dα i Obvious mathematical factorization. Huge practical importance. J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

26 Discrete adjoint method Discrete adjoint mesh method (2/3) Solve for adjoint vectors CFD gradient computation code computes only dj k dx = J k X + R λt k X The functional outputs sensitivities dj k (α)/dα i are calculated later by a mesh/geometrical tool Pros : CFD has no knowledge of parametrization. Huge memory savings [Nielsen, Park 2005] Try several parametrization. Check (dj k /ds) with engineers Cons : Matrix ( R/ X ) has to be explicitely computed (instead of R X computable by finite differences) Hard work... dx dα i J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

27 Discrete adjoint method Discrete adjoint mesh method (3/3) Solve for adjoint vectors. Compute only dj k dx = J k X + R λt k X Cons : Matrix ( R/ X ) has to be explicitely computed (instead of R X computable by finite differences) Hard work... How to calculate (dj k /ds)? Explicit link beween X and S dj k dα i = [ djk dx ] dx ds ds dα i Implicit link between X and S [Nielsen, Park 2005] dx dα i J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

28 Outline Continuous adjoint method 1 Introduction 2 Discrete adjoint method 3 Continuous adjoint method 4 Discrete vs Continuous adjoint 5 Conclusions J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

29 Continuous adjoint method Bibliography Mathematical references [Pironneau 73,74] Mathematical aeronautical reference [Jameson 88] Simplest introduction [Giles, Pierce 99] An introduction to the adjoint approach to design ERCOFTAC Workshop on Adjoint Methods, Toulouse J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

30 Continuous adjoint method Continuous Adjoint for toy problems (1/6) From [Giles, Pierce 99] section (3.2) Toy problems without design parameters Solve du 2 dx ɛd u = f on [0, 1] u(0) = u(1) = 0 dx 2 before calculating J = (u, g) = Adjoint problem? Define (if it exists) 1 0 u g dx L λ = g on [0, 1] plus boundary conditions such that J = (λ, f ) = 1 0 λ fdx J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

31 Continuous adjoint method Continuous Adjoint for toy problems (2/6) Direct: solve du 2 dx ɛd u = f on [0, 1] u(0) = u(1) = 0 dx 2 before calculating J = (u, g) = Adjoint problem (if it exists): 1 0 u gdx L λ = g on [0, 1] plus boundary conditions such that J = (λ, f ) = 1 0 λ fdx Defining equation L J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

32 Continuous adjoint method Continuous Adjoint for toy problems (3/6) Defining equation L (λ, f ) = 1 (λ, f ) = λ fdx = 1 0 λ dλ dx u dx + [λ u]1 0 + ɛ 1 dλ (λ, f ) = 0 dx u dx + [λ u]1 0 ɛ Finally (λ, f ) = 1 0 ( dλ dx ɛ d ) 2 λ dx 2 Suitable adjoint equation. Solving ensures (λ, f ) = (u, g) 1 0 ( du 1 d 2 λ dx 2 0 dx ɛ d 2 u dx 2 dλ dx du dx ɛ ) dx du dx dx ɛ [ λ du dx [ λ du dx ] 1 0 ɛ ] 1 0 [ dλ dx u [ ] 1 [ udx + [λ u] 1 dλ 0 + ɛ dx u ɛ λ du ] 1 0 dx 0 dλ dx ɛ d 2 λ = g on [0, 1] λ(0) = λ(1) = 0 dx 2 ] 1 0 J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

33 Continuous adjoint method Continuous Adjoint for toy problems (4/6) In order to calculate J = (u, g) = u g dω, solve for u div(k grad(u)) = f on Ω u = 0 on Ω In order to calculate J as (λ, f ) = λ f dω, solve for λ Ω Ω div(k grad(λ)) = g on Ω λ = 0 on Ω Definition of adjoint operator comes from (λ, f ) = u div(k grad(λ))dω k u (grad(λ).n)ds + k λ (grad(u).n)ds Ω Ω Ω J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

34 Continuous adjoint method Continuous Adjoint for toy problems (5/6) Direct: solve u t 2 u = f on [0, L] [0, T ] x 2 u(0,.) = u(l,.) = 0 u(., 0) = 0 before calculating J = (u, g) = Adjoint: solve L T 0 0 u g dxdt λ t 2 λ = g on [0, L] [0, T ] x 2 λ(0,.) = λ(l,.) = 0 λ(., T ) = 0 before calculating J as (λ, f ) = L T 0 0 λ f dxdt J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

35 Continuous adjoint method Continuous Adjoint for toy problems (6/6) Time derivative u gets λ t t Backward time integration for unsteady adjoint Convection term u gets λ x x Backward propagation in adjoint steady state solutions Diffusion term 2 u x 2 gets 2 λ x 2 J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

36 Continuous adjoint method Continuous Adjoint for 2D Euler equations (1/11) 2D Euler equations avec w = ρ ρu ρv ρe w t f (w) = + f (w) x ρu ρu 2 + p ρuv ρhu + g(w) y = 0 g(w) = p = (γ 1)ρ(E u2 + v 2 ), ρh = ρe + p 2 ρv ρuv ρv 2 + p ρhv J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

37 Continuous adjoint method Continuous Adjoint for 2D Euler equations (2/11) Figure: Coordinate transformation for airfoil-fitted structured mesh Coordinate transformation Γ, C 1 diffeomorphism D ξη = [ξ min, ξ max ] [η min, η max ] en D w. { Dξη D Γ xy (ξ, η) (x, y) J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

38 Continuous adjoint method Continuous Adjoint for 2D Euler equations (3/11) Figure: Normal surface vectors J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

39 Continuous adjoint method Continuous Adjoint for 2D Euler equations (4/11) 2D Euler equations generalized coordinates W = K ρ ρu ρv ρe ( U V ) F (W ) = K W t K = ( x y ξ η y x ξ η ) y x = 1 η K + y ξ ρu ρuu + p ξ x ρuv + p ξ y ρuh F (W ) ξ η x ξ + G(W ) η ( u v ) G(W ) = K = 0 ρv ρvu + p η x ρvv + p η y ρvh J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

40 Continuous adjoint method Continuous Adjoint for 2D Euler equations (5/11) Steady state equation can also be rewritten as ( f y ξ η g x ) + ( f y η η ξ + g x ) ξ Jacobians per mesh directions : = 0 sur D ξη df (w) a(w) = dw ( a 1 (w, ξ, η) = a(w) y ) η b(w) x η b(w) = dg(w) dw ( a 2 (w, ξ, η) = a(w) y ) ξ + b(w) x ξ J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

41 Continuous adjoint method Continuous Adjoint for 2D Euler equations (6/11) Coordinate transformation now depending on a design parameter α (for the sake of simplicity scalar) { Dξη D Γ α D w (ξ, η)(α) (x(ξ, η, α), y(ξ, η, α)) (1) D w changes with α but not D ξη Equation for dw /dα i? J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

42 Continuous adjoint method Continuous Adjoint for 2D Euler equations (7/11) Variations induced by d α change f (w) f (w) + dw df dw d dα αi i x x η η + 2 x d η α αi i Fluid dynamics equations on the fixed domain D ξη ( α D α f y ξ η g x ) + ( f y η η ξ + g x ) = 0 on D ξη ξ Differentiate w.r.t. α J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

43 Continuous adjoint method Continuous Adjoint for 2D Euler equations (8/11) Continuous direct differention equation ( a ξ 1 (w, ξ, η) dw ) + dα η ( a 2 (w, ξ, η) dw ( ) dα f (w) 2 y ξ η α g(w) 2 x + η α η ) + ( f (w) 2 y ξ α + g(w) 2 x ξ α Objective function (fixed domain D ξη ) J (α) = J 1 (w)dη + ξ min J 2 (w)dξdη D ξη derivative of the objective function (fixed domain D ξη ) dj (α) dα = ξ min dj 1 (w) dw dw dα dη + D ξη dj 2 (w) dw dw dα dξdη ) = 0 J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

44 Continuous adjoint method Continuous Adjoint for 2D Euler equations (9/11) continuous direct differentiation equation is multiplied by ψ, C 1, periodic in η min, η max ψ C 1 (D ξη ) 4 ( ψ T D ξη ξ Integration by parts ψ T D ξη ξ ψ T D ξη ξ ψ T D ξη η D ξη ψ T ( ξ ( ( f (w) 2 y η α g(w) 2 x η α a 1(w, ξ, η) dα dw ) + η ( ) + η a1(w, ξ, η) dw dα dξdη ψ T D ( ξη ) η f (w) 2 y η α g(w) 2 x dξdη η α ( ) f (w) 2 y ξ α + g(w) 2 x dξdη ξ α a 2(w, ξ, η) dw dα )) dξdη+ ( f (w) 2 y ξ α + g(w) 2 x ξ α a2(w, ξ, η) dw dα dξdη+ + ξ min ψ T a 1(w, ξ, η) dw dα dη + ξ min ψ T ( f (w) 2 y η α g(w) 2 x η α ) dη = 0. )) dξdη = 0 J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

45 Continuous adjoint method Continuous Adjoint for 2D Euler equations (10/11) Gradient of objective function for all ψ function of C 1 η (D ξη ) 4 dj (α) dα D ξη ψ T ξ D ξη ψ T ξ D ξη ψ T η = dj 1(w) ξ min dw dw a1(w, ξ, η) dw dα dη + dj 2(w) D ξη dw dα dw dξdη dα dξdη ψ T D ( ξη ) η f (w) 2 y η α g(w) 2 x dξdη η α ( ) f (w) 2 y ξ α + g(w) 2 x dξdη ξ α a2(w, ξ, η) dw dα dξdη+ + ξ min ψ T a 1(w, ξ, η) dw dα dη + ξ min ψ T ( f (w) 2 y η α g(w) 2 x η α ) dη ψ chosen so as to cancel all flow sensitivity terms dj 2(w) dw ψt ψt a1(w, ξ, η) a2(w, ξ, η) = 0 ξ η over ψ T a 1(w, ξ, η) + dj1(w) = 0 on ξ dw min D ξ,η J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

46 Continuous adjoint method Continuous Adjoint for 2D Euler equations (11/11) Final form of objective gradient (ψ being the solution of continuous adjoint equation) dj (α) dα = ψ T (f 2 ) y (w) g(w) 2 x dη i ξ min η α i η α i ψ T ( 2 ) y f (w) g(w) 2 x dξdη D ξη ξ η α i η α i ψ T ( ) f (w) 2 y + g(w) D ξη η ξ α 2 x dξdη i ξ α i Just as for discrete adjoint, one adjoint field for one function of interest and design parameters Partial differential equation which derivation exceeds level of maths ordinarly used by engineers Equation to be discretized to get numerical values (2) J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

47 Continuous adjoint method Some intuitions about adjoint vector? (1/7) Could I get some intuition about adjoint vector? Try again with continuous adjoint! Rewrite flow equation locally, neglecting metric derivatives terms ( f y ξ η g x ) + ( f y η η ξ + g x ) = 0 sur D ξη ξ df (w) a(w) = dw ( a 1 (w, ξ, η) = a(w) y ) η b(w) x η b(w) = dg(w) dw ( a 2 (w, ξ, η) = a(w) y ) ξ + b(w) x ξ J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

48 Continuous adjoint method Some intuitions about adjoint vector? (1/7) Could I get some intuition about adjoint vector? Try again with continuous adjoint! Rewrite flow equation locally, neglecting metric derivatives terms ( f y ξ η g x ) + ( f y η η ξ + g x ) = 0 sur D ξη ξ df (w) a(w) = dw ( a 1 (w, ξ, η) = a(w) y ) η b(w) x η b(w) = dg(w) dw ( a 2 (w, ξ, η) = a(w) y ) ξ + b(w) x ξ J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

49 Continuous adjoint method Some intuitions about adjoint vector? (2/7) Could I get some intuition about adjoint vector? Trying again based on continuous adjoint Rewrite flow equation locally, neglecting metric derivatives terms Reminder adjoint equation a 1(w, ξ, η) w w + a2(w, ξ, η) ξ η = 0 dj 2(w) dw ψt ξ ψt a1(w, ξ, η) a2(w, ξ, η) = 0 η Change of sign, transposed jacobians, source term. Hyperbolic system. Same conditions for existence of simple wave solutions ψ(ξ, η) = Ψ(aξ + bη)v, propagation par convection. Number of solutions for subsonic/supersonic flow... J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

50 Continuous adjoint method Some intuitions about adjoint vector? (3/7) Supersonic inviscid flow M = 1.5 AoA = 1 o Figure: mesh J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

51 Continuous adjoint method Some intuitions about adjoint vector? (5/7) Supersonic inviscid flow M = 1.5 AoA = 1 o Figure: iso-lines of Mach number J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

52 Continuous adjoint method Some intuitions about adjoint vector? (6/7) Supersonic inviscid flow M = 1.5 AoA = 1 o Figure: First component of adjoint vector for CDp J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

53 Continuous adjoint method Some intuitions about adjoint vector? (6/7) Supersonic inviscid flow M = 1.5 AoA = 1 o Figure: First component of adjoint vector for CDp (close view) J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

54 Continuous adjoint method Some intuitions about adjoint vector? (7/7) Supersonic inviscid flow M = 1.5 AoA = 1 o Figure: Fourth component of adjoint vector for CDp J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

55 Continuous adjoint method Some intuitions about adjoint vector? (7/7) Supersonic inviscid flow M = 1.5 AoA = 1 o Figure: Fourth component of adjoint vector for CDp (close view) J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

56 Outline Discrete vs Continuous adjoint 1 Introduction 2 Discrete adjoint method 3 Continuous adjoint method 4 Discrete vs Continuous adjoint 5 Conclusions J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

57 Discrete adjoint Discrete vs Continuous adjoint Assets calculates what you want = sensitivity of your code can deal with all types of functions code can be partly built by AD (automatic differentiation) higher order derivatives simple (not too complex) in a discrete framework Drawbacks no understanding of underlying physics (Euler flows...) numerical consistency with a set of pde? Dissipative scheme for this set of pde? J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

58 Discrete adjoint Discrete vs Continuous adjoint Assets get physical understanding of underlying equations (with all following restrictions) codes a dissipative discretization of underlying equation the code is shorter and simpler than the one of discrete adjoint Drawbacks does not calculate the sensitivity of your direct (steady state) code no reason that continuous adjoint equations would exist for all types of initial pde can not deal with far-field functions J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

59 Discrete vs Continuous adjoint Coexistence of continuous and discrete adjoint Coexistence comes from the fact that their assets are balanced Continuous more suitable for theoretical mechanics Probably discrete more suitable for practical applications J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

60 Outline Conclusions 1 Introduction 2 Discrete adjoint method 3 Continuous adjoint method 4 Discrete vs Continuous adjoint 5 Conclusions J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60

61 Conclusion Conclusions More material can be found in Numerical sensitivity analysis for aerodynamic optimization: A survey of approaches. Computers and Fluids 39 J.P. & RP Dwight 2010 Second order derivatives, frozen turbulence and other approximations, discretization of the continuous adjoint equation... Twenty-six years after [Jameson 88] famous article... All large CFD code in aeronautics have an adjoint module Some robustness issues to be solved Compatibility with some complex options of direct code possibly missing Integration in automated local shape optimization requires adjoint enhanced robustness and CAD/parametrization issue to be solved Numerous successful adjoint-based local optimizations and goal-oriented adaptations J. Peter (ONERA DMFN) Adjoint method for compressible CFD October 2, / 60