Discrete and continuous adjoint method for compressible...

Discrete and continuous adjoint method for compressible CFDJ. Peter ONERA

J. Peter1

1ONERA DMFN

October 2, 2014

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

Introduction

Outline

1 Introduction

2 Discrete adjoint method

3 Continuous adjoint method

4 Discrete vs Continuous adjoint

5 Conclusions


Introduction

Introduction (1/4)

Well-known aerodynamic optimization problems of the utmost importance

Aircraft drag reductionReduction of total pressure losses of a blade row.

Strongly constrained problems (from aerodynamics, structure...)

Several approaches for researches and studies in external aerodynamics

Flight testsWind tunnel experiments (with flight Re/lower than flight Re)Numerical simulation

Numerical simulation most adapted


Introduction

Introduction (2/4)

(Non solvable) pde → numerical simulation. Finite-volume simulation in thistalk.

Infinite dimension possible deformation → parametrization

Finite dimensional maths

Which type of optimization method ?

Local or global optimization ?


Introduction

Introduction (3/4)

Global optimization

genetic/evolutionnary algorithms, particle swarm, aunt colony, CMA-ES...large number of function evaluations requiredcombined with surrogate modelsin particular used for design space exploration with low fidelity models

Local optimization

very valuable when starting from pre-optimized shapespattern methods. e.g. simplex methodgradient-based methods. e.g. steepest descent, conjugate gradient

Popular and efficient descent methods require objective and constraintsensitivities w.r.t. design parameters


Introduction

Introduction (4/4)

Needed sensitivities w.r.t. design parameters

...not a trivial task in numerical simulation as state variables change withshape via the equations of the mechanical problem

Sensitivty calculation

70’s 80’s finite differences. Scaling with number of shape parametersControl theory [Lions 71, Pironneau 73,74] aerodynamics shape optimization[Jameson 88] adjoint method. Scaling with the number of functions to bedifferentiated

Other applications of adjoint method: understanding zones of influence forfunction value, goal-oriented mesh refinement


Discrete adjoint method

Outline

1 Introduction




5 Conclusions




Framework: compressible flow simulation using finite volume method.Discrete approach for sensitivity analysis

Notations

Volume mesh X , flowfield W (size na)Wall surface mesh SResidual R, C 1 regular w.r.t. X and W – steady state: R(W ,X ) = 0Vector of design parameters α (size nd), X (α) S(α) C 1 regular

Assumption of implicit function theorem

∀ (Wi ,Xi ) / R(Wi ,Xi ) = 0 (∂R/∂W )(Wi ,Xi ) 6= 0Unique steady flow corresponding to a mesh



IntroductionDiscrete gradient calculation methods

Functions of interest

Jk(α) = Jk(W (α),X (α)) k ∈ [1, nf ]Flowfield and volume mesh linked by flow equations R(W (α),X (α)) = 0

Sensitivities dJk/dαi k ∈ [1, nf ] i ∈ [1, nd ] to be computed

Discrete gradient computation methods

Finite differences – 2nd flow computations (non linear problems, size na)Direct differentiation method – nd linear systems (size na)Adjoint vector method – nf linear systems (size na)



Finite difference method

Choose steps δαi . Get shifted meshes X (α + δαi ), X (α− δαi )

Solve flows

R(W (α + δαi ),X (α + δαi )) = 0 R(W (α− δαi ),X (α− δαi )) = 0

dW

dαi (FD)

=W (α + δαi )−W (α− δαi )

2δαi

Compute outputs sensitivities

dJkdαi (FD)

=Jk(W (α + δαi ),X (α + δαi ))− Jk(W (α− δαi ),X (α + δαi ))

2δαi

Two issues: definition of δαi , cost of shifted flow solves



Direct differentiation method (1/2)

Discrete equations for mechanics (set of na non-linear equations )

R(W (α),X (α)) = 0

Differentiation with respect to αi i ∈ [1, nd ]. Derivation of nd linear systemof size na

∂R

∂W

dW

dαi= −(

∂R

∂X

dX

dαi)

Calculation of derivatives

dJkdαi

=∂Jk∂X

dX

dαi+∂Jk∂W

dW

dαi



Direct differentiation method (2/2)

Gradient vectors

∇αJk(α) =∂Jk∂X

dX

dα+∂Jk∂W

dW

dα

Check the flow sensitivities using finite differences

R(W (α + δαi ),X (α + δαi )) = 0 R(W (α− δαi ),X (α− δαi )) = 0

dW

dαi? ' W (α + δαi )−W (α− δαi )

2δαi

Check the outputs sensitivities

dJkdαi

? ' Jk(W (α + δαi ),X (α + δαi ))− Jk(W (α− δαi ),X (α + δαi ))

2δαi



Mathematical game (1/4)

Mathematical game in Rn to understand adjoint method

given (f , bi ) ∈ Rn (i ∈ 1, nd), given A ∈M(Rn)

Calculate the values of xi .f A xi = bi i ∈ 1, nd

Solution solving one linear system instead of nd linear systems ???




Linear algebra reminder: the inverse of the transpose is the transpose of theinverse

MT (M−1)T = (M−1M)T = IT = I

(M−1)TMT = (MM−1)T = IT = I

The notation M−T is suitable for (MT )−1 / (M−1)T





given (f , bi ) ∈ Rn (i ∈ 1, nd), given A ∈M(Rn)

Calculate the values of f .xi A xi = bi i ∈ 1, nd

f .xi = f .(A−1bi ) = ((A−1)T f ).bi = (A−T f ).bi efficient solution

Solve AT λ = f Calculate λ.bi i ∈ 1, nd





given (fj , bi ) ∈ Rn (i ∈ 1, nd j ∈ 1, nf ), given A ∈M(Rn)

Calculate the values of xi .fj A xi = bi i ∈ 1, nd

Solution solving nd linear systems

Solution solving nf linear systems



Discrete adjoint parameter method (1/5)

Several ways of deriving the equations of discrete adjoint method. Thefollowing also helps understanding continuous adjoint

Following equalities hold ∀λk ∈ Rna

λTk∂R

∂W

dW

dαi+ λTk (

∂R

∂X

dX

dαi) = 0

dJk(α)

dαi=∂Jk∂X

dX

dαi+∂Jk∂W

dW

dαi+ λTk

∂R

∂W

dW

dαi+ λTk (

∂R

∂X

dX

dαi)

dJk(α)

dαi= (

∂Jk∂W

+ λTk∂R

∂W)dW

dαi+∂Jk∂X

dX

dαi+ λTk (

∂R

∂X

dX

dαi)




Vector λk defined in order to cancel the factor of the flow sensitivity dWdαi

...

the adjoint equation. λk actually appears to be linked to functions Jk

∂Jk∂W

+ λTk∂R

∂W= 0


∀i ∈ [1, nd ]dJk(α)

dαi=∂Jk∂X

dX

dαi+ λTk (

∂R

∂X

dX

dαi)


dX

dα+ λTk (

∂R

∂X

dX

dα)

Method with nf and not nd linear systems to solve




Other ways to derive the discrete adjoint equation

Introduce a LagrangianManipulate direct differentiation gradient expression (like in the mathematicalgame)

From direct method gradient expression


dX

dα+∂Jk∂W

dW

dα


dX

dα− ∂Jk∂W

(dR

dW

)−1dR

dX

dX

dα


dX

dα−

(∂Jk∂W

(dR

dW

)−1)

dR

dX

dX

dα

Define λk column vector




From direct method gradient expression


dX

dα−

(∂Jk∂W

(dR

dW

)−1)

dR

dX

dX

dα

Define λk

λTk = −∂Jk

∂W

(dR

dW

)−1

or λTk (dR

dW) = −

∂Jk

∂Wor

(dR

dW

)T

λk = −∂Jk

∂W

T

Expresion of sensitivity


dX

dα+ λTk

dR

dX

dX

dα



Iterative solution of direct and adjoint equation (1/3)

CFD teams tend to mimic the solution of steady state flow altough flowequations are non-linear whereas direct/adjoint equation are linear

Storing the jacobian of the scheme and sending to direct solver has beendone but is rare and is not tractable for large cases

Iterative resolution is much more common. Newton/relaxation algorithm(∂R

∂W

)(APP) T (λ

(l+1)k − λ(l)

k

)= −

((∂R

∂W)Tλ

(l)k + (

∂Jk∂W

)T)




Common Newton/relaxation algorithm for adjoint(∂R

∂W

)(APP) T (λ

(l+1)k − λ(l)

k

)= −

((∂R

∂W)Tλ

(l)k +

∂Jk

∂W)T)

Common Newton/relaxation algorithm for direct(∂R

∂W

)(APP) ((dW

dαi)(l+1) − (

dW

dαi)(l)

)= −

((∂R

∂W)dW

dαi

(l)

+∂R

∂X

dX

dαi

)

Defining an approximate Jacobians ( ∂R∂W )(APP) is an old subject incompressible CFD (definition of implicit stages for backward-Eulerschemes...)

upwind approximate linearization of convective fluxneglecting cross derivatives in linearization of viscous fluxes...

Possibly adapting implicit stages and mutigrid algorithm (flow solver toadjoint solver)




Common Newton/relaxation algorithm for adjoint(∂R

∂W

)(APP) T (λ

(l+1)k − λ(l)

k

)= −

((∂R

∂W)Tλ

(l)k +

∂Jk

∂W)T)

Accuracy of adjoint vector only depends on ( ∂R∂W ). Only minor simplifcationsare allowed at this stage to perserve an acceptable accuracy

Convergence towards solution of the linear system depends on ( ∂R∂W ),

( ∂R∂W )(APP), multigrid (if active), other operations like smoothing (if active)




Checking adjoint method... much more difficult than checking directdifferentiation method. If

dJkdαi

<>Jk (W (α+ δαi ),X (α+ δαi ))− Jk (W (α− δαi ),X (α+ δαi ))

2δαi

no easy checking procedure

In the iterative resolution method, the gradient accuracy depends on the

( ∂R∂W )Tλ(l)k operation

If direct mode is coded, duality checks between direct and adjoint code areuseful. (U,V ) two column vectors of Rna

UT (∂R

∂W)V =

(UT (

∂R

∂W)

)adj−code

.V = UT .

((∂R

∂W)V

)lin−code

Valid for individual fluxes routine. Valid for part of the interfaces (border,joins...)



Discrete adjoint mesh method (1/3)

Vector λk defined by∂Jk∂W

+ λTk∂R

∂W= 0


∀i ∈ [1, nf ]dJk(α)

dαi=∂Jk∂X

dX

dαi+ λTk (

∂R

∂X

dX

dαi)

∀i ∈ [1, nf ]dJk(α)

dαi= (

∂Jk∂X

+ λTk∂R

∂X)dX

dαi

Obvious mathematical factorization. Huge practical importance.




Solve for adjoint vectors

CFD gradient computation code computes “only”

dJkdX

=∂Jk∂X

+ λTk∂R

∂X

The functional outputs sensitivities dJk(α)/dαi are calculated later by amesh/geometrical tool

Pros : CFD has no knowledge of parametrization. Huge memory savings[Nielsen, Park 2005] Try several parametrization. Check (dJk/dS) with engineers

Cons : Matrix (∂R/∂X ) has to be explicitely computed (instead of ∂R∂X

dXdαi

computable by finite differences) Hard work...




Solve for adjoint vectors. Compute “only”

dJkdX

=∂Jk∂X

+ λTk∂R

∂X

Cons : Matrix (∂R/∂X ) has to be explicitely computed (instead of ∂R∂X

dXdαi

computable by finite differences) Hard work...

How to calculate (dJk/dS) ?

Explicit link beween X and S

dJkdαi

=

[dJkdX

dX

dS

]dS

dαi

Implicit link between X and S [Nielsen, Park 2005]


Continuous adjoint method

Outline

1 Introduction




5 Conclusions



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 onAdjoint Methods, Toulouse 1999.



Continuous Adjoint for toy problems (1/6)

From [Giles, Pierce 99] section (3.2)

Toy problems without design parameters

Solvedu

dx− εd

2u

dx2= f on [0, 1] u(0) = u(1) = 0

before calculating

J = (u, g) =

∫ 1

0

u g dx

Adjoint problem ? Define (if it exists)

L∗λ = g on [0, 1] plus boundary conditions

such that

J = (λ, f ) =

∫ 1

0

λ fdx




Direct: solve

du

dx− εd

2u

dx2= f on [0, 1] u(0) = u(1) = 0

before calculating

J = (u, g) =

∫ 1

0

u gdx

Adjoint problem (if it exists):

L∗λ = g on [0, 1]

plus boundary conditions such that

J = (λ, f ) =

∫ 1

0

λ fdx

Defining equation L∗




Defining equation L∗

(λ, f ) =

∫ 1

0

λ fdx =

∫ 1

0

λ

(du

dx− εd

2u

dx2

)dx

(λ, f ) = −∫ 1

0

dλ

dxu dx + [λ u]1

0 + ε

∫ 1

0

dλ

dx

du

dxdx − ε

[λ

du

dx

]1

0

(λ, f ) = −∫ 1

0

dλ

dxu dx + [λ u]1

0 − ε∫ 1

0

d2λ

dx2du dx − ε

[λdu

dx

]1

0

− ε[dλ

dxu

]1

0

Finally

(λ, f ) =

∫ 1

0

(dλ

dx− εd

2λ

dx2

)udx + [λ u]1

0 + ε

[dλ

dxu

]1

0

− ε[λdu

dx

]1

0

Suitable adjoint equation. Solving

−dλ

dx− εd

2λ

dx2= g on [0, 1] λ(0) = λ(1) = 0

ensures (λ, f ) = (u, g)




In order to calculate J = (u, g) =

∫Ω

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 ) =

∫Ω

λ f dΩ, solve for λ

div(k grad(λ)) = g on Ω λ = 0 on∂Ω

Definition of adjoint operator comes from

(λ, f ) =

∫Ω

u div(k grad(λ))dΩ−∫∂Ω

k u (grad(λ).n)dS +

∫∂Ω

k λ (grad(u).n)dS




Direct: solve

∂u

∂t− ∂2u

∂x2= f on [0, L]× [0,T ] u(0, .) = u(L, .) = 0 u(., 0) = 0

before calculating

J = (u, g) =

∫ L

0

∫ T

0

u g dxdt

Adjoint: solve

−∂λ∂t− ∂2λ

∂x2= g on [0, L]× [0,T ] λ(0, .) = λ(L, .) = 0 λ(.,T ) = 0

before calculating J as

(λ, f ) =

∫ L

0

∫ T

0

λ f dxdt




Time derivative∂u

∂tgets −∂λ

∂tBackward time integration for unsteady adjoint

Convection term∂u

∂xgets −∂λ

∂x“Backward propagation” in adjoint steady state solutions

Diffusion term∂2u

∂x2gets

∂2λ

∂x2



Continuous Adjoint for 2D Euler equations (1/11)

2D Euler equations∂w

∂t+∂f (w)

∂x+∂g(w)

∂y= 0

avec

w =

ρρuρvρE

f (w) =

ρu

ρu2 + pρuvρHu

g(w) =

ρvρuv

ρv2 + pρHv

p = (γ − 1)ρ(E − u2 + v2

2), ρH = ρE + p




Figure: Coordinate transformation for airfoil-fitted structured mesh

Coordinate transformation Γ, C 1 diffeomorphism Dξη =[ξmin, ξmax ]× [ηmin, ηmax ] en Dw .

Γ

Dξη → Dxy

(ξ, η) → (x , y)




Figure: Normal surface vectors




2D Euler equations generalized coordinates

K = (∂x

∂ξ

∂y

∂η− ∂y

∂ξ

∂x

∂η)

(UV

)=

1

K

∂y∂η

− ∂x∂η

−∂y∂ξ

∂x∂ξ

( uv

)

W = K

ρρuρvρE

F (W ) = K

ρU

ρUu + p ∂ξ∂x

ρUv + p ∂ξ∂y

ρUH

G(W ) = K

ρV

ρVu + p ∂η∂x

ρVv + p ∂η∂y

ρVH

∂W

∂t+∂F (W )

∂ξ+∂G(W )

∂η= 0




Steady state equation can also be rewritten as

∂

∂ξ

(f∂y

∂η− g

∂x

∂η

)+

∂

∂η

(−f ∂y

∂ξ+ g

∂x

∂ξ

)= 0 sur Dξη

Jacobians per mesh directions :

a(w) =df (w)

dwb(w) =

dg(w)

dw

a1(w , ξ, η) =

(a(w)

∂y

∂η− b(w)

∂x

∂η

)a2(w , ξ, η) =

(−a(w)

∂y

∂ξ+ b(w)

∂x

∂ξ

)




Coordinate transformation now depending on a design parameter α (for thesake of simplicity scalar)

Γ

DξηDα → Dw

(ξ, η)(α) → (x(ξ, η, α), y(ξ, η, α))(1)

Dw changes with α but not Dξη

Equation for dW /dαi ?




Variations induced by dα change f (w) → f (w) + dfdw

dwdαi

dαi

∂x∂η

→ ∂x∂η

+ ∂2x∂η∂αi

dαi

Fluid dynamics equations on the fixed domain Dξη

∀ α ∈ Dα∂

∂ξ

(f∂y

∂η− g

∂x

∂η

)+

∂

∂η

(−f ∂y

∂ξ+ g

∂x

∂ξ

)= 0 on Dξη

Differentiate w.r.t. α




Continuous direct differention equation

∂∂ξ

(a1(w , ξ, η)dwdα

)+ ∂∂η

(a2(w , ξ, η)dwdα

)+

∂∂ξ

(f (w) ∂2y

∂η∂α− g(w) ∂2x

∂η∂α

)+ ∂∂η

(−f (w) ∂

2y∂ξ∂α

+ g(w) ∂2x∂ξ∂α

)= 0

Objective function (fixed domain Dξη)

J (α) =

∫ξmin

J1(w)dη +

∫Dξη

J2(w)dξdη

derivative of the objective function (fixed domain Dξη)

dJ (α)

dα=

∫ξmin

dJ1(w)

dw

dw

dαdη +

∫Dξη

dJ2(w)

dw

dw

dαdξdη




continuous direct differentiation equation is multiplied by ψ, C 1, periodic inηmin, ηmax

∀ψ ∈ C 1(Dξη)4∫Dξη

ψT(∂∂ξ

(a1(w , ξ, η) dw

dα

)+ ∂∂η

(a2(w , ξ, η) dw

dα

))dξdη+∫

DξηψT

(∂∂ξ

(f (w)

∂2y

∂η∂α− g(w) ∂

2x∂η∂α

)+ ∂∂η

(−f (w)

∂2y

∂ξ∂α+ g(w) ∂

2x∂ξ∂α

))dξdη = 0

Integration by parts

−∫Dξη

∂ψT

∂ξa1(w , ξ, η) dw

dαdξdη −

∫Dξη

∂ψT

∂ηa2(w , ξ, η) dw

dαdξdη+

−∫Dξη

∂ψT

∂ξ

(f (w)

∂2y

∂η∂α− g(w) ∂

2x∂η∂α

)dξdη

−∫Dξη

∂ψT

∂η

(−f (w)

∂2y

∂ξ∂α+ g(w) ∂

2x∂ξ∂α

)dξdη

+∫ξmin

ψT a1(w , ξ, η) dwdα

dη +∫ξmin

ψT

(f (w)

∂2y

∂η∂α− g(w) ∂

2x∂η∂α

)dη = 0.




Gradient of objective function for all ψ function of C 1η (Dξη)4

dJ (α)dα =

∫ξmin

dJ1(w)dw

dwdα

dη +∫Dξη

dJ2(w)dw

dwdα

dξdη

−∫Dξη

∂ψT

∂ξa1(w , ξ, η) dw

dαdξdη −

∫Dξη

∂ψT

∂ηa2(w , ξ, η) dw

dαdξdη+

−∫Dξη

∂ψT

∂ξ

(f (w)

∂2y

∂η∂α− g(w) ∂

2x∂η∂α

)dξdη

−∫Dξη

∂ψT

∂η

(−f (w)

∂2y

∂ξ∂α+ g(w) ∂

2x∂ξ∂α

)dξdη

+∫ξmin

ψT a1(w , ξ, η) dwdα

dη +∫ξmin

ψT

(f (w)

∂2y

∂η∂α− g(w) ∂

2x∂η∂α

)dη

ψ chosen so as to cancel all flow sensitivity termsdJ2(w)dw

− ∂ψT

∂ξa1(w , ξ, η)− ∂ψ

T

∂ηa2(w , ξ, η) = 0 over Dξ,η

ψT a1(w , ξ, η) +dJ1(w)dw

= 0 on ξmin




Final form of objective gradient (ψ being the solution of continuous adjointequation)

dJ (α)dαi

=∫ξmin

ψT

(f (w)

∂2y

∂η∂αi− g(w) ∂

2x∂η∂αi

)dη

−∫Dξη

∂ψT

∂ξ

(f (w)

∂2y

∂η∂αi− g(w) ∂

2x∂η∂αi

)dξdη

−∫Dξη

∂ψT

∂η

(−f (w)

∂2y

∂ξ∂αi+ g(w) ∂

2x∂ξ∂αi

)dξdη

(2)

Just as for discrete adjoint, one adjoint field for one function of interest anddesign parameters

Partial differential equation which derivation exceeds level of maths ordinarlyused by engineers

Equation to be discretized to get numerical values



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

∂

∂ξ

(f∂y

∂η− g

∂x

∂η

)+

∂

∂η

(−f ∂y

∂ξ+ g

∂x

∂ξ

)= 0 sur Dξη

a(w) =df (w)

dwb(w) =

dg(w)

dw

a1(w , ξ, η) =

(a(w)

∂y

∂η− b(w)

∂x

∂η

)a2(w , ξ, η) =

(−a(w)

∂y

∂ξ+ b(w)

∂x

∂ξ

)




Could I get some intuition about adjoint vector ?Trying again based on continuous adjoint

Rewrite flow equation locally, neglecting metric derivatives terms

a1(w , ξ, η)∂w

∂ξ+ a2(w , ξ, η)

∂w

∂η= 0

Reminder adjoint equation

dJ2(w)

dw−∂ψ

T

∂ξa1(w , ξ, η)−

∂ψT

∂ηa2(w , ξ, η) = 0

Change of sign, transposed jacobians, source term.

Hyperbolic system. Same conditions for existence of simple wave solutionsψ(ξ, η) = Ψ(aξ + bη)V , propagation par convection. Number of solutions forsubsonic/supersonic flow...




Supersonic inviscid flow M∞ = 1.5 AoA = 1o

Figure: 513 × 513 mesh





Figure: iso-lines of Mach number





Figure: First component of adjoint vector for CDp





Figure: First component of adjoint vector for CDp (close view)





Figure: Fourth component of adjoint vector for CDp





Figure: Fourth component of adjoint vector for CDp (close view)


Discrete vs Continuous adjoint

Outline

1 Introduction




5 Conclusions



Discrete adjoint

Assets

calculates what you want = sensitivity of your codecan deal with all types of functionscode 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 ofpde ?



Discrete adjoint

Assets

get physical understanding of underlying equations (with all followingrestrictions)codes a dissipative discretization of underlying equationthe code is shorter and simpler than the one of discrete adjoint

Drawbacks

does not calculate the sensitivity of your direct (steady state) codeno reason that continuous adjoint equations would exist for all types of initialpdecan not deal with far-field functions



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


Conclusions

Outline

1 Introduction




5 Conclusions


Conclusions

Conclusion

More material can be found in Numerical sensitivity analysis for aerodynamicoptimization: A survey of approaches. Computers and Fluids 39 J.P. & RPDwight 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 moduleSome robustness issues to be solvedCompatibility with some complex options of direct code possibly missingIntegration in automated local shape optimization requires adjoint enhancedrobustness and CAD/parametrization issue to be solvedNumerous successful adjoint-based local optimizations and goal-orientedadaptations


Discrete and continuous adjoint method for compressible...

Documents

Transcript of Discrete and continuous adjoint method for compressible...