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Ninth International Conference on CFD in the Minerals and Process Industries

CSIRO, Melbourne, Australia

10-12 December 2012

TOPOLOGICAL DERIVATIVE FORMULATION FOR SHAPE SENSITIVITY IN

INCOMPRESSIBLE TURBULENT FLOWS

Aleksandar JEMCOV1∗, Darrin STEPHENS2

1Aerospace and Mechanical Engineering and Center for Research Computing, University of Notre

Dame, Indiana, USA

2Applied CCM Pty Ltd, PO BOX 2220, Dandenong North, Victoria, 3175, Australia

∗Corresponding author, E-mail address: ajemcov@nd.edu

ABSTRACT

Shape derivative based on topological consideration is presented in

this work. The resulting derivative has simpler form compared to

corresponding classical shape derivative due to topological deriva-

tive formulation. Shape derivative is based on topological deriva-

tive in the limit of inﬁnitesimally weak source terms in momentum

equation approaching the boundary of the computational domain.

The consistency of two derivatives is demonstrated and computa-

tional example of the ﬂow in curved duct is used for the illustration

of the derivative computations.

Keywords: CFD, adjoint, incompressible, turbulent, topological

derivative, shape derivative

.

NOMENCLATURE

APDE operator

BBoundary operator

CLinear operator

α

Arbitrary test function

DkTurbulent kinetic energy diffusivity [m2/s]

D

ε

Turbulent dissipation diffusivity [m2/s]

GTurbulence production

HDomain height

J,JOutput functional

LLagrangian

NNumber

PParameters

QState vector

BΓBilinear concommitant

bBilinear boundary operator

c1,c2,c

µ

Turbulence model parameters

ck,c∗

µ

k,c

ε

,c∗

µε

Adjoint turbulence model parameter

u∗

tau Adjoint wall friction velocity[m

/s]

csMultiplicative parameter

fMomentum source

kTurbulent kinetic energy, [m2/s2]

nboundary normal

tboundary tangent

pDensity normalized pressure, [m2/s2]

uVelocity, [m/s]

y+Non-dimensional wall distance

ΩComputational domain

ΓBoundary of computational domain

Γb,Γ0Boundary segments of computational domain

δ

Variational derivative

ε

Turbulent dissipation, [m2/s3]

η

Indictor function

ν

Kinematic viscosity, [m2/s]

ν

tTurbulent Kinematic viscosity, [m2/s]

ξ

Local direction

φ

General variable

Sub/superscripts

fface

iCartesian component of a vector or tensor i

jCartesian component of a vector or tensor j

∗Adjoint variable ∗

′Derivative of a variable ′

INTRODUCTION

Computing the shape derivatives in turbulent incompress-

ible ﬂows is a basic requirement for any design optimisa-

tion gradient based algorithm. Solution of continuous adjoint

Navier-Stokes equations is a method of choice for many ap-

plications of design optimisation problems where gradient

information is required due to its efﬁciency in computing

gradients (Nadarajah and Jameson,2000), (Jameson et al.,

2008), (Anderson and Venkatakrishnan,1997). Adjoint in-

compressible equations lead to two different formulations of

the sensitivity gradients: shape derivatives and topological

derivatives (Othmer,2008). Both approaches have been used

in design optimisation to produce optimal shapes and opti-

mal domains. Despite their differences, Othmer (Othmer,

2008) have shown that they produce consistent gradients

close to the boundary of the domain. Therefore, the inter-

esting prospect of using topological derivatives in shape op-

timisation is investigated in this paper.

In this work, continuous adjoint Navier-Stokes equations that

account for the effect of turbulence in computation of the ad-

joint ﬁeld (Castro et al.,2007), (Bueno-Orovio et al.,2012),

(Zymaris et al.,2010) are used. In addition, a new way of

computing the shape sensitivity derivatives is proposed. The

new approach is based on the topological formulation of the

sensitivity derivative in the limit of vanishing source term in

Navier-Stokes equations. The source term used in the for-

mulation is equivalent to a porosity source term with van-

ishing porosity coefﬁcient. This formulation allows for the

classical deﬁnition and the topological sensitivity using the

adjoint and primal velocity ﬁelds that can be transformed to

the shape sensitivity derivative through a process of local in-

terpolation. Given the weak source term (porosity asymptot-

ically approaching zero), the Navier-Stokes equations are not

modiﬁed in the limit of zero strength of source term while the

adjoint system of equations can still be used to compute the

topological derivative. This is possible due to the fact that

adjoint system is formulated with arbitrarily small porosity

coefﬁcients and the passage to the zero limit is performed

after Lagrangian duality for the Navier-Stokes system is en-

forced.

GOVERNING EQUATIONS

Consider a CFD problem in abstract formulation:

A(Q) = 0 in Ω(1)

B(Q) = 0 on Γ=Γb∪Γo

Deﬁne functional of interest that we would like to minimize:

J(P,Q(P)) = 0 on Γo(2)

The output functional Jusually represents a measure of

Ω

Γo

b

Γ

Figure 1: Computational domain.

the performance of an engineering device and it can be de-

ﬁned on the boundary Γor within the computational domain

Ω. Examples of functionals include integral of the force on

the surface, dissipated power in the domain, uniformity of the

proﬁle of the velocity on the outlet, to name a few. Output

functionals Jdepend on the independent variable Qand

some parameters P. The nature of parameters Pis such

that they can represent quantities used to input data for the

model such as material properties, model constants and/or

boundary condition values. They can also deﬁne the shape

of the computational domain and here we are mostly con-

cerned with functionals that depend on parameters describing

the shape of the domain. In other words, we are interested in

computing the shape derivatives of functionals Jwith re-

spect to parameters Pdeﬁning the shape of the domain.

Computing derivative of output functionals Jiwith respect

to parameter Pjis deﬁned analytically as

δ

PjJj(Pj,Q(Pj)) = lim

β

→0

∂

∂β

Ji(Pj+

β

Rj)(3)

Rules of differentiation are well deﬁned for directional

derivatives that are similar to gradient calculations. In other

words, chain rule of differentiation can be applied to Eq. (3)

resulting in the need to compute derivatives of the indepen-

dent variable Qwith respect to parameter P. In this work

direct computation of derivatives of independent variable Q

with respect to shape parameters Pis avoided through the

introduction of adjoint variables, thus simplifying the prob-

lem signiﬁcantly.

Incompressible turbulent system of Navier-Stokes equations

is given by the following expression:

A=

A1

A2

A3

A4

(4)

=

∂

iui

uj

∂

jui+

∂

ip−

∂

j[(

ν

+

ν

t)(

∂

jui+

∂

iuj)] + fi

uj

∂

jk−

∂

j(Dk

∂

jk)−G+

ε

uj

∂

j

ε

−

∂

j(D

ε∂

j

ε

)−c1G

ε

k+c2

ε

2

k

The system of equations deﬁned in Eq. (4) consist of incom-

pressible continuity and momentum equations supplemented

by the k−

ε

turbulence model equations responsible for the

transport of the turbulent kinetic energy and dissipation. Ap-

propriate boundary conditions and transport properties must

accompany the system of equations (4).

Our goal is to derive adjoint system of equations based on

Eq. (4) and this is accomplished by introducing the La-

grangian L

L=J+hQ∗,Ai(5)

Vector of adjoint variables Q∗plays the role of Lagrangian

multipliers whereas integral Jrepresents the quantity that

is being optimized.

L=J+ZΩp∗u∗

ik∗

ε

∗

A1

A2

A3

A4

dΩ(6)

In order to deﬁne the system of adjoint equations, we seek

to compute the total variation of Lagrangian Lby following

the rules of the variational calculus:

δ

L=

δ

PL+

δ

uiL+

δ

pL+

δ

kL+

δε

L(7)

The total variation includes variations of all ﬁelds that de-

pend on parameter(s) P. In principle, expression for the to-

tal variation in Eq. (7) can be used to deﬁne the shape deriva-

tive if all individual variations in that expression are known.

However, computing the variation of the Lagrangian with re-

spect to primitive variables p,ui,k, and

ε

involves computing

the derivatives of the Navier-Stokes system of equations with

respect to those variables since they are known only implic-

itly thorough the solution of the system of equations deﬁned

in Eq. (4). Therefore, in order to compute total variation of

Lagrangian with Nparameters Pj,Nindependent problems

deﬁned through derivatives of the Navier-Stokes system of

equations have to be solved. This is computationally very

expensive even though the resulting system of equations is

linear. Therefore, a cheaper way of computing variations of

the Lagrangian with respect to primitive variables is required.

One way of achieving this goal is by simply requiring that

variations with respect to state variables vanish. This can be

stated trivially as the following expression:

δ

uiL+

δ

pL+

δ

kL+

δε

L=0 (8)

With this requirement the total variation of the Lagrangian

takes a simpliﬁed form

δ

L=

δ

PJ+

δ

PZΩQ∗AdΩ(9)

Since the variation of the Lagrangian with respect to param-

eter Pexplicitly is trivial, the only remaining unknown part

is ﬁnd the way to compute the integral:

δ

PZΩQ∗AdΩ(10)

Topological derivative formulation for shape sensitivity in incompressible turbulent ﬂows

Computation of the total variation of Lagrangian requires

that we know the values of Lagrange multipliers Q∗. Ad-

joint variables are computed through the introduction of the

Lagrange duality principle. Lagrange duality principle for a

general linear operator Cacting on a general variable

φ

is

given by the following expression:

φ

∗,C

φ

′Ω=

φ

′,C∗

φ

∗Ω+BΓ(11)

or in explicit integral form

ZΩ

φ

∗C

φ

′dΩ=ZΩ

φ

′C∗

φ

∗dΩ+ZΓb(

φ

′,

φ

∗)dΓ(12)

where the notation

δ

P

φ

=

φ

′was used for simplicity.

Eq. (12) and Eq. (8) are used to deﬁne the expression

δ

pJ+ZΩQ∗

δ

pAdΩ

+

δ

uiJ+ZΩQ∗

δ

uiAdΩ

+

δ

kJ+ZΩQ∗

δ

kAdΩ

+

δε

J+ZΩQ∗

δε

AdΩ=0 (13)

Variational differentiation of operator Adeﬁnes the lineari-

sation of non-linear operator Ain some suitably deﬁned

neighbourhood deﬁned by a particular value of the state vec-

tor Q. In other words, if the linearisation of the operator A

in Eq. (4) is known, then Eq. (13) can be used to deﬁne ad-

joint equations that will be used to compute adjoint variables

Q∗needed for the computation of the derivatives in Eq. (10).

Following the rules of the differentiation, the differentiated

continuity equation with respect to parameter Pis given by

δ

PA1=

∂

i

δ

Pui(14)

while differentiated momentum equation is given by expres-

sion

δ

PA2=uj

∂

j

δ

Pui+

δ

Puj

∂

jui+

∂

i

δ

Pp

−

∂

j[(

ν

+

ν

t)(

∂

j

δ

Pui+

∂

i

δ

Puj)]

−

∂

j[

δ

P

ν

t(

∂

jui+

∂

iuj)] +

δ

Pfi(15)

and the differentiated turbulent kinetic and dissipation equa-

tions in k−

ε

turbulence model is given by the following two

expressions:

δ

PA3=

δ

Puj

∂

jk+uj

∂

j

δ

Pk−

∂

j(Dk

∂

j

δ

Pk)

−

∂

j(

δ

PDk

∂

jk)−

δ

PG+

δ

P

ε

(16)

δ

PA4=

δ

Puj

∂

j

ε

+uj

∂

j

δ

P

ε

−

∂

j(D

ε∂

j

δ

P

ε

)

−

∂

j(

δ

PD

ε∂

j

ε

)c1

δ

PG

ε

k−c1G

δ

P

ε

k

+c1G

ε

k2

δ

Pk+2c2

ε

k

δ

P

ε

−c2

ε

2

k2

δ

Pk(17)

Application of the Lagrange duality principle leads to the fol-

lowing integral that is equivalent to expression in Eq. (13)

ZΩ(

∂

pJΩ+A∗

1)

δ

PpdΩ+ZΓ(

∂

pJΓ+B1)

δ

PpdΓ

+ZΩ(

∂

uiJΩ+A∗

2)

δ

PuidΩ+ZΓ(

∂

uiJΓ+B2)

δ

PuidΓ

+ZΩ(

∂

kJΩ+A∗

3)

δ

PkdΩ+ZΓ(

∂

kJΓ+B3)

δ

kPdΓ

+ZΩ(

∂ε

JΩ+A∗

4)

δ

P

ε

dΩ

+ZΓ(

∂ε

JΓ+B4)

δ

P

ε

dΓ=0 (18)

Where functional Jwas split into boundary and interior

contributions

J=JΓ+JΩ(19)

Requiring that each term in Eq. (18) is equal to zero in the

domain we deﬁne the set of adjoint equations:

Adjoint continuity:

A∗

1=−

∂

iu∗

i+

∂

pJΩ=0 (20)

Adjoint momentum:

A∗

2=−uj

∂

ju∗

i−uj

∂

iu∗

j−

∂

j

ν

(

∂

ju∗

i+

∂

iu∗

j)+

∂

ip∗]

+k∗

∂

ik+

ε

∗

∂

i

ε

+2

∂

jhk∗+c1

ε

∗

ε

k

ν

t(

∂

jui+

∂

iuj)i

+f∗

i+

∂

uiJΩ=0,(21)

Adjoint k equation:

A∗

3=−uj

∂

jk∗−

∂

j(Dk

∂

jk∗) + c∗

µ

k

∂

jk

∂

jk∗−c∗

µ

kG∗k∗

+c∗

µ

k

∂

j

ε∂

j

ε

∗+ck

ε

∗+c∗

µ

k(

∂

jui+

∂

iuj)

∂

ju∗

i

+

∂

kJΩ=0,(22)

Adjoint

ε

equation:

A∗

4=−uj

∂

j

ε

∗−

∂

j(Dk

∂

j

ε

∗)−c∗

µε∂

j

ε∂

j

ε

∗+c∗

εε

∗

+c∗

µε∂

jk

∂

jk∗+c

µε

G∗k∗+k∗−c∗

µε

(

∂

jui+

∂

iuj)

∂

ju∗

i

+

∂ε

JΩ=0,(23)

c∗

µ

k=2c

µ

k

ε

,

ck=−c1c

µ

G∗−c2

ε

2

k2,

c∗

µε

=c

µ

k2

ε

2,

c

ε

=2c2

ε

k.

Similar requirement for the boundary terms yields bound-

ary conditions for adjoint equations where unbalanced sur-

face integral terms must be perfectly balanced by terms in

the original output functional Jand its derivatives. In ad-

dition, adjoint wall functions were deﬁned in (Zymaris et al.,

2010) for the case of adjoint k−

ε

turbulence model. The

wall conditions for the adjoint

ε

conditions are deﬁned in a

similar way as the primal

ε

equation throuh the deﬁnition of

of the adjoint wall velocity u∗

τ

:

u∗

τ

= (

ν

+

ν

t)

∂

u∗

i

∂

xj+

∂

u∗

j

∂

xinjti.(24)

Eq. (24) is used to compute the adjoint viscous ﬂuxes at the

wall in the same way as the primal viscous ﬂuxes are com-

puted.

TOPOLOGICAL DERIVATIVE

Directional derivative previously was deﬁned to be

δ

L=

δ

PJ+

δ

PZΩQ∗AdΩ(25)

Applying the rules of differentiation

δ

PZΩQ∗AdΩ=ZΩQ∗

δ

PAdΩ+ZΩ

δ

PQ∗AdΩ(26)

Variation of adjoint variables is second order contribution

and will be ignored

δ

PZΩQ∗AdΩ=ZΩ

δ

PQ∗AdΩ(27)

We can now compute the derivative

δ

PZΩQ∗AdΩ=ZΩp∗u∗

ik∗

ε

∗

δ

PA1

δ

PA2

δ

PA3

δ

PA4

dΩ

(28)

If the forcing term in momentum equation is given by

fi=Pui

η

(29)

then the derivative becomes

δ

PZΩQ∗AdΩ=ZΩp∗u∗

ik∗

ε

∗

0

ui

η

0

0

dΩ

(30)

Total variation of the Lagrangian is

δ

L=

δ

PJ+ZΩ

δ

PQ∗AdΩ(31)

Since there is no explicit dependence of Jon P, direc-

tional derivativeat each cell becomes

δ

PLj=uiu∗

i

η

Ωj.(32)

Therefore, in order to evaluate this directional derivative we

need to compute primal and adjoint velocities. Topological

derivative in Eq. (31) depends only on the primal and adjoint

ﬁelds, volume and characteristic function

η

that deﬁnes the

location of the source term within the domain. There are

no restrictions on where that location can be as long as the

source term is within the computational domain Ω. There-

fore, we can consider the source term distribution deﬁned

very close to the boundary Γ0that is being modiﬁed.

Ω

Γo

b

Γ

fi

Figure 2: Sources distribution.

The connection between the topological derivative and the

shape derivative can be seen if we consider the limit of the

source term moving closer and closer to the boundary of the

domain. In Figure 2we consider one such situation in which

we would like to compute the topological derivative inﬁnites-

imally close to the boundary Γ0. Using the values of the com-

puted sensitivity in its current location, linear extrapolation is

used to deﬁne the value close to the boundary:

(uiu∗

j)f= (ui+

ξ

j

∂

jui△ui)(u∗

i+

ξ

j

∂

ju∗

i△u∗

i)

η

(33)

△ui=ui−uif,△u∗

i=ui−u∗

if(34)

fi

Γo

ξ

Ω

n

Figure 3: Sources distribution.

If the source term is approaching the surface in the direction

of the local normal, then local direction

ξ

can be expressed

as

ξ

=||

ξ

||nthus leading to the expression

(uiu∗

i)f=||

ξ

||2(nj

∂

jui△ui)(nj

∂

ju∗

i△u∗

i)

η

(35)

Taking into account boundary conditions close to the wall

and the following inequalities

||

∂

juiui|| >||ui||,||

∂

ju∗

iu∗

i|| >||u∗

i|| (36)

gives an expression for topological derivative inﬁnitesimally

close to the boundary of the domain

(uiu∗

i)f≈ ||

ξ

||2(nj

∂

juiui)(nj

∂

ju∗

iu∗

i)

η

(37)

Equation (37) is consistent with the deﬁnition of the shape

derivative deﬁned as (Othmer,2008)

δ

PLS=−(

ν

+

ν

t)nj

∂

juiuinj

∂

ju∗

iu∗

iAj(38)

up to a multiplicative parameter csand neglecting the contri-

butions of adjoint turbulent ﬁelds to shape derivative value.

Using the expression in Eq. (37) the shape derivative is de-

ﬁned as

δ

PLSj=csuiu∗

iΩj

η

(39)

Multiplicative parameter cscan be shown to correspond to

the following expression

cs=

ν

+

ν

t

||

ξ

||2(40)

Equation (39) will be used to move the boundary Γ0in order

to optimise the shape. Therefore, action of moving boundary

replaces the source term distribution in momentum equation

and for small changes in the shape the source term becomes

negligible. This corresponds to vanishing source term limit

in the momentum equation.

COMPUTATIONAL ALGORITHM

Computational algorithm for computing the shape deriva-

tives consists of following steps:

1. Compute primal ﬁeld Qusing governing equations de-

ﬁned in Eq. (4) in the limit fi→0

2. Compute adjoint ﬁeld Q∗using adjoint equations de-

ﬁned in Eq. (20), Eq. (21), Eq. (22), and Eq. (23) in the

limit f∗

i→0

3. Compute shape derivative using the deﬁnition from the

equation Eq. (39)

Topological derivative formulation for shape sensitivity in incompressible turbulent ﬂows

NUMERICAL RESULTS

In order to demonstrate the shape derivative computation,

turbulent incompressible ﬂow in a two-dimensional S-shaped

duct is used as an example. The computational mesh is given

in ﬁgure (4). The height of the ﬁrst layer of cells is se-

lected so that the value of y+is within the range of 30 to

50 enabling the use of wall functions for near wall mod-

elling of turbulent ﬁelds. The inlet boundary condition is

speciﬁed as a uniform velocity value equal to 5m/swhile

the outlet pressure boundary condition correspond to zero

gauge pressure condition. Kinematic viscosity is speciﬁed

to be

ν

=1×10−5[m2/s]while the height of the domain is

H=0.1mgiving the Reynolds number of 50,000. Results

of the computation of the prime ﬁeld are given in ﬁgures (5)

and (6).

Output functional that was used to compute shape derivatives

corresponds to dissipated power functional given by the fol-

lowing expression:

J=−ZΓp+1

2||U||2uinidΓ(41)

This functional represents a measure of the dissipated energy

within the duct and this particular form of the functional is

selected so that admissible boundary conditions are possi-

ble for adjoint equations. Shape parameters Piare positions

of centroids of ﬁnite volume faces on all wall boundaries.

Computed shape sensitivity is given in the ﬁgure (7). The

computed sensitivity ﬁeld indicates that in order to decrease

losses based on Eq. (41) the face centroids should be moved

in such way so that duct becomes straight. This is intuitively

correct result since the losses due to total pressure changes

within the duct will be at their local minimum if the duct is

straight. It should be also observed that the shape deriva-

tives computed here are given in their raw form without any

smoothing. Before these derivatives can be used in any gra-

dient based optimisation algorithm, a smoothing procedure

should be applied in order to control the roughness of the re-

sulting new shape. However, this was not the subject of the

current work.

CONCLUSIONS

Shape derivative based on topological arguments was derived

in this paper. The newly proposed way of computing shape

derivatives results in a simple expression involving only the

primal and adjoint ﬁelds. It was also shown that topologi-

cal derivative is consistent with the deﬁnition of the shape

derivatives when source terms in momentum equation are in-

ﬁnitesimally close to the boundary of the domain in the limit

of vanishing source term intensity. An example of the com-

putation of shape derivatives using topological arguments

demonstrates consistency with the classical formulation of

shape derivatives.

REFERENCES

ANDERSON, W. and VENKATAKRISHNAN, V. (1997).

“Aerodynamic design optimization on unstructured grids

with continuous adjoint formulation”. ICASE Report,9.

BUENO-OROVIO, A. et al. (2012). “Continuous adjoint

approach for the spalart-allmaras model in aerodynamic op-

timization”. AIAA Journal,50.

CASTRO, C. et al. (2007). “Systematic continuous ad-

joint approach to viscous aerodynamic design on unstruc-

tured grids”. AIAA Journal,45.

JAMESON, A. et al. (2008). “An unstructured adjoint

method for transonic ﬂow”. AIAA Journal,46.

NADARAJAH, S. and JAMESON, A. (2000). “A com-

parison of the continuous and discrete adjoint approach to

automatic aerodynamic optimization”. AIAA paper. AIAA.

OTHMER, C. (2008). “A continuous adjoint formulation

for the computation of topological and surface sensitivities

of ducted ﬂows”. Numerical Methods in Fluids,58.

ZYMARIS, A. et al. (2010). “Adjoint wall functions: a

new concept for use in aerodynamic shape optimization”.

Journal of Computational Physics,13.

Figure 4: Computational mesh.

Topological derivative formulation for shape sensitivity in incompressible turbulent ﬂows

Figure 5: Pressure ﬁeld.

Figure 6: Magnitude of velocity ﬁeld.

Topological derivative formulation for shape sensitivity in incompressible turbulent ﬂows

Figure 7: Sensitivity Vectors.