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# TOPOLOGICAL DERIVATIVE FORMULATION FOR SHAPE SENSITIVITY IN INCOMPRESSIBLE TURBULENT FLOWS

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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 derivative formulation. Shape derivative is based on topological derivative 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 computational example of the flow in curved duct is used for the illustration of the derivative computations.
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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
Computational 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
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 (1)
B(Q) = 0 on Γ=ΓbΓo
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
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
. 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 Qplays the role of Lagrangian
multipliers whereas integral Jrepresents the quantity that
is being optimized.
L=J+Zpu
ik
ε
A1
A2
A3
A4
d(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
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
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
=
φ
,C
φ
+BΓ(11)
or in explicit integral form
Z
φ
C
φ
d=Z
φ
C
φ
d+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+ZQ
δ
+
δ
uiJ+ZQ
δ
+
δ
kJ+ZQ
δ
+
δε
J+ZQ
δε
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
Qneeded 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
kc1G
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)
Z(
pJ+A
1)
δ
Ppd+ZΓ(
pJΓ+B1)
δ
PpdΓ
+Z(
uiJ+A
2)
δ
Puid+ZΓ(
uiJΓ+B2)
δ
PuidΓ
+Z(
kJ+A
3)
δ
Pkd+ZΓ(
kJΓ+B3)
δ
kPdΓ
+Z(
ε
J+A
4)
δ
P
ε
d
+ZΓ(
ε
JΓ+B4)
δ
P
ε
dΓ=0 (18)
Where functional Jwas split into boundary and interior
contributions
J=JΓ+J(19)
Requiring that each term in Eq. (18) is equal to zero in the
domain we deﬁne the set of adjoint equations:
A
1=
iu
i+
pJ=0 (20)
A
2=uj
ju
iuj
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+
uiJ=0,(21)
A
3=uj
jk
j(Dk
jk) + c
µ
k
jk
jkc
µ
kGk
+c
µ
k
j
ε
j
ε
+ck
ε
+c
µ
k(
jui+
iuj)
ju
i
+
kJ=0,(22)
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
Gk+kc
µε
(
jui+
iuj)
ju
i
+
ε
J=0,(23)
c
µ
k=2c
µ
k
ε
,
ck=c1c
µ
Gc2
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
Applying the rules of differentiation
δ
δ
δ
Variation of adjoint variables is second order contribution
and will be ignored
δ
δ
We can now compute the derivative
δ
ik
ε
δ
PA1
δ
PA2
δ
PA3
δ
PA4
d
(28)
If the forcing term in momentum equation is given by
fi=Pui
η
(29)
then the derivative becomes
δ
ik
ε
0
ui
η
0
0
d
(30)
Total variation of the Lagrangian is
δ
L=
δ
PJ+Z
δ
Since there is no explicit dependence of Jon P, direc-
tional derivativeat each cell becomes
δ
PLj=uiu
i
η
j.(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 . There-
fore, we can consider the source term distribution deﬁned
very close to the boundary Γ0that is being modiﬁed.
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
juiui)(u
i+
ξ
j
ju
iu
i)
η
(33)
ui=uiuif,u
i=uiu
if(34)
fi
Γo
ξ
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
juiui)(nj
ju
iu
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
ij
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 fi0
2. Compute adjoint ﬁeld Qusing adjoint equations de-
ﬁned in Eq. (20), Eq. (21), Eq. (22), and Eq. (23) in the
limit f
i0
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
=1×105[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.
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CASTRO, C. et al. (2007). “Systematic continuous ad-
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JAMESON, A. et al. (2008). “An unstructured adjoint
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OTHMER, C. (2008). “A continuous adjoint formulation
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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.