Shape optimization in ﬂuid mechanics with
eigenvalue constraints
Vincent Heuveline Frank Strauß
Numerical Methods in High Performance Computing
Steinbuch Center for Computing
Karlsruhe Institute of Technology
Outline
Introduction
Numerical scheme
Extension of optimization scheme
Numerical results
Outlook
Motivation
Goal:Minimization of certain objective functional by modifying
shape of underlying geometry
Application:Hydrodynamic stability control in CFD
→Control stability by optimization of underlying geometry
Stability represented by eigenvalues
Important issues
Formulation and numerical solution of physical system
Representation of geometry
Numerical solution of optimization problem
Model problem
Benchmark ﬂow around body in channel
→Optimization of shape of body to reduce drag and ensure
hydrodynamic stability
min
q
J(q) = F
D
(q)
s.t.
Re(λ
min
(A(q))) ≥ 0
V(q) = V
∗
q
≤ q ≤ ¯q
Physical model
Flow governed by stationary NavierStokes equations describing
viscous incompressible Newtonian ﬂuid ﬂow
Ω ⊂ R
2
,ν kinematic viscosity,(ρ = 1)
−νΔˆv +ˆv ∇ˆv +∇ˆp = f in Ω
∇ ˆv = 0 in Ω
Boundary conditions:
Noslip condition along rigid parts:ˆv
Γ
rigid
= 0
Inﬂow condition:ˆv
Γ
in
= ˆv
in
Freestream outﬂow condition:νδ
n
ˆv −ˆpn
Γ
out
= 0
Drag force:F
D
=
S
ρν
∂u
t
∂n
n
y
−pn
x
dS
Linear stability
Linearization about ˆu = {ˆv,ˆp},
v
(x,t) = ˆv(x) +v(x,t),p
(x,t) = ˆp(x) +p(x,t)
∂v
∂t
−νΔv +ˆv ∇v +v ∇ˆv +v ∇v +∇p = 0
∇ v = 0
A(ˆu)(u) = −νΔv +ˆv ∇v +v ∇ˆv +∇p = λv
∇ v = 0
Operator A is non selfadjoint!
If Re(λ
min
(A)) < 0,base solution ˆv is unstable
Parameterization of body
Parameterization by spline function consisting of cubic B´ezier
curves (on [t
0
,t
2
])
x(t) =
3
j=0
b
3i +j
B
3
j
t −t
i
t
i +1
−t
i
t ∈ [t
i
,t
i +1
] i = 0,1.
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
Length
Height
b
0
b
1
b
2
b
3
b
4
b
5
b
6
1
0.5
0
0.5
1
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Length
Height
Design variable vector q = (l,h,m,c
l
,c
h
),where b
3
= (c
l
l,c
h
h).
Solution process for hydrodynamic and eigenvalue problem
Forward problem
Discretization:FEM,TaylorHood Q
2
/Q
1
Fully parallel multigrid method
Eigenvalue problem
Highly tedious task since non selfadjoint
Davidson method coupled with multigrid method
Goaloriented a posteriori estimation for both linearization
error and eigenvalue discretization error
www.hiﬂow.de
Solution of optimization problem
Solution of optimization problem by SQP method with BFGS
update
active set strategy
derivatives by ﬁnite diﬀerences
Lagrange function:L(q,µ) = J(q) −µλ
min
(q)
→KKT system in iteration step k
∇
2
qq
L
(k)
−(∇
q
λ
(k)
min
)
T
∇
q
λ
(k)
min
0
p
(k)
µ
(k+1)
=
−∇
q
J
(k)
−λ
(k)
min
.
Complete algorithm
Step 1.Determine initial design variables q
(0)
= (q
1
,...,q
n
),
set k:= 0.
Step 2.Create mesh depending on design variable vector q
(k)
.
Step 3.Solve ﬂow problem to obtain drag force F
D
(q
(k)
).
Step 4.Solve eigenvalue problem to obtain Re(λ
min
(q
(k)
)).
Step 5.Determine derivatives of F
D
(q
(k)
) and Re(λ
min
(q
(k)
))
with respect to design variables by ﬁnite diﬀerences.
Step 6.Solve KKT system of the SQP algorithm to obtain
direction p
(k)
.
Step 7.Determine suitable step length τ
(k)
.
Step 8.Update design variable vector
q
(k+1)
= q
(k)
+τ
(k)
p
(k)
.
Step 9.If termination criterion is reached then STOP else set
k:= k +1 and goto Step 2.
Extension
Diﬃculties
Function evaluation expensive
Eigenvalue function may not be diﬀerentiable
Only local optimum may be found
→Modiﬁcation of solution algorithm
Original model is replaced by metamodel
Derivativefree optimization method
Metamodel by radial basis functions (RBF)
RBF approximation for drag and smallest eigenvalue.
Training set:N points x
i
∈ R
5
˜
f (x) =
N
i =1
c
i
Φ(kx −x
i
k),
where Φ(x) is the Gaussian radial basis function
Φ(x) = e
−x
2
/a
2
Coeﬃcients c
i
are determined by
Ac = F,
where A
ji
= Φ(x
j
−x
i
) and F
i
= f (x
i
),i,j = 1,...,N.
Optimization of attenuation factor a
Testing set of n additional points (ˆx
j
)
j=1,...,n
Evaluation of error e
f
(a) between the value of the original function
f and the approximation
˜
f for diﬀerent attenuation factors,i.e.
e
f
(a) = kf (ˆx
j
) −
˜
f (ˆx
j
;a)k j = 1,...,n
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a
eFD(a)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
2
3
4
5
6
7
8
9
10
11
x 10
3
a
e
(a)
→Further computations with a
f
= 0.7 and a
λ
= 0.9.
Particle swarm optimization
Semistochastic derivativefree optimization method
Mimics social behaviour of bird ﬂocking
Trajectory of swarm of birds (=particles) is followed
Position of particle x
i
is changed by velocity vector v
x
k+1
i
= x
k
i
+v
k
i
Velocity vector is determined by
velocity in previous iteration step v
k−1
i
best position x
∗
i
of particle i
best ever reached position x
∗
v
k
i
= ω
k
v
k−1
i
+c
1
r
1
(x
∗
i
−x
k
i
) +c
2
r
2
(x
∗
−x
k
i
).
r
1
,r
2
random numbers
Initial design
Data for benchmark conﬁguration (u
in
= 0.195 m/s)
F
D
= 6.9961,Re(λ
min
) = −0.0199
→instable behaviour!
Optimized design
F
D
= 5.360,λ
min
= 0.0778
Reduction of drag by 23.3%
Transition from instable design to stable design
[V.Heuveline,F.S.,08]
Comparison of algorithm
Comparison of
30 iterations of gradientbased method
30 iterations of two realizations of PSO (64 particles)
0
5
10
15
20
25
30
5.2
5.4
5.6
5.8
6
6.2
6.4
6.6
6.8
Iteration number
Drag
PSO1
PSO2
GS
PSO is suited for parallelization.
→Signiﬁcant speedup possible.
Hybrid approach
Combination of evaluations of metamodel and original model
Combination of derivativefree and gradientbased
optimization algorithm
Idea:Determine good starting point for gradientbased algorithm
0
5
10
15
20
25
30
5.2
5.4
5.6
5.8
6
6.2
6.4
6.6
6.8
Iteration number
Drag
PSO+GS
GS
Summary and outlook
Summary
Numerical scheme for shape optimization problem in
hydrodynamics with eigenvalue constraints
Transition from instable to stable design possible by
modiﬁcation of geometry
Timeconsuming computations require extension of numerical
scheme
Outlook
Parallelization of optimization scheme
Multiobjective optimization
Thank you very much for the attention
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