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 Navier-Stokes 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:

No-slip condition along rigid parts:ˆv|

Γ

rigid

= 0

Inﬂow condition:ˆv|

Γ

in

= ˆv

in

Free-stream 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 self-adjoint!

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,Taylor-Hood Q

2

/Q

1

Fully parallel multigrid method

Eigenvalue problem

Highly tedious task since non self-adjoint

Davidson method coupled with multigrid method

Goal-oriented 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

Derivative-free 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

Semi-stochastic derivative-free 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 gradient-based 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 speed-up possible.

Hybrid approach

Combination of evaluations of metamodel and original model

Combination of derivative-free and gradient-based

optimization algorithm

Idea:Determine good starting point for gradient-based 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

Time-consuming computations require extension of numerical

scheme

Outlook

Parallelization of optimization scheme

Multiobjective optimization

Thank you very much for the attention

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