Topology optimization

Eddie Wadbro

Introduction to PDE Constrained Optimization, 2016

February 15–16, 2016

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (1 : 15)

Sample problems

Acoustics

Planar horn(Electro-Voice)

Musicalinstrument

Structural mechanics

I-beam Truss

I Horn loudspeakers, usually forauditoriums or outdoor use: highefficiency, directs the sound energy

I Horns comes in various shapes:cylindrical, planar, rectangular. . .

I Structural elements designedto withstand loads

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (2 : 15)

How does an acoustic horn work?Input

P(x , t) = Aeikxe−iωt

Output

P(x , t) = Be−ikxe−iωt

0

30

60

90

120

150

180

210

240

270

300

330

−30

−24

−18

−12

−6

0

p1(µ )

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (3 : 15)

Material distribution (Topology optimization)I Placement of sound hard material arbitrarily in region ΩD

I Material distribution function α, ρ constant in each element

ΩD

f

ΩD

I α = 0 if solid and 1 if fluidI Want to solve:

minα

J(α) (reflections)

s.t. α(1 − α) = 0 a.e.governing PDE

I ρ = 0 if void and 1 if solidI Want to solve:

minρ

J(ρ) (compliance)

s.t. ρ(1 − ρ) = 0 a.e.∫ρ ≤ V

governing PDEEddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (4 : 15)

Numerical algorithms

I Gradients obtained by solving adjoint equations

I Acoustic problem: Formulate optimization problem as constrainedleast squares problems

I Utilize the structure: provide gradients for each residual term (notjust one gradient of full objective function)

I Optimization algorithms:I Svanberg’s MMA (Method of Moving Asymptotes)I OC (Optimality Criterion based algorithm)

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (5 : 15)

Relaxation

minα

J(α)

s.t. 0 < ε ≤ α ≤ 1governing PDE

minρ

J(ρ)

s.t. 0 < ε ≤ ρ ≤ 1∫ρ ≤ V

governing PDE

Theorem For the linear elasticity (that is, the beam) case, there exists asolution to the relaxed minimal compliance problemEddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (6 : 15)

Penalization

minα

J(α) + γ

∫(α− ε)(1 − α)

s.t. 0 < ε ≤ α ≤ 1governing PDE

minρ

J(ρp)

s.t. 0 < ε ≤ ρ ≤ 1∫ρ ≤ V

governing PDE

Theorem The problem is ill-posed (it lacks solutions within the set offeasible designs)Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (7 : 15)

Penalization

minα

J(α) + γ

∫(α− ε)(1 − α)

s.t. 0 < ε ≤ α ≤ 1governing PDE

minρ

J(ρp)

s.t. 0 < ε ≤ ρ ≤ 1∫ρ ≤ V

governing PDE

Theorem The problem is ill-posed (it lacks solutions within the set offeasible designs)Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (8 : 15)

Problems and features

I The penalty function is not closed with respect to the minimizingsequence ρn∞n=1 of `(u)

I The admissible designs represent isotropic (same properties in eachdirection) material with density ρ ∈ L∞(Ω)

I Each of the discrete problems is well posedI The compliance decreases with refinementI The desings shows more and more detailsI The numerics indicate that material with microstructures possessing

anisotrppic material behavior is advantageousI That is, the optimization tries to (in some uncontrolled sense) create

composite materials

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (9 : 15)

Problems and features

There are two ways to fix the problem: relaxation and restriction

I Relaxation: To obtain a closed set of admissible designs, extend theset to allow a composite material with mictrostructure at each point

I Homogenization theory can be used to determine a macto leveleffective stiffness and density of the composite material.

I Restriction: Introduce a parameter to limit the amount of detailallowed in the structure

I Limit the oscillations by imposing a constraint ‖∇ρ‖Lp(Ω) ≤ C ,1 ≤ p ≤ ∞

I Introduce a filter. There are many variations of this idea, below onlyone of these will be presented.

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (10 : 15)

Filtered densities

Introduce the filter radius τ and define the density by

ρ(x) =

∫ΩK (x , y)ρ(y) dy ,

where, for instance,

K (x , y) = σ(x)max

1 − |x − y |τ

, 0,

with σ(x) selected such that∫ΩK (x , y) dy = 1 for all x ∈ Ω.

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (11 : 15)

Filtered densities

ρ(x) =

∫ΩK (x , y)ρ(y) dy .

I Can be written as the convolution ρ = K ∗ ρI ρ is the physical density in the state equationI ρ is the design variableI τ is a measure on the levels of detail

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (12 : 15)

Filtering

minα

J(K ∗ α)+

γ

∫(K ∗ α− ε)(1 − K ∗ α)

s.t. 0 < ε ≤ α ≤ 1governing PDE

minρ

J((K ∗ ρ)p

)s.t. 0 < ε ≤ ρ ≤ 1∫

ρ ≤ V

governing PDE

Theorem For the linear elasticity (that is, the beam) case, there exists asolution to the relaxed minimal compliance problemEddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (13 : 15)

Sharpening/Postprocessing

minα

J(K ∗ α)+

γ

∫(K ∗ α− ε)(1 − K ∗ α)

s.t. 0 < ε ≤ α ≤ 1governing PDE

minρ

J((K ∗ ρ)p

)s.t. 0 < ε ≤ ρ ≤ 1∫

ρ ≤ V

governing PDE

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (14 : 15)

Non-linear filtering

By using a more elaborate filtering method, the following design can beobtained without any use of sharpening or postprocessing.

Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15–16, 2016 (15 : 15)