Topology Optimization Using the SIMP Method

Fabian Wein

Introductary Talk @ LSE29.10.2008

Fabian Wein Topology Optimization Using the SIMP Method

Optimization vs. Optimization

• Common claim

Engineers improve a system and call this ”optimizing”.But the optimum can only be found with optimization

methods.

• Modelling optimization problems is nontrivial• Design space (dimensions, topology, material, . . . )• Multiple criterions

• Different optimization methods

• Optimization results are guidelines for designers

Fabian Wein Topology Optimization Using the SIMP Method

Basic Optimization Problem

• Design vector x (e.g. dimensions, topology, shape, material)

• Problem

minx

J (x)

subject to

equality constraints

inequality constraints

box constraints

• Objective function J (x) 7→ R

Fabian Wein Topology Optimization Using the SIMP Method

Ingredients for the Optimization Problem

• Parametrization

• Iteration xk+1 = xk + td• starting point/ initial guess x0

• descent direction• step length• stopping criteria, optimality criteria

• Problems• existence• uniqueness• convergence• local optima

Fabian Wein Topology Optimization Using the SIMP Method

Optimization Approaches

• Gradient-free algorithms• stochastic algorithms (particle swarm optimization)• genetic algorithms• . . .

• Deterministic algorithms/ find descent directions• finite differences• automatic differentiation• sensitivity analysis

• Optimization domain• parameter optimization• shape optimization• topology optimization

Fabian Wein Topology Optimization Using the SIMP Method

Linear elasticity

Hooke’s law

[σσσ ] = [c0][S] (in Voigt notation: σσσ = [c0]Bu)

with

• [σσσ ],σσσ : Cauchy stress tensor

• [c0] : tensor of elastic modului

• [S],S : linear strain tensor

• u : displacement

• B =

∂

∂x 0 0 0 ∂

∂z∂

∂y

0 ∂

∂y 0 ∂

∂z 0 ∂

∂x

0 0 ∂

∂z∂

∂y∂

∂x 0

T

: differential operator

Fabian Wein Topology Optimization Using the SIMP Method

Strong Formulation

PDE

Find

u : Ω→ R3

fulfilling

BT [c0]Bu = f in Ω

with the boundary conditions

u = 0 on Γs

nT[σσσ ] = 0 on ∂ΩΓs

Fabian Wein Topology Optimization Using the SIMP Method

Discrete FEM Formulation

Solve

Global System

Ku = f

with

Assembly

K =ne∧

e=1

Ke; Ke = [kpq]; kpq =∫Ωe

(B)T [c0]BdΩ

Fabian Wein Topology Optimization Using the SIMP Method

Proportional Stiffness Model

Parametrization by design variable

• Model structure by local stiffness (full and void).

• Define local stiffness (finite) element wise: ρρρ = (ρ1 · · · ρne )T

• Continuous interpolation with ρmin ≤ ρe ≤ 1.

Introduce pseudo density ρρρ

[ce](ρρρ) = ρe [c0]; Ke(ρρρ) = ρeKe; K(ρρρ)u(ρρρ) = f

Fabian Wein Topology Optimization Using the SIMP Method

Minimal Mean Compliance

Different interpretations• Maximize stiffness• Minimize mean compliance• Minimize stored mechanical energy

Minimize compliance

minρρρ

J(u(ρρρ)) = minρρρ

fTu(ρρρ) = minρρρ

u(ρρρ)TK(ρρρ)u(ρρρ)

Fabian Wein Topology Optimization Using the SIMP Method

Find Derivative

General optimization procedure

• Evaluate objective function

• Find descent direction δδδ (e.g.gradient)

• Find step length along δδδ (linesearch)

Techniques to find descent direction

• Use gradient free methods

• Use finite differences

• Analytical first derivative

• Analytical second derivative

Fabian Wein Topology Optimization Using the SIMP Method

Sensitvity Analysis

• Sensitivity analysis provides analytical derivatives

• Abbreviate ∂(·)∂ρe

by (·)′

Derive mean compliance fTu

J ′ = f ′Tu+ fTu′ = fTu′

Find J ′ by deriving state condition Ku = f

Solve for every u′

Ku′ =−K′u

Fabian Wein Topology Optimization Using the SIMP Method

Adjoint Method

The adjoint method is based on the fixed vector λλλ

J = fTu+λλλT(Ku− f)

J ′ = fTu′+λλλT(K′u+ Ku′)

= (fT +λλλTK)u′+λλλ

TK′u

Solve: Kλλλ = −f =∂J

∂u

J ′ = −uTK′u

• The compliance problem is self-adjoint

• The general adjoint problem can be efficiently solved by(incomplete) LU decomposition

Fabian Wein Topology Optimization Using the SIMP Method

Naive Approach

Minimize compliance: straight forward, initial design 0.5

minρρρ

fTu s.th.: Ku = f ρe ∈ [ρmin : 1] note: Ke = ρeKe, K′e = Ke

The optimal topology is the trivial solution full material

Fabian Wein Topology Optimization Using the SIMP Method

Naive Approach

Minimize compliance: straight forward, initial design 0.5

minρρρ

fTu s.th.: Ku = f ρe ∈ [ρmin : 1] note: Ke = ρeKe, K′e = Ke

The optimal topology is the trivial solution full material

Fabian Wein Topology Optimization Using the SIMP Method

Add Constraint

Minimize compliance: volume constraint 50%

minρρρ

fTu s.th.:∫Ω

ρρρ ≤ 1

2V0

“Grey” material has no physical interpretation

Fabian Wein Topology Optimization Using the SIMP Method

Add Constraint

Minimize compliance: volume constraint 50%

minρρρ

fTu s.th.:∫Ω

ρρρ ≤ 1

2V0

“Grey” material has no physical interpretation

Fabian Wein Topology Optimization Using the SIMP Method

Third Try

Minimize compliance: penalize ρρρ by ρρρp with p = 3

minρρρ

fTu note: Ke = ρ3eKe, K

′e = 3ρ

2eKe

We have a desired 0-1 pattern but checkerboard structure

Fabian Wein Topology Optimization Using the SIMP Method

Third Try

Minimize compliance: penalize ρρρ by ρρρp with p = 3

minρρρ

fTu note: Ke = ρ3eKe, K

′e = 3ρ

2eKe

We have a desired 0-1 pattern but checkerboard structure

Fabian Wein Topology Optimization Using the SIMP Method

Forth Try

Minimize compliance: use averaged gradients

minρρρ

fTu note: K′e =

∑iHiρiρe

3ρ2eKe

∑iHiwith Hi = rmin−dist(e, i)

No checkerboards and no mesh dependency (view movie)

Fabian Wein Topology Optimization Using the SIMP Method

Forth Try

Minimize compliance: use averaged gradients

minρρρ

fTu note: K′e =

∑iHiρiρe

3ρ2eKe

∑iHiwith Hi = rmin−dist(e, i)

No checkerboards and no mesh dependency (view movie)Fabian Wein Topology Optimization Using the SIMP Method

Comparison of Different Optimizers

• SCPIP (MMA implementation by Ch. Zillober)

• Optimality Condition (heuristic for SIMP)

• IPOPT (general second order optimizer)

Fabian Wein Topology Optimization Using the SIMP Method

Performance

Fabian Wein Topology Optimization Using the SIMP Method

Optimality Condition

Optimality Condition: fix-point type update scheme

ρek+1=

max(1−ζ )ρek

,ρmin if ρekBη

ek≤max(1−η)ρek

,ρminmin(1+ζ )ρek

,1 if min(1+ζ )ρek,1 ≤ ρek

Bηek

ρekBη

ekelse

With

• Bek= Λ−1K

′e

• Λ to be found by bisection

• Step width ζ e.g. 0.2

• Damping η e.g. 0.5

Fabian Wein Topology Optimization Using the SIMP Method

Combined Load vs. Multiple Load Cases

For multiple loadcases several problems are averaged

Figure: Two loads applied simultaniously (left) and seperatly (right)

The left case is instable if the loads are not applied simultaniously

Fabian Wein Topology Optimization Using the SIMP Method

Problem Specific Optimization

Now only the left load is applied to the optimized structures

Figure: The scaling of the displacement is the same

Fabian Wein Topology Optimization Using the SIMP Method

Synthesis of Compliant Mechanisms - aka ”no title”

Generalizing the compliance to J = lTu with l = (0 · · · 0 1 0 · · ·)T.

For this case one has to apply springs to the load and output nodes

Fabian Wein Topology Optimization Using the SIMP Method

Synthesis of Compliant Mechanisms - aka ”no title”

Generalizing the compliance to J = lTu with l = (0 · · · 0 1 0 · · ·)T.

For this case one has to apply springs to the load and output nodes

Fabian Wein Topology Optimization Using the SIMP Method

Harmonic Optimization

Two common approaches

• Optimize for eigenvalues

• Perform SIMP with forced vibrations

Harmonic excitation

• Excite with a single frequency

• Gain steady-state solution in one step

• Complex numbers

Complex FEM system

(K+ jωC−ω2M)u = f

S(ω)u = f ST

= S

Fabian Wein Topology Optimization Using the SIMP Method

Harmonic Objective Functions: J(u(ρρρ))→ R

Compliance

J = |uT f| J ′ =−R(sign(J)uT S′u)

J = (uT f)2 J ′ =−2(uT f)uT S′u

J = uTRfI−uT

I fR J ′ = 2R(λλλT S′u) Sλλλ =− j

2f

J = uT u J ′ = 2R(λλλT S′u) Sλλλ =−u

Optimize for output

J = uTLu J ′ = 2R(λλλT S′u) Sλλλ =−LTu

• Optimize for velocity• Optimize for coupled quantities

Fabian Wein Topology Optimization Using the SIMP Method

Harmonic Interpolation Functions

Classical SIMP converges faster than mass to zero

µPedersen(ρe) =

ρ3e if ρ > 0.1

ρe

100if ρ ≤ 0.1

µRAMP(ρe) =ρe

1+q(1−ρe)

0e+000

2e-001

4e-001

6e-001

8e-001

1e+000

0 0.2 0.4 0.6 0.8 1

Con

trib

utio

n

Design variable

SIMPidendity

RAMP

Fabian Wein Topology Optimization Using the SIMP Method

(Global) Dynamic Compliance

We optimize for uT u and uTLu with L selecting f

This illustrates general optimization problems

• One has to know what one wants

• One might not want what one gets

Fabian Wein Topology Optimization Using the SIMP Method

Dynamic Compliance cont.

Do it again for a higher frequency (80 Hz) We optimize for uT uand uTLu with L selecting f

(a) uTu (b) uTLu

Note: the second example did not converge and is stopped after300 iterations. Movie

Fabian Wein Topology Optimization Using the SIMP Method

Thee Dimensions

(a) Single load (b) Surface load

Problems

• Requires iterative solvers (ILUPACK)

• Difficult to visualize (greyness!)

• Neither GiD not GMV are optimal

Fabian Wein Topology Optimization Using the SIMP Method

Volume Constraints

Volume constraint

• Avoids trivial solution (full or void)

• Removes greyness by penalization

Choice of volume constraint

• Engineering requirements (weight, price)

• Well optimization behaviour

• Nice pictures

Fabian Wein Topology Optimization Using the SIMP Method

Multicriterial optimization

Constrained Optimization

• Mathematically objective function and constraints are similar

• Consider the volume constraint as design variable

Multicriterial optimization

• Pareto efficiency:No criterion can be improved without worse another one.

• Solution is a Pareto front

• Requires user choice

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Com

plia

nce

Volume fraction

Compliance

Fabian Wein Topology Optimization Using the SIMP Method

Hypothesis

Heretical Hypothesis

A volume constraint can remove greyness only forsolutions 6= global solution.

Fabian Wein Topology Optimization Using the SIMP Method

Initial Guess

Initial guess

• Homogeneous intermediate material

• Chosen to match volume constraint

• Mathematical feasible, physcial unfeasible

• 6= traditional engineering solution

• Impressive objective over iterations charts

Fabian Wein Topology Optimization Using the SIMP Method

The End

Last comments

• The optimal solution lays inside the PDE (plus adjoint RHS)

• Optimization helps to understand systems better

• Optimization is the next step after simulation

• Ole Sigmund: A 99 Line Topology Optimization Code writtenin MATLAB; 2001

• Thanks for your time!

Fabian Wein Topology Optimization Using the SIMP Method