Topology Optimization Using the SIMP Methodsites.google.com/site/fabianwein/simp_presentation.pdf · Introduction Optimization General optimization • Objective function (scalar)

Topology Optimization Using the SIMP Method

Fabian Wein

August 2008

August 2008 Topology Optimization Using the SIMP Method

Introduction

Introduction

• About this document• This is a fragment of a talk given interally• Intended for engineers and mathematicians

• SIMP basics• Detailed introduction (based on linear elasticity)• Optimization vs. simulation• Do it yourself with Sigmund’s ”99-line code”

• Piezoelectric Loupspeaker• Extended SIMP model• Results


Introduction

Optimization

General optimization

• Objective function (scalar)

• Design variable

Structural optimization

• Topology design of truss structures

• Shape optimization• Parametrization• Level set metheod (eventually topology gradient)

• Topology optimization• Homogenization• SIMP


Classical SIMP Background

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



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



Discrete FEM formulation

Solve

Global System

Ku = f

with

Assembly

K =ne∧

e=1

Ke; Ke = [kpq]; kpq =∫Ωe

(B)T [c0]BdΩ


Classical SIMP Incredients

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



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)



Find derivative

General optimization procedure

• Evaluate objective function

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

• Find step length along δδδ (line search)

Techniques to find descent direction

• Use gradient free methods

• Use finite differences

• Analytical first derivative

• Analytical second derivative



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



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


Classical SIMP Application of 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



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



Add constraint

Minimize compliance: volume constraint 50%

minρρρ

fTu s.th.:∫Ω

ρρρ ≤ 1

2V0

“Grey” material has no physical interpretation



Add constraint

Minimize compliance: volume constraint 50%

minρρρ

fTu s.th.:∫Ω

ρρρ ≤ 1

2V0

“Grey” material has no physical interpretationAugust 2008 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



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 structureAugust 2008 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



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 dependencyAugust 2008 Topology Optimization Using the SIMP Method

Classical SIMP Optimizers

Comparison of different optimizers

• SCPIP (MMA implementation by Ch. Zillober)

• Optimality Condition (heuristic for SIMP)

• IPOPT (general second order optimizer)



Performance



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


Extensions to SIMP Multiple loads

Complex 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


Extensions to SIMP Multiple loads

Problem specific optimization

Now only the left load is applied to the optimized structures

Figure: The scaling of the displacement is the same


Extensions to SIMP Optimization for arbitrary nodes

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


Extensions to SIMP Optimization for arbitrary nodes

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


Extensions to SIMP Harmonic optimization

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



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



Harmonic tranfer 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



(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


Piezoelectricity Model

Piezoelectric Model

We couple linear elasticity with electrostatic

Material law

[σσσ ] = [cE0 ][S]− [e0]

TE,

D = [e0][S]+ [εεεS0 ]E.

E : electric field intensity in V/mD : electric displacement field C/m2

[σσσ ] : Cauchy stress tensor[S] : linear strain tensor[cE

0 ] : tensor of elastic moduli[cm] : tensor of elastic moduli[εεεS

0 ] : tensor of dielectric permittivities[e0] : tensor of piezoelectric moduli



Strong formulation

Grounded Electrode

Loaded Electrode

Plate

Piezoelectric Material

Support

Strong formulation

Find up : Ωp → R3, um : Ωm → R3, φ : Ωp → Rfulfilling

BT([cE

0 ]Bup +[e0]TBφ

)= 0 in Ωp,

BT

([e0]Bup− [εεεS

0 ]Bφ

)= 0 in Ωp,

BT [cm]Bum = 0 in Ωm



Strong formulation

Boundary conditions

um = 0 on Γs ,

nTp [σσσp] = 0 on ∂Ωp \Γg ,

nTm[σσσm] = 0 on ∂Ωm \ (Γg ∪Γs),

nTp [σσσp] =−nT

m[σσσm] on Γg ,

np =−nm on Γg ,

up = um on Γg ,

φ = 0 on Γg ,

φ = φl on Γl ,

nTp D = 0 on ∂Ωp \ (Γl ∪Γg )



FEM formulation

Bilinearforms

Keuu = [kuu

pq ]; kuupq =

∫Ωe

(Bu

p

)T[cE

0 ]BuqdΩ,

Keuφ = [kuφ

pq ]; kuφpq =

∫Ωe

BTp [e0]BqdΩ,

Keφφ = [kφφ

pq ]; kφφpq =−

∫Ωe

BT

p [εεεS0 ]Bq dΩ.

Global system Kumum Kumup 0KT

umupKupup Kupφ

0 KTupφ

Kφφ

um

up

φφφ

=

0qu

qφ


SIMP optimization of piezoelectric devices Model

Piezoelectric SIMP model

Design variables

• pseudo density ρρρ

• pseudo polarization ρρρp

Extended material tensors

[cEe ] = µc(ρe)[c

E0 ],

[ee ] = µe(ρe)[e0],

[εεεSe ] = µε(ρe)µp(ρ

pe )[εεεS

0 ].

ρe ∈ [ρmin;1] ρpe ∈ [−1;1]

Adjoint PDE for inhomogeneous forward problem

• No ’electric’ excitation but ’load’ from objective

• Gradient reduces with first order elements toJ ′ = wT

e K′euuue +wT

e K′euφ φφφe


SIMP optimization of piezoelectric devices Transduction

Transduction basics

Reciprocal theorem in elasticity∫Γta

tTa ubdΓ =∫Γtb

tTb uadΓ

“. . . by knowing the body response for one load case, we can calculate thedisplacement at any point of the body caused by another load case.”

Extension to piezoelectricity promisses (Kogl and Silva, 2005):

“. . . the conversion of electrical into elastic energy and vice versa.”



Transduction basics

Reciprocal theorem in elasticity∫Γta

tTa ubdΓ =∫Γtb

tTb uadΓ

“. . . by knowing the body response for one load case, we can calculate thedisplacement at any point of the body caused by another load case.”

Extension to piezoelectricity promisses (Kogl and Silva, 2005):

“. . . the conversion of electrical into elastic energy and vice versa.”



Transduction in piezoelectricity

• Loadcase a: fa 6= 0 Qa = 0• Loadcase b: fb = 0 Qb 6= 0• Loadcase c : fc 6= 0 grounded electrodes

Extension to piezoelectricity

Lab = φφφTb KT

uφua +φφφTa Kφφ φφφb

L′ab = −uTa

K′ub.

Difference to Kogl and Silva (2005)

• We have fixed supporting mechanical plate• Kogl and Silva have loadcase c and volume constraint

Original objective

J(ρρρ) = w lnLab− (1−w) lnLcc , 0≤ w ≤ 1,



Results of mean transduction

The result is the trivial result “vanishing material”

• We excite with fixed charge and fixed nodal force

• Loadcase a (Qa = 0): minimal stiffening (u) and maximal bending (φ)

• Loadcase b (fb = 0): surface charge density and E tend to infinity

5.0e-006

1.0e-005

1.5e-005

2.0e-005

2.5e-005

3.0e-005

3.5e-005

4.0e-005

4.5e-005

5.0e-005

10 20 30 40 50 60 70 80 90 100

Dis

plac

emen

t (m

)

Area void/piezo (%)

mechanical excitationelectric excitation

1.0e-001

1.0e+000

1.0e+001

1.0e+002

1.0e+003

1.0e+004

10 20 30 40 50 60 70 80 90 100

Vol

tage

(V

)

Area void/piezo (%)

mechanical excitationelectric excitation

Figure: Parameter study with varying area covered by piezoelectric material


SIMP optimization of piezoelectric devices Results

Optimize for maximum displacement of ground plate

Thickness of layers

• Piezoelectric layer: 50 µm

• Supporting layer: 10, 20, 50, 100, 200, 500 µm

• No volume constraint and no filtering required!


SIMP optimization of piezoelectric devices Results

Locking effects

• Very thin elements• Shear locking occures when using linear finite elements• Straight forward extension of incompatible modes by SIMP

Solve local system of higher order(Ke

uu Keuα

Keαu Ke

αα

)(ue

αααe

)=

(fe0

)

0.0e+000

5.0e-005

1.0e-004

1.5e-004

2.0e-004

2.5e-004

3.0e-004

5 10 15 20 25 30 35 40

Dis

plac

emen

t

Discretization of edge

linear, with lockinglinear, locking free

quadratic

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

5 10 15 20 25 30 35 40

Vol

ume

frac

tion

Discretization of edge

linear, with lockinglinear, locking free

quadratic


End

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

• Thanks for your time!


Documents

Topology Optimization Using the SIMP Methodsites.google.com/site/fabianwein/simp_presentation.pdf · Introduction Optimization General optimization • Objective function (scalar)