Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Self-Penalization in Topology Optimization
Fabian Wein
extended version of the talk held atEUROMECH522
October 10th-12th, 2011
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Introduction
Anecdote
• starting piezoelectric topology optimization during my PhDthesis maximizing mean transduction
• w/o min compliance term• w/o volume constraint• → vanishing optimal design
• vanishing design as validity test for problem formulation
• 10 maximization problems with linear interpolation w/ovolume constraint
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Compliance Minimization
• solid material is the trivial optimal solution
• volume constraint solution is known to be gray (VTS)
• implicit penalization by power law and volume constraintBendsøe; 1989
• regularization Sigmund, Petersson; 1998 . . . .• ill-posedness• checkerboards• mesh dependency
• most regularization approaches enforce some grayness
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Definition
• ersatz material topology optimization problems
Definition of self-penalization
• linear continuous design variable
• only box constraints on the design variable
• sufficiently distinct black and white solution
• term ’self-penalization’ suggested by Ole Sigmund incommunication at WCSMO-08
• !!! self-penalization is an effect/ phenomenon and not amethod !!!
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
References
• observable since mechanism design Sigmund; 1997
• mentioned for wave guiding Sigmund, Jensen; 2003
• proof extremal piezoelectric polarization Donoso, Bellido;
2008/2009
• short discussion in piezoelectricity Rupp; 2009
• remarks at multiphysics talks at ECCM-2010
• piezoelectric self-penalization W. et al.; 2011
• . . . ?
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Motivation
• standard penalization based on volume constraint→ fails when volume constraint is not active
• cost for volume constraint, regularization, penalization?
• actual weight for some applications of secondary interest
• self-penalization occurs→ why?
• early and basic considerations only!
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Examples
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Piezoelectric Topology Optimization Silva et al.; 1997
• mechanical-electrical coupling
• elec. energy → displ. (actuator)
• straining → electric field (sensor)
• ersatz material: ρ [cE ], ρ [e ], ρ [εS ] a) T > T
Pb ZrO
−+
2+ 4+2−
3
cb) T < T
c
Constitutive equations and FEM
σ = [cE ]S− [e ]TE
D = [e ]S + [εS ]E
→(
Suu(ρ) Kuφ (ρ)Kuφ (ρ)T −Kφφ (ρ)
)(uφ
)=
(fq
)
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Gedankenexperiment (Actuator)
• single design variable
• apply ρ separately on [cE ], [e ] and [εS ]
• contradicting effects of ρ → optimal ρ∗
• grayness only for ρmin < ρ∗ < ρmax
σ = [cE ]S− [e ]TE
D = [e ]S + [εS ]E
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 0.2 0.4 0.6 0.8 1 1.2 1.4
dis
pla
ce
me
nt
in m
m
pseudo density
mechelec
mech+coupl+eleccoupling
• ρ∗ > ρmax = 1 → no grayness but solid as optimal design
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Gedankenexperiment (Sensor)
• three nonlinear effects on[cE ], [e ] and [εS ] σ = [cE ]S− [e ]TE
D = [e ]S + [εS ]E
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 0.2 0.4 0.6 0.8 1 1.2 1.4
ele
c.
po
ten
tia
l in
V
pseudo density
mech+coupl+elecelec
mechcoupling
• ρ∗→ ρmin : no grayness but void as optimal design
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Boundness
• model is arbitrary choice of geometry, thickness and material
• actuator: ρ∗ > 1 but ρ∗→ ∞ ?
10-6
10-5
10-4
10-3
10-2
10-1
100
10-4
10-2
100
102
104
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
dis
pla
ce
me
nt
in m
m
ele
c.
po
ten
tia
l in
V
pseudo density
actuator, displacement sensor, elec. potential
• actuator: ρ∗ bounded; grayness may occur
• sensor: ρ∗→ ρmin; no grayness expected
• only static problems maxuz and maxφ
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
General Static Elastic Problem
Static problem
minJ(u(ρ));∂ J
∂ρe= λT
e
∂Ke
∂ρeue ; KTλ =−∂ J
∂u
Optimal grayness ρ∗ 6∈ {ρmin;ρmax}
〈λe ,∂Ke
∂ρeue〉= 0
⇔ 〈Bλe , [c ]B ue〉= 0
⇔ 〈Sλe , [c ]Sue 〉= 0
⇔ 〈Sλe ,σue 〉= 0
Conditions
‖ue‖ = 0 (1)
‖λe‖ = 0 (2)
‖Sue‖ = 0 (3)
‖Sλe‖ = 0 (4)
Sλe ⊥ σue (5)
• Sλe ⊥ σue meant for vectors ‖Sλe‖> 0 and ‖σue‖> 0
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Again because it is easy but important!
Static problem
minJ(u(ρ));∂ J
∂ρe= λT
e
∂Ke
∂ρeue ; KTλ =−∂ J
∂u
Optimal grayness ρ∗ 6∈ {ρmin;ρmax}
〈λe ,∂Ke
∂ρeue〉= 0
⇔ 〈Bλe , [c ]B ue〉= 0
⇔ 〈Sλe , [c ]Sue 〉= 0
⇔ 〈Sλe ,σue 〉= 0
Conditions
‖ue‖ = 0 (6)
‖λe‖ = 0 (7)
‖Sue‖ = 0 (8)
‖Sλe‖ = 0 (9)
Sλe ⊥ σue (10)
• Sλe ⊥ σue meant for vectors ‖Sλe‖> 0 and ‖σue‖> 0
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Force Inverter
• mechanism design example Sigmund; 1997
• ρmin = 0.001 chosen carefully
• some “free” gray regions
kin kout
fin uout
?
(a) density (b) forward (c) adjoint
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Analysing the Gradient
(a) ∂Jmd∂ρe
<−0.00001 (b) ∂Jmd∂ρe
> 0.00001 (c) | ∂Jmd∂ρe| ≤ 0.00001
• (a) desire for ρ > ρmax at support, loads, bars
• (b) desire for ρ < ρmin to improve mechanism
• (c) large region | ∂Jmd∂ρe| ≤ 0.00001 out of [−0.0073;0.0734]
→ mostly solid and void
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Identifying Actual Grayness Conditions
(a) Sλe⊥ σue (b) strain ‖Sue‖ (c) strain ‖Sλe
‖
• only ‖Sue‖= 0 and ‖Sλe‖= 0 → rigid displacement
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Varying Penalization for Static Elasticity Force Inverter
• with linear ρ interpolation ρmin is crucial
• void material at load points is a local optimum
• 2400 combinations of ρmin and p for power law ρp
→ SNOPT and SCPIP (MMA) have similar similar results
• grayness measure: g(ρ) = 1N ∑
Ne 4ρe (1−ρe)
• strong self-penalization for proper (p,ρmin)
(a) solution
10-6
10-5
10-4
10-3
10-2
lower bound ρminp
1.0
1.5
2.0
2.5
3.0
3.5
pe
na
lty p
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
(b) objective
10-6
10-5
10-4
10-3
10-2
lower bound ρminp
1.0
1.5
2.0
2.5
3.0
3.5
penalty p
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(c) grayness
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Dynamic Elasticity
Wave guiding Sigmund, Jensen; 2003
J(u(ρ)) = uTLu∗ with∂ J
∂ρe= 2Re
{λT ∂S
∂ρeu
}
• time-harmonic S = K + jω C−ω2M with C = αKK + αMM
• also self-penalization reported in Sigmund, Jensen; 2003
• ∂J∂ρe
= 2(λTR
∂SR∂ρe
uR−λTR
∂SI∂ρe
uI−λTI
∂SR∂ρe
uI−λTI
∂SI∂ρe
uR
)• arbitrary combinations for ∂J
∂ρe= 0 ∧ ρe 6∈ {ρmin;ρmax}
• not much self penalization observed for wave guiding
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Pamping
Pamping
Carte (ρe ,q) = q ρe (1−ρe) M0
• penalized damping Jensen, Sigmund; 2005
• vanishes at solid-void solution
S =ρK0 + jω (αK ρ K + (αM + q (1−ρ))ρ M0)−ω2
ρM
∂ J
∂ρe=〈BλR, [c ]B uR〉−ω
2〈λR,M0uR〉−ω αK〈BλR, [c ]BuI〉
−ω αM〈λR,M0 uI〉−ω q (1−2ρ)〈λR,M0 uI〉+ 〈BλI, [c ]B uI〉−ω
2〈λI,M0uI〉−ω αK〈BλI, [c ]BuR〉−ω αM〈λI,M0 uR〉−ω q (1−2ρ)〈λI,M0 uR〉
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
?
(a) wave guiding (b) no pamping (c) pamping q=7
0.60.81.01.21.41.61.82.0
24 26 28 30 32
-20-15-10 -5 0 5 10 15 20
ob
jective
in
10
5
damping αM
pamping q
qαM
0.10.20.20.20.30.30.40.4
24 26 28 30 32
-20-15-10 -5 0 5 10 15 20
gra
yness
damping αM
pamping q
qαM
• example for fixed αK: varying αK vs. varying qFabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Pamping: Discussion
• pamping is not self-penalization
• pamping might reduce objective value
• physical interpretation: adds dissipative material
• optimal design ρ∗ depends on q→ ∃q : ρ∗(q) 6∈ {ρmin; ρmax}
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Self-Penalization Occurs!
piezoelectricity
• counteracting effects ρ∗ possibly 6∈ {ρmin; ρmax}• additional effects for dynamic case?
• better results (by appropriate inital designs) show strongerself-penalization
static elasticity
• conditions for grayness ’unlikely’ to occur→ if, then likely no or rigid displacement
• mechanisms are intuitively solid-void based
dynamic elasticity
• wave guiding: no explanation, not observed
• more likely to occur with mechanisms
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Categories of Grayness
negligible relevance
• initial value due to rigid displacement
• insensitive to solid-void mapping
• remedies (post optimization)• grayness constraint• min vol penalty term• min vol problem with argmin J constraint
beneficial damping
• ρ corresponds to stiffness, mass, damping
• pamping experiment: optimal damping → ρ∗ ∈ {ρmin; ρmax}beneficial springs
• physical interpretation of grayness?
. . .
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Summarizing Words
• self-penalization occurs for many problems→ this is a fact, independent if you believe it or not
• this presentation attempts to explain reasons self-penalization
• is self-penalization a “phenomenon” or an “effect”?
• scientific curiosity shall be reason enough to think about theproblem
• we want to motivate you to consider your own topologyoptimization problem w/o volume constraints and penalization→ what is the difference to the constrained solution?
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
Where Shall it Lead To?
• a better understanding of self-penalization
• question optimality and methods of “conventional” solid-voidsolutions
• a collection of methods to solve problems w/o volumeconstraints
• projection methods (Heaviside type density filters)• rigorous feature size control (MOLE constraints)• pamping for dynamic systems (with care)• . . .
• consider categories of grayness• develop methods to avoid negligible grayness
• are there drawbacks in exploiting self-penalization?
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
Introduction Piezoelectricity Static Elasticity Dynamic Elasticity Conclusions
The End
Thank you for listening and for questions!
Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization
