präsentation alexander hasse
TRANSCRIPT
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Synthesis of Compliant Mechanisms via topology optimization
Alexander Hasse
Summer School ASME 2016
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Outline
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Introduction What is topology optimization? A short introduction to structural optimization Topology Optimization of compliant
Mechanisms Design examples
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© ENGINEERING DESIGNFriedrich-Alexander-Universität Erlangen-NürnbergProf. Dr.-Ing. Sandro WartzackProf. Dr. sc. ETHZ Alexander Hasse
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Introduction
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Prof. Dr. sc. ETH Alexander HasseFriedrich-Alexander-UniversityD-91052 ErlangenGermany
Phone: 0049 9131 85 23663Email: [email protected]
Alexander Hasse
2001-2007 Diploma Degree in Mechanical Engineering (TU Dresden)
2007-2011 Doctoral Work in MechanicalEngineering (ETH Zurich)
2011-2012 Post-Doc (ETH Zurich)
2012-2014 Head Engineering (Monolitix AG)
Since 2014 Professor for Mechatronic Systems at FAU Erlangen-Nuermberg
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IntroductionRobotic grippers
4Source: Monolitix
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IntroductionMedical instruments
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surgical handle – conventional
surgical handle – compliant
Source: Empa
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IntroductionShape adaptive structures
6
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What is topology optimization?The design problem in general
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What is topology optimization?Definition of topology, shape and size
8Source: Wikipedia, Bendsoe and Sigmund
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What is topology optimization?Example force inverter
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inF outF
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Structural optimizationDifferent modules
10
Structural modelling
Opti. formulation
Req
uire
men
ts
Optimization algorithm
Parameterization
min ( )
( ) 0
dfmitV K
,
x
φx
Source: Kress
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Structural optimizationParameterization and structural modelling
11Source: Bendsoe and Sigmund
0( )
1
mi
ii
xk k
Ground-structure approach
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Structural optimizationParameterization and structural modelling
12Source: Bendsoe and Sigmund
0( )
1
mp ii
i
xk k
Solid Isotropic Material with Penalisation (SIMP)
0( )
1
mi
ii
m x v
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Optimization formulation„Spring method“ according to Bendsoe and Sigmund
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Aktor
in inF u
outu outu
ink outk
inF
0( )
1
min
min ( )
0 1 , 1,...,
out
mi
ii
i
u
mit
x v V
x x i m
xx
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Preprocessing
1. Build up ground structure
2. Generate „ground“ stiffness matrix for each ground-structure member
„Spring method“Example force inverter
14
symmetry conditionfixed stiffness ground-element fixed stiffness ground-element
0( ) , 1i i mk
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„Spring method“Example force inverter
15
Optimization
Initial design variables
Built stiffness matrix
Caculate structural response
Update design variables by a proper optimization algorithm
0( )
1
mi
i in outi
xk k k k
0x
1inu k f
outu
newx
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„Spring method“Example force inverter
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Update design variables by a proper optimization algorithmoutu
Sensitivity analysis
Calculate sensitivities
newx
out
i
dudx
( ) ( )out out i out i
i
du u x x u xdx x
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„Spring method“Example force inverter
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Calculate sensitivities analytically
ku f
d d ddx dx dxu f kk u
1T Toutdu d d ddx dx dx dx
u f kz z k u
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Structural optimizationProcedure
18
Structural modelling
Opti. formulation
Req
uire
men
ts
Optimization algorithm
Parameterization
min ( )
( ) 0
dfmitV K
,
x
φx
Source: Kress
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Structural optimization
19
Recommendation
inF outF
“A 99 line topology optimization code written in Matlab” written by Sigmundwith small modifications described in the book „Topology Optimization -Theory, Methods, and Applications”
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Optimization formulation„MPE/SE“ according to Ananthasuresh
20
inF
outu
inF
(a) (b) (c)virtuellF
TMPE v ku
virtuellkv f
inku f
TSE u ku
inku fmin MPEf
SE
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Optimization formulation„Characteristic stiffness formulation“ according to Wang
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inF
outu(a)
0aa ac a a
ca cc c
é ù é ù é ùê ú ê ú ê ú=ê ú ê ú ê úë ûë û ë û
k k u fk k u
1( )aa ac cc ca a a a-- = =k k k k u ku f
11 12
21 22
in in
out out
F uk kF uk k
2( )11 22( ) e dGA GAf k kx
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Formulation for mechanisms with multiple outputs
NACA 0012
NACA 2412
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6Design criterion
kφ λφ
1Tφ φ
1 1 12 2 2
T Tφ kφ λφ φ λ
u φ
2 12
TSE φ kφ
2
12
T SEφ kφ
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6Design criterion
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6Design criterion
1 0 00 0
0 01 1 10 02 2 2
0 00 0
T
p
l
l
é ùê úê úê úê úê ú= = ê úê úê úê úê úê úë û
Φ kΦ K
1k
2 3, , , nk
1 1φ χ
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6Design criterion
1 , 1
1 , 1
Ti iTi i
i m
i m p
= " =
= " = +
χ χ
χ χ
1
1
0 00
1 1 12 2 2
00 0
mT
m
p
k
kk
k
+
é ùê úê úê úê úê úC C = = ê úê úê úê úê úê úë û
k K
1 1m m p+é ùC = ë ûχ χ χ χ
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6Optimization formulation
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6Optimization formulation
aa ac a a
ca cc c c
k k u fk k u f
1( )aa ac cc cak k k k k
kφ λφ
1 dφ φ
Problem
mitkψ wψ 1 dψ φ
Solution
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6Optimization formulation
1 1 1ˆ | | ... | | | ... |d j j qΦ φ φ φ φ φ
with1...
maxdj dii qaa
1 dψ φ
1 22 2 1
1 1
T
T
ψ kφψ φ ψψ kψ
1
1
Tqi q
q q iTi i i
ψ kφψ φ ψ
ψ kψ
1 2| | ... | qΨ ψ ψ ψ
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6Optimization formulation
1 0 00 0
0 00 0
T
q
Λ Ψ kΨ
withkψ wψ 1 dψ φ
1
2..
( )min q
f x
Objective function
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6Design example – force inverterProblem statement and parameterization
31
symmetry conditionfixed stiffness ground-element fixed stiffness ground-element
Design domain with support and master DoFs; arrows define the directions of positive displacement in the master DoFs
Parameterized design domain
1
( )n
i ii
xk x k
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6Design example – force inverterResults and discussion
32
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6Design example – force inverterEigenvalue analysis of the force inverters
33
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6Design example – force inverterDeformation plots
34
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6Design Example – shape adaptive structureProblem statement
35
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6Design Example – shape adaptive structureResults
36
Load case 1, F =-140 N Load case 2, Fx =140 N, Fy =-140 N
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6Design example – Morphing wing
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d
=k Eigenvalue problem
= ( )Belt Stiffk k k xStiffness matrix
Design criterion
1 d
1 2
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6Design example – Morphing wing
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NACA 0012
NACA 2412
d
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6Design example – Morphing wing
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S
IP
LE
T
xx
xxx
= ( )Belt Stiffk k k x
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6Design example – Morphing wingOptimization procedure
40
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6Design Example – Morphing wing
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Output from the design routine
Prototype as hybrid design: CFRP belt and inner stiffening structure in Polyamide
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6Design example – Morphing wing
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Single Belt Complete Belt‐Rib Structure
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Applied displacement at the trailing edge
Applied displacement at the leading edge
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© LEHRSTUHL FÜR KONSTRUKTIONSTECHNIKFriedrich-Alexander-Universität Erlangen-NürnbergProf. Dr.-Ing. Sandro WartzackH
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Comparison between the desired and the actual deformed shapes by an applied displacement of 3.5 mm
Comparison between the desired and the actual deformed shapes by an applied displacement of 3.5 mm
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Thank you for your attention.