topology optimization - university of michigankikuchi/gmcontents/topology.pdf · 1 topology...
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Topology OptimizationMathematics for Design
Homogenization Design Method(HMD)
Why topology ?
Change in shape & size may not lead our design criterion for reduction of
structural weight.
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Structural Design3 Sets of Problems
Sizing Optimization thickness of a plate or
membrane height, width, radius of the
cross section of a beam
Shape Optimization outer/inner shape
Topology Optimization number of holes configuration
Shape of the Outer Boundary
Location ofthe ControlPoint of aSpline
thickness distribution
hole 2hole 1
Sizing OptimizationStarting of Design Optimization
1950s : Fully Stressed Design
1960s : Mathematical Programming ( L. Schmit at UCLA )
= allowable in a structure
minmax
allowableu
Total Weightu
Design Sensitivity Analysis
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Ku = f
1
Design VelocitySensitivity
Dg g gD
Dg g gD
= +
= + =
= + +
ud d d u
K u fKu f u Kd d d
K fK ud d d d u
Equilibrium : State Equation
Design SensitivityPerformanceFunctions g
Typical Performance FunctionsStrain Energy Density
For Structural Design (This must be constant !)
Mises Equivalent StressFor Strength Design and Failure Analysis
Mean Compliance & Maximum DisplacementFor Stiffness Design
Maximum StrainFor Formability Study of Sheet Metals
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Hemp in 1950sSize to Topology
Eliminate unnecessary bars by designing the cross sectional area.
An Optimization Algorithm
P1 P2
E, Amin
max
max
Ku f
u
=
=
e allowablei u
e e ee
N
A L1
K u K u f
D B uD B u
u uu
u
=
+
=
=
=
A A A
A A A
A A
e e e
e
e
e e e
ee e
e
e
i
e
i
i
i
e
b g
Design Sensitivity
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Prager in 1960sDesign Optimization Theory
Maximizing the minimum total potential energy
1 1
12
e eN NT T
e e e e e ee e= =
= = d K d d f
max mine
e
designA
Leads Equilibrium
d
Why Total Potential ?Maximizing the Global Stiffness
Minimizing the mean compliance (Prager) when forces are applied
Maximizing the mean compliancewhen displacement is specified
min T sdesign u f
max Tsdesign u f
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1 1
Weight Constraint
12
E EN NT T
e e e e e e e ee e
Total Potential Energy
L A L W = =
= +
d K d d f
( )1
1
12
E
E
NT T e
e e e e e e e e ee e
N
e e ee
L L AA
A L W
=
=
= +
+
Kd K d f d d
Lagrangian
Variation
1
1 02
0E
e e e
T ee e e e
eN
e e ee
LA
A L W
=
=
+ =
K d f
Kd d
Optimality Condition
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T ee e
e e eL A
= Kd d
Something must beConstant !
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Physical MeaningStrain Energy Density Must be Constant
1 12
1 12
T ee e
e e e
Te e e
e e e
Weight Average of the Stiffness
L A
A L
=
=
Kd d
d K d
Pragers Condition
Example 1
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Example 2
(a) Single Loading
(b) Multiple Loading
Design Domain
Example 3
Applying Torque
Design Domain
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TOPODANUKIA Topology Optimization Soft
Toyota Central R&D Labs.
Making up a grand-structure
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Set up support and load conditions
Only a bending load is applied
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Two Loads are applied
Further DevelopmentFirst Order Analysis in Toyota Central R&D
Microsoft EXCEL Based Software
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Extension to ContinuumCharacteristic Function
D
= unknown optimum domainD = specified fixed domain
xxx x D
a f =
RST10
ifif i.e. \
What can we get from this ?Optimal Material Distribution
1 12 2
new
T T
DU d dD
= =
= = D
D D
Strain Energy of a Body
Shape Design
Find the best Find the best Dnew
Material Design
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Homogenization Design Method Shape and Topology Design of Structures is
transferred to Material Distribution Design(Bendsoe and Kikuchi, 1988)
X
Y
b
t
t
g
Unit cell
Unit cell
Review Under the assumption of periodic microstructures which can be
represented by unit cells.
Using the asymptotic expansion of all variables and the average technique to determine the homogenized material properties and constitutive relations of composite materials.
Homogenization Method : Mathematics
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HDM Test Problem
P
Design Domain
Nondesign Domain
support
40
55
R10
15
20
20
Starting from Uniform Perforation
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Design ProcessStructural Formation Process
2.800
3.000
3.200
3.400
3.600
3.800
4.000
4.200
4.400
0 10 20 30 40 50Ieration
Convergence History of Iteration
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Mesh Refinement
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Change Volumes
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Design Constraint
Design area
No design area
(Full material)
Load case 1
Load case 2
Load case 3
Displacement fixed along circle
Design area
No design area
(Full material)
No design area
(No material)
(Same boundary condition)
100
40
20
102
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Result of Design Constraint
Influence of Design Domain
Non-design Domain
Design Domain
Design Domain
1.25
1.25
5
2
2
1.25
1.25
5
0.5
12
12
0.5
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Different Topology
Shape Design Example
20
10
60
30
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Shape to Topology
Extension to ShellsRib Formation
P
20
20
20
10
30
h 0 =0.1 h 1 =1.0
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Commercialization of HMDFrom University to Industry
Three-dimensionalshaping of a structure forOptimum without any spline functions
OPTISHAPE Development1986~1989
AcceptanceTopology Optimization Methods
Commercial Codes have been developed in USA, Europe, and Pacific Regions
OPTISHAPE@Quint Corporation, Tokyo, Japan, 1989
OPTISTRUCT@Altair Computing, Troy, USA, 1996
MSC/CONSTRUCT@MSC German, 1997 And Others (OPTICON, ANSYS, ..)
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MSC/NASTRAN-OPTISHAPE
Quint/OPTISHAPE + MSC/NASTRAN Shape and Topology Optimization
Static Global Stiffness Maximization Maximizing the Mean Eigenvalues
Frequency Control for Free Vibration Increase of the Critical Load
MSC/PATRAN integration Developed by MSC Japan and Quint Corp.
Static/Dynamic Stiffness Maximization
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MSC/PATRANGUI Environment
MSC/NASTRAN Solver
Design Example by MSC.NASTRAN-OPTISHAPE
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Integration withShape Optimization
Prof. Azegamis Method
Shape Design Optimization by MSC.NASTRAN-OPTISHAPE
Optimized
Initial Design
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Compliant Mechanism Design by QUINT/OPTISHAPE
Application of QUINT/OPTISHAPE @ Kanto Automotive
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Surface GeometrySurface GeometryGenerationGeneration
TopologyTopologyOptimizationOptimization
Package SpacePackage Space
Parametric Shape VectorsParametric Shape VectorsSize and Shape Size and Shape
OptimizationOptimization
System Level RequirementsSystem Level Requirements
Finite ElementFinite ElementModelingModeling
Control Arm Development ExampleControl Arm Development Example
Altair: Concept Design Environment Product Design Synthesis
Altair: Concept Design Environment Product Design Synthesis
Altair/OptiStruct Input:
FE model of design space Load cases, frequencies, constraints Mass target
Output: Optimal material distribution via density plot CAD geometry interpretation : using OSSmooth
Thenuse to create optimal design
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OptiStruct Version 3.4 Expanded Objective function
Minimize Mass, Stiffness or Frequency Constraints on Mass, Stiffness, Freq, Disp
Now available on Windows NT FE improvements, faster solution time HTML/Windows on-line documentation Improved integration with HyperMesh3.0
OptiStruct Case StudyVolkswagen Bracket
Minimize Mass of Engine Bracket Subject to stiffness/frequency constraints
7 loadcases: operating, pulley, transport
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OptiStruct Case StudyVolkswagen Bracket Results
Mass reduced by 23% Original mass 950g ; Final mass 730g
Performance targets were met
OptiStruct: Topography Designfor Future Automotive Body Engineering
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ALTAIR/OPTISTRUCT Results
Extension of HDMTopology Optimization Method
Structural Design Static and Dynamic Stiffness Design Control Eigen-Frequencies Design Impact Loading Elastic-Plastic Design
Material Microstructure Design Youngs and Shear Moduli, Poissons Ratios Thermal Expansion Coefficients
Flexible Body Design (MEMS application) Piezocomposite and Piezoelectric Actuator Design
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QUINT/OPTISHAPE Application to Contro Frequencies
Material DesignSpecial Mechanism : Negative
Special Mechanism
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Compliant Mechanism DesignProfessor S. Kota @ UM
Negative Thermal ExpansionBing-Chung Chens Design
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