Topology Optimization Mathematics for Design Homogenization Design Method (HMD) Why topology? Change in shape & size may not lead our design criterion for reduction of structural weight. 1
Structural Design 3 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 hole 2 hole 1 thickness distribution Location of the Control Point of a Spline Sizing Optimization Starting of Design Optimization 1950s : Fully Stressed Design σ = σ allowable in a structure 1960s : Mathematical Programming ( L. Schmit at UCLA ) min σ σ u u allowable max Total Weight Design Sensitivity Analysis 2
Equilibrium : State Equation Design Sensitivity Ku = f Dg g u g = + Dd d d u Design Velocity Sensitivity Performance Functions g K u f Ku = f u + K = d d d Dg g 1 K f g = + K u+ Dd d d d u Typical Performance Functions Strain Energy Density For Structural Design (This must be constant!) Mises Equivalent Stress For Strength Design and Failure Analysis Mean Compliance & Maximum Displacement For Stiffness Design Maximum Strain For Formability Study of Sheet Metals 3
Hemp in 1950s Size to Topology Eliminate unnecessary bars by designing the cross sectional area. An Optimization Algorithm Ku= f σe σ u u E, A min i allowable max N max e= 1 ρ AL e e e P1 P2 Design Sensitivity K u K f = u + A A A σ DBu = DB u e e e e e = e e A A A e u i A e b e e e e u u = i i u A i e g e 4
Prager in 1960s Design Optimization Theory Maximizing the minimum total potential energy 1 Π= Π = Ne Ne T T e e e e e e e= 1 e= 1 2 d K d d f max design A e min Π de Leads Equilibrium Why Total Potential? Maximizing the Global Stiffness Minimizing the mean compliance (Prager) when forces are applied T min ufs design max design T s u f Maximizing the mean compliance when displacement is specified 5
Lagrangian Variation NE NE 1 T T L= de Kede de fe + λ ρeaele W 2 e= 1 e= 1 Total Potential Energy Weight Constraint NE T 1 T K e δl= δd ( K d f ) + d d λρ L δ A e= 1 2 Ae e e e e e e e e e NE + δλ ρeal e e W e= 1 Kd = f e e e Optimality Condition 1 d 2 N E e= 1 K T e e de e e Ae ρ AL e e e + λρ L = 0 W 0 1 T 1 de 2 ρ L Something must be Constant! K A e e e e d e = λ 6
Physical Meaning Strain Energy Density Must be Constant 1 T 1 de 2 ρ L K A e e e 1 T 1 d K d 2 ρeal e e e d e e e Weight Average of the Stiffness e = λ Prager s Condition = λ Example 1 7
Example 2 160 Design Domain 100 (a) Single Loading (b) Multiple Loading Example 3 200 Design Domain 100 Applying Torque 100 8
TOPODANUKI A Topology Optimization Soft Toyota Central R&D Labs. Making up a grand-structure 9
Set up support and load conditions Only a bending load is applied 10
Two Loads are applied Further Development First Order Analysis in Toyota Central R&D Microsoft EXCEL Based Software 11
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Extension to Continuum Characteristic Function Ω Ω = unknown optimum domain D = specified fixed domain D χ Ω axf = RST 1 0 if x Ω if x Ω i.e. x D \ Ω What can we get from this? Optimal Material Distribution Strain Energy of a Body 1 T 1 T U = 2 ε DεdΩ= χ 2 D Ω dd Ω ε Dε Shape Design = σ Material Design = D new Find the best Ω Find the best D new 15
Homogenization Design Method Shape and Topology Design of Structures is transferred to Material Distribution Design (Bendsoe and Kikuchi, 1988) Homogenization Method : Mathematics t Γ t Y Unit cell Ω b Γ X Γ g 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. 16
HDM Test Problem Design Domain 40 Nondesign Domain R10 P 20 15 support 20 55 Starting from Uniform Perforation 17
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4.400 4.200 4.000 3.800 3.600 3.400 3.200 3.000 Design Process Structural Formation Process Convergence History of Iteration 2.800 0 10 20 30 40 50 Ieration 20
Mesh Refinement 21
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Change Volumes 23
Design Constraint Displacement fixed along circle Design area Load case 1 No design area (Full material) Load case 3 Load case 2 100 Design area 20 2 10 40 No design area (Full material) (Same boundary condition) No design area (No material) 24
Result of Design Constraint Influence of Design Domain 12 2 2 1.25 Design Domain 5 1.25 Non-design Domain 1.25 Design Domain 5 1.25 0.5 12 0.5 25
Different Topology Shape Design Example 20 60 10 30 26
Shape to Topology Extension to Shells Rib Formation P 20 20 30 20 10 h 0 =0.1 h 1 =1.0 27
Commercialization of HMD From University to Industry Three-dimensional shaping of a structure for Optimum without any spline functions OPTISHAPE Development 1986~1989 Acceptance Topology 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,..) 28
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 29
MSC/PATRAN GUI Environment MSC/NASTRAN Solver Design Example by MSC.NASTRAN-OPTISHAPE 30
Integration with Shape Optimization Prof. Azegami s Method Initial Design Optimized Shape Design Optimization by MSC.NASTRAN-OPTISHAPE 31
Compliant Mechanism Design by QUINT/OPTISHAPE Application of QUINT/OPTISHAPE @ Kanto Automotive 32
Altair: Altair: Concept Concept Design Design Environment Environment Product Product Design Design Synthesis Synthesis System Level Requirements Package Space Topology Optimization Control Arm Development Example Surface Geometry Generation Size and Shape Optimization Parametric Shape Vectors Finite Element Modeling 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 Then use to create optimal design 33
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 Study Volkswagen Bracket Minimize Mass of Engine Bracket Subject to stiffness/frequency constraints 7 loadcases: operating, pulley, transport 34
OptiStruct Case Study Volkswagen Bracket Results Mass reduced by 23% Original mass 950g ; Final mass 730g Performance targets were met OptiStruct: Topography Design for Future Automotive Body Engineering 35
ALTAIR/OPTISTRUCT Results Extension of HDM Topology Optimization Method Structural Design Static and Dynamic Stiffness Design Control Eigen-Frequencies Design Impact Loading Elastic-Plastic Design Material Microstructure Design Young s and Shear Moduli, Poisson s Ratios Thermal Expansion Coefficients Flexible Body Design (MEMS application) Piezocomposite and Piezoelectric Actuator Design 36
QUINT/OPTISHAPE Application to Contro Frequencies Material Design Special Mechanism : Negative ν Special Mechanism 37
Compliant Mechanism Design Professor S. Kota @ UM Negative Thermal Expansion Bing-Chung Chen s Design β α H H 8.01 = 0 52.7 = 0 0 7.89 0 58.9 38
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