Topology Optimization OPTISHAPE

Transcription

1 Topology Optimization OPTISHAPE 1. Topology Optimization 2. Homogenization Design Method 3. Background of the New Approach 4. Mathematical Formulation 5. Optimality Criteria Method 1

2 What is Topology Design? Shape design keeps the initial topology, while the shape of exterior/interior domain is designed. If an extra hole is generated, or if two holes are merged to a single one, we say topology has changed. Finding the number, location, and shape of the holes is a typical topology problem. 2

3 OPTISHAPE The key idea is to transfer shape/topology design to Optimum material distribution with on/off switch condition 3

4 Idea in OPTISHAPE Extension Ω to the fixed domain D Ω Ω D The elasticity tensor E χ Ω E χ Ω ( x) = 1 if x Ω 0 if x Ω χ Ω E L ( Ω) 4

5 Examples by OPTISHAPE 5

6 Topology Optimization has become an important and well recognized sub-area of structural optimization Design Sensitivity Analysis (1960s & 70s) linear and nonlinear problems Sizing Optimization (1960s) Shape Optimization (1970s & 80s) Topology ( Layout ) Optimization (90s) Discrete and Continuum Topology Optimization Material Based Optimization Extension to MEMS area 6

7 Evidence Last Two : 1st and 2nd World Congress on Structural and Multi-disciplinary Optimization ( Germany95 & Poland97) There are numerous sessions on topology optimization related Commercial Codes OPTISHAPE(Japan), OPTISTRUCT(US) MSC-NASTRAN, ANSYS ---- Fall 97 7

8 OPTISHAPE : Present Maximization of the global stiffness of an elastic structure Maximization of the mean eigenvalue problems for free vibration Combination of the above two Maximization of the dynamic stiffness for frequency response problems Heat Conduction/Thermal Loading 8

9 OPTISHAPE : Near Future include SHAPE OPTIMIZATION capability based on Azekami and Shimoda s Method (at Mitsubishi Motor) for detailed shape design after the standard topology optimization include sensitivity analysis for sizing TOPOLOGY + SHAPE + SIZING 9

10 OPTISHAPE : Future Compliant Mechanism, Mechanism, and Flexible Body Design to control deformation and motion of structures, flexible multi-bodies, compliant mechanisms, and even mechanisms to have integrated synthesis study of mechanical systems toward smart structure design with control Material Design Young s and Shear moduli and Poisson s ratios Piezo-electric material design for MEMS 10

11 for more information 11

12 Typical Procedure 1 (1) Define a design domain which contains the final optimum structure geometric restriction for the on/off condition on-flag : solid structure always exists off-flag : void (hole) must be assigned (2) Define the loading and displacement constraint multiple loadings and multiple constraints are possible in OPTISHAPE 12

13 Multiple Loadings & Multiple Constraints The University of Michigan Loading #1 Loading #3 Constraint #2 Constraint #1 Loading #2 Loading #4 13

14 Typical Procedure 2 (3) Define the volume ( or weight ) constraint z ρdω W 1 Ω (4) Applying OPTISHAPE, and obtain the Optimum Layout ( Topology & Shape ) the Maximum Mises Equivalent Stress the Mean Compliance and Strain Energy Density 14

15 Typical Procedure #3 (5) Repeat the above steps for two more different weight constraints W & W 2 3 (6) Obtain the maximum Mises stress and the mean compliance (7) Using the quadratic interpolation of Maximum Mises Stresses & Weights compute the weight for the allowable stress constraints 15

16 Determination of the Weight Constraint to enforce the stress constraint if exist Maximum Mises Stress Three-Point Method σ max W1 W2 W3 Weight Constraint Weight for the Upper Bound Mises Equivalent Stress 16

17 Internal Virtual Work zω Extended Domain by.χ Ω zd b g T b g b g b g T Ω e v Ee u dω = e v χ Ee u dd New Material Constants ( Extended Elasticity Matrix ) χ Ω E L b g D is very discontinuous Impossible to take its derivative that is, no design sensitivity analysis 17

18 Possible Approximation Elasticity Matrix This interval can be small as we want Possibly Rapidly Varying Ee Design Domain in the x Direction Introduce Two Scales at an arbitrary point No Way to Approximate by a Standard Way with a distribution function x 18

19 Relaxation The University of Michigan Very Rapidly Varying Function χ Ω E cannot be approximated by a differentiable function of position x in the standard way x, y = Introduce the two scales ε and the micro-scale perforation, and then χ Ω E is approximated by the homogenized average elasticity matrix E H F HG x I K J 19

20 Origin of the Idea G. Cheng and N. Olhoff in plate thickness optimization smoothly varying thickness is not optimum optimum involves rapidly changing ribs Homogenization is required 20

21 Mathematicians The University of Michigan Lurier, Cherkaev,and Fedrov (1981) Notion of G-convergence that is in the specially designed average sense convergence Kohn and Strang (1984) Microscale performation and specialized variational principles Murat and Tartar (1983) Homogenization Theory from Hadamard Shape Design Problem 21

22 Hadamard v.s. Homogenization Hadamard Rapidly Varying Boundary movement Murat & Tartar Formation of Microstructure 22

23 Bendsoe & Kikuchi 1986 a a = b = 1 = b = 0 void / hole solid structure b a Rotation Micro-Scale Perforation by infinitely many rectangular holes Design Variables are b a, b,θ g at every where 23

24 The Homogenization Design t Ω t y Ω d Ω f solid microstructure x macrostructure y y void microstructure porous microstructure 24

25 Why Homogenization? Small Scale Rapidly Varying Heterogeneity This idealization is regarded as the homogenization in theoretical mechanics If the exact heterogeneity is used in mechanics, we must introduce so fine finite elements to represent all the detail. This is a difficult task. Equivalent Homogeneous Material 25

26 Microstructures 2D 1 3D φ ϕ α θ γ 1 β 1 α β 1 θ 26

27 Many Choices The University of Michigan The key feature is an approximation of the extended elasticity matrix χ Ω E L D There are infinitely many ways to approximate this by using Generalized Porous Media Constitutive Equations (bio-mechanics, Geo-mechanics) Power Low of Density/Elasticity Constants Rank 1 & Rank 2 Orthotropic Materials Others b g 27

28 Power Low The University of Michigan Most popular approach at present (Meljek, Yang,...) Altair/OPTISTRUCT is now assuming this approach χ Ω E ρ p E For p=2 or 4, the design variable becomes the density ρ such that ρ = R S T 1 0 if if solid structure void / hole 28

29 Homogenized Average Young's Modulus E s b H E = 1 ρ E + ρe g v s Upper Bound Mixture Theory (1D) E H 1 = 1 ρ ρ + E E v s Lower Bound Mixture Theory (1D) E v ρ = 1 Material Density 29

30 Short Coming of Power Low Un isotropic principal stress distribution Density approch can make only isotropic performation This requires a lot of meshing to have reasonable layout However, easy programing and handy design variable 30

31 Rank 1 & Rank 2 Materials Rank 1 Lamination Zero E in 2 Zero Shear Rank 2 Lamination Zero Shear Rectangular hole is very close to Rank 2 Material

32 Elasticity Matrix The University of Michigan Rank 1 Material E H = Lb NM g 1 a E s O QP Rank 2 Material E H = Es c hl q 2 1 ν 1 ab L c hb gb g b g b g b g b g ν NM ν 1 ab 1 a + ν 1 b ν 1 b 0 1 b 1 b O QP 32

33 Advantage of Rank 2 Rank 2 elasticity matrix can be computed in the closed form while rectangular hole requires FE calculation 33

34 Rank 2 and Rectangular Hole Rank 2 material is regarded as the optimum orthotropic material ( since its shear modulus becomes zero ) Then this yield the stable optimum solution mathematically verified in the sense that FE mesh dependency cannot be observed However, this results considerably lots of gray perforated medium in the optimum 34

35 Penalization The University of Michigan We would like to have clear segregation of solid and void portions to define precise shape and topology, that is No gray scale solution is desirable χ Ω E R S FH G T ρ E p E H E E H I KJ E H Density Power Low Rank2 & Rectangular hole... Penalization 35

36 Inequality Relation Multi-Loading 1 The University of Michigan b g b g b g i i i 1 1 max min a u, u f u = max min a u, u ba, b, θg i= 1, 2,..., i i i i i i max 2 ba, b, θg i= 1, 2,..., i max 2 1 b g 1 b g b g max am um, um = max am um, um f m um ba, b, θg 2 ba, b, θg 2 Minimum Principle to the I-th Load ( Equilibrium ) 1 2 b g b g b g b g 1 ai ui, ui fi u = min ai vi, vi fi v vi 2 36

37 a The University of Michigan Multiple Loading 2 Formtion of a Single Objective Function z T m um, um = max e ui χωe max e ui D i,..., imax i,..., imax b g z = 1 = 1 b g z T m m i Ω 0 D i= 1,..., imax D i= 1,..., i b g b g b g T f u = max e u χ Es dd + max u χ ρbdd+... Approximated Design Problem dd max 1 b g b g max min am v, v f m v b g a, b, θ v 2 subject to ρdd W zd 0 i Ω 37

38 Weighted Sum Approach Alternate The University of Michigan imax 1 max b g b g w min a v, v f v b g i i i a, b, θ v 2 subject to i= 1 ρdd W zd 0 Diaz and Bendsoe OPTISHAPE can do both ways by user s choice 38

39 Lagrangian The University of Michigan Optimization Problem zd 1 b g b g max min a v, v f v ba, b, θg v 2 subject to ρdd W 0 Lagrangian 1 z L = a v, v f v λ ρdd W 2 D b g b g e j 0 δl = a v, δv f δv δλ ρdd W z z z D D D F HG F HG F HG e Taylor & Prager in 1967 First Variation T T χωe a λ zd b g b g e j 0 b g b I T v v g e λ K JδadD b g e v b g e v χωe e b χωe e θ b v g b I v g K J ρ a ρ b δθdd I K J δbdd 39

40 Equilibrium a The University of Michigan Optimality Condition b g b g v,δv = f δv δv Weight Constraint : Kuhn-Tucker Condition zd z 0 0 D e j,, λ ρdd W = 0 λ 0 ρdd W 0 Optimality Condition z z e D D b g b gt χ b g a a ρ ΩE e v e v λ a a dd 0, 0 a 1 c T b bh b g χ b g ρ ΩE e v e v λ b b dd 0, 0 b 1 b g v T F HG I K J F HG I K J χωe e θ b v g = 0 40

41 zd The University of Michigan Optimality Criteria Method 1 b g b gt χ b g a a ρ ΩE e v e v λ a a dd 0, 0 a 1 e e b g v b g v T T k + 1 a = a F HG I K J χωe e a χωe e a ρ λ a b g bkg L M N b v g b v g λ ρ a a a = 0 if 0 and 1 = 1 if a 0 and a 1 b g b g b g b g b g e j T k χ e k k k k a b j e j ΩE e v,, θ e v a bkg ρ e a a bkg b b j kg bk λ,, θ g O P Q α, 0 < α 1 41

42 Optimality Criteria Method 2 With design constraint 0 a 1 F R S T L T k χ k k k k a b k k a b G + 1 g = 0 1 a b g e j e j e j ΩE e v,, θ e v max,min, a G M bkg ρ e a a bkg b bkg bk λ,, θ gj H N b g b g b g b g b g O P Q α UI V J W KJ Choice of Parameter Algorithm of the optimality criteria method is very similar with the fully stressed design α

43 Lagrange Multiplier λb k g is computed by the bisection method z ρeab g k k, bb gjdd = W 0 D based the implicit function theorem Volume constraint is always saturated. This is correct, but for eigenvalue related problems. this is not true. 43

44 Optimum Angle : Pedersen Noting that we have b g b g b g b g c h b g T T Ω Ω R l q b g T c E h b gs σ T 1 σ1 σ2 0 φ χω T φ σ T 1 e v χ E e v = s v χ E s v = U V W e b g E b v e v g T χω = 0 θ θ b g c T h b g 1 T φ χ E T φ = 0 φ = 0 Ω Optimum angle is the one for the principal stress! 44

45 Engineering Idea Optimality Condition can be explained as σ 2 σ 1 in the principal stress direction Rectangular hole should be aligned Large hole can be assumed in the small principal stress direction Small hole must be placed in the large principal stress direction 45

46 Elaboration The University of Michigan Using the relation ρ = 1 ab 1 a = ρ b we can change the design variable b g b g a, b, θ ρ, b, θ and then we can show 1 b g b g max min a v, v f v bρ, b, θg v 2 1 max min max a v, v ρ v bb, θg 2 b g f b g v Habor Jog Bendsoe Solve b and angle analytically, then apply optimality criteria method only to the density 46

47 Monotonic Convergence Optimality criteria method is monotonically converging to a local optima The local optima obtained may be strongly dependent of the initial condition Uniformly Biased Initial Condition in OPTISHAPE The local optima may depend on the FE mesh 47

48 Many Problems but It is so powerful! 48

49 Excersise #11 The University of Michigan Supporting Rollers for Sliding Big Sliding Doors Strong Wind Force (distributed) 49

50 Thin Plate Find a pattern of reinforcement of a thin plate subjected to a strong wind force by using OPTISHAPE Reinforcement Beams 50

51 Topology Optimization -2- OPTISHAPE 1. Extension of HMD 2. Free Vibration Problem 3. Frequency Response Problem 4. Buckling Problem 5. Flexible Bodies 51

52 Extension of OPTISHAPE Free Vibration Problem Maximization of lowest frequency Maximization of the distance of two frequencies Inverse frequency identification problem Frequency Response Problem Buckling Problem for Stability Flexible Multi-Body Design Material Microstructure Design 52

53 Eigenvalue Problem Maximizing the lowest eigenvalue max b a, b, θ g λ min Ku = λmu Several eigenvalues are crashing while optimization is performed One eigenvalue with n number of eigenvectors Ultimate Optima 53

54 Free Vibration The University of Michigan Discrete Free Vibration Problem Separation of variable b g x t b g = exp iωt u b g 2 d M x Kx 0 2 dt + = b g 2 ω M exp iωt u + K exp iωt u = 0 c 2 ω Mu + Kuh b g exp iωt = 0 2 ω + = Mu Ku 0 2 Ku = ω Mu = λmu 54

55 Modification by Ma, Cheng, Kikuchi The University of Michigan Eigenvalues max λ max b g min b g a, b, θ a, b, θ imax i= 1 w i λ mini Mathematics can be constructed by using sub-differentiability Number of Iteration ---- Bendsoe and Rodriges i max w = weights, i = 1,..., i i = number of crashing eigenvalues max 55

56 Further Modification max λ max w λ + z λ min ba, b, θg ba, b, θg λ mini = λ l = 2nd 3rd,..., imax jmax i mini i= 1 j= 1 minimum crashing eigenvalue higher eigenvalues We shall maximize not only the lowest frequency but also several eigenvalues at once... yields sub-optima, but make sense in engineering j l 56

57 Maximization of Distance jmax imax max w jλ j b g a, b, θ j= 1 i= 1 w i λ i j max j= 1 i max i= 1 w j w λ i λ i j = higher eigenvalue = lower eigenvalue 57

58 Inverse Frequency Problem imax 1 p max c h w λ λ target b g i i i a, b, θ 2 i= 1 Idendify the structural configuration so that it has specified eigenvalues for the first small set of eigenvalues 58

59 Frequency Response Problem max min b g a, b, θ v v T Kv ω 2 v T Mv v T b 2 K = stiffness matrix M = mass matrix ω = specified frequency b = excited force with ω frequency 59

60 Discrete Equilibrium c h K ω 2 M u = b b g b t b g u t b g b g = exp iωt b = = exp iωt u = exciting force dynamical response bg b t =exp iωt b g b 60

61 Buckling Problem max b a, b, θ g K K λ G = min = stiffness matrix Ku K u = λ G geometric stiffness matrix Rodrigues and Gedes possibility of local buckling modes 61

62 Linear Combinations of the above cases OPTISHAPE 62

63 OPTISHAPE in Practice OPTISHAPE is currently integrated into SDRC/I-DEAS. In future an icon of OPTISHAPE will be added into I-DEAS option menu OPTISHAPE design models can be developed by MSC/PATRAN and the input and output data are fully compatible with MSC/NASTRAN 63

64 64

65 OPTISHAPE to CAD/CAM OPTISHAPE can translate the image data of the optimum topology/configuration into a STL file (as well as SLC file in near future) so that smoothed 3D surfaces can be plotted to show the concept of the design (NC Link) OPTISHAPE can produce a set of wire frame models of sliced models perpendicular to a specified direction (CAD Link) 65

66 OPTISHAPE More in Practice 66

67 Current Development Research OPTISHAPE is constantly enhanced 67

68 Flexible Body Design compliant mechanism design by Mary Frecker & Shinji Nishiwaki Shinji Ejima 68

69 Mechanical Design Structure Design Mechanism Design New Design Based on Flexibility 69

70 Structure Design Flexible Structure Stiff Structure 70

71 Flexible Body Design Design a structure that moves to the specified direction as much as possible when input forces are given Input Force Motion/Kinematics Control Desired Motion 71

72 Compliant Mechanism MEMS(Micro Electro Mechanical System) Basic Ideas & Clues for Rigid Link Mechanism Design Microcompliant crimping mechanism ~250 µm 72

73 Kinematic Synthesis Based on traditional rigid body kinematics Lumped compliance (Pivot) I. Her and A. Midha (1986), L. L.Howell and A. Midha (1994), (1996) 73

74 Continuum Synthesis Distributed compliance O. Sigmund (1995), (1996) U. D. Larsen, O. Sigmund and S. Bouswstra (1996) 74

75 Design of Flexible Structures Truss Approach Configuration by Trusses Continuous Approach Distribution of Materials 75

76 Truss Approach Based on Ground Structure Design Compliant Mechanism Design By Mary I. Frecker 76

77 Problem Statement 1 Kinematic Function fa= applied force A B fb= dummy load max (mutual energy) max (v B T K u A ) subject to : K u A = f A K v B = f B 77

78 Problem Statement 2 Structural Function fixed point A B fb= resistance of workpiece min( strain energy ) min (u B T K u B ) subject to : K u B = - f B 78

79 Multicriteria Optimization max mutual energy strain energy max v T B K u A u T B K u B subject to : K u A = f A K v B = f B K u B = - f B total resource constraint lower and upper bounds 79

80 A Simple Design Problem F Figure 5a. Design Problem. Figure 5b. Initial Guess. F F Figure 5c. Compliant Mechanism Solution. Figure 5d. Stiffest Structure Solution. 80

81 Verification of Function Deformed Shape F This can satisfy the original objective 81

82 Compliant Gripper F Design Problem D Initial Guess F D Solution and Finite Element Model Compliant Grippers 82

83 3D Compliant Gripper D F Design Problem Initial Guess Solution and Finite Element Model Three Dimensional Compliant Mechanism 83

84 Continuous Approach Based on Homogenized Design Method Fixed Grid / Voxel Mesh Method Homogenization Method 84

85 Formulation of Flexibility t 1 u 1 u 1 Γt 1 t 2 Maximize L 2 ( u 1 )= t 2 u 1 Γt dγ 2 Flexibility at Stiffness at Γt 2 (Mutual Mean Compliance) Γt 2 Minimize L 1 ( u 1 )= t 1 u 1 dγ (Mean Compliance) Γt 1 Γt 1 85

86 Design of Flexible Structures Kinematic function Flexibility Structural function Stiffness + + Defamation Constraint Reaction force Applied force Constraine d Motion 86

87 Multicriteria Optimization Flexibility Max Mutual Mean Compliance Stiffness Trade Off Min Mean Compliance Compromise Solutions 87

88 Multi-Objective Functions (1) Max Mutual Mean Compliance Mean Compliance (2) Max w Log(Mutual Mean Compliance) -(1-w)Log( Mean Compliance) where w is a weighting Coefficient 88

89 Weighting Method vs. Ratio Formulation Max w1 Mutual Mean Compliance -(1-w1) Mean Compliance where w1 is a weighting Coefficient Max Mutual Mean Compliance Mean Compliance 80 A(Γ t1 ) Example 40 B(Γ t2 ) 89

90 Weighting Method vs. Ratio Formulation w1 =0.001 w1 =0.01 w1 =0.1 Ratio Formulation 90

91 Optimization Procedure Start Calculate homogenized elasticity coefficients No Convergence Yes End Calculate sensitivities using FEM Update angle θ Solve SLP problem with respect α and β Filtering 91

92 SLP vs. OC The University of Michigan Sequential Linear Programming (SLP) Linear Approximation + Simplex Method Slow Convergence Easy Implementation for Any Objective Functions Optimality Criteria Method (OC) KKT-Conditions + Heuristics Quick Convergence Difficult to Construct Heuristics for General Objective Functions 92

93 Checkerboard Pattern ρ 1 =ρ 4 = 1 ρ 2 =ρ 3 = 0 > Artificially stiffer ρ 1 =ρ 2 =1/2 ρ 3 =ρ 4 = 1/2 Filtering scheme ρ 1 = 1 ( 4 3ρ + ρ + ρ ρ ) ρ 2 = 1 ( 4 ρ + 3ρ ρ + ρ ) ρ 3 = 1 ( 4 ρ ρ + 3ρ + ρ ) ρ 4 = 1 ( 4 ρ + ρ + ρ + 3ρ ) ρ 1 ρ 2 ρ 3 ρ 4 where ρ i = 1 α i β i (i = 1,...,4) One group 93

94 Elimination of Checkerboard Eigen-frequency problem No Filtering Filtering 94

95 Compliant Clamp A(Γ t 1 ) Ω d 10 B(Γ t 2 ) Design domain Large displacement analysis 95

96 Compliant Gripper 60 Ω d A(Γ t1 ) 10 B(Γ t2 ) 10 Original Shape Design domain Deformed Shape Large displacement analysis 96

97 Compliant Pliers A(Γ t 1 ) Hinge Ω d 10 B(Γ t 2 ) Design domain Optimal configurations 97

98 Final Configuration Optimal configuration Workpiece Final configuration 98

99 Compliant Internal Gripper B 20 Ω d A Design domain Optimal configuration Workpiece 99

100 Constrained Motion Compliant Gripper Adding Mutual Stiffness Transitional Motion 100

101 Suspension Design B Ωd A B Ωd A Design domains Optimal configurations 101

102 Interpretation The University of Michigan Optimal Configuration + Deformed Shape Strut Type Rigid Link Mechanisms Double wishbone Type 102

103 Two Displacement Outputs 40 B1(Γ t2 ) A(Γ t1 ) Ω d 30 B2(Γ t2 ) Design domain Optimal configuration 103

104 Extension to Multi-Flexibility Necessary for Many Performance Criteria Automotive Body Design 2 t 1 n Flexibilities Required 1 t 1 1 Γt 1 2 Γt 1... to n n Mutual Mean Compliances Should be Positive 2 Γt 2 2 t 2 1 Γt 2 1 t 2 104

105 Multi-criteria Optimization Max i-th Mutual Mean Compliance ( immc) (for i=1,...,n) Min i-th Mean Compliance ( i MC) (for i=1,...,n) Multi-Objective Function Max -1/Cf Log( 1/Cs Log( Exp(-Cf MMC)) i Exp(Cs MC)) i 105

106 Multi-flexiblity Design A1 B2 10 A1 B2 Design Domain Ωd 20 Design Domain Ωd A2 B1 A2 B1 10 First flexible mode Second flexible mode 106

107 Optimal Configurations Cf =Cs = 3 Cf=Cs =

108 Verification of Performance Deformed Shape A 1 A 2 108

109 Extension 3D Compliant Mechanism Design New 3D element is used Complaint Mechanism Design with a Displacement Constraint New formulation is introduced + Image based design 109

110 Design of Simple Model 24 Applied Force 2 Dummy Force Constraint 8 Design Domain 110

111 Optimal Configurations (1) Unconstrained Case Total Volume =20% 111

112 One Constraint Case (Total Volume =30%) 112

113 Two Direction Constraint (Total Volume =30%) 113

114 Design of Gripper 5 24 Symmerty Condtion Applied Force Symmerty Condtion Constraint 4 4 Dummy Force 8 Design Domain 114

115 Optimal Configurations (1) Unconstrained Case Total Volume =20% 115

116 Optimal Configurations (2) Constrained Case Total Volume =20% 116

117 Design of Clamp Applied Force 5 24 Symmerty Condtion Dummy Force Design Domain 117

118 Optimal Configurations Unconstrained Case Total Volume =20% 118

119 Design of Pliers Applied Force 5 24 Symmerty Condtion Dummy Force Design Domain 119

120 Optimal Configurations Unconstrained Case Total Volume =20% 120

121 Design of Torsion Bar Applied Force Flexibility Design Domain 121

122 Optimal Configuration Total Volume =10% 122

123 Design of Torsion Plate (1) Applied Force Flexibility Design Domain 123

124 Optimal Configuration (1) Total Volume =10% 124

125 Design of Torsion Plate (2) Applied Force Flexibility Design Domain 125

126 Optimal Configurations (2) Total Volume =10% 126

127 Design of Sus-Like Structure Applied Force Flexibility Design Domain 127

128 Optimal Configuration Total Volume =10% 128

129 Design of Another Gripper Applied Force 5 Flexibility Design Domain 129

130 Optimal Configuration Total Volume =10% 130

131 Design of Tensile Model Applied Force Flexibility Design Domain 131

132 Optimal Configurations Total Volume =10% 132

133 Complaint Mechanism Design with a Displacement Constraint 133

134 Multi-Objective Function (a) Phase 1 Phase 2 Max Flexibility Max Stiffness St. Flexibility Pr escribed Flexibility? 134

135 Optimal Configurations (a)

136 Multi-Objective Function (b) Exterior penalty function method Min L NM b g log Stiffness b g b g + log Flexibility log Pr escribed Flexibility m O 2P P r Q 136

137 Optimal Configurations (b)

138 Image Based Design Optimal Topology Practical Structure Image processing 138

139 Verification of Displacement Digital Mesh (Voxelcon) Threshold=100 Threshold=80 Gray scale (0~256) 139

140 Material Micro-structure Design Jun Fonseca : Brazil and O. Sigmund : Denmark 140

141 Material Design Design negative Poisson s ratio by the homogenization design method 141

142 Solution Method 1 1. Specify the Material Constants Desired D 1 = 1 v 12 0 E 1 E 1 ν 21 E 2 1 E G Define the Unit Cell for Microstructural Design Design Variable = Holes a = α 1 α

143 Solution Method 2 3. Apply the Homogenization Method to find the material constants for a design 4. Solve the Inverse Problem Based on the Least Squares Method min design 1 2 D d D 2 with symmetry and periodic conditions 143

144 Bendsoe s Material D =

145 Negative Poisson s Ratio Plane stress Cubic Material (3 independent elastic constants) E= E

146 Application of a Filtering isotropic negative Poisson s ratio microstructures non filtered 60x60 filtered 60x60 Easier Interpretation of the topology without changing of the properties 146

147 Isotropic Material Design Poisson s ratio

148 Jun Fonseca & Anne Marsan 148

149 Three Dimensional Design Tensor Product Orthotropic Two negative and one positive Poisson s ratios E=

150 3D Microstructure Cubic symmetry (3 independent elastic coefficients) Negative Poisson s ratio E = 10 4 E

151 Another 3D Microstructure Full 3D design smaller mesh: 20x20x20 isotropic with negative Poisson s ν=1/4 151

152 Future Material Design Jose M. Gedes : Portugal and J.E. Taylor : USA 152

153 Direct Use of Elasticity Matrix 1 max min a b v, v g f b v g v 2 E subject to ρdd W zd 0 More design variables, more optimum 153

154 Still More Research on Homogenization Design Continuous Development and Enhancement of OPTISHAPE 154

155 Practice Demands many new feature of OPTISHAPE 155

156 Request from Practice Automatic surface recognition Applying the image based modeling method developed by Minako Sekiguchi,we can define a smoothed three dimensional body Then convert to STL format Automatic Mesh Generation for Detailed FE analysis Applying VOXELCON to extract Wire Frame Model, then go to a CAD system & FE soft 156