Mest 16888 Topology Optimization ESD.J7 (1)The evolutionary method Xie and Steven (1993) The homogenization method Bendsoe and Kikuchi (1988)-UniV of Michigan (3)Density approach Yang and Chuang(1994) cell-based approach Ndv>1000 o Massachusetts Institute of Technology-Dr Il Yong Kim
12 © Massachusetts Institute of Technology – Dr. Il Yong Kim (1) The evolutionary method Xie and Steven (1993) (2) The homogenization method Bendsoe and Kikuchi (1988) – Univ. of Michigan (3) Density approach Yang and Chuang (1994) * cell-based approach Topology Optimization Ndv > 1000
Mest 16888 Topology optimization ES077 8 Homogenization method Density approach (1)Design variables: density of each cell (2)The constitutive equation is expressed in terms of Young's modulus 「。or(=)s(2=JrFd How to define the relation between the density and Young's modulus? o Massachusetts Institute of Technology-Dr Il Yong Kim
14 © Massachusetts Institute of Technology – Dr. Il Yong Kim Topology optimization Homogenization method / Density approach (1) Design variables: density of each cell (2) The constitutive equation is expressed in terms of Young’s modulus ³ ³ : * : * adm ij ij i i V (z)H (z)d F z d z Z How to define the relation between the density and Young’s modulus? U ? E
Mest 16888 Topology optimization ESD.J7 Homogenization method Infinitely many micro cells with voids The porosity of this material is optimized using an optimality criterion procedure Each material may have different void size and orientation o Massachusetts Institute of Technology-Dr Il Yong Kim
15 © Massachusetts Institute of Technology – Dr. Il Yong Kim Topology optimization Homogenization method - Infinitely many micro cells with voids - The porosity of this material is optimized using an optimality criterion procedure - Each material may have different void size and orientation
Mest 16888 Topology optimization ES077 Homogenization method Relationship between density and elastic modulus Design variables: a1, a2, 0 For 2-d elastic problem Solid part area: 34s J2a,a,)dQ2 D1D,0 D2D20 00D D=D(a1,a2,6) Review papers: Hassani B and Hinton E(1998) o Massachusetts Institute of Technology-Dr Il Yong Kim
16 © Massachusetts Institute of Technology – Dr. Il Yong Kim Topology optimization Homogenization method - Relationship between density and elastic modulus - Design variables : a1, a2, T 1 2 11 11 12 11 22 12 22 22 12 66 12 1 2 For 2-D elastic problem, Solid part area : (1 ) 0 0 0 0 ( , ,) s aa d D D D D D D Da a V H V H V H T : : : ½ ª º ½ ° ° °° « » ® ¾ ®¾ « » ° ° °° « » ¯ ¿ ¬ ¼¯ ¿ ³ * Review papers : Hassani B and Hinton E (1998) 1 1 y Y X x θ 1-a2 1-a1
Mest 16888 Topology optimization ESD.J7 ☆ Density approach Artificial material Design variable: density u3oEmgE3 Low computational cost Simple in its idea Density p E E 0≤p≤1 o Massachusetts Institute of Technology-Dr Il Yong Kim
18 © Massachusetts Institute of Technology – Dr. Il Yong Kim Artificial material - Design variable : density Topology optimization Density approach U o , 0 d U d1 n E E Density U Young’s modulus E/E 0 Material cost 0 1 1 Low computational cost Simple in its idea