Exploring Frame Structures with Negative Poisson s Ratio. via Mixed Integer Programming. University of Tokyo

Transcription

1 Exploring Frame Structures with Negative Poisson s Ratio via Mixed Integer Programming Rui Kureta, Yoshihiro Kanno University of Tokyo ( Currently: Tokyo Keiki, Inc.) July 21, 2014 (WCCM XI, ECCM V, ECFD VI)

2 structures with negative Poisson s ratio...expand transversely when stretched longitudinally.

3 mixed-integer programming m-i linear prog.: mixed : min c T x+r T y s.t. a T i x+g T i y b i x {0,1} n, y R l x j : integer (discrete) variable y l : real (continuous) variable replace x {0,1} n with 0 x 1 linear prog. relaxation (i = 1,...,m), can be solved with, e.g., a branch-and-bound method

4 materials with NPR (= auxetic materials) naturally occurred materials cadmium [Li 76] single crystal of arsenic [Gunton & Saunders 72] layered ceramics [Song, Zhou, Xu, Xu, & Bai 08] artificial materials polymer foam [Lakes 87] re-entrant structure [Friis, Lakes, & Park 88] [Evans, Alderson, & Christian 95]

5 materials with NPR (= auxetic materials) naturally occurred materials cadmium [Li 76] single crystal of arsenic [Gunton & Saunders 72] layered ceramics [Song, Zhou, Xu, Xu, & Bai 08] artificial materials polymer foam [Lakes 87] re-entrant structure [Friis, Lakes, & Park 88] [Evans, Alderson, & Christian 95] (possible) applications tunable filters [Alderson et al. 00] fasteners [Choi & Lakes 91] artificial intervertebra discs [Martz, Lakes, Goel, & Park 05]

6 optimization to achieve NPR existing methods: truss model [Sigmund 94] continuum & homogenization method [Larsen, Sigmund, & Bouwstra 97] [Schwerdtfeger et al. 11] continuum & genetic alg. [Matsuoka, Yamamoto, & Takahara 01]

7 optimization to achieve NPR existing methods: truss model [Sigmund 94] continuum & homogenization method [Larsen, Sigmund, & Bouwstra 97] [Schwerdtfeger et al. 11] continuum & genetic alg. [Matsuoka, Yamamoto, & Takahara 01] Local stress constraints were not considered. Post-processing before manufacturing: gray areas & hinges.

8 optimization to achieve NPR existing methods: truss model [Sigmund 94] continuum & homogenization method [Larsen, Sigmund, & Bouwstra 97] [Schwerdtfeger et al. 11] continuum & genetic alg. [Matsuoka, Yamamoto, & Takahara 01] our method: periodic frame structure stress constraints & pre-determined beam sections no hinges, no thin members

9 optimization to achieve NPR existing methods: truss model [Sigmund 94] continuum & homogenization method [Larsen, Sigmund, & Bouwstra 97] [Schwerdtfeger et al. 11] continuum & genetic alg. [Matsuoka, Yamamoto, & Takahara 01] our method: periodic frame structure stress constraints & pre-determined beam sections manufacturability (no post-processing) global optim. no hinges, no thin members an idea MILP for truss [Rasmussen & Stolpe 08] [K. & Guo 10]

10 problem setting assume periodicity & symmetry unit cell: planar frame structure design variables: sections of members 2L 2L 2L 2L unit cell 2L 2L 2L 2L periodicity

11 problem setting assume periodicity & symmetry unit cell: planar frame structure design variables: sections of members x i : integer variable x i = 1 Member i has pre-determined section. x i = 0 Member i is removed. sym. L/2 2L member i L/2 2L unit cell L/2 L/2 design domain

12 problem setting assume periodicity & symmetry unit cell: planar frame structure design variables: sections of members more general setting: catalog = {ā 1,ā 2,...,ā P } x ip = 1 Member i has pre-determined section ā p. x i1 = = x ip = 0 Member i is removed. x ip 1 p (section) = p x ip ā p

13 optimization problem max u out s.t. equilibrium eq. specifying u in stress constraints avoiding member intersection u in u out u out u in can be reduced to m-i linear prog.

14 fictitious boundary cond. u in max. u out s. t. u in is specified. null structure is optimal u out = + L/2 L/2 u out L/2 L/2

15 fictitious boundary cond. u in max. u out s. t. u in is specified. null structure is optimal u out = + L/2 L/2 u out L/2 L/2 f out fictitious cond.: fix node in apply f out at node out require internal forces satisfying force-balance eq.

16 reduction to MIP (1) equilibrium eq.: Ku = f stiffness matrix: K = m 3 k ij b ij b T ij (b ij : const. vec.) i=1 j=1 member stiffnesses: k ij = k ij x i ( k ij : const.) integer variable: x i = { 1 if member i exists 0 if member i is removed

17 reduction to MIP (1) equilibrium eq.: Ku = f stiffness matrix: K = m 3 k ij b ij b T ij (b ij : const. vec.) i=1 j=1 member stiffnesses: k ij = k ij x i ( k ij : const.) integer variable: x i = { 1 if member i exists 0 if member i is removed Reformulate all constraints as linear constraints.

18 reduction to MIP (2) equil. eq. Ku = f m 3 k ij v ij b ij = f i=1 v ij = j=1 { b T iju if x i = 1 ( ) 0 if x i = 0 ( ) (force-balance) (compatibility) stress constraints: q i (u) q y i + m(e) i (u) m y i 1 ( )

19 reduction to MIP (2) equil. eq. Ku = f m 3 k ij v ij b ij = f i=1 v ij = j=1 { b T iju if x i = 1 ( ) 0 if x i = 0 ( ) (force-balance) (compatibility) stress constraints: q i (u) q y i + m(e) i (u) m y i 1 ( ) ( ) & ( ) k i1 q y i v i1 + l i 2 k i2 m y i v i2 + k i3 m y i v i3 x i ( ) v ij b T iju M(1 x i ) (M 0 : const.)

20 goal: MIP formulation max u out 3 s.t. k ij v ij b ij = f, i E j=1 v ij b T iju M(1 x i ), j, i, k i1 q y i v i1 + l i 2 x ip {0,1}, k i2 m y i v i2 + k i3 m y i v i3 x ip, i, i. avoiding member intersection: x i +x i 1 i i only linear constraints (and integer constraints).

21 ex.) global optimization 66 candidate members Timoshenko beam element solver: CPLEX ver mm 12 mm beam cross-section (width) (thickness) = mm (width) (thickness) = mm

22 ex.) global optimization 66 candidate members Timoshenko beam element solver: CPLEX ver mm 12 mm beam cross-section (width) (thickness) = mm ν = (width) (thickness) = mm ν =

23 ex.) global optimization 12 mm optimal base cell 12 mm beam cross-section (width) (thickness) = mm ν = (width) (thickness) = mm ν =

24 fabricated optimal structure fabricated by photo-etching stainless steel thickness of beams: 0.5 mm, width: 0.75 mm

25 fabricated optimal structure fabricated by photo-etching stainless steel thickness of beams: 0.5 mm, width: 0.75 mm undeformed state deformed state

26 local search: a heuristics MIP approach global optim. limitation of prob. size towards large probs. local search with MIP [Stolpe & Stidsen 07] [Svanberg & Werme 07] solve MIP within neighborhood N(x,r) { m } N(x,r) = x x i x i r r : radius x : incumbent solution i=1

27 ex.) local search 12 mm 12 mm candidate members initial solution ν = local search 748 members r = 4 (radius of neighborhood) no guarantee of global optimality

28 ex.) local search 12 mm 12 mm candidate members convergence history: initial solution ν = final design ν = st step 2nd step 3rd step

29 conclusions design of counterintuitive structures Optimization might be a helpful tool. structures with negative Poisson s ratio topology optimization of frame structures max. the output displacement mixed-integer programming selection of member cross-sections integer variables from a catalog of available sections (incl. void) stress constraints no hinges, no thin members, no post-processing