Topology Optimization of Tensegrity Structures Based on Nonsmooth Mechanics. Yoshihiro Kanno. November 14, 2011 ACOMEN 2011

Transcription

1 Topology Optimization of Tensegrity Structures Based on Nonsmooth Mechanics Yoshihiro Kanno November 14, 2011 ACOMEN 2011

2 tensegrity definition tension + integrity [Fuller 75] [Emmerich], [Snelson] pin-jointed structure strut: compressive force cable: tensile force self-equilibrium condition with prestresses discontinuity of struts each node has at most one strut

3 tensegrity definition tension + integrity [Fuller 75] [Emmerich], [Snelson] pin-jointed structure strut: compressive force cable: tensile force self-equilibrium condition with prestresses discontinuity of struts each node has at most one strut

4 tensegrity definition tension + integrity pin-jointed structure strut compressive force cable tensile force application cable dome & civil engineering structure deployable structure [Tibert & Pellegrino 02] [Sultan & Skelton 03] antenna-mast structure [Djouadi, Motro, Pons & Crosnier 98] cell cytoskeleton model [Stamenović & Coughlin 00] [Volokh, Vilnay & Belsky 00] [Wandling, Cañadas & Chabrand 03] discrete mathematics [Jórdan, Recski & Szabadka 09]

5 exploring new topology topology connectivity label of member strut or cable

6 exploring new topology topology connectivity label of member strut or cable given a topology find locations of nodes group-theoretic symmetry [Connelly & Terrell 95] [Connelly & Back 98] rotational symmetry [Sultan, Corless & Skelton 02] [Masic, Skelton & Gill 05] optimization [Zhang, Ohsaki & Kanno 06] [Masic, Skelton & Gill 06] given locations of nodes find a topology (!)

7 aim topology optimization prestresses discontinuity of struts upper & lower bounds equilibrium state under external load assuming linear elasticity slack/taut behaviors of cables nonsmooth mechanics upper & lower bounds for axial forces compliance constraint

8 aim topology optimization prestresses discontinuity of struts upper & lower bounds equilibrium state under external load assuming linear elasticity slack/taut behaviors of cables nonsmooth mechanics upper & lower bounds for axial forces compliance constraint NM & convex optimization (, CRC Press, 2011) cable, membrane, masonry structure, etc.

9 label of member label (x i,y i ) = (1,0) i S (strut) (x i,y i ) = (0,1) i C (cable) (x i,y i ) = (0,0) i N (none) ground structure method

10 label of member label (x i,y i ) = (1,0) i S (strut) (x i,y i ) = (0,1) i C (cable) (x i,y i ) = (0,0) i N (none) discontinuity constraint on struts each node has at most one strut

11 label of member label (x i,y i ) = (1,0) i S (strut) (x i,y i ) = (0,1) i C (cable) (x i,y i ) = (0,0) i N (none) discontinuity constraint on struts x i 1, i E(n j ) n j (node) (4) (3) (2) E(n j ) = {1,2,3,4} n j (1)

12 label of member label (x i,y i ) = (1,0) i S (strut) (x i,y i ) = (0,1) i C (cable) (x i,y i ) = (0,0) i N (none) discontinuity constraint on struts x i 1, i E(n j ) n j (node) cross-sectional area

13 label of member label (x i,y i ) = (1,0) i S (strut) (x i,y i ) = (0,1) i C (cable) (x i,y i ) = (0,0) i N (none) discontinuity constraint on struts x i 1, i E(n j ) n j (node) cross-sectional area ξ s if i S a i = ξ c if i C 0 if i N a i = ξ s x i +ξ c y i

14 constraints on prestress prestress: q i q s q s i S q i q c i C q c q i i N q i (x i,y i ) = (1,0) i S (strut) (x i,y i ) = (0,1) i C (cable) (x i,y i ) = (0,0) i N (none)

15 constraints on prestress prestress: q i q s q s i S q i q c i C q c q i i N q i (q s +q c )(1 y i )+q c q i q c y i q s x i q i (q c +q s )(1 x i ) q s linear inequalities (with 0 1 variables) (x i,y i ) = (1,0) i S (strut) (x i,y i ) = (0,1) i C (cable) (x i,y i ) = (0,0) i N (none)

16 constraints under the external load axial force: s i Hs = f (force-balance) (x i,y i ) = (1,0) i S (strut) (x i,y i ) = (0,1) i C (cable) (x i,y i ) = (0,0) i N (none)

17 constraints under the external load axial force: s i Hs = f (force-balance) constitutive law (M 1 0: constant) s i = k si c si +k ci c ci M 1 x i c si, M 1 y i c ci (x i,y i ) = (1,0) i S (strut) (x i,y i ) = (0,1) i C (cable) (x i,y i ) = (0,0) i N (none)

18 constraints under the external load axial force: s i Hs = f (force-balance) constitutive law (M 1 0: constant) compatibility relation idea: remark: s i = k si c si +k ci c ci M 1 x i c si, M 1 y i c ci M 2 (1 x i ) c si h T i u, M 2 (1 y i ) c ci h T i u, [Stolpe & Svanberg 03] [Rasmussen & Stolpe 08] [Kanno & Guo 10] M 1 is related to the lower & upper bounds for forces

19 constitutive law of cable slack/taut behavior s [slack] 0 [taut] 1 k c c s i : axial force c i : elongation k i : stiffness

20 constitutive law of cable slack/taut behavior s [slack] 0 [taut] 1 k c c s i = k i e i s i : axial force c i : elongation k i : stiffness (e i : elastic elongation) complementarity form s i 0, e i c i 0, s i (e i c i ) = 0

21 constitutive law of cable slack/taut behavior s [slack] 0 [taut] 1 k c c s i = k i e i s i : axial force c i : elongation k i : stiffness (e i : elastic elongation) using 0 1 variable (z i {0,1})

22 constitutive law of cable slack/taut behavior s [slack] 0 [taut] 1 k c c s i = k i e i s i : axial force c i : elongation k i : stiffness (e i : elastic elongation) using 0 1 variable (z i {0,1}) 0 e i c i M(1 z i ), 0 s i Mz i z i = 0 : slack s i = 0 z i = 1 : taut e i = c i

23 MIP formulation min # of cables s.t. constraints on prestresses (self-equilibrium condition) (discontinuity of struts) (lower & upper bounds) constraints under the external load (force-balance equation) (discrete cross-sectional areas) (compatibility relation) (slack/taut behaviors of cables) constraint on compliance avoiding intersecting members 0 1 variables: indicating (S), (C), or (N) indicating slack or taut

24 mixed integer programming (MIP) global optimization application min c T x 1x+c T 2t s.t. A 1 x+a 2 t b t {0,1} m variables : x (continuous) t i = 0 or 1 (integer)

25 mixed integer programming (MIP) global optimization min c T x 1x+c T 2t s.t. A 1 x+a 2 t b t {0,1} m variables : x (continuous) t i = 0 or 1 (integer) branch-and-bound method, cutting plane method branch-and-cut method software packages [CPLEX], [GUROBI], [SCIP], [GLPK], etc application

26 mixed integer programming (MIP) global optimization application min c T x 1x+c T 2t s.t. A 1 x+a 2 t b t {0,1} m variables : x (continuous) combinatorial optimization t i = 0 or 1 (integer) discrete optimization of structures [Stolpe & Svanberg 03], [Rasmussen & Stolpe 08] worst-scenario analysis of uncertain structures [Kanno & Takewaki 07], [Guo, Bai & Zhang 08]

27 ex.) 3-layer example 10 nodes 45 members (perfect graph) CPLEX (ver. 11.2) equilateral triangle square equilateral triangle

28 ex.) 3-layer example 10 nodes 45 members 3 struts 10 cables (w, f) = (10J,10N), (d s,d k ) = (1,0)

29 ex.) 3-layer example 10 nodes 45 members 4 struts 15 cables (w, f) = (10J,100N), (d s,d k ) = (1,0)

30 ex.) 3-layer example 10 nodes 45 members 4 struts 17 cables (w, f) = (20J,200N), (d s,d k ) = (3,0)

31 ex.) 3-layer example 10 nodes 45 members 4 struts 17 cables 3 cables slacken

32 ex.) 3-layer example 10 nodes 45 members 5 struts 24 cables (w, f) = (10J,200N), (d s,d k ) = (5,0)

33 ex.) 3-layer example 10 nodes 45 members 5 struts 24 cables 4 cables slacken

34 ex.) 5-layer example 16 nodes 93 members CPLEX (ver. 11.2) equilateral triangle equilateral triangle square equilateral triangle equilateral triangle

35 ex.) 5-layer example 16 nodes 93 members 7 struts 31 cables (w, f) = (50J,50N) (d s,d k ) = (2,0) 1 cable slackens

36 ex.) 5-layer example 16 nodes 93 members 7 struts 31 cables (w, f) = (50J,50N) (d s,d k ) = (2,0) 1 cable slackens

37 ex.) optimization against self-weight load 18 nodes 116 members 2 triangles 3 squares

38 ex.) optimization against self-weight load 18 nodes 116 members 9 struts 41 cables min. compliance under self-weight load (d s,d k ) = (2,0)

39 ex.) optimization against self-weight load 18 nodes 116 members 9 struts 42 cables min. compliance under self-weight load (d s,d k ) = (3,0)

40 conclusions tensegrity self-equilibrium structure discontinuity constraint on struts topology: connectivity of struts & cables optimization MIP formulation label of member (S) strut, (C) cable, or (N) none integer variable stress constraints & compliance constraint cable taut or slack integer variable requires no information of topology in advance