Approximate method for cutting pattern optimization of membrane structures

Transcription

1 Proceedngs of the IASS Annual Syposu 217 Interfaces: archtecture.engneerng.scence 25-28th Septeber, 217, aburg, Gerany Annette Bögle, Manfred Grohann (eds.) Approxate ethod for cuttng pattern optzaton of ebrane structures Maoto OSAKI*, Shun SABURI a, Fuyosh AKEDA b *Departent of Archtecture and Archtectural Engneerng, Kyoto Unversty, Kyoto-Dagau Katsura, Nshyo, Kyoto , Japan, ohsa@arch.yoto-u.ac.jp a Departent of Archtecture and Archtectural Engneerng, Kyoto Unversty, Japan b ayo Kogyo Corporaton, Japan Abstract A coputatonally effcent ethod s presented for approxate optzaton of cuttng pattern of ebrane structures. he plane cuttng sheet s generated by nzng the error fro the shape obtaned by reducng the stress fro the desred curved shape. he equlbru shape s obtaned solvng a nzaton proble of total stran energy. he external wor done by the pressure s also ncorporated for analyss of pneuatc ebrane. An approxate ethod s also proposed for analyss of an Ethylene etrafluoroethylene (EFE) fl, where elasto-plastc behavor s odeled as a nonlnear elastc ateral under onotonc loadng condton. Effcency of the proposed ethod s deonstrated through exaples of a frae-supported PolyVnyl Chlorde (PVC) ebrane structure and an ar pressured square EFE fl. Keywords: Mebrane structure, cuttng pattern optzaton, energy nzaton, PVC, EFE, pneuatc ebrane 1. Introducton In the process of desgnng a ebrane structure, t s portant to acheve a oderately unfor stress dstrbuton to prevent fracture and slacenng. Dffculty arses fro the fact that the curved surface s generated by connectng plane sheets (Ohsa and Fujwara [1]). Although there are any ethods for cuttng pattern optzaton, ost of the ethods should carry out fnte eleent analyss any tes (Ohsa and Uetan [2], Bletznger et al. []). For shape desgn of ebrane structures, the ateral s usually supposed to have orthotropc or sotropc elastc behavor. herefore, the equlbru shape analyss for specfed cuttng pattern can be forulated as a forced dsplaceent proble, whch can be solved by nzaton of the total stran energy. owever, Ethylene etrafluoroethylene (EFE) fl has an elasto-plastc property; therefore, t s dffcult to optze the shape usng a gradent-based optzaton algorth. In ths study, we present a coputatonally effcent teratve ethod for approxate optzaton of cuttng pattern of ebrane structures. he plane cuttng sheet s generated by nzng the error fro the shape obtaned by reducng the stress fro the desred curved shape, whch s dscretzed nto trangular fnte eleents. he equlbru shape correspondng to the specfed cuttng pattern s obtaned by energy nzaton. he external wor done by the ar pressure s also ncorporated for analyss of pneuatc ebrane structures. he proposed ethod s extended to desgn of EFE fl. Effcency of the proposed ethod s deonstrated through exaples of a frae-supported PolyVnyl Chlorde (PVC) ebrane structure and an ar-pressured square EFE fl. 2. Energy nzaton for equlbru shape analyss Consder a curved ebrane structure dscretzed by trangular fnte eleents wth constant stress n plane stress state. Let D denote the consttutve atrx defnng the sotropc or orthotropc Copyrght 217 by Ohsa, Sabur, and aeda Publshed by the Internatonal Assocaton for Shell and Spatal Structures (IASS) wth persson.

2 Proceedngs of the IASS Annual Syposu 217 Interfaces: archtecture.engneerng.scence elastc ateral property of the ebrane. In the followng, stress s evaluated as the force per unt length of secton. Relaton between the stran vector ε ( 1, 2, ) and the stress vector σ (,, ) of the th eleent s wrtten as 1 2 σ Dε (1) where the subscrpts 1 and 2 n stress and stran coponents ndcate the values n two prncpal drectons. Fgure 1:Local coordnates, node nubers, prncpal drectons, and deforaton of trangular eleent. 2 Let u (,, ) u u v denote the relatve dsplaceents of nodes, as shown n Fg. 1, n local (x,y)- coordnates of eleent, and ( x, y ) defnes the prncpal drectons. he stran-dsplaceent p p relaton s wrtten usng atrx C as ε Cu (2) he vecor consstng of the global (, ) -coordnates of the nodes of cuttng sheet s denoted by 2n, where n s the nuber of nodes. he process of fndng the equlbru shape s regarded as a forced dsplaceent proble of the sheet to the specfed boundary of the curved surface. herefore, the stran energy S( ) of the ebrane s regarded as a functon of as 1 1 S( ) Aε( ) Dε( ) () 2 where s the nuber of eleents, and A ( ) s the area of the th eleent. he equlbru shape s obtaned by nzng S( ) under an approprate boundary condtons. If ar pressure p s gven, the pressure potental energy W ( ) s gven as (Bouzd and Le van [4], Fscher [5]) W( ) pv( ) (4) where V ( ) s the volue of ebrane structure. Usng the dvergence theore, Bouzd and Le van [4] derved the followng expresson W * ( ) for a ebrane dscretzed by trangular fnte eleents: 2

3 Proceedngs of the IASS Annual Syposu 217 Interfaces: archtecture.engneerng.scence * p A 1 W ( ) n ( ) ( ) (5) where n ( ) and ( ) are the unt noral vector and coordnate vector of the center of gravty of the th eleent. owever, the ter 1/ n the rght-hand-sde of Eq. (5) s not necessary, because we should consder shape varaton only n the noral drecton of surface. Actually they dd not use 1/ n the nuercal exaples. Let denote the coordnate vector of node. Varaton of W ( ) s drectly coputed as W( ) pv( ) p 1 I 1 I V ( ) A p n( ) ( ) pan( ) 1 I (6) where I s the set of nodes of eleent, and 1 I (7) has been used. he equlbru shape s obtaned by nzng the total potental energy ( ) defned as ( ) S( ) W( ) (8) Dfferentaton of ( ) wth respect to leads to ( ) ε( ) ( ) Aε( ) D pan( ) 1I 1I n ( ) pa ( ) 1 I ( ) p n( ) A A p A 1I 1 1I ε ( ) D n ( ) ( ) By contrast, the equlbru equaton s gven as ε ( ) p Aε( ) D An( ) 1I 1 herefore, the thrd ter, denoted by e, n the rght-hand-sde of Eq. (9) reans as an addtonal ter, whch s rewrtten as (9) (1)

4 Proceedngs of the IASS Annual Syposu 217 Interfaces: archtecture.engneerng.scence n ( ) epa ( ) 1 I n n ( ) pa ( ) 1 K where K s the set of eleents connected to node. Although detals are otted, the absolute values of coponents of e are suffcently sall, f the surface s dscretzed nto suffcently any eleents.. Approxate optzaton of cuttng pattern We propose a sple update rule of the stress paraeters called reducton stress for approxate cuttng pattern optzaton. he ethod s based on the nverse process of generatng a plane sheet fro a curved surface by reducng the stress (Ohsa and Fujwara [1]). he algorth s llustrated n Fg. 2, and suarzed as follows: (1) arget stress Equlbru shape analyss Stress at equlbru Update Convergence Ideal target stress Fgure 2: Schee of approxate cuttng pattern optzaton. Step 1: Assgn the target equlbru shape, boundary condton, target stress, and generate trangular eshes on the target surface. Copute the edge lengths L 1, L 2, and L of the th trangle of * * the target equlbru shape. Specfy the deal target stresses 1, 2, and * ( ), and ntalze the step counter s. Step 2: Specfy the reducton stresses ˆ s 1, ˆ s s 2, and ˆ ( ) n prncpal drectons. Reove the stress fro the trangular eleents on the equlbru shape, and copute the unstressed edge lengths L 1, L 2, and L. Step : Assgn a plane P near the target surface, and project the trangular esh on P to generate the P ntal esh of the cuttng panel. Let L1 ( ) P, L 2 ( ), and L P ( ) denote the edge lengths of the trangular eleents on P, whch are functons of the vector of nodal coordnates of the trangular esh of P. Solve the followng proble to nze the error n the edge lengths: Mnze 1 1 P 2 F( ) ( L ( ) L ) where s a weght paraeter. In the followng exaples, 1/ L to prevent reversal of short edge. 4

5 Proceedngs of the IASS Annual Syposu 217 Interfaces: archtecture.engneerng.scence Step 4: Carry out equlbru shape analyss by nzng the total stran energy or the total potental s s s energy to fnd the nodal coordnates on surface and the stresses 1, 2, and. Step 5: Let s s 1, and odfy the reducton stresses 1 and 2 as ˆ ˆ ( ), s 1 s * s 1 1 c 1 1 ˆ s ˆ s ˆ ˆ ( ), s 1 s * s 2 2 c 2 2 s 1 ˆ ( ) Also update the target surface by the equlbru surface obtaned n Step 4. Step 6: Go to Step 2, f ternaton condton s not satsfed. (11) 4. Analyss of EFE fl EFE fl s usually odeled as elasto-plastc ateral wth von Mses yeld crteron (Coelho et al. [6], oshno and Kato [7]). he relaton between stress and stran n unfor tenson s often odeled as blnear relaton, whch s dentfed by experents as shown n Fg., where and are the * yeld stress and stran. If the target stress s larger than, alost unfor stress dstrbuton can be expected, because the stffness after yeldng s saller than the ntal elastc stffness. σ * σ σ ε Fgure : Relaton between stress and stran of EFE sheet under unfor tenson. ε We consder a onotonc loadng process ncreasng the pressure to reach the equlbru shape. Although the stffness after yeldng depends on the stress rato between 1 and 2, we assue the deal state satsfyng 1 2 for whch the relaton between the equvalent stress and equvalent stran s obtaned by experent. Snce we consder a onotonc loadng process, the equlbru shape of an ar-pressured EFE fl can be obtaned by nzng ( ) S( ) W( ) wth 1 1 S( ) A ε σ ε ε σ σ (12) 2 5. Nuercal exaples he proposed algorth of approxate cuttng pattern optzaton s appled to a frae-supported PVC ebrane and an ar pressured EFE fl. he optzaton probles are solved usng sequental quadratc prograng pleented n SNOP Ver. 7 (Gll et al. [8]). 5

6 Proceedngs of the IASS Annual Syposu 217 Interfaces: archtecture.engneerng.scence 5.1. Frae supported PVC ebrane Consder an P-type frae-supported ebrane (Model 1) as shown n Fg. 4. he proporton of the odel s W1 1.W, W2 1.W, and.2w. he ateral property s assued to be orthotropc elastc. oung s odulus n warp and fll drectons are N/ and N/, respectvely. he shear odulus s 24.2 N/, and Posson s ratos are.51 and.55. he ebrane s dvded nto two cuttng sheets as shown n Fg. 5(a). he total nubers of nodes and eleents are 16 and 24, respectvely. W 2 D C Z Z W 1 A B W 1 W 1 W 2 W 2 Fgure 4: An P-type frae supported ebrane structure (Model 1). (a) (b) Fgure 5: Cuttng sheets of Model 1: (a) trangular esh, (b) trangular esh projected to -plane before optzaton (blue) and cuttng sheet after optzaton (red). he target stress s. N/ n both warp and weft drectons. he hstory of average, axu, nu values and standard devaton of stress s lsted n able 1, where 1 and 2 denote the drectons of warp and weft, respectvely. As seen fro the table, the nu value ncreases fro a negatve value to a postve value. he average value gradually converges to the target value. If we stop at the 2th step, the cuttng pattern s as shown n Fg. 5(b). Note that the cuttng pattern s close to the trangular plan of the half part of surface, whch eans that the area of cuttng sheet s saller than the surface area. he stress dstrbuton at the eghth step s shown n Fg. 6. As seen fro the fgure, the stresses n warp and weft drectons are alost unfor except n the area near corners. 6

7 Proceedngs of the IASS Annual Syposu 217 Interfaces: archtecture.engneerng.scence able 1: story of average, axu, nu values and standard devaton of stress (N/) of Model 1. Step Step 5 Step 1 Step 15 Step 2 Drecton Average Max. Mn. Std. Dev. (a) (b) Fgure 6: Stress dstrbuton after 2 steps of optzaton: (a) warp drecton, (b) weft drecton Ar pressured EFE fl Consder a square ar-pressured EFE sheet as shown n Fg. 7, where the rato of to W s.58. he elastc ateral property s sotropc. oung s odulus s N/, hardenng coeffcent s 1.4 N/, elastc shear odulus s 55.2 N/, shear odulus after yeldng s.6, Posson s rato s.45, and the stress and stran at yeldng s.2 N/ and.2, respectvely. he specfed ar pressure s 1. N/ 2, and the target stress s 4. N/. In ths case, the radus of curvature s 2 4./1. = 8., f the surface s sphercal. Z B C W A D W Fgure 7: Ar-supported EFE sheet (Model 2). 7

8 Proceedngs of the IASS Annual Syposu 217 Interfaces: archtecture.engneerng.scence able 2: story of average, axu, nu values and standard devaton of stress of Model 2. Step Step 2 Step 4 Step 7 Step 1 Drecton Average Max Mn Std. Dev he hstory of average, axu, nu values and standard devaton of stress s lsted n able 2, where 1 and 2 denote the drectons n - and -drectons on the global coordnates of the cuttng sheet. As seen fro the table, EFE has a better accuracy than PVC, because the stffness at the target stress of EFE s saller than that of PVC. he cuttng pattern and stress dstrbuton after 1 steps are shown n Fgs. 8(a) and (b), respectvely. It s seen fro these results that the cuttng pattern s qute dfferent fro the trangular shape, because the curvature of the surface s very large. (a) (b) Fgure 8: Cuttng sheets and stress dstrbuton of Model 2: (a) trangular esh, (b) trangular esh projected to -plane before optzaton (blue) and cuttng sheet after optzaton (red). 6. Conclusons An approxate ethod has been presented for cuttng pattern optzaton of ebrane structures. he conclusons obtaned fro ths study are suarzed as follows: 1. Approxate plane cuttng pattern for the curved surface wth specfed target stress can be obtaned by reovng the stress of each trangular eleent and nzng the error of edge length for connectng the trangular eleents on a plane. 2. By adjustng the stress paraeter called reducton stress, approxate optal cuttng pattern can be obtaned after several teratons of cuttng pattern generaton and equlbru shape analyss, whch s forulated as an optzaton proble of nzng the stran energy under forced dsplaceents at the boundary. 8

9 Proceedngs of the IASS Annual Syposu 217 Interfaces: archtecture.engneerng.scence. he equlbru shape of a pneuatc ebrane structure can also obtaned by nzng the total potental energy ncludng the wor done by the ar pressure. 4. he ateral property of an EFE sheet can be odeled as blnear nonlnear elastc n the process of onotoncally ncreasng the pressure to for the equlbru shape. Acnowledgeents hs study s partally supported by JSPS KAKENI No. 16K148. References [1] M. Ohsa and J. Fujwara, Developablty condtons for prestress optzaton of a curved surface, Cop. Meth. Appl. Mech. Engng., Vol. 192, pp , 2. [2] M. Ohsa and K. Uetan, Shape-stress trade-off desgn of ebrane structures for specfed sequence of boundary shapes, Cop. Meth. Appl. Mech. Engng., Vol. 182, pp. 7-88, 2. [] K.-U. Bletznger, R. Wüncher, F. Daoud and N. Caprub, Coputatonal ethods for for fndng and optzaton of shells and ebranes, Coput. Meth. Appl. Mech. Eng., Vol. 194, pp , 25. [4] R. Bouzd and A. Le van, Nuercal soluton of hyperelastc ebranes by energy nzaton, Cop. Struct., Vol. 82, pp , 24. [5] D. Fscher, Confguraton dependent pressure potentals, J. Elastcty, Vol. 19, pp , [6] M. Coelho, D. Roehl and K.-U. Bletznger, Nuercal sulaton of burst-test of an EFE ebrane, Proc. IASS Syposu, Brasla, 215. [7]. oshno and S. Kato, Vscous characterstcs of EFE fl sheet under equal baxal tensons, Proceda Eng., Vol. 155, pp , 216. [8] P. E. Gll, W. Murray and M. A. Saunders, SNOP: An SQP algorth for large-scale constraned optzaton. SIAM J. Opt., Vol. 12, , 22. 9