Design of Optimal Hygrothermally Stable Laminates with Extension-Twist Coupling by Ant Colony Optimization

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1 d Iteratioal Coferece o Egieerig Optimizatio September 6-9, 010, Lisbo, Portugal esig of Optimal Hygrothermally Stable Lamiates with Extesio-Twist Couplig by t Coloy Optimizatio ditya P. pte 1, Robert. Hayes, o Pig Wag 3, Eria. rmaios 4 1 ept. of Radiatio Ocology, Washigto Uiversity School of edicie, St. Louis, US, aditya.apte@gmail.com ept. of erospace Egieerig, Georgia Istitute of Techology, tlata, US, robert.hayes@gatech.edu 3 ept. of echaical ad erospace Egieerig, The Uiversity of Texas, rligto, US, bpwag@uta.edu 4 ept. of echaical ad erospace Egieerig, The Uiversity of Texas, rligto, US, armaios@uta.edu bstract Lamiated composite materials allow for tailorig of their elastic properties through deliberate selectio of the orietatio of their costituet lamias (referred to as ply agles). mog the achievable properties is a couplig betwee i-plae ad out-of-plae deformatio modes. Extesio-Twist is a specific type of couplig which has applicatios i rotor blades to chage the agle of attac with a chage i rotor speed. The optimal Ply-agles for E-T couplig have bee obtaied i previous wor usig Sequetial Quadratic Programmig. Sice the Objective ad Costraits are highly oliear, there exist multiple local ad global solutios. This wor tried to explore through the use of t Coloy Optimizatio (CO), a global optimizer, whether there are ay better desigs tha those obtaied i previously wor. It was verified that the E-T coupligs obtaied i previous wor was global optimal. I additio, CO produced ew global optimal desigs which were similar to the previous desigs i terms of objective ad costrait satisfactio. Keywords: t Coloy Global Composite Couplig 1. Itroductio Couplig of deformatio modes ca be used advatageously to elastically tailor the respose of a composite structure. For example, extesio-twist couplig has applicatios ragig from wid turbie blades to tilt-rotor aircraft. Other types of couplig iclude extesio-bedig, bedig-twist, ad aticlastic couplig. Extesio-twist couplig is oly achievable usig asymmetric stacig sequeces, which ofte produce hygrothermal istabilities, meaig temperature or moisture chages cause bedig or warpig [1]. Past desig methods ivolve obtaiig the desired couplig properties the attemptig to miimize hygrothermal effects []. Seequetial Quadratic Programmig (SQP) has previously bee used to satisfy hygrothermal stability as the primary costrait the ivestigates the achievable elastic properties. The first established family of extesio-twist-coupled composites that retai hygrothermal stability was detailed by Wicler [3] ad give by the stacig sequece [θ / (θ-90) / θ / -θ / (90-θ) / -θ]. (1) These Wicler-type lamiates have bee the idustry stadard sice their itroductio. review of their usage has bee published previously [4]. It has bee discovered, however, that this family does ot cotai all hygrothermally stable lamiates. Further, it has also bee foud that Wicler s family does ot maximize extesio-twist couplig amog lamiates cosistig of eight plies. This wor uses a global optimizer, t Coloy Optimizatio to explore better desigs with hygrothermal stability eforced as give i previous wor [5]. t Coloy Optimizatio is described followed by optimizatio problem formulatio ad results.. t Coloy Optimizatio I the last decade various algorithms have bee researched to solve optimizatio problems based o prey catchig behavior of ats. Such algorithms fid applicatio i robotics, objects clusterig, commuicatio etwor ad combiatorial optimizatio. The ats whose behavior was modeled i this wor are foud i exica desert ad are ow as Pachycodyla apicalis [8]. These ats use visual cues as feedbac mechaism for catchig preys. They use simple priciples to search their preys, both from global ad local viewpoit. Startig from their est, they globally cover a give surface by partitioig it ito may hutig sites. Each at performs a local radom exploratio of its hutig sites ad choice of site is sesitive to the success previously met o this site. These priciples have bee traslated ito a algorithm to obtai global optimal solutio of ucostraied ad costraied optimizatio problems [8, 9]. Followig is a brief algorithm based o the behavior of these ats. 1

2 (1) Choose radomly the iitial est locatio N. This is the startig poit to the algorithm. () For each at a i, i 1...: If a i has less tha p hutig sites i memory, the Create a ew site i the eighborhood of N ad explore the created site Else If the previous site exploratio was successful the Explore agai the same site Else Explore a radomly selected site (amog the p sites i memory) (3) Remove from the ats memories al sites which have bee explored usuccessfully more tha P local a i cosecutive times. (4) Perform recruitmet (best site copyig betwee two radomly selected ats) (5) If more tha T iteratios have bee performed The Chage the est locatio ad Reset memories of all ats (6) Go to () or Stop if a stoppig criteria is satisfied Figure 1 illustrates the global ad local hutig strategy. The parameters which gover the spread of global ad local sites are site ad local. ased o the actual behavior of ats these parameters are set such that site/local = 10. Figure illustrate the covergece of algorithm by movig est to more iterestig areas of the desig space. Readers are ecouraged to refer to [8] for i depth iformatio about recruitmet process ad the performace of algorithm. Lagrage multiplier approach was used to solve the costraied optimizatio problem. Figure 1 Illustratio of Global ad Local search mechaism: (a) Sites s1, s ad s3 are radomly geerated (their maximum distace from the est N is give by site), (b) Squares represet local exploratios of site s1 at a maximum distace local from s1.

3 (a) (b) (c) (d) Figure Illustratio of algorithm. I (a), the est is radomly placed i the search space. The, i (b), hutig sites are radomly created aroud the est with the distributio geerated by the site parameters. I (c), due to the local exploratios, hutig sites move towards more iterestig areas of the search space (here the ceter of the space i our example). The, i (d), the est moves to the best geerated poit so far. Hutig sites are the created agai as i (b), ad so o. 3. Optimizatio Problem Formulatio s give i classical lamiatio theory [6] for specially orthotropic plies, a lamiate s o-mechaical i-plae forces ad shears ad out-of-plae momets ad curvatures are related to the i-plae strais ad out-of-plae curvatures as N N N T, H () Sice hygrothermal stability ca be defied has havig the out-of-plae curvatures equal to zero for ay chage i temperature or moisture, this ca be expressed as 3

4 N N N T, H (3) where ij ad ij, are the i-plae ad couplig stiffess coefficiets respectively ad (_) (T,H) idicates o-mechaical quatities. The o-mechaical i-plae forces ad shears ad out-of-plae momets ad curvatures for a -ply lamiate are give by N N N T, H 1 T 1 T cos cos si T, H T 1 1 cos cos 1 si (4) where θ is the agle of the th ply, ad T 1, T, ad T 3 are solely fuctios of the material properties ad temperature ad moisture chages. Usig (), it ca be prove that the ecessary ad sufficiet coditios to esure hygrothermal stability are either N (T,H) =N (T,H) ad N (T,H) = (T,H) = (T,H) = (T,H) =0 or (5a, b) ij =0. This leads to the followig objective ad costraits for our problem. Objective is to maximize Extesio-Twist term: Costraits for Hygrothermal Stability ca be expressed as: cos 0 (7a) 1si 0 (7b) si 0 (7c) cos 0 (7d) : i 1 N g i (6) 4

5 Figures 3 ad 4 show cotours for objective ad costraits. It ca be see that the objective ad costraits are highly oliear which merit the use of global optimizer lie CO. Figure 3 Variatio of Objective fuctio (eq. 6) w.r.t. d ad 3rd ply agles. Figure 4 Variatio of Costrait fuctios (eq. 7 a-d) w.r.t. d ad 3 rd ply agles 4. Results CO produced multiple global solutios. It was verified that the results obtaied from the previous wor were global optimal. These multiple global solutios would be valuable if ay other desig or maufacturig criteria eeds to be imposed further. Table 1 lists the desigs from previous wor ad the oes obtaied from the curret wor. Table 1: E-T couplig desigs from previous ad curret wor Lamiate Optimal Stacig Sequece (deg): Previous wor 5-Ply [-58.7 /.4 /45 / 78.6 / -31.3] 6-Ply [1. / /-48.7 / 48.7 / 63.8 / -1.] 8-Ply 10-Ply [1.5 / -7.1 / / 9.6 / -9.6 / 57.9 / 7.1 / -1.5] [. /-69.0 /-65.3 / 31.8 / 4.1 / -4.1 / / 65.3 / 69.0 / -.] Optimal Stacig Sequece (deg): ore desigs from CO No ew desig. CO produced same solutio as previous wor [-.95 /8.98 / / / / 68.73] [73.67 / /43.87 / -39. / / 9.09 /33.55 / ] [-75. /7.71 /30.1/ / / 43.9/ -.61/73.41] [15.09 / /4.7/ /-51.8 /53.81 / 61.8 / -.38 / / -14.9] 5

6 I order to compare the ew desigs with the previous oes, robustess of the all the desigs was computed by varyig ply-agles from - to + degrees. Figure 5 shows histograms for objective fuctios (of previous ad ew solutios) with the desigs perturbed by degrees aroud the optimal for 8-ply desig. Figure 6 shows histograms for sum of squared costrait fuctios (of previous ad ew solutios) with the desigs perturbed by degrees aroud the optimal for 8-ply desig. It ca be observed that both the solutios are almost equally robust oe does better with objective ad the other with costraits. Objective fuctio Objective fuctio Figure 5 Robustess of Objective. Previous wor (Left) ad Curret wor (right). Y-axis represets couts umber of Costraits i1 costrait umber of Costraits costrait i Figure 6 Robustess of Costraits. Previous wor (Left) ad Curret wor (right). Y-axis represets couts i1 i 5. Coclusio 1>It was verified that the E-T coupligs obtaied i previous wor were global optimal. > I additio, CO produced ew global optimal desigs which were similar to the previous desigs i terms of objective ad costrait satisfactio. Refereces [1] Hyer,.W., Some Observatios o the Curved Shape of Thi Usymmetric Lamiates, Joural of Composite aterials, Vol. 15, (), 1981, pp [] rmaios, E.., Hooe,., Kamat,., Palmer,. ad Li, J., esig ad Testig of Composite Lamiates with Optimum Extesio-twist Couplig, Composite aterials: Testig ad esig, edited by E.T. Campoeschi Jr, Vol., merica Society for Testig ad aterials, Philadelphia, P, 1993, pp

7 [3] Wicler, S. I., Hygrothermally Curvature Stable Lamiates with Tesio-torsio Couplig, Joural of the merica Helicopter Society, Vol. 31, (7), [4] Hayes, R., Carey, R., ad rmaios, E.., New Class of Hygrothermally Stable Lamiates with Extesio-twist Couplig, i Proceedigs of the HS 65 th ual Forum, Grapevie, TX, US, ay, 009. [5] Cross, R.J., Hayes, R.., ad rmaios, E.., Families of Hygrothermally Stable symmetric Lamiated Composites, Joural of Composite aterials, Vol. 4, (7), 008, pp [6] Joes, R.., echaics of Composite aterials, d Ed., Taylor & Fracis, Philadelphia, P, 1999, pp [7] rmaios, E.., aeev,., ad Hooe,., Fiite-isplacemet alysis of Lamiated Composite Strips with Extesio-Twist Couplig, Joural of erospace Egieerig, Vol. 9, (3), [8] N omarche, G Veturii, Slimae, O how Pachycodyla apicalis ats suggest a ew search algorithm, Future Geeratio Computer Systems (000) [9]. pte ad. P. Wag, Usig power of ts for Optimizatio, I , 45 th I/ SE/ SCE/ HS/ SC Structures, Structural yamics ad aterials Coferece, Palm Sprigs, Califoria, pr. 19-,