Comparison of parameterization schemes for solving the discrete material optimization problem of composite structures

Transcription

1 10 th World Congress on Structural and Multidiscilinary Otiization May 19-24, 2013, Orlando, Florida, USA Coarison of araeterization schees for soling the discrete aterial otiization roble of coosite structures Pierre Duysinx 1, Maria Guillero 1, Tong Gao 2, Michael Bruyneel 3 1 Uniersity of Liege (Ulg), Belgiu, {.duysinx, guillero}@ulg.ac.be 2 Northwestern Polytechnical Uniersity (NWPU), Xi an, China, gaotong@nwu.edu.cn 3 LMS-SAMTECH, Belgiu, ichael.bruyneel@lsint.co 1. Abstract Otial design of coosite structures can be forulated as an otial selection of aterial in a list of different lainates. Based on the seinal work by Stegann and Lund, the otial roble can be stated as a toology otiization roble with ultile aterials. The research work carries out a large inestigation of different interolation and enalization schees for the otial aterial selection roble. Besides the classical Design Material Otiization (DMO) schee and the recent Shae Function with Penalization (SFP) schee by Bruyneel, the research introduces a generalization of the SFP aroach using a bi-alue coding araeterization (BCP) by Gao, Zhang and Duysinx. The aer roides a coarison of the different araeterization aroaches. It also rooses alternatie enalization schees and it inestigates the effect of the ower enalization. Finally, we discuss the solution asects in the ersectie of soling large-scale industrial alications. The conclusions are illustrated by a nuerical alication for the coliance axiization of an in-lane coosite ly. 2. Keywords: Coosite Structure Otiization, Toology Otiization, Discrete Material Otiization Sequential Conex Prograing. 3. Introduction Taking the best of coosite aterial high strength and stiffness to weight ratios is essential to iroe the efficiency of airlanes, ground ehicles, wind turbines and renewable energy systes. To this end, the discrete otial orientation otiization is a fundaental roble of coosite structure otiization, which can be alied to sole different robles of interest, for instance, the otial orientation distribution roble of lies, or the otial stacking sequence of ultile-layer lainated structures. The Discrete Material Otiization (DMO) aroach roosed by Stegann and Lund [10] has oened a breakthrough in coosite otiization. The fundaental idea is to forulate the coosite otiization roble as an otial aterial selection roble in which the different lainates and ly orientations are considered as different aterials and to sole it as a toology otiization roble using continuous ariables. This aroach can be regarded as a generalization of the ulti-hase toology otiization roosed in Thosen [12] and in Sigund and Torquato [9]. To transfor the discrete roble into a continuous one, one introduces a suitable araeterization to exress the aterial roerties as a weighted su of the candidate aterial roerties. Soe difficulties of the discrete aterial selection using toology otiization are 1/ to find efficient interolation and enalization schees of the aterial roerties and 2/ to be able to hae efficient solution algoriths to handle ery large scale otiization robles with any design ariables. Besides the seinal work by Stegann and Lund [10], we extend and generalize the work by Bruyneel [2] with the alternatie SFP schee by using a bi-alue coding araeterization (BCP) by Gao et al. [6]. The resent research work carries out a large inestigation of different interolation and enalization schees for the otial aterial selection roble. In articular, the work considers the solution asects in the ersectie of soling large-scale industrial alications. 4. Discrete Material Otiization Models The discrete otial orientation design of the lainate can be treated as an otiization aterial selection roble with ultile aterials. Following the idea by Lund and Stegann [10], the Discrete Material Otiization (DMO) consists in writing the linear anisotroic aterial stiffness atrix C i of a coosite ly noted i as a weighted su oer the stiffness of soe candidate aterials {j} (i.e. lies with different orientations): j i wij i j1 C C (1) where the weighting function w ij associated with the jth aterial hase should satisfy 1

2 (2) 0 w 1 w 1 w 0 k j when w 1 ij ij ij j1 Fro the conditions (2), it coes that no additional constraint is needed to ensure the resence of a single aterial hase at each design eleent if one end u with a 0/1 design satisfying the constraints. This is achieed by using a enalization of the interediate densities 4.1. Discrete Material Otiization (DMO) Stegann and Lund [10] resented seeral Design Material Otiization (DMO) interolation schees, aong which the ost usual one (usually called DMO4) is: ij ij i ij 1 j w x 1 x with 0 x 1 (3) In this schee, the nuber of design ariables attached to each designable eleent or region just equals the nuber of candidate aterial hase, i.e., =. The design ariables range fro 0 to 1, eaning the resence or absence of aterial i. As in the SIMP ethod, the enalization factor is alied to ush the design ariables to their extree alues 0 and Shae Function with Penalization More recently, Bruyneel [2] resented an alternatie araeterization odel naed SFP based on the finite eleent shae functions. For a design roble with 0, 90, -45 and 45 lies, the shae functions of four-node finite eleents are introduced as: 1 1 wi1 1 xi1 1 xi 2 wi 2 1 xi11 xi wi 3 1 xi11 xi 2 wi 4 1 xi11 xi x 1, j 1,2,3,4 ij (4) Obiously, the SFP interolation schee also satisfies the conditions (2). As in the SIMP ethod, the enalization factor is alied to ush the design ariables to their extree alues +/-1. When coared to the DMO schee, SFP introduces only two ariables for four fiber orientations. In SFP, the resence of one aterial hase is characterized by a secific cobination of design ariables taking bi-alues of +1 and/or -1. The saller nuber of design ariables in SFP is an adantage oer the DMO schees to reduce the size of the otiization roble. As indicated in Ref. [2], een if it ay be quite difficult, it is ossible, in rincile, to extend the SFP to ore than four aterials by building colex shae functions related to n node finite eleents satisfying the conditions (2) Bi-alue coding araeterization (BCP) The bi-alued coding araeterization (BCP) schee generalizes the SFP schee and roides an alternatie to the classical DMO interolation schee. To oercoe the shortcoing of the SFP schee, one can abandon the idea of finite eleent shae functions and kee in ind only the idea of defining the shae function using bi-alues of +1 and -1. Thus, a new BCP schee is roosed here as the aterial araeterization odel for aterial hases, 1 i j 1 with 1 1 and 1,, 2 jk (5) k1 w s x x k where, the nuber of design ariables is an integer defined by the ceiling function of =log 2. In other words, ( 1) the BCP schee akes it ossible to interolate between 2 1 and 2 aterial hases with design ariables. For exale, for =3 one can interolate between aterials with 5 8. The s jk alues are gien at Tables 1 and 2 for 2 and 3 binary coding ariables. The alues of s jk are equal to 1 or -1. For =2 obiously the BCP aterial araeterization recoers exactly the SFP schee (4). To illustrate the coding clearly, a sketch is shown in Fig. 1. Each candidate aterial hase locates at the ertex of the square or of a cube in the 2D or 3D saces. 2

3 k j Table 1: s jk alues ( =2, =4) k j Table 2: s jk alues ( =3, =8) M4 x 2 1 M3 M8 M5 x 3 M7 M6 x x 1 x 1 M1-1 M2 (a) =2, =4 (b) =3, =8 Figure 1: Illustrations of the BCP schee M1 M4 M2 M Penalization of interediate densities In eq. (3), (4) and (5), the ower enalization of interediate is used to reent the interediate alues of the design ariables at the solution, and therefore to aoid any ixture of candidate aterials in the final design. The ower enalization with an exonent [1] is ery conenient but this choice is not unique. Other enalization schees hae been exlored successfully by the authors: If the interediate alues of a ariable ust be enalized, the following schees hae been inestigated: -SIMP [1] -RAMP [11] -Halin Tsai [7] -Polynoial [13] f ( ) (6) f ( ) (7) 1 (1 ) r f ( ) (1 r) 1 1 f ( ) (9) Basically, one can find equialent enalizations of interediate densities by a roer choice of the enalization araeter in each schee. For instance, the araeters = 3 for SIMP, r = for Halin-Tsai and = 16 for the olynoial schee roide siilar enalization schees. Our nuerical exerients showed that the different schees conduct to siilar results for equialent enalization rofiles. Howeer when considering density deendent loads, one has to consider schees with non-zero deriaties in zero density as ointed out in Ref. [3]. (8) 3

4 The authors also inestigated continuation rocedures in which the enalization is rogressiely increased. Because of the resence of any local otia, the idea is to use a classic continuation strategy to increase rogressiely the enalization araeter. Howeer, the continuation strategy gies no guarantee to aoid the local otia. It is just reduces the tendency to be traed in a local configuration. 5. Lainate stiffness otiization roble 5.1. Miniization of structural coliance Here, the iniu coliance design of a lainated coosite is considered with fiber angles to be otiized. With a discrete aterial araeterization, the otiization roble of a lainate can be stated as follows: x i n k find: 1,, ; 1,, iniize: C T Fu subject to: F Ku One notices that no olue constraint is included because we consider an otiu orientation roble. For fixed loads, the sensitiity of the coliance can be generally exressed as: C 2 F K K x x x x (10) T T T u u u u u (11) For each finite eleent, the eleent stiffness atrix is calculated using one of the interolation schees (3), (4) or (5) so that the artial deriatie can be calculated with: K x i j1 wij j K i (12) x Obiously, fro the sensitiity exression, the sensitiity C/x ight be ositie or negatie due to the suation exression of the stiffness interolation schee, which eans the objectie function can be non-onotonous and any local solutions ight exist. The large-scale otiization roble is soled by alying the well-known concet of sequential conex rograing (SCP), in which one resorts to a sequence of conex subrobles of (10). In this aer, the structural analysis is carried out using SAMCEF finite eleent software and the MMA faily otiizer [4] is adoted to seek the otial solution of each subroble Maxiization of natural frequency The roble of axiization of natural frequency is stated as follows: x i n k find: 1,, ; 1,, 2 iniize: 2 subject to: K M u 0 1 x 1 where K and M are the stiffness and ass atrix of the whole structure, resectiely. ω is one of the circular natural frequencies and u the corresonding ode shae. Lewise, the sensitiities can be deried by differentiating the eigenalue equation so that K M u u u u 2 x u Mu T i 2 T i i i i i x x T where M i is the eleent ass atrix. Generally, the natural frequency is a non-onotonous function of design ariables because the sensitiity in eq. (14) ight be ositie or negatie. Here, it is iortant to notice that both eleent stiffness and ass atrices should be araeterized. Siilarly to the situation of stiffness atrix, the ass atrix can be written as follows if using for instance the BCP interolation schee: (13) (14) M j 1 i M ij i i j 1 s x j1 2 (15) k1 Notice that the enalty factors K and M in both interolations ay take different alues. Howeer generally one choses M =1. M 4

5 5.3. Introduction of a olue constraint In fact, the BCP schee resented aboe can only be used to attribute a certain solid aterial hase of all candidates to each finite eleent while no oid is allowed. To reduce the structural weight, the following interolation odel was roosed by Bruyneel et al. [5] to allow the selection of discrete aterials and the resence of oid siultaneously. q j i i ij i i j1 c y w c 0 y 1 (16) where y i refers to the additional toology ariable that identifies the resence of the solid aterial (y i =1;w ij =1;w =0, k j) and oid (y i =0) oer eleent i. q is the enalization factor intending to ush y i toward 0 or 1. Corresondingly, the olue constraint can be stated as i i (17) i V yv V Here, V denotes the whole olue of the structure accuulated by each eleent olue V i full of solid aterials. V refers to the uer bound of the olue constraint. Note that the olue is controlled without distinction between solid aterial hases. Howeer, if the olue constraint is concerned with secific aterial hases for a general layout design of inhoogeneous aterials, the aboe exression is no longer suitable. Suose ξ and ζ indicate two different sets of solid aterial hases, e.g., orous aterials and fiber reinforced coosites, the following araeterization odel of general for is roosed to distinguish the contributions of secific aterial hases q (18) c y w c y w c y q i i i i i i i i In this case, the olue control of aterial set ξ still holds the exression of eq. (14). Clearly, the araeterization odel of eq. (18) will autoatically degenerate into eq. (16) if only one set of solid aterial hase exists. To guarantee the unifor weights, i.e. exactly fair starting guess of each candidate aterial, the initial alue of the toology ariable should be set to be y i =0.5 1/q, deending uon the enalization factor q. An alternatie general interolation odel using the 1bi-alue concet is written as: q q 1 yi 1 yi i i i i i i 2 2 (19) c w c w c 1 y 1 Corresondingly, the olue constraint used to control aterial set ξ is exressed as 1 yi V Vi V 1 yi 1 (20) i 2 In fact, aterial set ζ is indirectly controlled when the olue constraint is iosed to aterial set ξ. Lewise, a unifor weighting can be achieed when the initial alue is set to be y i =0. 6. Nuerical alications In this section, we consider seeral nuerical alications to illustrate and coare the different interolation schees Structural coliance iniization The axiu in-lane coliance roble (10) is soled by selecting the otial orientation of the ly. An orthotroic coosite aterial whose roerties are listed in Table 4. The local ly orientation can be searched in a list of discrete orientation angles (see Table 3). A square structural doain consisting of a single ly is considered (see Fig. 2). The odel is eshed with quadrangular finite eleents. The structure is claed along the left edge and a in oint ertical load is alied at the lower right corner. Besides, 16 searate designable atches are considered. This eans that all eleents of each atch hae the sae orientation, while the orientations ight be different between atches. The DMO, SFP and BCP schees are adoted to araeterize the aterial roerties. 5

6 Nuber of aterial hases () Design odel with 4 4 atches Loads and boundary conditions Figure 2: Model of the square late under ertical force Nuber of design ariables for each region ( ) Table 3: Orientations Discrete orientation angle ( ) /45/0/ /60/40/20/0/-20/-40/-60/ /75/60/45/30/15/0/-15/-30/-45/-60/-75 Table 4 Material roerties (Material 1) E x E y G xy xy GPa 10.62GPa 5.45GPa 0.33 (a) DMO (C= ) =4 (b) SFP (C= ) =2 (c) BCP (C= ) =2 Figure 3: Otiization results of the square late under ertical force (=4) Figure 4: Iteration histories of the weight for atch 16 (BCP =4) The case of four orientations (=4) is considered. For this roble, four design ariables er atch are needed using the DMO schee; while only two ariables are required for SFP and BCP schees. The otiization results by DMO, SFP and BCP schees are gien in Fig. 3. All solutions are nearly the sae een though sall differences exist. Actually, BCP and SFP schees result in exactly the sae solution because both schees are identical in this articular case. The otiu coliance using the SFP/BCP schee is a little better. Howeer, the gradient-based algoriths used in the sequential conex rograing otiization algoriths cannot guarantee the global otiu conergence. 6

7 Using the BCP schee, the iteration histories of the weights w ij for atch 16 are lotted in Figure 4. At the starting oint, all weights are exactly the sae. Finally, the orientation -45 eerges as the otiu choice for this atch with a unit weight, while the weights of the other orientations gradually diinish to zero for the eliination of their effect. Figure 5: Influence of the enalization factor of the BCP schee uon the otiization results The influence of the enalization factor on the otiization results is inestigated in Fig. 5. For different alues of, the otiization iterations are quite stable, but the coliances and orientation layouts are different in the otiization results. As in toology otiization, a saller enalization factor leads to stiffer design otius. Howeer, a too sall enalization factor akes the otiization iteration conerge quite slowly. In the cases of =2 and =1.5, the otiization rocesses hae not conerged after 30 iterations, while the other tests need about 10 to 15 iterations. Besides, there are still soe atches consisting of ixed aterial for these two tests after een 50 iterations, as shown in Fig. 5. As a conclusion, the suggested alue for the enalization factor 2.5, 4. is 6.2. Natural frequency axiization We consider a 4-layer square late whose four corners are sily fixed (see Figure 6). The late size is 4 4 and the total thickness is 0.1. Each layer is eshed into quadrangular solid shell eleents and 4 4 atches. As a result, there are totally 64 designable atches. Figure 6: Model of a 4-layer square lainate Assue Material 1 (see Table 4) is adoted and four candidate orientations (90/45/0/-45) are used. First, the fundaental frequency is axiized without olue constraint. The otial layout is obtained ery quickly after 4 iterations. As shown in Fig.7, all layers hae the sae layout. Here, layer 1 refers to the botto layer and layer 4 the to one. Now, the olue constraint related to eq. (18) is added into the otiization odel and suose only 75% atches can be filled with aterial 1. As shown in Fig. 8(a), both botto and to layers are exactly the sae as those without olue constraint in Fig.7. For the iddle layers, 8 atches near the edges are oid while the filled atches hae the sae fiber orientations as those in Fig.7. The otiization iteration cures are lotted in Fig. 8(b). 7

8 (a) Layout of the orientations (b) Iteration history Figure7: Otiization results of the fundaental frequency axiization (a) Layout of the orientations (b) Iteration history Figure 8: Otiization results of the fundaental frequency axiization with olue constraint 6.3. Four-layer lainated U-bea The bea is shown in Figure 9a. The thickness of the lainate is 1. Quadrangular ulti-layer solid shell eleents in Sacef are used to discretize the lainate bea with a basic esh size of and all eleents can be designed indeendently. The eleent stiffness atrix related to each layer of each candidate orientation is extracted for sensitiity analysis. The bea is claed at one end and a unifor line force is alied on the other end, as shown in Figure 9b. Suose both flanges hae a syetrical fiber orientation layout and only one flange is shown for all otiization results. 8

9 Figure 9: Model of a 4-layer lainated bea. a/ geoetry. b/ load case Table 5 Material roerties (MC2) glass-eoxy E x E y G xy xy 54GPa 18GPa 9GPa 0.25 olyer-foa E 0.125GPa 0.3 Suose now both orthotroic glass-eoxy with 4 candidate orientations and isotroic olyer-foa are aailable (see Table 5). The olue fraction of glass-eoxy is assued to be less than 80% of the whole structure. According to eq. (18), the starting oint is feasible for the enalization factor V =1. As shown in Fig. 11(a), the otiization rocess is stable and conerges after 31 iterations. The otial orientation layout is resented in Fig. 11(b). Layer 1 refers to the inner layer and layer 4 indicates the outer one. It is seen that the olue constraint is less than its uer bound at the starting oint and stably increases to the uer bound. Meanwhile, glass-eoxy is laced at the loaded end of the bea, esecially the ertical rib, while the olyer-foa, denoted by gray, occuies the fixed end and inner layer. Meanwhile, glass-eoxy of orientation 0/-45 degree is not used in the final layout. 7. Conclusions In this aer, we resent a noel araeterization schee based on a bi-alue coding for soling the discrete aterial otiization of coosite structures. With a reduced nuber of design ariables, the BCP schee [6] generalizes the SFP schee [2] and is a challenger to the classic DMO [11] for large-scale robles. Furtherore, the BCP forulation roides a well-osed roble for an efficient solution using sequential conex rograing algoriths. Different enalization functions of interediate densities hae been roosed and the choice of the enalization araeters has been discussed. The on-going work is deoted to extend the alication of this noel araeterization schee to larger robles inoling industrial coosite structures including coliance, dislaceent, stress constraints but also buckling and erieter constraints. 8. Acknowledgeents This work was suorted by the Walloon Region of Belgiu and SKYWIN (Aerosace Cluster of Wallonia), through the roject VIRTUALCOMP. Figure11 Otiization result of the 4-layer lainated bea under line force with olue constraint iteration history 9

10 Layer 4 Layer 3 Layer 2 Layer 1 Figure12 Otiization result of the 4-layer lainated bea under line force with olue constraint Orientation layout 9. References [1] M.P. Bendsøe, M. P. Otial shae design as a aterial distribution roble. Structural Otiization, 1 (4), , 1989 [2] M. Bruyneel. SFP a new araeterization based on shae functions for otial aterial selection: alication to conentional coosite lies. Structural and Multidiscilinary Otiization. 43 (1), 17-27, [3] M. Bruyneel and P. Duysinx. Note on toology otiization of continuu structure including self-weight. Structural and Multidiscilinary Otiization. 29 (4), , [4] M. Bruyneel, P. Duysinx, and C. Fleury. A faily of MMA aroxiations for structural otiization. Structural and Multidiscilinary Otiization, 24 (4), , [5] M. Bruyneel, P. Duysinx, C. Fleury and T. Gao. Extension of the Shae Functions with Penalization for Coosite-Ply Orientation. AIAA Journal, 49 (10), , [6] T. Gao, W. Zhang, and P. Duysinx. A bi-alue coding araeterization schee for the discrete otial orientation design of the coosite lainate. Int. J. for Nu. Methods in Engng. 91 (1), , 2012 [7] J.C. Halin and S.W. Tsai. Effect of enironental factors on coosite aterials. AFML-TR, , June [8] E. Lund, and J. Stegann. On structural otiization of coosite shell structures using a discrete constitutie araetrization. Wind Energy, 8, , 2005 [9] O. Sigund and S. Torquato. Design of aterials with extree theral exansion using a three-hase toology otiization ethod. Journal of the Mechanics and Physics of Solids, 45 (6), , 2000 [10] J. Stegann and E. Lund. Discrete aterial otiization of general coosite shell structures. Int. J. for Nu. Methods in Engng. 62 (14), , 2005 [11] M. Stole and K. Sanberg. An alternatie interolation schee for iniu coliance toology otiization. Structural and Multidiscilinary Otiization, 22 (2), , [12] J. Thosen. Toology otiization of structures coosed of one or two aterials, Structural Otiization, 5(1-2): , [13] J. Zhu, W. Zhang and P. Beckers. Integrated layout design of ulticoonent systes. Int. J. for Nu. Methods in Engng, 78 (6), ,