Composite Structures 73 (2006) 403–412 www.elsevier.com/locate/compstruct
A strength-based multiple cutout optimization in composite plates using fixed grid finite element method Yi Liu a, Feng Jin a, Qing Li a
b,*
Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, PR China b School of Engineering, James Cook University, Townsville QLD 4811, Australia Available online 25 April 2005
Abstract The design of interior cutouts in laminated composite panels is of great importance in aerospace, automobile and structural engineering. Based on the Tsai–Hill failure criterion of the first ply, this paper presents a newly developed Fixed (FG) Grid Evolutionary Structural Optimization (ESO) method to explore shape optimization of multiple cutouts in composite structures. Different design cases with varying number of cutouts, ply orientations and lay-up configurations are taken into account in this study. The examples demonstrate that the optimal boundaries produced by FG ESO are much smoother than those by traditional ESO. The results show the remarkable effects of different opening numbers and various lay-up configurations on resulting optimal shapes. The paper also provides an in-depth observation in the interactive influence of the adjacent cutouts on the optimal shapes. 2005 Elsevier Ltd. All rights reserved. Keywords: Fixed grid; Finite element method; Evolutionary structural optimization; Shape optimization; Laminated composites; Multiple cutouts
1. Introduction Laminated composite structures are playing an imperative role in aerospace, automotive, naval, mechanical and civil industries when the ratio of strength to weight, specific stiffness and directional properties are crucial. As a typical structural feature, interior cutouts are often indispensable in meeting such special technical requirements as laying fuel lines and electrical cables in aircraft wing spars, opening access holes for the service of interior parts in machine, ventilating the air of tubes, and passing the liquid at the bottom of container. The presence of these holes, however, may significantly change the stress intensity and structural performance, which could to a certain extent affect the operational life of the composite structures. For this reason, one of the foremost design objectives can be to
*
Corresponding author. Tel.: +61 7 4781 5762; fax: +61 7 4781 4660. E-mail address:
[email protected] (Q. Li).
0263-8223/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.02.014
minimize the resulting stress-induced failure due to the introduction of cutouts. Typically, there are two solutions to such a technical problem, namely reinforcement design and geometry design. The former seeks for an optimal reinforcement in the vicinity of the hole such that the usage efficiency of the reinforcement is maximized while without violating the stress constraints, as the representative work done by Engels and Becker [1,2]. This does provide an excellent solution to the design cases where the modification of hole shape is not allowed. However, the elite reinforcement pattern may require a special fabrication process and could to a certain extent increase the manufacturing cost. For this reason, the latter solution, i.e. the geometry design of the cutouts, may be considered as an important alternative to the above-mentioned design objective when possible. This class of problem has drawn a certain attention from the facet of shape optimization. In literature, Backlund and Isby optimized the cutout shape in a sheared composite panel by using a spline curve, where the weight is minimized subject to
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a constraint in the maximum Tsai–Hill value [3]. Vellaichamy et al. attempted to minimize the failure index, where the orientation and aspect ratio of elliptical cutouts were optimized for different lay-ups and load cases [4]. To consider the openings in the cylindrical composite pressure vessel, Ahlstrom and Backlund optimized the hole shapes with different ply configurations to minimize the Tsai–Hill indices [5]. By employing a growthstrain method, Han et al. recently considered two types of optimization problems on: (1) making a uniform Tsai–Hill index distribution for each element subject to a volume constraint [6] and (2) minimizing the volume under a constraint in the highest Tsai–Hill level [7]. In their studies, both single and multiple cutouts were presented in a plane stress panel. Different from the aboveadopted Tsai–Hill index, Muc and Gurba took the uniformity of Huber–Mises–Hencky stress as a failure criterion for cutout optimization in the composite structures with the genetic algorithm used [8]. In addition to the strength criterion, other structural characteristics like dynamics and stability have been also considered in the geometrical optimization of the cutouts. Taking reference axis and aspect ratio of the elliptical cutouts as design variables, Sivakumar et al. explored the size optimization problem for maximizing the fundamental natural frequency and minimizing weight, where a genetic algorithm was employed [9,10]. Bailey and Wood investigated the effect of cutout shape on buckling and postbuckling behaviours for a quasiisotropic composite panel with the lay-ups of [±45/0/90]2s [11]. Hu and Lin also studied the influence of the cutout size and ply orientation on optimal buckling loads, where the typically sequential linear programming was exploited [12]. To a certain extent, these results further add to the importance and feasibility of solutions to cutout design optimization. However, one common point of the abovementioned shape optimization techniques is that a remeshing step may be needed to cope with the potential severe mesh distortion during the design processes. This has stimulated the development of various alternative methods without remeshing, which includes, but not restricted to, the microstructure-based homogenisation methods [13], the uses of artificial material model method as SIMP (solid isotropic microstructure with penalty) [14], heuristic methods as genetic algorithm, and more intuitive methods as ESO (evolutionary structural optimization) [15]. Among these non-remeshing methods, ESO has been successfully employed by Falzon et al. for solving the shape design problem of single cutout [16] and multiple cutouts [17] in the composite structures, where the effects of lay-up orientation were considered. However, one of the major drawbacks in these optimization methods is the jagged boundary and necessitation of an additional post-processing step to smooth out before sending to the downstream [18]. More impor-
tantly, the jagged boundaries usually result in some unanticipated stress concentration and may lead to misinterpretation of the intermediate results in some cases, where the stress distribution is a predominant factor of the design optimization [19]. It is noticed that the Fixed Grid (FG) finite element framework is of an important potential to take both the advantages of non-remeshing in the analysis domain and smoothing in the optimal boundary. The FG FE method has been primarily developed to account for the modelling problems where the geometry or physical properties of the domain change with time, e.g. phase change between liquids and solids [20]. Garcia and Steven extended FG FEA to elasticity problems in 1999 [21] and then applied it to the shape optimization problems by using either mathematical programming [22] or evolution strategies [23]. In their work, the boundary was represented by B-spline functions. Kim et al. introduced FG FEA technique into the ESO method and successfully solved a range of topology optimization problems, which enables a smooth change in the newly generated topological boundaries [24,25]. More recently, Woon et al. combined FG FEA with the genetic algorithm to improve the computing efficiency [26,27]. However, these attempts in applying FG FEA to structural optimization have been limited to the isotropic materials. It would be practically significant to extend this technique to the aforementioned cutout design problems in anisotropic composite structures. This paper thus aims at developing the FG ESO method for the shape optimization problems of multiple cutouts in composite structures.
2. Tsai–Hill strength design criterion From the standpoint of structural integrity and material usage efficiency, it is expected to achieve an equal strength around the boundaries of cutouts. This necessitates an optimality criterion to quantitatively identify the failure in the laminate plates. In this paper, the Tsai–Hill failure criterion with plane stress assumption is chosen. In a plane stress finite element framework, the Tsai–Hill index can be computed for each node and ply [28] as, r 2 r r r 2 s 2 x x y y xy #j;l ¼ 2 þ þ ðl ¼ 1; 2; . . . ; MÞ X Y S X ð1Þ where M is the total number of layers of laminate composite plate, j the nodal number of finite elements, rx, ryx, sxy, are the direct and shear stresses in x, y plane and X, Y, S are the corresponding failure strengths. It is clear that at each node, the layer with the maximum Tsai–Hill value is of the highest risk of failure (i.e. the first ply failure criterion). Hence quantity
Y. Liu et al. / Composite Structures 73 (2006) 403–412
#j ¼ maxð#j;l Þ l
ð2Þ
is considered as the design Tsai–Hill value for node j. The fundamental idea of the FG ESO method is to progressively remove the least efficient material from the design region so that as a uniform Tsai–Hill index as possible can be achieved in the design boundaries of the cutouts. Based on the traditional ESO method [15], the threshold of material removal can be written in terms of rejection ratio RR(k) and mean Tsai–Hill value # as Z 1 ðkÞ ^ #ðx; yÞ dV RRðkÞ # ¼ # RR ¼ V V ! n 1X #j RRðkÞ ð3Þ n j¼1 where superscript k stands for the current number of steady states (k = 0, 1, 2, . . .), #^ denotes the current threshold of material removal, V is the total volume and n the total number of nodes in the design domain. For the shape optimization, all the candidate nodal Tsai–Hill values, #j (j = 1, . . ., n) in the newly created boundary will be examined against the threshold #^ to determine whether this node should be degraded in the structure. In such a way, the boundary will evolve towards an optimum [15,19], where all the nodal Tsai–Hill indices will gradually approach to uniform.
3. Fixed grid based evolutionary structural optimization Unlike the traditional finite element method, the fixed grid finite element method does not need a fitted mesh to discretize the analysis domain [21]. Instead, a rectangular grid with equally sized elements is superimposed on the structureÕs geometry. The material properties of each element are then determined according to the amount of material it contains. Fig. 1 shows a typical example of the analysis domain discretized by a fixed grid, where the elements are categorized as three subsets of I, N and O. Subsets I and O denote the elements inside
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and outside the structure respectively, while subset N stands for those neither-in-nor-out elements [21–25]. The material properties of these three subsets of elements can be determined in terms of volume fraction ae by solid volume VI (inside the structure) to whole volume Ve in element e as, VI ð4Þ ae ¼ Ve In a two-dimensional plane stress case, Eq. (4) can be replaced by an area ratio, ae ¼
AI Ae
ð5Þ
where AI and Ae are respectively the elemental areas inside the structure and total area. As a unified form, the material elastic tensor De of element e in the fixed grid can be computed as, De ðNÞ ¼ ae De ðIÞ þ ð1 ae ÞDe ðOÞ
ð6Þ
It should be noted that an N element in FG represents a partially inside element where its material property value is proportional to its volume (area) fraction. By using Eq. (6), such a bi-material N element is consequently transformed into a homogeneous isotropic material element. The original ESO method was presented in a binary operation where an element either does (I element) or does not exist (O element) [15]. In a FE framework, this does not ensure a smoothing boundary. Unlike the original ESO method the FG ESO method incorporates an elegant material model in those N elements. More specifically, the FG ESO method allows a partial removal of elemental material by continuously altering the volume (area) fraction while maintaining the smoothness of the new boundary. Without loss of generality, let us take a four node quadrilateral grid as an example. To determine the area fraction, the nodal Tsai–Hill indices #j1 ; #j2 ; #j3 and #j4 in those four nodes j1, j2, j3 and j4 of element e are compared against the optimality criterion #^ in Eq. (3). This leads to categorizing candidate element e to these three subsets as 8 ^ > < I; if 8#j 2 ð#j1 ; #j2 ; #j3 ; #j4 Þ > # ð7Þ e 2 O; if 8#j 2 ð#j ; #j ; #j ; #j Þ 6 #^ 1 2 3 4 > : N; otherwise The area fractions for those elements in subsets I and O can be given straightforward as, 1; e 2 I ae ¼ ð8Þ 0; e 2 O
Fig. 1. Fixed grid modelling of the design domain.
For these elements falling into subset N, the area fraction is determined by computing the Tsai–Hill index contour against the threshold within the element. In the FG finite element framework, the Tsai–Hill value
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(a)
(b)
(c) Fig. 2. Calculation of the various area fractions in N element (a) with three low nodal Tsai–Hill indices; (b) with two low nodal Tsai–Hill indices and (c) with one low nodal Tsai–Hill index.
distribution is interpolated in terms of shape functions Nj and nodal indices #j as #e ðn; gÞ ¼
N X
N j ðn; gÞ#j
ð9Þ
j¼1
Therefore the threshold curve across the element can be implicitly defined as ce ðn; gÞ ¼ #e ðn; gÞ #^ ¼
N X
N j ðn; gÞ#j #^ ¼ 0
ð10Þ
j¼1
which provides a mathematical representation of the newly formed piecewise boundary. In the bilinear quadrilateral element, Eq. (10) stands for a straight line in the element. Fig. 2(a)–(c) illustrate three different scenarios of N element, where one, two and three nodal Tsai–Hill indices are respectively accounted for below the thresh^ In each case, two intersection points in the old level #. edges can be determined by two length parameters (‘a, ‘b), as in Fig. 2.
a FG ESO steady state is claimed if all nodal Tsai–Hill values are higher than the threshold at the current step. Unlike the element-based ESO method, however, FG ESO may allow infinitely small boundary modifications. As pointed out by Kim et al. [25], a small boundary modification can still perturb the nodal Tsai–Hill values, which leads to another small modification. In such a way, FG ESO often keeps removing a small percentage of material before reaching a meaningful ESO ‘‘steady state’’, which has the optimization process stuck in a localized non-optimum status. To enable a more significant modification, therefore, the FG ESO steady state needs to be redefined by checking if the total removal material volume DVi during the ith iteration is smaller than a prescribed minimum volume required for each iteration, i.e. DV i < DV min
When a steady state is reached, the rejection ratio (threshold) is increased to a higher level by ER as RRðkÞ ¼ RRðk1Þ þ ER
4. FG ESO procedure In the typical ESO procedure, a steady state is reached when no more elements can be removed with the current threshold. If the optimum has not achieved yet, the level of removal criterion must be increased so as to continue the process of element removal. Similarly, in a FG ESO procedure a steady state is defined when no more modifications can be made. Strictly speaking,
ð11Þ
ð12Þ
It is convenient to set up RR(0) at zero and RR(1) be automatically assigned once the material alteration process getting start. During the evolutionary optimization, the stress concentration and accordingly the highest Tsai–Hill level will be reduced. To monitor such an improvement, a normalized deviation function between the maximum and the minimum Tsai–Hill indices in the design boundaries will be computed as,
Y. Liu et al. / Composite Structures 73 (2006) 403–412
f
ðkÞ
#ðkÞ #ðkÞ max min ¼ ð0Þ # #ð0Þ max min
407
ð13Þ
To quantify an optimum shape, a reasonably small tolerance s may be prescribed for determining the convergence on the objective function as ðkþ1Þ f f ðkÞ ð14Þ 6s f ðkÞ For convenience, the FG based ESO procedure for shape optimization problem of composite cutouts is systematically re-organized as follows: Step 1. Lay an appropriately dense FG mesh over the domain. Set up an evolutionary rate ER and the minimum removal volume DVmin; Step 2. determine volume (area) fraction from Eqs. (4) or (5) and compute homogeneous material properties as Eq. (6); Step 3. carry out an FG FEA to obtain the Tsai–Hill index of each node as Eq. (1); Step 4. check for the normalized deviation function (Eq. (13)) to justify whether an optimum has reached (Eq. (14)). If the optimum has been identified, output the results and terminate the iteration. Otherwise, go to Step 5; Step 5. compare all the nodal Tsai–Hill indices in the design domain against the threshold in Eq. (3). If a steady state is reached, increase the rejection ratio by an evolution step as Eq. (12) and repeat this current step; Step 6. determine the material removal amount as Eq. (7) and form a new boundary through a series of the piecewise representations, e.g. Eq. (10). Go back to Step 2.
5. Design examples To demonstrate the capabilities of the proposed FG ESO procedure, a number of design cases are presented herein. All design cases are based on an L · L = 1 m · 1 m thin plate, which is discretized by 100 · 100 uniform square grid as shown in Fig. 3. To start with, some equally sized (0.04 m · 0.04 m) small cutouts are initially seeded at the preselected positions in the design
Fig. 3. Fixed grid model of the design domain.
domain. The laminate is made of carbon fibre graphite– epoxy composite with a ply thickness of 1.6 · 104 m. The material parameters of the composite are summarized in Table 1 [16,17]. In all the evolutionary optimization processes below, an initial rejection ratio of RR0 = 0 and an evolution rate of ER = 1% are set. The minimum removal volume for each iteration is given as one element, i.e. DVmin = Ve. In all the design cases presented below, the laminated plates are loaded with both the shear and normal tensile stresses as s = /r = 0.3r and the ratio of normal stresses as k = 1.0 (refer to Fig. 3). It should be pointed out that although there may be a risk of buckling under shear loads as studied in the references [11,12], the Tsai– Hill failure is considered predominating and a strength criterion is adopted in this study. Indeed, it has been examined that the buckling does not occur under such a tensile-shear loading ratio in the following examples. 5.1. Optimal shapes and the effects of cutout number In this example, a quasi isotropic lay-up with [±45/0/ 90]s is utilized to observe the optimal shape of cutouts in the laminate plate. Four different design cases with one, distant two, close two and four cutouts are taken into account, where the initial cutouts are positioned at the
Table 1 Composite material data Constitutive parameters Young modulus E1 Young modulus E2 Shear modulus G12 PoissonÕs ratio m Ply thickness t
Strengths 128 GPa 11.3 GPa 6.0 GPa 0.30 1.6 · 104 m
Longitudinal tensile Longitudinal compressive Transverse tensile Transverse compressive Shear
1450 MPa 1250 MPa 50 MPa 100 MPa 93 MPa
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Fig. 4. Optimal opening shapes and the effects of cutout number (Ac = 0.1148 m2) (the jagged boundaries from ESO and the smooth boundaries from FG ESO): (a) Case 1: one cutout (steady state 15); (b) Case 2: two distant cutouts (steady state 16); (c) Case 3: two close cutouts (steady state 16) and (d) Case 4: four cutouts (steady state 55).
(x, y) coordinates of Case 1: (0.5L, 0.5L); Case 2: (0.25L, 0.25L)/(0.75L, 0.75L); Case 3: (0.375L, 0.375L)/ (0.625L, 0.625L) and Case 4: (0.25L, 0.25L)/(0.25L, 0.75L)/(0.75L, 0.25L)/(0.75L, 0.75L) respectively, as depicted in Fig. 4(a)–(d). Fig. 4(a)–(d) presents the optimal shapes of the cutouts. For the convenience of comparison, the final cutout areas in all these four design cases are set in the same value of Ac = 0.1148 m2. From the plots, it can be seen that the FG ESO results are reasonably close to the ESO results by Falzon et al. [17]. For such a quasi isotropic material property, the symmetry of optimal shape can be evidently observed in both the ESO and FG ESO results. The elliptically shaped cutouts in Fig. 4(a), (b) and (d) provide solid evidence in verifying the present FG ESO method. The exemption in Fig. 4(b) is due to an interactive effect between two closely positioned cutouts on the evolutionary optimization and this problem will be addressed in Section 5.3. The slight differences between the ESO and FG ESO solutions can be claimed as the degree of stress concen-
tration in the newly created boundaries. As a result of the introduction of N type elements, the stress concentrations in FG ESO are to a great extent reduced. More remarkably, in all design cases, FG ESO always presents much smoother optimal boundaries. This clearly demonstrates one of the most notable advantages of the present FG ESO method over the traditional ESO method, which is of particular significance in shape optimization problems. To monitor the process of the presented four design cases, the evolutionary histories of the normalized deviation functions are plotted in Fig. 5. As the design process progresses, the deviations in all the cases reduce consistently. It is interesting to note that during the course of the optimization, the more the number of the cutouts, the more the steady states experienced to reach the same total areas of the cutouts given the same evolution rate. The Tsai–Hill results are summarized in Table 2. Looking at the maximum Tsai–Hill indices in these different design cases, it is found that the lowest level oc-
Normalized Deviation Function f
Y. Liu et al. / Composite Structures 73 (2006) 403–412 1.0 Case 1 Case 2 (2 distant holes) Case 3 (2 close holes) Case 4 (4 holes)
0.9
0.8
0.7
0.6
0
10
20 30 40 Number of Steady States
50
60
Fig. 5. Evolutionary history of the normalized objective function.
curs in the cases with four cutouts, followed by two distant cutouts and single cutout. The exemption of Case 2 is again due to the interactive effect between two forming holes. As a result of the comparison, it can be concluded that without considering the mutual effect between the holes, the increase in opening numbers would improve stress concentration and consequentially reduce the cutout-induced failure. 5.2. Investigation into the effects of different lay-ups To explore the effects of different material configurations on the optimal shapes, several different arrangements of lay-ups are given herein as Case 5: isotropic material; Case 6: angle-ply with [(±30)2]s; Case 7: angleply with [(±60)2]s; Case 8: cross-ply with [(90/0)2]s. In all these design cases, four initial cutouts are seeded at the positions (0.25L, 0.25L)/(0.25L, 0.75L)/(0.75L, 0.25L)/ (0.75L, 0.75L), similarly to Fig. 4(d). It is clearly observed that the orientations of lamination are of evident effect on the formation of cutouts as depicted in Figs. 4(d) and 6(b)–(d). In the quasi-isotropic (Case 4 in Fig. 4(d)) and isotropic (Case 5 in Fig. 6(a)) materials, all the resulting cutouts are almost elliptically shaped under biaxial stresses and shear. The shape (ellipse), orientation of the principal axes (b = 45) and the aspect ratio (j = 2) of these cutouts (Figs. 4(d) and 6(a)) strongly resemble to the results by Falzon et al. [17]. This also demonstrates that the composite with lay-up of [±45/0/90]s is of an excellent isotropic characteristic to accommodate the interior cutouts.
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It is interesting to note that the angle-ply lay-ups with [(±30)2]s in Fig. 6(b) and [(±60)2]s in Fig. 6(c) exhibit a nice diagonal symmetry one another, which reasonably follows the diagonal symmetrical pattern in these two angle-ply lay-ups. In each design case, the orientations of the cutout principal axes tend to rotate the same angle for all four holes and also the resulting shapes of the optimized cutouts highly resemble each other. The cross-ply lay-up with [(90/0)2]s, however leads the optimization to some diamond-like cutouts as in Fig. 6(d). It is noted that the cutout themselves are also diagonally symmetric. The major and minor axes of the each cutout excitingly display an orthogonal relationship, which well reflects the orthogonal material behaviour under the direct and shear loading. Table 3 presents a summarized comparison between the maximum and mean Tsai–Hill indices in the optimized cutouts for different lay-up orientations. Since the limiting Tsai–Hill criterion is adopted in this paper, the mean Tsai–Hill index in such a plate with more angle plies as [±45/0/90]s appears higher. This implies that under the specific loading condition, anisotropic composites can better accommodate the cutouts than the isotropic composite does. Furthermore, the maximum Tsai–Hill indices in angle-ply lay-ups with [(±30)2]s (Case 6) and with [(±60)2]s (Case 7) are lower than those in isotropic material and the cross-ply lay-ups with [(90/ 0)2]s (Case 8). Once again, this indicates that these angleply lay-ups seem more suitable to the cutouts. 5.3. Multiple cutout interaction in the quasi-isotropic plate To observe the interactive effect of two cutouts, the evolutionary history of optimization process is explored in the quasi-isotropic plate. As in Fig. 4(c), the two initial cutouts are placed at (0.375L, 0.375L)/(0.625L, 0.625L) respectively. Fig. 7(a)–(d) exhibits the interaction process between these two cutouts. In the beginning, the size of these two holes is relatively small compared to the distance of their centres. The cutouts are formed as two perfect ellipses and there is almost no effect on the cutout shapes observed, as in Fig. 7(a). As the optimization progresses, these two cutout areas evolve towards closer and closer. As a result of this, the boundaries become interfering each other from the facing edges where the head of ellipses are getting flatter and flatter (Fig. 7(b)) until almost
Table 2 Comparison of the Tsai–Hill indices in the different number of cutouts Case 1 Maximum Tsai–Hill indices #max Mean Tsai–Hill indices #min Ratios #max/#min
Case 2 3
1.10 · 10 5.30 · 102 2.075
Case 3 3
1.16 · 10 5.50 · 102 2.109
Case 4 3
1.21 · 10 5.44 · 102 2.224
1.15 · 103 5.43 · 102 2.117
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Fig. 6. The effects of different material configurations on the shapes of cutouts (the jagged boundaries from ESO and the smooth boundaries from FG ESO): (a) Case 5: isotropic material (steady state 27); (b) Case 6: angle-ply lay-up with [(±30)2]s (steady state 11); (c) Case 7: angle-ply lay-up with [(±60)2]s (steady state 11) and (d) Case 8: cross-ply with [(90/0)2]s (steady state 55).
Table 3 Comparison of the Tsai–Hill indices in the different material configurations
Maximum Tsai–Hill indices #max Mean Tsai–Hill indices #min Ratios #max/#min
Case 5
Case 6
Case 7
Case 8
1.10 · 103 5.30 · 102 2.075
1.02 · 103 4.93 · 102 2.069
1.02 · 103 4.93 · 102 2.069
1.17 · 103 4.25 · 102 2.753
parallel at some stage (Fig. 7(c)), which is in an excellent agreement with Han et al.Õs work [6,7]. The further optimization will have two boundaries merge to a single hole as in Fig. 7(d). Moreover, the single cutout presents in a perfect ellipse though it experiences a complex evolutionary optimization process. From Table 4, it can be seen that as the increase in the hole area and interactive effects of these two cutouts, the ratio of the maximum to the mean Tsai–Hill indices are getting greater. This implies that the degree of stress concentration is worsened when the cutouts interact. When these two cutouts are merged into a single hole,
the ratio is lowering, which to a certain extent, shows the some improvement in the stress concentration.
6. Concluding remarks This paper develops a Tsai–Hill strength-based fixed grid evolutionary structural optimization method for the shape optimization of multiple interior cutouts in a laminate composite plate. It is found that the FG ESO method can always present a smooth boundary in all the resulting optimal shapes, which demonstrates a
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Fig. 7. Interactive history of two cutouts: (a) Iteration 2 (Ac = 0.0239 m2); (b) Iteration 200 (Ac = 0.0631 m2); (c) Iteration 300 (Ac = 1.358 m2) and (d) Iteration 360 (Ac = 2.295 m2).
Table 4 Interactive history of Tsai–Hill indices in two cutouts 2
Area Ac (m ) Maximum Tsai–Hill indices #max Mean Tsai–Hill indices #min Ratios #max/#min
a
b
c
d
0.02393 7.71 · 102 4.07 · 102 1.894
0.06308 9.01 · 102 4.61 · 102 1.954
0.13575 1.37 · 103 5.87 · 102 2.334
0.22949 1.70 · 103 8.09 · 102 2.101
notable advantage over the traditional element-based ESO method. This is considered particularly useful for such design problems as interior cutout optimization where there usually is high stress concentration around the newly generated boundary. The effects of number of cutouts on the shape optimization are investigated in this paper. The relatively distant multiple cutouts are of very similar optimal shape to the single cutout (i.e. ellipse). When there is no interactive effect between the cutouts, the increase in the number of holes can appropriately reduce the stress concentration.
The lay-up patterns of laminate composites have a strong influence on the optimal shapes. It is found that under certain tension and shear, the angle-ply composites, e.g. with [(±30)2]s and [(±60)2]s, tend to better accommodate multiple cutouts than those quasi-isotropic (e.g. [±45/0/90]s) and cross-ply (e.g. [(90/0)2]s) do. This implies that the angle-ply composites behave less sensitive to stress concentration. This paper also explores the interactive effect of two close cutouts. When two boundaries approach each other during the evolution process, the effect becomes more and more sizeable. As a result of this, the stress concentration
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is getting worse. At some stage, two cutouts may amalgamate together to form a new single boundary. From the example, it is observed that the stress concentration is then released in the newly merged boundary. Apart from the reduction of boundary stress concentration caused by element induced jagged-edge, the present Fixed Grid ESO method yields a much more manufacturable optimal geometry, which may avoid a non-trivial post-processing step before forwarding the optimal results to a CAD/CAM platform. This makes the FG ESO method particularly attractive to various practical applications. Acknowledgments The constructive suggestion from Professors Chuhan Zhang and Guanglun Wang in Tsinghua University are grateful. References [1] Engels H, Becker W. Optimization of hole reinforcements by doublers. Struct Multidiscip Optim 2000;20:57–66. [2] Engels H, Hansel W, Becker W. Optimal design of hole reinforcements for composite structures. Mech Compos Mater 2002;38:417–28. [3] Backlund J, Isby R. Shape optimization of holes in composite shear panels. In: Rozvany GIN, Karihaloo BL, Structural optimization, Dordrecht, 1988. p. 9–16. [4] Vellaichamy S, Prakash BG, Brun S. Optimum design of cutouts in laminated composite structures. Comput Struct 1990;37:241–6. [5] Ahlstrom LM, Backlund J. Shape optimization of openings in composite pressure-vessels. Compos Struct 1992;20:53–62. [6] Han SY, Bae SS, Jung SJ. Shape optimization in laminated composite plates by growth-strain method, Part I—Volume control. Key Eng Mater 2004;261–263:833–8. [7] Han SY, Park JY, Ma YJ. Shape optimization in laminated composite plates by growth-strain method, Part II—Stress control. Key Eng Mater 2004;261–263:839–44. [8] Muc A, Gurba W. Genetic algorithms and finite element analysis in optimization of composite structures. Compos Struct 2001;54: 275–81. [9] Sivakumar K, Iyengar NGR, Deb K. Optimum design of laminated composite plates with cutouts using a genetic algorithm. Compos Struct 1998;42:265–79.
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