Topology optimization of support structure of ... - CyberLeninka

Chinese Journal of Aeronautics, (2013),26(6): 1422–1429

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

Topology optimization of support structure of telescope skin based on bit-matrix representation NSGA-II Liu Weidong, Zhu Hua *, Wang Yiping, Zhou Shengqiang, Bai Yalei, Zhao Chunsheng State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Received 19 September 2012; revised 11 December 2012; accepted 20 March 2013 Available online 1 August 2013

KEYWORDS Bit-matrix representation; Finite element method; Flexible skin; Matrix-based NSGA-II; Structural optimization; Topology optimization

Abstract Non-dominated sorting genetic algorithm II (NSGA-II) with multiple constraints handling is employed for multi-objective optimization of the topological structure of telescope skin, in which a bit-matrix is used as the representation of a chromosome, and genetic algorithm (GA) operators are introduced based on the matrix. Objectives including mass, in-plane performance, and out-of-plane load-bearing ability of the individuals are obtained by finite element analysis (FEA) using ANSYS, and the matrix-based optimization algorithm is realized in MATLAB by handling multiple constraints such as structural connectivity and in-plane strain requirements. Feasible configurations of the support structure are achieved. The results confirm that the matrix-based NSGA-II with multiple constraints handling provides an effective method for two-dimensional multi-objective topology optimization. ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.

1. Introduction Morphing wings conduct better aerodynamic performance because of the ability of changing their shapes during flights.1–5 Telescope skin is suitable for a one-dimensional (1D) morphing wing as it can help the wing maintain a smooth appearance and transfer aerodynamic loads during morphing, as shown in Fig. 1. * Corresponding author. Tel.: +86 25 84896131. E-mail addresses: [email protected] (W. Liu), hzhu103@nuaa. edu.cn (H. Zhu). Peer review under responsibility of Editorial Committee of CJA.

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In the Smart Wing Program, Kudva6 proposed a multi-layer structure with elastomer membrane skin supported by a honeycomb core. The design manages an airfoil shape under aerodynamic loads. However, the mechanical properties and the specification details of the honeycomb have not been released yet, as mentioned by Thill et al.7 Sandwich semi-rigid skin, consisting of inner elastic support and a highly elastic surface layer, has drawn much attention in recent research. Olympio and Gandhi8,9 suggested a hybrid cellular honeycomb with flexible face-sheets to avoid the bulge effect of the flexible skin which would affect the wing’s aerodynamic performance. Bubert et al.10 studied a passive 1D morphing skin which was made up of a V-shaped accordion honeycomb substructure and an elastomer-fiber-composite surface layer. The analytical results show that the hybrid honeycomb is able to support certain out-of-plane aerodynamic pressure, but usually with heavy

1000-9361 ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. http://dx.doi.org/10.1016/j.cja.2013.07.046

Topology optimization of support structure of telescope skin based on bit-matrix representation NSGA-II

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2. Presentation of the multi-objective problem The support structure of telescope skin should be light-weight, flexible enough for in-plane morphing, and able to support certain out-of-plane pressure. 2.1. In-plane performance

Fig. 1

Telescope skin.

weight and high in-plane stiffness. The V-shaped accordion honeycomb has light weight and low in-plane stiffness. It is suitable for in-plane deformations but weak in out-of-plane performance; hence carbon rods were adopted in their work to enhance the out-of-plane load-bearing ability. Therefore, it becomes a key problem in the design of telescope skin to find support structures with light weight, low in-plane stiffness, and high bending stiffness for out-of-plane deformations. Structural topology optimization has received enough attention recently as a generalized shape optimization method. Sigmund11 presented a 99-line topology optimization code in MATLAB for compliance minimization of statically loaded structures. As an important approach, the solid isotropic micro-structure with penalization (SIMP) method which was originally introduced by Bendsøe12, has gotten general acceptance. Though computationally effective, both of these two methods can neither perform multi-objective search nor deal with the problem of compliance maximization under certain constraints. Recently, genetic algorithms (GAs) have been increasingly adopted in structural topology optimization.13–15 Jakiela et al.16,17 used a GA with a binary-string representation in continuum structural topology design. Wang and Tai18 suggested a GA with a bit-array representation to deal with structural topology design optimization. A violation penalty method was also proposed in Ref.18 to drive the GA search towards topologies with better structural performances. It should be noticed that the optimum algorithms of the two approaches did not demonstrate anything other than singleobjective problems. Olympio and Gandhi19 introduced a hybrid GA for the optimization of telescope skin’s support structure and obtained satisfactory results. However, the possibility of practical engineering applications of the structure should be taken into account. In addition, there is still work for the representation method (the definition of the search space) to be further improved. As mentioned by Schoenauer15, the choice of a representation method is vital to GAs. In our work, a bit-matrix representation method which is more intuitive and straightforward is proposed. To coincide with the bit-matrix representation method and the multi-objective problem, an NSGA-II with matrix-based operators is proposed to accomplish topology optimization in MATLAB. Combination of the algorithm and the engineering problem is also discussed. The objectives of feasible individuals are obtained by ANSYS and transmitted into MATLAB for non-dominated sorting. With the matrix-based GA operators, the algorithm is finally applied to the topology optimization of telescope skin and succeeds in providing feasible and effective results.

Wmorph and LGS are selected as two objectives to be minimized, where Wmorph indicates the work required by the structure to morph in plane which represents the in-plane morphing ability, and LGS represents the ratio of local strain to global strain which concentrates more on the selection range of materials, as done in Ref.19 It should be noticed that not only the geometrical appearance but also the material used for manufacturing should be considered. Gandhi and Anusonti-Inthra20 indicated that strain capability of about 2% of telescope skin would be more than adequate for airfoil camber applications. However, existing materials such as metals and polymers do not have such large strain capabilities. For most metals, the elastic limit strain is about 0.2% and LGS 6 0.1 is required to reach a global strain of 2%. For polymers, the elastic limit strain is about 1% and LGS 6 0.5 is required. In our study, LGS 6 0.5 is selected as a constraint. 2.2. Out-of-plane performance Assuming that p0 is the air-load pressure applied to the structure, d is selected as another objective to be minimized, which is the out-of-plane displacement of the structure under p0. 2.3. Objectives and constraints Besides the three objectives mentioned above, the mass of the structure is also considered as an objective to be minimized. From the analysis above, in this study, the objectives and the constraints of the topology optimization problem are set as follows: 8 MðxÞ > > > < dðx; p Þ 0 Min ð1Þ > LGSðx; FÞ > > : Wmorph ðx; FÞ s:t:

LGSðx; FÞ 6 0:5

ð2Þ

where x is the solution matrix of the design domain, F is the inplane morphing load applied to the structure, and M(x) is the mass of the structure. The objective values of the structures are obtained by finite element analysis (FEA) using ANSYS and then transmitted into MATLAB for further processes such as non-dimensionalization and non-dominated sorting. The constraint-handling method will be further discussed later. 3. Implementation of bit-matrix representation NSGA-II GAs based on the Darwinian survival-of-fitness principle are stochastic search methods that operate on populations to obtain better approximations approaching the best solutions

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step by step. A population is formed by a set of individuals called chromosomes and GA operators are nothing more than selection, crossover, and mutation. New chromosomes forming a new population are produced by GA operators. Recently, a lot of GAs have been developed, yet most of which focus on single-objective optimization. Deb21 suggested a multi-objective evolutionary algorithm (MOEA) based on non-dominated sorting, called NSGA-II, which is able to find much better spread of solutions and better convergence near the true Pareto-optimal front. However, there were no operators designed for NSGA-II with bit-matrix chromosomes. In this study, matrix-based crossover and mutation operators are introduced to coincide with our representation of chromosomes. Fig. 3

Crossover method.

3.1. Bit-matrix representation 3.3. Mutation operator Traditional chromosomes of topology optimizations based on a GA are usually encoded data: decimal numbers, binary strings, or bit arrays.14,16–18,22 Coding and decoding processes accomplish the mapping between the design domain and the searching domain of the algorithm. The design domain is evenly meshed into discrete units in two-dimensional topology optimization. Here, it is assumed that the units can only exist in two possible forms: full of material or empty. The corresponding value of the former is assumed to be 1, and the latter 0. Then a bit-map of the design domain is obtained, as shown in Fig. 2. The bit-map can be easily associated with a kind of data format: the bit-matrix. Considering MATLAB has powerful matrix-processing ability, the bit-matrix is adopted as the representation of the chromosome. 3.2. Crossover operator Crossover is the main GA operator and produces new chromosomes that inherit parents’ genetic information. For realcoded and binary-coded GAs, different crossover strategies were reported, such as intermediate recombination, linear recombination, single-point or multi-point crossover, uniform crossover, and so on. These crossover methods are mainly suitable for one-dimensional chromosomes. For two-dimensional bit-matrix chromosomes, it would be more convenient to adopt a two-dimensional graphical crossover operator. In this study, a random-sub-matrix crossover operator is introduced. Two random sub-matrices at the same position of the parents are selected to interchange with each other, as shown in Fig. 3, where r, s, q, and t are random natural numbers and restricted to 1 6 r 6 q 6 m and 1 6 s 6 t 6 n. This kind of crossover operator is beneficial to the treatment of twodimensional structures than the conventional one-dimensional operators and performs effectively in generating new chromosomes.

Fig. 2

Bit-matrix representation of the design domain.

Mutation is mainly used as an auxiliary operator of a GA. For bit-representation GAs, a flip bit method is usually utilized as the mutation operator. In this study, random elements of the parent chromosome are selected to flip. In the approach, a random bit-matrix is generated with the same size of the chromosome according to certain pre-defined variable scale. The child chromosome is generated by using the XOR operator between the random bit-matrix and the parent chromosome, as shown in Fig. 4. 3.4. Other operators The times of the chromosome chosen to reproduce is determined by the selection operator. The classic binary-tournament selection is adopted in the process in which two individuals directly compete for selection. The fast non-dominated sorting procedure and the elitist strategy introduced in Ref.21 are adopted. 3.5. Stopping criteria In this study, the algorithm stops when the number of total iterations reaches a pre-defined maximum generation number. 4. Approaches of the topology optimization The material distribution method and the FEA are adopted in this study. The units of the design domain are assumed to be either full of material or void. No intermediate values between 1 and 0 are considered. The material distribution method usually has problems in mesh-dependence and checkerboard. Filtering and connectivity analysis are used as the treatments.

Fig. 4

Mutation method.


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(a) Original image

(b) Result of erosion

Fig. 7

Connectivity analysis.

4.2. Connectivity analysis (c) Result of dilation

(d) Result of open

(e) Result of close

Fig. 5

Demonstration of basic image morphology operators.23

The chromosomes are nearly randomly obtained by GA operators. Some of them are feasible for the FEA while others are not, because the topology structure of a feasible chromosome should contain both boundary and load conditions. Another expectation is that the structure has only one single object, which means the structure which is considered to be connected should only have one load path. As shown in Fig. 7, the left edge of the design domain is simply supported, and a load is applied to the right edge. The object which connects both the left and right edges is considered to be a load path, so Object 1 is a load path, while Objects 2–4 are not. Other structures with no load path or more than one load path are regarded as disconnected and penalized with the method proposed by Wang and Tai.18 For structures having more than one load path, the violation value violmolp(x) is e violmolpðxÞ ¼ Cc ðnc ðxÞ 1Þ þ Ca A

Fig. 6

Filter radius R.

where Cc and Ca are penalty coefficients for the number of load e respecpaths nc(x) and the total area of unusable objects A, tively. Cc = A0 and Ca = 1 are set in the procedure, where A0 equals the area of the whole design domain. For structures having no load path, the violation value violnlp(x) is violnlpðxÞ ¼ A20

4.1. Filtering To alleviate the mesh-dependence of the solution and make it more suitable for manufacturing, image morphology filters which can prevent gray transition regions were introduced to topology optimization by Sigmund.23 As shown in Fig. 5, the original image is shown in Fig. 5(a). Erosion filter (Fig. 5(b)) and dilation filter (Fig. 5(c)) are the two basic filters. Two new filters are created by combining these two filters: the open filter (erosion followed by dilation as shown in Fig. 5(d)) and the close filter (dilation followed by erosion as shown in Fig. 5(e)). In this study, the close filter is used after the open filter, as Olympio and Gandhi19 did in their approach. MATLAB provides toolbox functions of these filters which can be directly called by users: IMDILATE, IMERODE, IMOPEN, and IMCLOSE. The operators have a radius R, which is the lower limit of the dimension of void and material regions. Here R is set to a 3 by 3 all-1 matrix, as shown in Fig. 6.

ð3Þ

ð4Þ

All the solutions will then be processed by constraints handling.

Fig. 8

Two constraints applied to solutions.

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4.3. Multiple constraints handling In Ref. 19, insufficient attention was paid to constraints handling. In this study, two levels of constraints are mainly considered, as shown in Fig. 8: (1) connectivity constraint; (2) strain constraint. The penalty method for the strain constraint is similar to that of the connectivity constraint. The violation value violstrn(x) is formulated as follows: violstrnðxÞ ¼ A0 LGSðx; FÞ

ð5Þ

There are three different kinds of solutions: (1) disconnected; (2) connected but violating strain constraint; (3) feasible with no violation. Blum et al.24 introduced three ways to deal with infeasible chromosomes: (1) discarding them; (2) applying a high penalty in the fitness function so that they are unlikely to survive; or (3) repairing them. A constraintdomination principle without any penalty parameter was first introduced by Deb et al.21 It was based on the penalty function approach for single-constraint handling. In our study, it is improved to be suitable for this multiple constraints problem. This method employs binary tournament selection to decide which solution is better. The rules for the selection process are listed as follows:

Fig. 10

Boundary and load conditions for in-plane morphing.

(1) If both solutions belong to the same kind of 1 or 2, the one with a smaller violation value is selected. (2) If both solutions belong to kind 3, a better solution is selected by non-dominated sorting. (3) If two solutions belong to different kinds, the one in the larger kind number is selected.

4.4. Boundary and load conditions of the structure for FEA The support structure of telescope skin is assumed to be cellular, as shown in Fig. 9. For the case of in-plane performance, left, top, and bottom edges of the skin are simply supported; a displacement load is applied to the right edge. One of the cells is chosen as the object for analysis. The cell can be divided into four equal parts owing to the symmetric boundary and load conditions. To reduce calculation complexity, a quarter of the cell is then taken into consideration. The quadrant is assumed to be square. As shown in Fig. 10, the length of the horizontal edge equals the vertical edge, i.e., Lx = Ly. Similar to the whole structure,

Fig. 11 Boundary and load conditions for out-of-plane performance.

three edges of the quarter of the cell are simply supported and the right edge bears a displacement load. For the case of out-of-plane performance, four edges of the cell are all simply supported. A pressure of p0 is applied to the structure in the normal direction, as shown in Fig. 11. 4.5. Algorithm implementation scheme The algorithm process of matrix-based NSGA-II with multiple

constraints handling is shown in Fig. 12. The initial population size is set to be 20. The size of the bit-matrix is set to be 20 · 20, which means that the design domain is evenly meshed into 400 elements. The crossover and mutation probabilities are both set to be 100%, meaning that all selected chromosomes have to undergo the operations of crossover and mutation. The mutation scale is set to be a function with a variable of current generation number: Rm ¼ 0:1ð1 0:5curGen=maxGenÞ

Fig. 9

Cellular skin structures.

ð6Þ

where curGen is the current generation number and maxGen is the maximum generation number. In this study, the maxGen is set to be 105.


Fig. 12

Algorithm process.

Fig. 14

Fig. 13

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Solutions obtained by the presented algorithm.

5. Results and discussion Twenty solutions are obtained after iteration and listed in the sequence of mass in Fig. 13. Corresponding objective values of the solutions are shown in Fig. 14.

Objective values of solutions.

It is noticed that the algorithm of the bit-matrix representation NSGA-II with multiple constraints handling conducts effectively, and all these 20 solutions satisfy the constraints with relative optimization. It can be seen from Fig. 13 that structures with small out-of-plane displacements usually have large LGS values. However, these objectives conflict in certain degree, so no solution can obviously be considered as the best selection. Then in practical engineering, reasonable trade-off decision should be made according to practical requirements. It can be seen from the objective values in Fig. 14 that the ellipse-like Solution 1 in Fig. 13 is the lightest structure, while Solution 20 is the heaviest one and has the minimum

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W. Liu et al. out-of-plane displacement; Solution 7 has the minimum LGS value and Solution 5 has the lowest in-plane stiffness, both of which present honeycomb-like structures. These four solutions are represented in Figs. 15–18 for a structure of 4 by 4 cells respectively. The simulated structures show that the algorithm proposed in this study can efficiently provide general information of material distributions. Specific structure parameters may be further considered by sizing optimization before manufacturing. 6. Conclusions Fig. 15

Fig. 16

Structure with the lightest mass.

Structure with minimum out-of-plane displacement.

Fig. 17

Structure with minimum LGS value.

(1) A bit-matrix representation NSGA-II with multiple constraints handling is proposed for the multi-objective problem of telescope-skin support structure topology optimization. Objectives of individuals are obtained by FEA using ANSYS and then transmitted into MATLAB for next-step processing. Results show that this approach provides a feasible solution for the multiobjective problem with multiple constraints. (2) Compared with traditional methods, bit-matrix representation has various advantages in two-dimensional topology optimization. Firstly, the bit-matrix is an intuitionistic and natural description of the design domain. It contains both the numerical values of the corresponding units and the geometrical information. In addition, it can be directly adopted as a chromosome of the GA without any coding or encoding process. Secondly, two-dimensional GA operators based on bit-matrix representation concern directly the geometrical distribution of material and can always keep their geometrical information during genetic processes. In addition, there is no boundary overflow of variables, because the new chromosomes obtained by GA operators are always in the design domain. (3) A strain requirement is included considering the structural material for engineering applications. Along with the connectivity requirement, it turns the optimization to be a problem with two-level constraints. The method for multiple constraints handling is proposed and provides an effective treatment for the multi-objective problem. (4) Feasible solutions for support structure of telescope skin are obtained by the algorithm. These solutions are all relatively optimal, considering these objectives are conflicting. The structure for the skin can be selected from them according to practical requirements. (5) The telescope-skin support structure topology optimization is currently for static objectives. The objective values are obtained by FEA using ANSYS. With the powerful finite element processing capacity of ANSYS, this approach can be conveniently transplanted to deal with more complicated problems such as the topology optimization of dynamic objectives.

Acknowledgements

Fig. 18

Structure with lowest in-plane stiffness.

The authors would like to thank the anonymous reviewers for their critical and constructive review of the manuscript. This

Topology optimization of support structure of telescope skin based on bit-matrix representation NSGA-II study was co-supported by the National Natural Science Foundation of China (Nos. 50905085 and 91116020) and the National Science Foundation for Post-doctoral Scientists of China (No. 2012M511263). References 1. Bartley-Cho JD, Wang DP, Martin CA, Kudva JN, West MN. Development of high-rate, adaptive trailing edge control surface for the smart wing phase 2 wind tunnel model. J Intel Mater Syst Struct 2004;15(4):279–91. 2. Sofla AYN, Meguid SA, Tan KT, Yeo WK. Shape morphing of aircraft wing: status and challenges. Mater Des 2010;31(3):1284–92. 3. Barbarino S, Bilgen O, Ajaj RM, Friswell MI, Inman DJ. A review of morphing aircraft. J Intel Mater Syst Struct 2011;22(9):823–77. 4. Icardi U, Ferrero L. Preliminary study of an adaptive wing with shape memory alloy torsion actuators. Mater Des 2009;30(10):4200–10. 5. Diaconu CG, Weaver PM, Mattioni F. Concepts for morphing airfoil sections using bi-stable laminated composite structures. Thin Wall Struct 2008;46(6):689–701. 6. Kudva JN. Overview of the DARPA Smart Wing Project. J Intel Mater Syst Struct 2004;15(4):261–7. 7. Thill C, Etches J, Bond I, Potter K, Weaver P. Morphing skins. Aeronaut J 2008;112(1129):117–39. 8. Olympio KR, Gandhi F. Flexible skins for morphing aircraft using cellular honeycomb cores. J Intel Mater Syst Struct 2010;21(17):1719–35. 9. Olympio KR, Gandhi F. Zero Poisson’s ratio cellular honeycombs for flex skins undergoing one-dimensional morphing. J Intel Mater Syst Struct 2010;21(17):1737–53. 10. Bubert EA, Woods BKS, Lee K, Kothera CS, Wereley NM. Design and fabrication of a passive 1D morphing aircraft skin. J Intel Mater Syst Struct 2010;21(17):1699–717. 11. Sigmund O. A 99 line topology optimization code written in MATLAB. Struct Multi Optim 2011;21(2):120–7. 12. BendsØe MP. Optimal shape design as a material distribution problem. Struct Optim 1989;1(4):193–202. 13. Jenkins WM. Structural optimization with the genetic algorithm. Struct Eng 1991;69(24):418–22. 14. Chapman C, Saitou K, Jakiela MJ. Genetic algorithms as an approach to configuration and topology design. J Mech Des 1994;116(4):1005–12.

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15. Schoenauer M. Shape representations and evolutionary schemes. In: Fogel LJ, Angeline PJ, Ba¨ck T, editors. Proceedings of the fifth annual conference on evolutionary programming. Berlin: Springer; 1996, p. 121–29. 16. Chapman C, Jakiela MJ. Genetic algorithm-based structural topology design with compliance and topology simplification considerations. J Mech Des 1996;118(1):89–98. 17. Jakiela MJ, Chapman C, Duda J, Adewuya A, Saitou K. Continuum structural topology design with genetic algorithms. Comput Method Appl Mech Eng 2000;186(2–4):339–56. 18. Wang SY, Tai K. Structural topology design optimization using genetic algorithms with a bit-array representation. Comput Method Appl Mech Eng 2005;194(36–38):3749–70. 19. Olympio KR, Gandhi F, Skin designs using multi-objective topology optimization. In: Proceedings of the 49th structures, structural dynamics, and materials conference. Berlin: Springer; 2008, p. 1025–50. 20. Gandhi F, Anusonti-Inthra P. Skin design studies for variable camber morphing airfoils. Smart Mater Struct 2008;17(1):015025–15033. 21. Deb K. A fast and elitist multiobjective genetic algorithm: NSGAII. IEEE Trans Evol Comput 2002;6(2):182–97. 22. Wang SY, Tai K. Graph representation for structural topology optimization using genetic algorithms. Comput Struct 2004;82(20–21):1609–22. 23. Sigmund O. Morphology-based black and white filters for topology optimization. Struct Multi Optim 2007;33(4–5):401–24. 24. Blum C, Roli A. Metaheuristics in combinatorial optimization: overview and conceptual comparison. ACM Comput Surv 2003;35(3):268–308. Liu Weidong received his B.S. degree in aircraft design and engineering from Nanjing University of Aeronautics and Astronautics in 2008 and became a Ph.D. student from then on. His main research interests are mechanical design and theory, morphing aircraft design, and so on. Zhu Hua is an associate professor in the College of Aerospace Engineering at Nanjing University of Aeronautics and Astronautics, Nanjing, China. He received his Ph.D. degree from the same university in 2007. His current research interests are mechanical electronic engineering and piezoelectric actuators.