A two-step optimization scheme based on equivalent stiffness parameters for forcing convexity of fiber winding angle in composite frames

  • Zunyi Duan
  • Jun YanEmail author
  • Ikjin LeeEmail author
  • Erik Lund
  • Jingyuan Wang
Research Paper


For stiffness design optimization of composite frame structures, one of the major problems when using fiber winding angles as design variables directly is the lack of convexity of the objective function, which may lead to different local optima depending on initial designs when a traditional gradient-based optimization algorithm is applied. Therefore, the present paper adopts a gradient-based two-step optimization scheme to cope with the difficulty and search for a better optimal design of composite frames in which the fiber winding angles are taken as design variables. To realize the two-step optimization scheme, the equivalent stiffness parameters of a composite beam with circular cross-section are derived in explicit expressions and used to force the convexity of the design optimization of the composite frame. The stiffness matrices are linearly expressed in terms of the stiffness parameters, which guarantee the convexity of the design variable feasible region in the stiffness parameter space. The equivalent stiffness parameters are adopted to keep invariance of physical quantities between fiber winding angle and equivalent stiffness parameter spaces. In the two-step optimization scheme, the minimum identification problem with the constraint that the objective function at the new starting point is less than or equal to the previous objective function at the optimum point in fiber winding angle space is established. Then, the two-step optimization scheme can be implemented in the fiber winding angle and structural equivalent stiffness parameter spaces, respectively, until the minimum identification problem is not possible to identify a new starting point. The proposed two-step optimization scheme for composite frames fully takes advantage of the stiffness parameters in convexity and fiber winding angles as practically physical quantities, respectively. The sensitivity information of the objective function with respect to fiber winding angles and equivalent stiffness parameters is derived by the analytical sensitivity analysis method. Numerical examples show that the two-step optimization scheme can effectively force convexity of the optimization model and help to eliminate the initial design dependency. The effectiveness of the proposed two-step scheme is further verified through the particle swarm optimization (PSO) algorithm which is an evolutionary algorithm with global optimization capability.


Composite frame structures Fiber winding angle optimization Initial design dependency Equivalent stiffness parameters Analytical sensitivity analysis 



The authors thank Prof. Bin Niu for checking the correctness of the formula derivation.

Funding information

Financial supports for this research were provided by the National Natural Science Foundation of China (Nos. 11672057,11711530018 and 11732004), the National Key R&D Program of China (2017YFC0307203), Program (LR2017001) for Innovative Talents at Colleges and Universities in Liaoning Province, the 111 project (B14013), the Korea Institute of Energy Technology Evaluation and Planning, and the Ministry of Trade Industry & Energy of the Republic of Korea (No. 20172010000830).


  1. Abdalla MM, Setoodeh S, Gürdal Z (2007) Design of variable stiffness composite panels for maximum fundamental frequency using lamination parameters. Compos Struct 81(2):283–291Google Scholar
  2. An H, Chen S, Huang H (2015) Simultaneous optimization of stacking sequences and sizing with two-level approximations and a genetic algorithm. Compos Struct 123:180–189Google Scholar
  3. Andersen ED, Roos C, Terlaky T (2003) On implementing a primal-dual interior-point method for conic quadratic optimization. Math Program 95(2):249–277MathSciNetzbMATHGoogle Scholar
  4. Aymerich F, Serra M (2008) Optimization of laminate stacking sequence for maximum buckling load using the ant colony optimization (ACO) metaheuristic. Compos A: Appl Sci Manuf 39(2):262–272Google Scholar
  5. Bakis CE, Bank LC, Brown VL, Cosenza E, Davalos JF, Lesko JJ, Machida A, Rizkalla SH, Triantafillou TC (2002) Fiber-reinforced polymer composites for construction—state-of-the-art review. J Compos Constr 6(2):73–87Google Scholar
  6. Bendsoe MP, Sigmund O (2003) Topology Optimization-Theory, Methods and Applications. Springer, New YorkGoogle Scholar
  7. Birge B (2003) PSOt - a particle swarm optimization toolbox for use with Matlab. SIS 3:973–990Google Scholar
  8. Bloomfield MW, Diaconu CG, Weaver PM (2009) On feasible regions of lamination parameters for lay-up optimization of laminated composites. Proceedings of the Royal Society of London a: Mathematical, Physical and Engineering Sciences, 465: 1123–1143Google Scholar
  9. Bohrer RZG, Almeida SFM, Donadon MV (2015) Optimization of composite plates subjected to buckling and small mass impact using lamination parameters. Compos Struct 120:141–152Google Scholar
  10. Bruyneel M (2011) SFP - a new parameterization based on shape functions for optimal material selection: application to conventional composite plies. Struct Multidiscip Optim 43(1):17–27Google Scholar
  11. Bruyneel M, Fleury C (2002) Composite structures optimization using sequential convex programming. Adv Eng Softw 33(7):697–711zbMATHGoogle Scholar
  12. Byrd RH, Gilbert JC, Nocedal J (2000) A trust region method based on interior point techniques for nonlinear programming. Math Program 89(1):149–185MathSciNetzbMATHGoogle Scholar
  13. Chang N, Wang W, Yang W, Wang J (2010) Ply stacking sequence optimization of composite laminate by permutation discrete particle swarm optimization. Struct Multidiscip Optim 41(2):179–187Google Scholar
  14. Cook RD, Malkus DS, Plesha ME, Witt RJ (2002) Concepts and applications of finite element analysis, Fourth Edition. John Wiley & Sons, New YorkGoogle Scholar
  15. Davalos JF, Kim Y, Barbero EJ (1994) Analysis of laminated beams with a layer-wise constant shear theory. Compos Struct 28(3):241–253Google Scholar
  16. Deng S, Pai PF, Lai CC (2005) A solution to the stacking sequence of a composite laminate plate with constant thickness using simulated annealing algorithms. Int J Adv Manuf Technol 26(5–6):499–504Google Scholar
  17. Diaconu CG, Sato M, Sekine H (2002) Buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads using lamination parameters. Compos Struct 58(4):423–433Google Scholar
  18. Duan ZY, Yan J, Lee IJ, Wang, JY, Yu T (2018) Integrated design optimization of composite frames and materials for maximum fundamental frequency with continuous fiber winding angles. Acta Mech Sinica 34(6):1084–1094Google Scholar
  19. Faria AR (2015) Optimization of composite structures under multiple load cases using a discrete approach based on lamination parameters. Int J Numer Methods Eng 104(9):827–843MathSciNetzbMATHGoogle Scholar
  20. Ferreira RT, Rodrigues HC, Guedes JM, Hernandes JA (2014) Hierarchical optimization of laminated fiber reinforced composites. Compos Struct 107:246–259Google Scholar
  21. Foldager J (1997) Design optimization of laminated composite plates divided into rectangular patches with use of lamination parameters, In: Gutkowski W and Mroz Z (eds): Proc WCSMO 2-second world congress of structural and multidisciplinary optimization, Inst of Fundamental Technological Research, Warsaw, Vol 2, 669–675Google Scholar
  22. Foldager J, Hansen JS, Olhoff N (1998) A general approach forcing convexity of ply angle optimization in composite laminates. Struct Optim 16(2–3):201–211Google Scholar
  23. Fukunaga H, Vanderplaats GN (1991) Strength optimization of laminated composites with respect to layer thickness and/or layer orientation angle. Comput Struct 40(6):1429–1439Google Scholar
  24. Fukunaga H, Sekine H (1992) Stiffness design method of symmetrical laminates using lamination parameters. AIAA J 30(11):2791–2793Google Scholar
  25. Ganguli R (2013) Optimal design of composite structures: a historical review. J Indian Inst Sci 93(4):557–570Google Scholar
  26. Gao T, Zhang WH, Duysinx P (2012) A bi-value coding parameterization scheme for the discrete optimal orientation design of the composite laminate. Int J Numer Methods Eng 91(1):98–114zbMATHGoogle Scholar
  27. Ghiasi H, Pasini D, Lessard L (2009) Optimum stacking sequence design of composite materials Part I: constant stiffness design. Compos Struct 90(1):1–11Google Scholar
  28. Ghiasi H, Fayazbakhsh K, Pasini D, Lessard L (2010) Optimum stacking sequence design of composite materials Part II: variable stiffness design. Compos Struct 93(1):1–13Google Scholar
  29. Gomes FA, Senne TA (2011) An SLP algorithm and its application to topology optimization. Comput Appl Math 30(1):53–89MathSciNetzbMATHGoogle Scholar
  30. Grenestedt JL, Gudmundson P (1993) Layup optimization of composite-material structures. Optimal design with advanced materials. Optimal Design with Advanced Materials 311–336Google Scholar
  31. Hammer VB, Bendsøe MP, Lipton R, Pedersen P (1997) Parametrization in laminate design for optimal compliance. Int J Solids Struct 34(4):415–434zbMATHGoogle Scholar
  32. Ibrahim S, Polyzois D, Hassan SK (2000) Development of glass fiber reinforced plastic poles for transmission and distribution lines. Can J Civ Eng 27(5):850–858Google Scholar
  33. Jones R M (2014) Mechanics of composite materials. CRC pressGoogle Scholar
  34. Kennedy J, Eberhart RC. (1995) Particle swarm optimization. In: Proceedings of the fourth IEEE international conference on neural networks, 1942–1948Google Scholar
  35. Khani A, Ijsselmuiden ST, Abdalla MM, Gurdal Z (2011) Design of variable stiffness panels for maximum strength using lamination parameters. Compos Part B 42(3):546–552Google Scholar
  36. Kim Y, Davalos JF, Barbero EJ (1996) Progressive failure analysis of laminated composite beams. J Compos Mater 30(5):536–560Google Scholar
  37. Kiyono CY, Silva ECN, Reddy JN (2017) A novel fiber optimization method based on normal distribution function with continuously varying fiber path. Compos Struct 160:503–515Google Scholar
  38. Lei F, Qiu RB, Bai YC, Yuan CF (2018) An integrated optimization for laminate design and manufacturing of a CFRP wheel hub based on structural performance. Structural and Multidisciplinary Optimization, 1–13Google Scholar
  39. Liu B, Haftka RT (2004) Single-level composite wing optimization based on flexural lamination parameters. Struct Multidiscip Optim 26(1–2):111–120Google Scholar
  40. Liu ST, Hou YP, Sun X, Zhang YC (2012a) A two-step optimization scheme for maximum stiffness design of laminated plates based on lamination parameters. Compos Struct 94(12):3529–3537Google Scholar
  41. Liu XF, Cheng GD, Yan J, Jiang L (2012b) Singular optimum topology of skeletal structures with frequency constraints by AGGA. Struct Multidiscip Optim 45(3):451–466Google Scholar
  42. Liu DZ, Toropov VV, Barton DC, Querin QM (2015) Weight and mechanical performance optimization of blended composite wing panels using lamination parameters. Struct Multidiscip Optim 52(3):549–562MathSciNetGoogle Scholar
  43. Lund E, Stegmann J (2005) On structural optimization of composite shell structures using a discrete constitutive parametrization. Wind Energy 8(1):109–124zbMATHGoogle Scholar
  44. Miki M (1982) Material design of composite laminates with required in-plane elastic properties. Proc Progress in Science and Engineering of Composites, ICCM IV, TokyoGoogle Scholar
  45. Nikbakt S, Kamarian S, Shakeri M (2018) A review on optimization of composite structures part I: laminated composites. Compos Struct 195:158–185Google Scholar
  46. Przemieniecki J S (1985) Theory of matrix structural analysis. Courier CorporationGoogle Scholar
  47. Riche RL, Haftka RT (1993) Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm. AIAA J 31(5):951–956zbMATHGoogle Scholar
  48. Schütze R (1997) Lightweight carbon fibre rods and truss structures. Mater Des 18(4–6):231–238Google Scholar
  49. Setoodeh S, Abdalla MM, Gürdal Z (2006) Design of variable–stiffness laminates using lamination parameters. Compos Part B 37(4–5):301–309Google Scholar
  50. Sigmund O (2001) On the usefulness of non-gradient approaches in topology optimization. Struct Multidiscip Optim 43(5):589–596MathSciNetzbMATHGoogle Scholar
  51. Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng 62(14):2009–2027zbMATHGoogle Scholar
  52. Svanberg K (1987) The method of moving asymptotes – a new method for structural optimization. Int J Numer Methods Eng 24:359–373MathSciNetzbMATHGoogle Scholar
  53. Thuwis GA, Breuker RD, Abdalla MM, Gürdal Z (2010) Aeroelastic tailoring using lamination parameters. Struct Multidiscip Optim 41(4):637–646Google Scholar
  54. Todoroki A, Terada Y (2004) Improved fractal branch and bound method for stacking-sequence optimizations of laminates. AIAA J 42(1):141–148Google Scholar
  55. Tsai SW, Pagano NJ (1968) Invariant properties of composite materials. No. AFML-TR-67-349. Air Force Materials Lab, Wright-Patterson AFB, OhioGoogle Scholar
  56. Venkataraman S, Haftka RT (1999) Optimization of composite panels - a review, proceedings of the American Society for Composites – 14th annual technical conference, Fairborn, Ohio, 479–488Google Scholar
  57. Xu YJ, Zhu JH, Wu Z, Cao YF, Zhao YB, Zhang WH (2018) A review on the design of laminated composite structures: constant and variable stiffness design and topology optimization. Advanced Composites and Hybrid Materials,1–18Google Scholar
  58. Yan J, Duan ZY, Lund E, Wang JY (2017) Concurrent multi-scale design optimization of composite frames with manufacturing constraints. Struct Multidiscip Optim 56(3):519–533Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, International Research Center for Computational MechanicsDalian University of TechnologyDalianChina
  2. 2.Department of Mechanical EngineeringKorea Advanced Institute of Science and TechnologyDaejeonRepublic of Korea
  3. 3.Harbin Electric Power Equipment Company LimitedHarbin Electric CorporationHarbinChina
  4. 4.Department of Materials and ProductionAalborg UniversityAalborgDenmark

Personalised recommendations