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Topology optimization of composite hyperelastic material using SPIMFO-method

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Fiber reinforced materials are used in assorted engineering application and for this reason, new additive manufacturing technologies have been developed for this type of materials. With these technologies, it is possible to construct composite structures with different shapes and desired fiber orientation. Therefore, reinforced composite structures can be designed based on an optimized fiber orientation. Because of it, several works propose different methods to tailor fiber directions. Albeit, the determination of fiber optimized orientation is a problem usually subjected to multiple local minima issues, unless discrete material optimization methods are used. In addition, in these works, usually a linear relation between stress and strain is considered, which limits simulations to small displacements, strains and rotations. Recently, a method named SPIMFO has been developed where the angle is considered a continuous variable and the local minima issues are circumvented. Thus, this work proposes to determine the optimized fiber orientation of a fiber reinforced composite structure by using the SPIMFO method with a constitutive equation in fully nonlinear range based on transversely isotropic neo-Hookean model. A new method to measure the fiber continuity named index of average continuity is proposed and implemented. The results obtained by using the proposed method are compared to results obtained by using a discrete model named NDFO-m, which is proposed in a previous work.

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  1. Awad ZK, Aravinthan T, Zhuge Y, Gonzalez F (2012) A review of optimization techniques used in the design of fibre composite structures for civil engineering applications. Mater Des 33:534–544

    Article  Google Scholar 

  2. Bendsøe M, Sigmund O (2003) Theory, methods and applications. Topology optimization. Springer, Berlin

    MATH  Google Scholar 

  3. Bonet J, Burton A (1998) A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations. Comput Methods Appl Mech Eng 162(1–4):151–164

    Article  Google Scholar 

  4. 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):17–27

    Article  Google Scholar 

  5. Dickson AN, Ross KA, Dowling DP (2018) Additive manufacturing of woven carbon fibre polymer composites. Compos Struct 206:637–643

    Article  Google Scholar 

  6. da Silva ALF, Salas RA, Silva ECN, Reddy J (2020) Topology optimization of fibers orientation in hyperelastic composite material. Compos Struct 231:111488

    Article  Google Scholar 

  7. Funke SW, Farrell PE (2013) A framework for automated pde-constrained optimisation. arXiv preprint arXiv:1302.3894

  8. Gao T, Zhang W, 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):98–114

    Article  Google Scholar 

  9. Gay D (2014) Composite materials: design and applications. CRC Press, Boca Raton

    Book  Google Scholar 

  10. Hou Z, Tian X, Zhang J, Li D (2018) 3d printed continuous fibre reinforced composite corrugated structure. Compos Struct 184:1005–1010

    Article  Google Scholar 

  11. Karush W (1939) Minima of functions of several variables with inequalities as side constraints. M. Sc. Dissertation. Dept. of Mathematics, Univ. of Chicago

  12. Kaw AK (2005) Mechanics of composite materials. CRC Press, Boca Raton

    Book  Google Scholar 

  13. Kiyono C, Silva E, Reddy J (2017) A novel fiber optimization method based on normal distribution function with continuously varying fiber path. Compos Struct 160:503–515

    Article  Google Scholar 

  14. Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In : Neyman J (ed) Proceedings of the second Berkeley symposium on mathematical statistics and probability (1951)

  15. Lazarov BS, Sigmund O (2011) Filters in topology optimization based on helmholtz-type differential equations. Int J Numer Meth Eng 86(6):765–781

    Article  MathSciNet  Google Scholar 

  16. Lee JW, Kim JJ, Yoon GH (2019) Stress constraint topology optimization using layerwise theory for composite laminates. Compos Struct.

    Article  Google Scholar 

  17. Lee J, Yoo J, Min S, Yoon M (2019) Topology optimization of anisotropic magnetic composites in actuators using homogenization design method. Struct Multidiscip Optim 60(4):1423–1436.

    Article  MathSciNet  Google Scholar 

  18. Logg A, Mardal KA, Wells G (2012) Automated solution of differential equations by the finite element method: The FEniCS book, vol 84. Springer, Berlin

    Book  Google Scholar 

  19. Mei C, Wang Q, Yu C, Xia Z (2020) IGA based bi-layer fiber angle optimization method for variable stiffness composites. Comput Model Eng Sci 124(1):179–202

    Article  Google Scholar 

  20. Naumann U (2012) The art of differentiating computer programs: an introduction to algorithmic differentiation, vol 24. SIAM, Philadelphia

    MATH  Google Scholar 

  21. Nikbakt S, Kamarian S, Shakeri M (2018) A review on optimization of composite structures part I: laminated composites. Compos Struct 195:158–185

    Article  Google Scholar 

  22. Ning F, Cong W, Qiu J, Wei J, Wang S (2015) Additive manufacturing of carbon fiber reinforced thermoplastic composites using fused deposition modeling. Compos B Eng 80:369–378

    Article  Google Scholar 

  23. Papapetrou VS, Patel C, Tamijani AY (2020) Stiffness-based optimization framework for the topology and fiber paths of continuous fiber composites. Compos Part B-Eng.

    Article  Google Scholar 

  24. Pejman R, Aboubakr SH, Martin WH, Devi U, Tan MHY, Patrick JF, Najafi AR (2019) Gradient-based hybrid topology/shape optimization of bioinspired microvascular composites. Int J Heat Mass Transf.

    Article  Google Scholar 

  25. Quan Z, Wu A, Keefe M, Qin X, Yu J, Suhr J, Byun JH, Kim BS, Chou TW (2015) Additive manufacturing of multi-directional preforms for composites: opportunities and challenges. Mater Today 18(9):503–512

    Article  Google Scholar 

  26. Salas RA, Ramírez FJ, Montealegre-Rubio W, Silva ECN, Reddy JN (2018) A topology optimization formulation for transient design of multi-entry laminated piezocomposite energy harvesting devices coupled with electrical circuit. Int J Numer Methods Eng 113(8):1370–1410.

    Article  MathSciNet  Google Scholar 

  27. Salas RA, Ramírez-Gil FJ, Montealegre-Rubio W, Silva ECN, Reddy J (2018) Optimized dynamic design of laminated piezocomposite multi-entry actuators considering fiber orientation. Comput Methods Appl Mech Eng 335:223–254

    Article  MathSciNet  Google Scholar 

  28. Salas Varela RA (2018) Projeto dinâmico de estruturas piezocompósitas laminadas (epla) utilizando o método de otimização topológica (mot). Ph.D. thesis, Universidade de São Paulo

  29. Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Meth Eng 62(14):2009–2027

    Article  Google Scholar 

  30. Tong X, Ge W, Gao X, Li Y (2019) Simultaneous optimization of fiber orientations and topology shape for composites compliant leading edge. J Reinf Plast Compos 38(15):706–716.

    Article  Google Scholar 

  31. Venkataraman, S., Haftka, R.T.: Optimization of composite panels-a review. In: Proceedings-American society for composites, pp 479–488 (1999)

  32. Wriggers P (2008) Nonlinear finite element methods. Springer, Berlin

    MATH  Google Scholar 

  33. Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57

    Article  MathSciNet  Google Scholar 

  34. Yin L, Ananthasuresh G (2001) Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct Multidiscip Optim 23(1):49–62

    Article  Google Scholar 

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ALF Silva thanks CAPES (Coordination of Superior Level Staff Improvement) and INCT/CEMTEC (National Institute on Advanced Eco-Efficient Cement-Based Technologies) for the financial during his Master’s Degree Under Grant 88887. 165790/2018-00. RA Salas gratefully acknowledge support of the RCGI - Research Centre for Gas Innovation, hosted by the University of São Paulo (USP) and sponsored by FAPESP - São Paulo Research Foundation (2014/50279-4) and Shell Brasil. ECN Silva thanks the financial support of CNPq (National Council for Research and Development) Under Grant 302658/2018-1.

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Correspondence to Emilio Carlos Nelli Silva.

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da Silva, A.L.F., Salas, R.A. & Silva, E.C.N. Topology optimization of composite hyperelastic material using SPIMFO-method. Meccanica 56, 417–437 (2021).

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