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Discrete Material and Thickness Optimization of laminated composite structures including failure criteria

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Abstract

This work extends the Discrete Material and Thickness Optimization approach to structural optimization problems where strength considerations in the form of failure criteria are taken into account for laminated composite structures. It takes offset in the density approaches applied for stress constrained topology optimization of single-material problems and develops formulations for multi-material topology optimization problems applied for laminated composite structures. The method can be applied for both stress- and strain-based failure criteria. The large number of local constraints is reduced by the use of aggregate functions, and the developed approach is demonstrated for optimization problems involving both constant and varying thickness laminated composites.

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References

  • Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Giga-voxel computational morphogenesis for structural design. Nature 550(7674):84–86

    Article  Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–203

    Article  Google Scholar 

  • Blasques JP, Stolpe M (2012) Multi-material topology optimization of laminated composite beam cross sections. Compos Struct 94(11):3278–3289

    Article  Google Scholar 

  • Bloomfield MW, Diaconu CG, Weaver PM (2009) On feasible regions of lamination parameters for lay-up optimization of laminated composites. Proc R Soc A Math Phys Eng Sci 465(2104):1123–1143

    Article  MathSciNet  MATH  Google Scholar 

  • Bruggi M (2008) On an alternative approach to stress constraints relaxation in topology optimization. Struct Multidiscip Optim 36(2):125–141

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Bruyneel M, Duysinx P (2006) Note on singular optima in laminate design problems. Struct Multidiscip Optim 31(2):156–159

    Article  Google Scholar 

  • Bruyneel M, Duysinx P, Fleury C, Gao T (2011) Extensions of the shape functions with penalization parameterization for composite-ply optimization. AIAA J 49(10):2325–2329

    Article  Google Scholar 

  • Cairns D, Mandell J, Scott M, Maccagnano J (1999) Design and manufacturing considerations for ply drops in composite structures. Compos Part B Eng 30(5):523–534

    Article  Google Scholar 

  • Cheng GD, Guo X (1997) 𝜖-relaxed approach in topology optimization. Struct Optim 13:258–266

    Article  Google Scholar 

  • Cheng GD, Jiang Z (1992) Study on topology optimization with stress constraint. Eng Optim 20:129–148

    Article  Google Scholar 

  • Cheng G, Olhoff N (1981) An investigation concerning optimal design of solid elastic plates. Int J Solids Struct 17(3):305–323

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng KT, Olhoff N (1982) Regularized formulation for optimal design of axisymmetric plates. Int J Solids Struct 18(2):153–170

    Article  MATH  Google Scholar 

  • Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43:1453–1478

    Article  MathSciNet  MATH  Google Scholar 

  • Duysinx P, Sigmund O (1998) New development in handling stress constraints in optimal material distribution. In: 7th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimization. A collection of technical papers (held in St. Louis, Missouri), vol 3, pp 1501–1509

  • ESAComp (2016) ESAComp - software for analysis and design of composites, Release 4.5. 2. http://www.esacomp.com

  • de 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):827–843

    Article  MathSciNet  MATH  Google Scholar 

  • 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  MATH  Google Scholar 

  • Ghiasi H, Pasini D, Lessard L (2009) Optimum stacking sequence design of composite materials Part I: Constant stiffness design. Compos Struct 90(1):1–11

    Article  Google Scholar 

  • 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):1–13

    Article  Google Scholar 

  • Gill PE, Murray W, Saunders MA (2005) SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM Rev 47(1):99–131

    Article  MathSciNet  MATH  Google Scholar 

  • Groenwold AA, Haftka RT (2006) Optimization with non-homogeneous failure criteria like {T}sai–{W}u for composite laminates. Struct Multidiscip Optim 32:183–190

    Article  Google Scholar 

  • Guo X, Cheng GD, Yamazaki K (2001) A new approach for the solution of singular optima in truss topology optimization with stress and local buckling constraints. Struct Multidiscip Optim 22(5):364–373

    Article  Google Scholar 

  • Gürdal Z, Haftka RT, Nagendra S (1994) Genetic Algorithms for the design of laminated composite panel. Mech Adv Mater Struct 30(3):29–35

    Google Scholar 

  • Gürdal Z, Haftka RT, Hajela P (1999) Design and Optimization of Laminated Composite Materials. Wiley, New York

    Google Scholar 

  • Haftka RT, Gürdal Z (1992) Elements of structural optimization, 3rd edn. Kluwer, Norwell

    Book  MATH  Google Scholar 

  • Hammer VB, Bendsøe MP, Lipton R, Pedersen P (1997) Parametrization in laminate design for optimal compliance. Int J Solids Struct 34:415–434

    Article  MATH  Google Scholar 

  • Herencia JE, Weaver PM, Friswell MI (2008) Optimization of anisotropic composite panels with T-shaped stiffeners including transverse shear effects and out-of-plane loading. Struct Multidiscip Optim 37(2):165–184

    Article  Google Scholar 

  • Holmberg E, Torstenfelt B, Klarbring A (2013) Stress constrained topology optimization. Struct Multidiscip Optim 48(1):33–47

    Article  MathSciNet  MATH  Google Scholar 

  • Hvejsel CF, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidiscip Optim 43(6):811–825

    Article  MATH  Google Scholar 

  • Hvejsel CF, Lund E, Stolpe M (2011) Optimization strategies for discrete multi-material stiffness optimization. Struct Multidiscip Optim 44(2):149–163

    Article  Google Scholar 

  • IBM ILOG (2015) IBM ILOG CPLEX Optimization Studio V12.6

  • IJsselmuiden ST, Abdalla MM, Gürdal Z (2008) Implementation of strength-based failure criteria in the lamination parameter design space. AIAA J 46(7):1826–1836

    Article  Google Scholar 

  • IJsselmuiden ST, Abdalla MM, Seresta O, Gürdal Z (2009) Multi-step blended stacking sequence design of panel assemblies with buckling constraints. Compos Part B Eng 40(4):329–336

    Article  Google Scholar 

  • Irisarri FX, Lasseigne A, Leroy FH, Le Riche R (2014) Optimal design of laminated composite structures with ply drops using stacking sequence tables. Compos Struct 107(Supplement C):559–569

    Article  Google Scholar 

  • Irisarri FX, Peeters DMJ, Abdalla MM (2016) Optimisation of ply drop order in variable stiffness laminates. Compos Struct 152(Supplement C):791–799

    Article  Google Scholar 

  • Johansen LS, Lund E (2009) Optimization of laminated composite structure using delamination criteria and hierarchical models. Struct Multidiscip Optim 38(4):357—-375

    Article  Google Scholar 

  • Johansen LS, Lund E, Kleist J (2009) Failure optimization of geometrically linear/nonlinear laminated composite structures using a two-step hierarchical model adaptivity. Comput Methods Appl Mech Eng 198:2421–2438

    Article  MATH  Google Scholar 

  • Kennedy GJ (2016) A full-space barrier method for stress-constrained discrete material design optimization. Struct Multidiscip Optim 54(3):619–639

    Article  MathSciNet  Google Scholar 

  • Kennedy GJ, Martins JRRA (2013) A laminate parametrization technique for discrete ply-angle problems with manufacturing constraints. Struct Multidiscip Optim 48(2):379–393

    Article  MathSciNet  Google Scholar 

  • Kennedy G, Martins J (2014) A parallel finite-element framework for large-scale gradient-based design optimization of high-performance structures. Finite Elem Anal Des 87:56–73

    Article  Google Scholar 

  • Khani A, IJsselmuiden ST, Abdalla MM, Gürdal Z (2011) Design of variable stiffness panels for maximum strength using lamination parameters. Compos Part B Eng 42(3):546–552

    Article  Google Scholar 

  • Kirsch U (1990) 𝜖-relaxed approach in topology optimization. Struct Optim 2(3):133–142

    Article  Google Scholar 

  • 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(Supplement C):503–515

    Article  Google Scholar 

  • Kogiso N, Watson LT, Gürdal Z, Haftka RT (1994) Genetic algorithms with local improvement for composite laminate design. Struct Optim 7(4):207–218

    Article  Google Scholar 

  • Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. In: International federation of active controls symposium on computer-aided design of control systems. Zurich, Switzerland, pp 113–117

  • Kristinsdottir BP, Zabinsky ZB, Tuttle ME, Neogi S (2001) Optimal design of large composite panels with varying loads. Compos Struct 51(1):93–102

    Article  Google Scholar 

  • Le Riche R, Haftka RT (1993) Optimization of laminate stacking sequence for buckling load maximization by generic algorithm. AIAA J 31:951–956

    Article  MATH  Google Scholar 

  • Le Riche R, Haftka RT (1995) Improved genetic algorithm for minimum thickness composite laminate design. Compos Eng 5(2):143–161

    Article  Google Scholar 

  • Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41(4):605–620

    Article  Google Scholar 

  • Liu B, Haftka RT (2001) Composite Wing Structural Design Optimization with Continuity Constraints. In: A01-25021, Structural Dynamics and Materials Conference and Exhibit; American Institute of Aeronautics & Astronautics, Seattle, WA, pp 1–12

  • Liu B, Haftka RT, Akgu̇n MA (2000) Two-level composite wing structural optimization using response surfaces. Struct Multidiscip Optim 20(2):87–96

    Article  Google Scholar 

  • Liu D, Toropov VV, Querin OM, Barton DC (2011) Bilevel optimization of blended composite wing panels. J Aircr 48(1):107–118

    Article  Google Scholar 

  • Liu D, Toropov VV, Barton DC, Querin OM (2015) Weight and mechanical performance optimization of blended composite wing panels using lamination parameters. Struct Multidiscip Optim 52:549–562

    Article  MathSciNet  Google Scholar 

  • Lund E, Stegmann J (2005) On structural optimization of composite shell structures using a discrete constitutive parametrization. Wind Energy 8(1):109–124

    Article  MATH  Google Scholar 

  • Miki M, Sugiyama Y (1993) Optimum design of laminated composite plates using lamination parameters. AIAA J 31:921–922

    Article  Google Scholar 

  • Mukherjee A, Varughese B (2001) Design guidelines for ply drop-off in laminated composite structures. Compos Part B Eng 32(2):153–164

    Article  Google Scholar 

  • Nagendra S, Jestin D, Gu̇rdal Z, Haftka RT, Watson LT (1996) Improved genetic algorithm for the design of stiffened composite panels. Comput Struct 58(3):543–555

    Article  MATH  Google Scholar 

  • Oest J, Lund E (2017) Topology optimization with finite-life fatigue constraints. Structural and Multidisciplinary Optimization. https://doi.org/10.1007/s00158-017-1701-9

  • Overgaard L, Lund E, Thomsen O (2010) Structural collapse of a wind turbine blade. Part A: Static test and equivalent single layered models. Compos A: Appl Sci Manuf 41(2):257–270

    Article  Google Scholar 

  • Pari̇s J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of continuum structures with local and global stress constraints. Struct Multidiscip Optim 39(4):419–437

    Article  MathSciNet  MATH  Google Scholar 

  • Pari̇s J, Navarrina F, Colominas I, Casteleiro M (2010) Block aggregation of stress constraints in topology optimization of structures. Adv Eng Softw 41(3):433–441

    Article  MATH  Google Scholar 

  • Peeters D, Abdalla M (2016) Optimization of Ply Drop Locations in Variable-Stiffness Composites. AIAA J 54(5):1760–1768

    Article  Google Scholar 

  • Rozvany GIN (1972) Grillages of maximum strength and maximum stiffness. Int J Mech Sci 14(10):651–666

    Article  Google Scholar 

  • Rozvany GIN, Birker T (1994) On singular topologies in exact layout optimization. Struct Optim 8(4):228–235

    Article  Google Scholar 

  • Schmit LA, Farshi B (1973) Optimum laminate design for strength and stiffness. Int J Numer Methods Eng 7(4):519–536

    Article  Google Scholar 

  • Seresta O, Gürdal Z, Adams DB, Watson LT (2007) Optimal design of composite wing structures with blended laminates. Compos Part B Eng 38(4):469–480

    Article  Google Scholar 

  • Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4-5):401–424

    Article  Google Scholar 

  • Sørensen R, Lund E (2015) Thickness filters for gradient based multi-material and thickness optimization of laminated composite structures. Struct Multidiscip Optim 52(2):227–250

    Article  Google Scholar 

  • Sørensen SN, Lund E (2013) Topology and thickness optimization of laminated composites including manufacturing constraints. Struct Multidiscip Optim 48(2):249–265

    Article  MathSciNet  Google Scholar 

  • Sørensen SN, Sørensen R, Lund E (2014) DMTO - a method for Discrete Material and Thickness Optimization of laminated composite structures. Struct Multidiscip Optim 50(1):25–47

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124

    Article  Google Scholar 

  • Sved G, Ginos Z (1968) Structural optimization under multiple loading. Int J Mech Sci 10(10):803–805

    Article  Google Scholar 

  • Tortorelli D, Michaleris P (1994) Design sensitivity analysis: overview and review. Inverse Prob Eng 1:71–105

    Article  Google Scholar 

  • Tsai SW, Pagano NJ (1968) Invariant properties of composite materials. In: Tsai SW et al (eds) Composite Materials Workshop, Technomic Publishing Co., Stamfort, Connecticut. Progress in Material Science, vol 1, pp 233–253

  • Wu C, Gao Y, Fang J, Lund E, Li Q (2017) Discrete topology optimization of ply orientation for a carbon fiber reinforced plastic (CFRP) laminate vehicle door. Mater Des 128:9–19

    Article  Google Scholar 

  • Yan J, Duan Z, Lund E, Wang J (2017) Concurrent multi-scale design optimization of composite frames with manufacturing constraints. Structural and Multidisciplinary Optimization. https://doi.org/10.1007/s00158-017-1750-0

  • Yang RJ, Chen CJ (1996) Stress-based topology optimization. Struct Optim 12(2):98–105

    Article  Google Scholar 

  • Zein S, Bruyneel M (2015) A bilevel integer programming method for blended composite structures. Adv Eng Softw 79:1–12

    Article  Google Scholar 

  • Zein S, Colson B, Grihon S (2012) A primal-dual backtracking optimization method for blended composite structures. Struct Multidiscip Optim 45(5):669–680

    Article  MathSciNet  MATH  Google Scholar 

  • Zein S, Madhavan V, Dumas D, Ravier L, Yague I (2016) From stacking sequences to ply layouts: An algorithm to design manufacturable composite structures. Compos Struct 141:32–38

    Article  Google Scholar 

  • Zhou M, Fleury R (2012) Composite optimization - ply drop-rate constraints for concepts and detailed design. In: Proceedings of the 23rd International Congress of Theoretical and Applied Mechanics (ICTAM). Beijing, China

  • Zhou M, Fleury R, Kemp M (2011) Optimization of composites - recent advances and application. The 7th Altair CAE Technology Conference. Altair, Troy

    Google Scholar 

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Acknowledgements

This work was partly supported by the Danish Research Council for Technology and Production Sciences (FTP), grant no. 10-082695, and the Innovation Fund Denmark project OPTI_MADE_BLADE, grant no. 75-2014-3. This support is gratefully acknowledged.

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  1. Department of Materials and Production, Aalborg University, Fibigerstraede 16, 9220, Aalborg East, Denmark

    Erik Lund

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Lund, E. Discrete Material and Thickness Optimization of laminated composite structures including failure criteria. Struct Multidisc Optim 57, 2357–2375 (2018). https://doi.org/10.1007/s00158-017-1866-2

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