Genetic programming-assisted multi-scale optimization for multi-objective dynamic performance of laminated composites: the advantage of more elementary-level analyses

  • Kanak Kalita
  • Tanmoy MukhopadhyayEmail author
  • Partha Dey
  • Salil Haldar
Original Article


High-fidelity multi-scale design optimization of many real-life applications in structural engineering still remains largely intractable due to the computationally intensive nature of numerical solvers like finite element method. Thus, in this paper, an alternate route of metamodel-based design optimization methodology is proposed in multi-scale framework based on a symbolic regression implemented using genetic programming (GP) coupled with d-optimal design. This approach drastically cuts the computational costs by replacing the finite element module with appropriately constructed robust and efficient metamodels. Resulting models are compact, have good interpretability and assume a free-form expression capable of capturing the non-linearly, complexity and vastness of the design space. Two robust nature-inspired optimization algorithms, viz. multi-objective genetic algorithm and multi-objective particle swarm optimization, are used to generate Pareto optimal solutions for several test problems with varying complexity. TOPSIS, a multi-criteria decision-making approach, is then applied to choose the best alternative among the Pareto optimal sets. Finally, the applicability of GP in efficiently tackling multi-scale optimization problems of composites is investigated, where a real-life scenario is explored by varying fractions of pertinent engineering materials to bring about property changes in the final composite structure across two different scales. The study reveals that a microscale optimization leads to better optimized solutions, demonstrating the advantage of carrying out a multi-scale optimization without any additional computational burden.


Multi-scale optimization Machine learning-based optimization Genetic programming Symbolic regression d-Optimal design Robust composite structures 



KK acknowledges the financial support from MHRD, India, through the award of Ph.D. Scholarship during the period of this research work.

Supplementary material

521_2019_4280_MOESM1_ESM.docx (22 kb)
Supplementary material 1 (DOCX 21 kb)


  1. 1.
    Koide RM, Ferreira AP, Luersen MA (2015) Laminated composites buckling analysis using lamination parameters, neural networks and support vector regression. Lat Am J Solids Struct 12(2):271–294Google Scholar
  2. 2.
    Reddy MRS, Reddy BS, Reddy VN, Sreenivasulu S (2012) Prediction of natural frequency of laminated composite plates using artificial neural networks. Engineering 4(06):329Google Scholar
  3. 3.
    García-Macías E, Castro-Triguero R, Friswell MI, Adhikari S, Sáez A (2016) Metamodel-based approach for stochastic free vibration analysis of functionally graded carbon nanotube reinforced plates. Compos Struct 152:183–198Google Scholar
  4. 4.
    Mukhopadhyay T, Naskar S, Dey S, Adhikari S (2016) On quantifying the effect of noise in surrogate based stochastic free vibration analysis of laminated composite shallow shells. Compos Struct 140:798–805Google Scholar
  5. 5.
    Mukhopadhyay T, Mahata A, Dey S, Adhikari S (2016) Probabilistic analysis and design of HCP nanowires: an efficient surrogate based molecular dynamics simulation approach. J Mater Sci Technol 32(12):1345–1351Google Scholar
  6. 6.
    Mahata A, Mukhopadhyay T, Adhikari S (2016) A polynomial chaos expansion based molecular dynamics study for probabilistic strength analysis of nano-twinned copper. Mater Res Express 3:036501Google Scholar
  7. 7.
    Dey S, Mukhopadhyay T, Spickenheuer A, Gohs U, Adhikari S (2016) Uncertainty quantification in natural frequency of composite plates - An Artificial neural network based approach. Adv Compos Lett 25(2):43–48Google Scholar
  8. 8.
    Dey S, Mukhopadhyay T, Sahu SK, Adhikari S (2016) Effect of cutout on stochastic natural frequency of composite curved panels. Compos Part B Eng 105:188–202Google Scholar
  9. 9.
    Ju S, Shenoi RA, Jiang D, Sobey AJ (2013) Multi-parameter optimization of lightweight composite triangular truss structure based on response surface methodology. Compos Struct 97:107–116Google Scholar
  10. 10.
    Heinonen O, Pajunen S (2011) Optimal design of stiffened plate using metamodeling techniques. J Struct Mech 44(3):218–230Google Scholar
  11. 11.
    Dutra TA, de Almeida SFM (2015) Composite plate stiffness multicriteria optimization using lamination parameters. Compos Struct 133:166–177Google Scholar
  12. 12.
    Passos AG, Luersen MA (2018) Multiobjective optimization of laminated composite parts with curvilinear fibers using Kriging-based approaches. Struct Multidiscip Optim 57(3):1115–1127Google Scholar
  13. 13.
    Ganguli R (2002) Optimum design of a helicopter rotor for low vibration using aeroelastic analysis and response surface methods. J Sound Vib 258(2):327–344Google Scholar
  14. 14.
    Dey S, Mukhopadhyay T, Khodaparast HH, Adhikari S (2016) A response surface modelling approach for resonance driven reliability-based optimization of composite shells. Period Polytech Civ Eng 60(1):103Google Scholar
  15. 15.
    Jafari R, Yousefi P, Hosseini-Hashemi S (2013) Vibration optimization of skew composite plates using the Rayleigh–Ritz and response surface methods. In: International conference on smart technologies for mechanical engineeringGoogle Scholar
  16. 16.
    Todoroki A, Suenaga K, Shimamura Y (2003) Stacking sequence optimizations using modified global response surface in lamination parameters. Adv Compos Mater 12(1):35–55Google Scholar
  17. 17.
    Todoroki A, Sasai M (2002) Stacking sequence optimizations using GA with zoomed response surface on lamination parameters. Adv Compos Mater 11(3):299–318Google Scholar
  18. 18.
    Todoroki A, Ozawa T, Mizutani Y, Suzuki Y (2013) Thermal deformation constraint using response surfaces for optimization of stacking sequences of composite laminates. Adv Compos Mater 22(4):265–279Google Scholar
  19. 19.
    Todoroki A, Ishikawa T (2004) Design of experiments for stacking sequence optimizations with genetic algorithm using response surface approximation. Compos Struct 64(3–4):349–357Google Scholar
  20. 20.
    Karsh PK, Mukhopadhyay T, Dey S (2018) Spatial vulnerability analysis for the first ply failure strength of composite laminates including effect of delamination. Compos Struct 184:554–567Google Scholar
  21. 21.
    Mukhopadhyay T, Naskar S, Dey S, Chakrabarti A (2019) Condition assessment and strengthening of aged structures: perspectives based on a critical case study. Pract Period Struct Design Constr 24(3):5019003Google Scholar
  22. 22.
    Mukhopadhyay T, Naskar S, Karsh PK, Dey S, You Z (2018) Effect of delamination on the stochastic natural frequencies of composite laminates. Compos Part B Eng 154:242–256Google Scholar
  23. 23.
    Sliseris J, Rocens K (2013) Optimal design of composite plates with discrete variable stiffness. Compos Struct 98:15–23Google Scholar
  24. 24.
    Cardozo SD, Gomes H, Awruch A et al (2011) Optimization of laminated composite plates and shells using genetic algorithms, neural networks and finite elements. Lat Am J Solids Struct 8(4):413–427Google Scholar
  25. 25.
    Marín L, Trias D, Badalló P, Rus G, Mayugo JA (2012) Optimization of composite stiffened panels under mechanical and hygrothermal loads using neural networks and genetic algorithms. Compos Struct 94(11):3321–3326Google Scholar
  26. 26.
    Bacarreza O, Aliabadi MH, Apicella A (2015) Robust design and optimization of composite stiffened panels in post-buckling. Struct Multidiscip Optim 51(2):409–422Google Scholar
  27. 27.
    Nik MA, Fayazbakhsh K, Pasini D, Lessard L (2012) Surrogate-based multi-objective optimization of a composite laminate with curvilinear fibers. Compos Struct 94(8):2306–2313Google Scholar
  28. 28.
    Koza JR (1992) Genetic programming. MIT Press, CambridgezbMATHGoogle Scholar
  29. 29.
    Hussain A, Sohail MF, Alam S, Ghauri SA, Qureshi IM (2018) Classification of M-QAM and M-PSK signals using genetic programming (GP). Neural Comput Appl. Google Scholar
  30. 30.
    Murata T, Ishibuchi H (1995) MOGA: multi-objective genetic algorithms. In: IEEE international conference on evolutionary computationGoogle Scholar
  31. 31.
    Mostaghim S, Teich J (2003) Strategies for finding good local guides in multi-objective particle swarm optimization (MOPSO). In: Proceedings of the swarm intelligence symposium, 2003. SIS’03. IEEEGoogle Scholar
  32. 32.
    Singh AP, Mani V, Ganguli R (2007) Genetic programming metamodel for rotating beams. Comput Model Eng Sci 21(2):133zbMATHGoogle Scholar
  33. 33.
    Jalal M, Ramezanianpour AA, Pouladkhan AR, Tedro P (2013) Application of genetic programming (GP) and ANFIS for strength enhancement modeling of CFRP-retrofitted concrete cylinders. Neural Comput Appl 23(2):455–470Google Scholar
  34. 34.
    Jones RM (1998) Mechanics of composite materials, 2nd edn. Taylor & Francis Ltd, LondonGoogle Scholar
  35. 35.
    Dey S, Mukhopadhyay T, Adhikari S (2017) Metamodel based high-fidelity stochastic analysis of composite laminates: a concise review with critical comparative assessment. Compos Struct 171:227–250Google Scholar
  36. 36.
    Chakraborty S, Mandal B, Chowdhury R, Chakrabarti A (2016) Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion. Compos Struct 135:236–249Google Scholar
  37. 37.
    Mukhopadhyay T, Dey TK, Chowdhury R, Chakrabarti A, Adhikari S (2015) Optimum design of FRP bridge deck: an efficient RS-HDMR based approach. Struct Multidiscip Optim 52(3):459–477Google Scholar
  38. 38.
    Vladislavleva EY (2008) Model-based problem solving through symbolic regression via pareto genetic programming. CentER, Tilburg University, TilburgGoogle Scholar
  39. 39.
    Sharifipour M, Bonakdari H, Zaji AH (2018) Comparison of genetic programming and radial basis function neural network for open-channel junction velocity field prediction. Neural Comput Appl 30(3):855–864Google Scholar
  40. 40.
    Koza JR (1994) Genetic programming as a means for programming computers by natural selection. Stat Comput 4(2):87–112Google Scholar
  41. 41.
    Barricelli NA et al (1954) Esempi numerici di processi di evoluzione. Methodos 6(21–22):45–68MathSciNetGoogle Scholar
  42. 42.
    Kalita K (2019) Design of composite laminates with nature-inspired optimization. PhD thesis. Indian Institute of Engineering Science and Technology Shibpur, India 711103.
  43. 43.
    Mukhopadhyay T, Dey T, Chowdhury R, Chakrabarti A (2015) Structural damage identification using response surface-based multi-objective optimization: a comparative study. Arab J Sci Eng 40(4):1027–1044MathSciNetzbMATHGoogle Scholar
  44. 44.
    Shi LM, Fang H, Tong W, Wu J, Perkins R, Blair RM, Branham WS, Dial SL, Moland CL, Sheehan DM (2001) QSAR models using a large diverse set of estrogens. J Chem Inf Comput Sci 41(1):186–195Google Scholar
  45. 45.
    Hawkins DM (2004) The problem of overfitting. J Chem Inf Comput Sci 44(1):1–12MathSciNetGoogle Scholar
  46. 46.
    Consonni V, Ballabio D, Todeschini R (2010) Evaluation of model predictive ability by external validation techniques. J Chemom 24:194–201Google Scholar
  47. 47.
    Goldberg DE (2006) Genetic algorithms. Pearson Education, BengaluruGoogle Scholar
  48. 48.
    Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the 6th international symposium on micro machine and human science, 1995. MHS’95Google Scholar
  49. 49.
    Diyaley S, Shilal P, Shivakoti I, Ghadai RK, Kalita K (2017) PSI and TOPSIS based selection of process parameters in WEDM. Period Polytech Eng Mech Eng 61(4):55Google Scholar
  50. 50.
    Raju B, Hiremath SR, Mahapatra DR (2018) A review of micromechanics based models for effective elastic properties of reinforced polymer matrix composites. Compos Struct 204:607–619Google Scholar
  51. 51.
    Naskar S, Mukhopadhyay T, Sriramula S (2018) Probabilistic micromechanical spatial variability quantification in laminated composites. Compos B Eng 151:291–325Google Scholar
  52. 52.
    Naskar S, Mukhopadhyay T, Sriramula S (2019) Spatially varying fuzzy multi-scale uncertainty propagation in unidirectional fibre reinforced composites. Compos Struct 209:940–967Google Scholar
  53. 53.
    Naskar S, Mukhopadhyay T, Sriramula S, Adhikari S (2017) Stochastic natural frequency analysis of damaged thin-walled laminated composite beams with uncertainty in micromechanical properties. Compos Struct 160:312–334Google Scholar
  54. 54.
    Dey S, Mukhopadhyay T, Sahu SK, Adhikari S (2018) Stochastic dynamic stability analysis of composite curved panels subjected to non-uniform partial edge loading. Eur J Mech A Solids 67:108–122MathSciNetzbMATHGoogle Scholar
  55. 55.
    Kumar RR, Mukhopadhyay T, Pandey KM, Dey S (2019) Stochastic buckling analysis of sandwich plates: the importance of higher order modes. Int J Mech Sci 152:630–643Google Scholar
  56. 56.
    Dey S, Mukhopadhyay T, Adhikari S (2018) Uncertainty quantification in laminated composites: a meta-model based approach. CRC Press, Boca Raton ISBN 9781315155593 zbMATHGoogle Scholar
  57. 57.
    Karsh PK, Mukhopadhyay T, Dey S (2019) Stochastic low-velocity impact on functionally graded plates: probabilistic and non-probabilistic uncertainty quantification. Compos B Eng 159:461–480Google Scholar
  58. 58.
    Dey S, Mukhopadhyay T, Naskar S, Dey TK, Chalak HD, Adhikari S (2019) Probabilistic characterization for dynamics and stability of laminated soft core sandwich plates. J Sandw Struct Mater 21(1):366–397Google Scholar
  59. 59.
    Karsh PK, Mukhopadhyay T, Dey S (2018) Stochastic dynamic analysis of twisted functionally graded plates. Compos B Eng 147:259–278Google Scholar
  60. 60.
    Maharshi K, Mukhopadhyay T, Roy B, Roy L, Dey S (2018) Stochastic dynamic behaviour of hydrodynamic journal bearings including the effect of surface roughness. Int J Mech Sci 142–143:370–383Google Scholar
  61. 61.
    Mukhopadhyay T (2018) A multivariate adaptive regression splines based damage identification methodology for web core composite bridges including the effect of noise. J Sandw Struct Mater 20(7):885–903Google Scholar
  62. 62.
    Dey TK, Mukhopadhyay T, Chakrabarti A, Sharma UK (2015) Efficient lightweight design of FRP bridge deck. Proc Inst Civ Eng Struct Build 168(10):697–707Google Scholar
  63. 63.
    Mukhopadhyay T, Chowdhury R, Chakrabarti A (2016) Structural damage identification: a random sampling-high dimensional model representation approach. Adv Struct Eng 19(6):908–927Google Scholar
  64. 64.
    Mukhopadhyay T, Dey TK, Dey S, Chakrabarti A (2015) Optimization of fiber reinforced polymer web core bridge deck—a hybrid approach. Struct Eng Int 25(2):173–183Google Scholar
  65. 65.
    Kalita K, Haldar S (2017) Eigenfrequencies of simply supported taper plates with cut-outs. Struct Eng Mech 63(1):103–113Google Scholar
  66. 66.
    Kalita K, Ramachandran M, Raichurkar P, Mokal SD, Haldar S (2016) Free vibration analysis of laminated composites by a nine node isoparametric plate bending element. Adv Compos Lett 25(5):108Google Scholar
  67. 67.
    Sayyad AS, Ghugal YM (2015) On the free vibration analysis of laminated composite and sandwich plates: a review of recent literature with some numerical results. Compos Struct 129:177–201Google Scholar
  68. 68.
    Xiang S, Wang K-M, Ai Y-T, Sha Y-D, Shi H (2009) Natural frequencies of generally laminated composite plates using the Gaussian radial basis function and first-order shear deformation theory. Thin Walled Struct 47:1265–1271Google Scholar
  69. 69.
    Aydogdu M (2009) A new shear deformation theory for laminated composite plates. Compos Struct 89:94–101Google Scholar
  70. 70.
    Zhen W, Wanji C (2006) Free vibration of laminated composite and sandwich plates using global–local higher-order theory. J Sound Vib 298:333–349Google Scholar
  71. 71.
    Akhras G, Li W (2005) Static and free vibration analysis of composite plates using spline finite strips with higher-order shear deformation. Compos B Eng 36:496–503Google Scholar
  72. 72.
    Ray MC (2003) Zeroth-order shear deformation theory for laminated composite plates. J Appl Mech 70:374–380zbMATHGoogle Scholar
  73. 73.
    Matsunaga H (2000) Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory. Compos Struct 48:231–244Google Scholar
  74. 74.
    Wu C-P, Chen W-Y (1994) Vibration and stability of laminated plates based on a local high order plate theory. J Sound Vib 177:503–520zbMATHGoogle Scholar
  75. 75.
    Cho KN, Bert CW, Striz AG (1991) Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory. J Sound Vib 145:429–442Google Scholar
  76. 76.
    Kant T, Manjunatha BS (1988) An unsymmetric FRC laminate C° finite element model with 12 degrees of freedom per node. Eng Comput 5:300–308Google Scholar
  77. 77.
    Pandya BN, Kant T (1988) Finite element analysis of laminated composite plates using a higher-order displacement model. Compos Sci Technol 32:137–155Google Scholar
  78. 78.
    Senthilnathan NR, Lim SP, Lee KH, Chow ST (1987) Buckling of shear-deformable plates. AIAA J 25:1268–1271Google Scholar
  79. 79.
    Phan ND, Reddy JN (1985) Analysis of laminated composite plates using a higher-order shear deformation theory. Int J Numer Meth Eng 21:2201–2219zbMATHGoogle Scholar
  80. 80.
    Reddy JN (1984) A simple higher-order theory for laminated composite plates. J Appl Mech 51:745–752zbMATHGoogle Scholar
  81. 81.
    Whitney JM, Pagano NJ (1970) Shear deformation in heterogeneous anisotropic plates. J Appl Mech 37:1031–1036zbMATHGoogle Scholar
  82. 82.
    Mindlin RD (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18:31–38zbMATHGoogle Scholar
  83. 83.
    Kirchhoff GR (1850) Uber das gleichgewicht und die bewegung einer elastischen Scheibe. Journal für die reine und angewandte Mathematik (Crelle’s Journal)Google Scholar
  84. 84.
    Kalita K, Shivakoti I, Ghadai RK (2017) Optimizing process parameters for laser beam micro-marking using a genetic algorithm and particle swarm optimization. Mater Manuf Process 32(10):1101–1108Google Scholar
  85. 85.
    Stehlík M, Střelec L, Thulin M (2014) On robust testing for normality in chemometrics. Chemometr Intell Lab Syst 130:98–108Google Scholar
  86. 86.
    Ragavendran U, Ghadai RK, Bhoi AK, Ramachandran M, Kalita K (2019) Sensitivity analysis and optimization of EDM process. Trans Can Soc Mech Eng 43(1):13–25Google Scholar
  87. 87.
    Shooshtari A, Razavi S (2010) A closed form solution for linear and nonlinear free vibrations of composite and fiber metal laminated rectangular plates. Compos Struct 92(11):2663–2675Google Scholar
  88. 88.
    Shivakoti I, Pradhan BB, Diyaley S, Ghadai RK, Kalita K (2017) Fuzzy TOPSIS-based selection of laser beam micro-marking process parameters. Arab J Sci Eng 42(11):4825–4831Google Scholar
  89. 89.
    Kalita K, Ragavendran U, Ramachandran M, Bhoi AK (2019) Weighted sum multi-objective optimization of skew composite laminates. Struct Eng Mech 69(1):21–31Google Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Kanak Kalita
    • 1
  • Tanmoy Mukhopadhyay
    • 2
    Email author
  • Partha Dey
    • 3
  • Salil Haldar
    • 1
  1. 1.Department of Aerospace Engineering and Applied MechanicsIndian Institute of Engineering, Science and TechnologyHowrahIndia
  2. 2.Department of Aerospace EngineeringIndian Institute of Technology KanpurKanpurIndia
  3. 3.Department of Mechanical EngineeringAcademy of TechnologyHooghlyIndia

Personalised recommendations