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An efficient approach for metaheuristic-based optimization of composite laminates using genetic programming

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Abstract

Optimization of the dynamic performance of composite laminates has become necessary in many design applications of structural engineering. Although finite element analysis-based metaheuristic search is a widely adopted approach for design optimization of composite laminates, recently there has been a tremendous impetus on the use of metamodel-based metaheuristic optimization, mainly due to significant saving in computational time as well as cost. In this paper, genetic programming-based symbolic regression (SR) metamodeling technique is introduced for predicting natural frequencies of composite laminates. The optimization performance based on SR metamodels is compared with the traditionally adopted polynomial regression (PR) metamodels. The SR and PR metamodels are further solved using three metaheuristic search algorithms, i.e. genetic algorithm (GA), repulsive particle swarm optimization with local search (RPSOLC) and co-evolutionary host-parasite (CHP) for single objective optimization problems. A comprehensive analysis reveals that the SR metamodels are more accurate and compact than the PR metamodels in addition to better interpretability. Furthermore, CHP algorithm is noticed to consistently outperform both the GA and RPSOLC algorithms.

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Acknowledgements

The authors acknowledge the help of Dr. Ranjan Ghadai and Dr. Salil Haldar for their constitutive comments on a previous draft.

Funding

This research has received no external funding.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, India

    Kanak Kalita

  2. Production Engineering Department, Jadavpur University, Calcutta, India

    Shankar Chakraborty

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  1. Kanak Kalita
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  2. Shankar Chakraborty
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Correspondence to Shankar Chakraborty.

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Appendices

Appendix A: Polynomial regression metamodels for the test problems (TP)

TP-01

\({\left({\lambda }_{1}\right)}^{2}=219.8785+213.5809\left(\frac{{E}_{1}}{{E}_{2}}\right)+439.0622\left(\frac{{G}_{12}}{{E}_{2}}\right)+279.9152\left(\frac{{G}_{23}}{{E}_{2}}\right)+250.7809\left({ \upsilon }_{12}\right)+4.9276\left(\frac{{E}_{1}}{{E}_{2}}\right)\left(\frac{{G}_{12}}{{E}_{2}}\right)+4.5119\left(\frac{{E}_{1}}{{E}_{2}}\right)\left(\frac{{G}_{23}}{{E}_{2}}\right)-246.4965\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\frac{{G}_{23}}{{E}_{2}}\right)-252.7662\left(\frac{{G}_{23}}{{E}_{2}}\right)\left({ \upsilon }_{12}\right)-0.0551{\left(\frac{{E}_{1}}{{E}_{2}}\right)}^{2}-152.6717{\left(\frac{{G}_{12}}{{E}_{2}}\right)}^{2}-147.8168{\left(\frac{{G}_{23}}{{E}_{2}}\right)}^{2}\)

\({\left({\lambda }_{21}\right)}^{2}=-27.926+263.398\left(\frac{{E}_{1}}{{E}_{2}}\right)+1102.0988\left(\frac{{G}_{12}}{{E}_{2}}\right)+855.4249\left(\frac{{G}_{23}}{{E}_{2}}\right)+637.1448\left({ \upsilon }_{12}\right)+13.1769\left(\frac{{E}_{1}}{{E}_{2}}\right)\left(\frac{{G}_{12}}{{E}_{2}}\right)+13.7958\left(\frac{{E}_{1}}{{E}_{2}}\right)\left(\frac{{G}_{23}}{{E}_{2}}\right)-743.6287\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\frac{{G}_{23}}{{E}_{2}}\right)-738.7692\left(\frac{{G}_{23}}{{E}_{2}}\right)\left({ \upsilon }_{12}\right)-0.1493{\left(\frac{{E}_{1}}{{E}_{2}}\right)}^{2}-423.1414{\left(\frac{{G}_{12}}{{E}_{2}}\right)}^{2}-468.0938{\left(\frac{{G}_{23}}{{E}_{2}}\right)}^{2}\)

TP-02

\(ln\left({\lambda }_{1}\right)= 4.4282-0.3614\left(\frac{b}{a}\right)-1.3347\left(\frac{h}{a}\right)-1.2572\left(\alpha \right)+0.1514\left(n\right)+8.6191\left(\frac{b}{a}\right)\left(\alpha \right)-0.0742\left(\frac{h}{a}\right)\left(\alpha \right)-0.0391\left(\frac{h}{a}\right)\left(n\right)-0.0002\left(\alpha \right)\left(n\right)+0.0428{\left(\frac{b}{a}\right)}^{2}+0.0004{\left(\alpha \right)}^{2}-0.0071{\left(n\right)}^{2}\)

\(ln\left({\lambda }_{21}\right)= 5.106-1.5926\left(\frac{b}{a}\right)-4.9656\left(\frac{h}{a}\right)-0.0055\left(\alpha \right)+5.4627\left(n\right)+1.2641\left(\frac{b}{a}\right)\left(\frac{h}{a}\right)-0.0047\left(\frac{b}{a}\right)\left(\alpha \right)-0.8106\left(\frac{h}{a}\right)\left(n\right)+0.1188{\left(\frac{b}{a}\right)}^{2}-0.0232{\left(n\right)}^{2}\)

TP-03

\({\lambda }_{1}=64.7921-0.0009\left({\theta }_{1}\right)-0.0131\left({\theta }_{2}\right)-0.0011\left({\theta }_{3}\right)+0.0097\left({\theta }_{4}\right)+0.0001\left({\theta }_{2}\right)\left({\theta }_{3}\right)-0.0001\left({\theta }_{3}\right)\left({\theta }_{4}\right)-0.0029{\left({\theta }_{1}\right)}^{2}-0.0017{\left({\theta }_{2}\right)}^{2}-0.0009{\left({\theta }_{3}\right)}^{2}\)

\({\lambda }_{21}=12.3146-0.0031\left({\theta }_{1}\right)-0.0136\left({\theta }_{2}\right)-0.0188\left({\theta }_{3}\right)-0.0263\left({\theta }_{4}\right)+0.0004\left({\theta }_{1}\right)\left({\theta }_{3}\right)+0.0003\left({\theta }_{1}\right)\left({\theta }_{4}\right)-0.0002\left({\theta }_{2}\right)\left({\theta }_{3}\right)-0.0002\left({\theta }_{2}\right)\left({\theta }_{4}\right)+0.0017{\left({\theta }_{1}\right)}^{2}+0.0008{\left({\theta }_{2}\right)}^{2}-0.0004{\left({\theta }_{3}\right)}^{2}\)

TP-04

\(\sqrt{{\lambda }_{1}}=13.4425+0.0218\left(\frac{{E}_{1}}{{E}_{2}}\right)-0.0123\left(\frac{{G}_{12}}{{E}_{2}}\right)-21.0942\left(\frac{{G}_{23}}{{E}_{2}}\right)+1.3446\left({ \upsilon }_{12}\right)-1.7704\left(\frac{b}{a}\right)+22.3928\left(\frac{h}{a}\right)+0.0295\left(\alpha \right)+0.0079\left({\theta }_{1}\right)+0.0033\left({\theta }_{2}\right)- 0.0144\left({\theta }_{3}\right)+0.0289\left({\theta }_{4}\right)+0.0548\left(\frac{{E}_{1}}{{E}_{2}}\right)\left(\frac{{G}_{12}}{{E}_{2}}\right)-0.5112\left(\frac{{E}_{1}}{{E}_{2}}\right)\left(\frac{h}{a}\right)+0.0004\left(\frac{{E}_{1}}{{E}_{2}}\right)\left(\alpha \right)-0.7894\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\frac{b}{a}\right)+0.0326\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\alpha \right)- 0.0148\left(\frac{{G}_{12}}{{E}_{2}}\right)\left({\theta }_{1}\right)+0.0162\left(\frac{{G}_{12}}{{E}_{2}}\right)\left({\theta }_{3}\right)-0.0145\left(\frac{{G}_{12}}{{E}_{2}}\right)\left({\theta }_{4}\right)+0.6889\left(\frac{{G}_{23}}{{E}_{2}}\right)\left(\frac{b}{a}\right)-0.0592\left({ \upsilon }_{12}\right)\left({\theta }_{4}\right)+1.2734\left(\frac{b}{a}\right)\left({\theta }_{3}\right)-1.7528\left(\frac{b}{a}\right)\left({\theta }_{4}\right)-1.3099\left(\frac{h}{a}\right)\left(\alpha \right)- 0.0769\left(\frac{h}{a}\right)\left({\theta }_{2}\right)+2.9684\times {10}^{-5}\left({\theta }_{3}\right)\left({\theta }_{4}\right)+18.8401\left(\frac{{G}_{23}}{{E}_{2}^{2}}\right)+0.1672{\left(\frac{b}{a}\right)}^{2}+0.0016{\left(\alpha \right)}^{2}-0.0001{\left({\theta }_{1}\right)}^{2}-0.0001{\left({\theta }_{3}\right)}^{2}\)

\(ln\left({\lambda }_{21}\right)=1.2618+0.0275\left(\frac{{E}_{1}}{{E}_{2}}\right)+1.3682\left(\frac{{G}_{12}}{{E}_{2}}\right)+3.9339\left(\frac{{G}_{23}}{{E}_{2}}\right)+10.8273\left({ \upsilon }_{12}\right)-1.2377\left(\frac{b}{a}\right)-24.1563\left(\frac{h}{a}\right)+0.0191\left(\alpha \right)+0.0041\left({\theta }_{1}\right)+0.0047\left({\theta }_{2}\right)+0.0124\left({\theta }_{3}\right)-0.0247-0.0014\left({\theta }_{4}\right)-0.0449\left(\frac{{E}_{1}}{{E}_{2}}\right)\left(\frac{{G}_{23}}{{E}_{2}}\right)-6.7378\times {10}^{-5}\left(\frac{{E}_{1}}{{E}_{2}}\right)\left({\theta }_{2}\right)-9.7069\times {10}^{-5}\left(\frac{{E}_{1}}{{E}_{2}}\right)\left({\theta }_{3}\right)-0.0288\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\alpha \right)-13.9501\left(\frac{{G}_{23}}{{E}_{2}}\right)\left({ \upsilon }_{12}\right)+17.2227\left(\frac{{G}_{23}}{{E}_{2}}\right)\left(\frac{h}{a}\right)+0.0095\left(\frac{{G}_{23}}{{E}_{2}}\right)\left({\theta }_{4}\right)-0.0247\left({ \upsilon }_{12}\right)\left({\theta }_{3}\right)+5.1894\left(\frac{b}{a}\right)\left(\frac{h}{a}\right)-0.0014\left(\frac{b}{a}\right)\left({\theta }_{4}\right)+0.1978\left(\frac{h}{a}\right)\left(\alpha \right)-8.2466\times {10}^{-5}\left(\alpha \right)\left({\theta }_{1}\right)-5.6789\left(\alpha \right)\left({\theta }_{3}\right)+2.3161\left({\theta }_{1}\right)\left({\theta }_{4}\right)-+0.0798{\left(\frac{b}{a}\right)}^{2}-4.8291\times {10}^{-5}{\left({\theta }_{1}\right)}^{2}\)

Appendix B: Genetic programming metamodels for the test problems

$$\begin{aligned} \begin{array}{lll} \text{TP-01}&&\qquad\qquad {{\lambda }_{1}=46.8303+1.1436\left(\frac{{E}_{1}}{{E}_{2}}\right)+1.0631\left(\frac{{G}_{12}}{{E}_{2}}\right)+0.3715\left(\frac{{G}_{23}}{{E}_{2}}\right)}\\ & &\qquad\qquad {{\lambda }_{21}=\frac{647.6556 + 51.4718\left(\frac{{E}_{1}}{{E}_{2}}\right) + 94.2528\left(\frac{{G}_{12}}{{E}_{2}}\right)}{18.5836+0.1788\left(\frac{{E}_{1}}{{E}_{2}}\right)}}\\ \text{TP-02}&&\qquad \qquad {{\lambda }_{1}=\frac{\left\{\left.\begin{array}{c}65.3499\left(\frac{h}{a}\right)-0.0005\left(\frac{b}{a}\right)\left(\alpha \right)+0.0012\left(\alpha \right)\left(n\right)+0.0003{\left(\alpha \right)}^{2}-\\ 0.0005\left(\frac{b}{a}\right)\left(\frac{h}{a}\right){\left(\alpha \right)}^{2}+0.0012\left(\frac{h}{a}\right){\left(\alpha \right)}^{2}\left(n\right)+0.0003\left(\frac{h}{a}\right){\left(\alpha \right)}^{3}\end{array}\right\}\right.}{0.995\left(\frac{h}{a}\right)}}\\ &&\qquad\qquad {{\lambda }_{21}=\frac{\left.\left\{\begin{array}{c}-22.3145\left(\frac{b}{a}\right)\left(\frac{h}{a}\right)\left(\alpha \right)-41.3423\left(\frac{b}{a}\right)\left(\frac{h}{a}\right)\left(n\right)-1594.3957{\left(\frac{b}{a}\right)}^{2}\left(\frac{h}{a}\right)+\\ 2.2614{\left(\frac{b}{a}\right)}^{2}\left(\alpha \right)+4.1897{\left(\frac{b}{a}\right)}^{2}\left(n\right)+55.0413{\left(\frac{h}{a}\right)}^{2}\left(\alpha \right)+{101.9757\left(\frac{h}{a}\right)}^{2}\left(n\right)-\\ 34.68{\left(\frac{b}{a}\right)}^{2}\left(\frac{h}{a}\right)\left(n\right)+326.4915{\left(\frac{b}{a}\right)}^{3}+7.1016{\left(\frac{b}{a}\right)}^{3}\left(n\right)-75.5412{\left(\frac{b}{a}\right)}^{4}\end{array}\right.\right\}}{2.9875{\left(\frac{b}{a}\right)}^{4}}}\\ \text{TP-03}&&\qquad \qquad {{\lambda }_{1}=\frac{\left\{\left.\begin{array}{c}846.8593-0.1368{\theta }_{2}-26.9210{\theta }_{3}+0.0044{\theta }_{2}{\theta }_{3}-0.0362{{\theta }_{1}}^{2}-\\ 0.0221{{\theta }_{2}}^{2}-0.0086{{\theta }_{3}}^{2}+0.0012{{{\theta }_{1}}^{2}\theta }_{3}+0.0007{{\theta }_{2}}^{2}{\theta }_{3}+0.0003{{\theta }_{3}}^{3}\end{array}\right\}\right.}{13.8227-0.4387{\theta }_{3}}}\\ &&\qquad\qquad {{\lambda }_{21}=\left\{0.638\left(\mathrm{cos}\left(0.0445{\theta }_{1}\right)\right)+\mathrm{cos}\left(0.0446{\theta }_{1}\right)+\mathrm{cos}\left(\frac{0.5377{\theta }_{2}}{10.2826}\right)-\mathrm{cos}\left(\mathrm{cos}\left(\frac{0.5578{\theta }_{3}}{\mathrm{cos}\left(0.0457{\theta }_{1}\right)}\right)\right)\right\}\left(-6.1035\right)+17.8864}\\ \text{TP-04}&&\qquad\qquad{{\lambda }_{1}=\left({C}_{1}{C}_{2}+{{C}_{1}C}_{3}+29.9123\right)+4.367}\\ &&\qquad\qquad {{\mathrm{C}}_{1}=-0.000008\left(\frac{{\mathrm{E}}_{1}}{{\mathrm{E}}_{2}}\right)\left({\alpha }\right)-0.0002\left({\alpha}\right)-0.0033}\\ &&\qquad\qquad {{\mathrm{C}}_{2}=\mathrm{cos}\left(\mathrm{cos}\left(1.7609\left({\theta }_{1}\right)-0.807\left(\alpha \right)\right)\right)-300.1912}\\ &&\qquad\qquad {{\mathrm{C}}_{3}=2.0607\left({\alpha}\right)/\left\{-4.6815\left(\frac{\mathrm{h}}{\mathrm{a}}\right).\mathrm{cos}\left(\mathrm{cos}\left(1.7664\left({\uptheta }_{1}\right)+0.8064\left({\alpha}\right)\right)\right)\right\}}\\ &&\qquad \qquad {{\lambda }_{21}=\frac{\left\{\left.\begin{array}{c}-26.5903\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\frac{b}{a}\right){\left(\frac{h}{a}\right)}^{3}-1.2061\left(\frac{b}{a}\right){\left(\frac{h}{a}\right)}^{3}\left({\alpha}\right)+4.4494\left(\frac{b}{a}\right){\left(\frac{h}{a}\right)}^{2}{\left({\alpha}\right)}^{2}-1846.1715\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\frac{b}{a}\right){\left(\frac{h}{a}\right)}^{2}+\\ 32.6238\left(\frac{{G}_{12}}{{E}_{2}}\right){\left(\frac{h}{a}\right)}^{3}+0.0338\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\frac{h}{a}\right){\left({\alpha}\right)}^{2}-279.9009\left(\frac{b}{a}\right){\left(\frac{h}{a}\right)}^{2}\left({\alpha}\right)+1.4797{\left(\frac{h}{a}\right)}^{3}\left({\alpha}\right)-\\ \begin{array}{c}3.7319{\left(\frac{h}{a}\right)}^{2}{\left({\alpha}\right)}^{2}+274.4143\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\frac{b}{a}\right)\left(\frac{h}{a}\right)+2265.0778\left(\frac{{G}_{12}}{{E}_{2}}\right){\left(\frac{h}{a}\right)}^{2}-1.49\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\frac{h}{a}\right)\left({\alpha}\right)+\\ 12.4472\left(\frac{b}{a}\right)\left(\frac{h}{a}\right)\left({\alpha}\right)-1.0894\left(\frac{b}{a}\right)\left(\frac{h}{a}\right)\left({\theta }_{1}\right)+267.2716{\left(\frac{h}{a}\right)}^{2}\left({\alpha}\right)-336.6804\left(\frac{{G}_{12}}{{E}_{2}}\right)\left(\frac{h}{a}\right)-\\ 0.0083\left(\frac{{G}_{12}}{{E}_{2}}\right)\left({\theta }_{1}\right)+122.2417\left(\frac{b}{a}\right)\left(\frac{h}{a}\right)-15.2715\left(\frac{h}{a}\right)\left({\alpha}\right)+0.9138\left(\frac{h}{a}\right)\left({\theta }_{1}\right)-149.9802\left(\frac{h}{a}\right)\end{array}\end{array}\right\}\right.}{20.2929\left(\frac{b}{a}\right)\left(\frac{h}{a}\right)-24.8975\left(\frac{h}{a}\right)}} \end{array} \end{aligned}$$

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Kalita, K., Chakraborty, S. An efficient approach for metaheuristic-based optimization of composite laminates using genetic programming. Int J Interact Des Manuf 17, 899–916 (2023). https://doi.org/10.1007/s12008-022-01175-7

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