Wasserstein distributionally robust co-design of a fractional-order dual-resonator elastic metamaterial for broadband vibration attenuation

  • Asokan  Vasudevan orcid

    Faculty of Business and Communications, INTI International University, Nilai 71800, Malaysia
  • Yogeesh Nijalingappa orcid

    Mathematics & Natural Sciences, Gulf University for Science & Technology, Safat 13060, Kuwait; Department of Mathematics, Government First Grade College, Tumkur 572102, India; Research Fellow, INTI International University, Nilai 71800, Malaysia
  • Rashed Abu  Hammour orcid

    Faculty of Technical Education, Al-Ahliyya Amman University, Amman 19328, Jordan; Hourani Center for Applied Scientific Research (HCASR), Al-Ahliyya Amman University, Amman 19328, Jordan
  • Poornachandran William orcid

    School of Computer Science and Technology, Karunya Institute of Technology and Sciences, Coimbatore 641114, India; Centre for Research and Consultancy, UNITAR International University, Petaling Jaya 47301, Malaysia
  • Suleiman Ibrahim  Mohammad orcid

    Faculty of Business and Communications, INTI International University, Nilai 71800, Malaysia
  • Torki M.  Al-Fawwaz

    College of Commerce and Business, Lusail University, Lusail P.O. Box 9717, Qatar
Article ID: 4698
DOI: https://doi.org/10.59400/sv4698
Keywords: Bloch–Floquet analysis; optimal transport; attenuation band; Kelvin–Voigt damping; uncertain dynamics; risk-sensitive design; quasi-Monte Carlo validation

Abstract

Locally resonant elastic metamaterials provide subwavelength attenuation but remain sensitive to uncertainty in stiffness, damping, and fractional order. This study develops a Wasserstein distributionally robust co-design framework that jointly selects the mass split, resonance tuning, loss scales, and fractional orders of a dual-resonator periodic cell. Bloch dispersion is condensed into a complex effective-mass model, and a feasibility-aware loss balances integrated attenuation, threshold coverage, internal-motion amplification, and resonance separation. Five bounded uncertainties in host stiffness, resonator stiffnesses, common loss scale, and fractional-order drift are represented in a finite 1-Wasserstein ambiguity set trained on 64 scrambled Sobol scenarios with 256 adversarial candidates. The proposed design is independently compared with equal-mass, nominal, and conditional-value-at-risk alternatives over 4,096 unseen scenarios. Across this validation set, the Wasserstein design reduced worst loss by 73.68% and increased fifth-percentile integrated attenuation by 32.50% relative to the equal-mass baseline while satisfying the internal-motion constraint. It allocated 56.03% of the resonator mass to the lower branch and selected tuning frequencies of 27.20 and 40.31 Hz. Exact antiresonance verification, uncertainty envelopes, finite-chain attenuation-equivalent insertion loss, and local sensitivity analysis support the numerical results. The framework provides a reproducible basis for robust broadband attenuation and subsequent specimen-level calibration and experimental validation.

Published
2026-09-01
How to Cite
Vasudevan, A., Nijalingappa, Y., Hammour, R., William, P., Mohammad, S., & Al-Fawwaz, T. (2026). Wasserstein distributionally robust co-design of a fractional-order dual-resonator elastic metamaterial for broadband vibration attenuation. Sound & Vibration, 60(6). https://doi.org/10.59400/sv4698
Issue
Vol. 60 No. 6 (2026): In Progress
Section
Article

Copyright (c) 2026 Asokan  Vasudevan, Yogeesh Nijalingappa, Rashed Abu  Hammour, Poornachandran William, Suleiman Ibrahim  Mohammad, Torki M.  Al-Fawwaz

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]Milton GW, Willis JR. On modifications of Newton’s second law and linear continuum elastodynamics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2007; 463(2079): 855–880. doi: 10.1098/rspa.2006.1795

[2]Liu Z, Zhang X, Mao Y, et al. Locally resonant sonic materials. Science. 2000; 289(5485): 1734–1736. doi: 10.1126/science.289.5485.1734

[3]Failla G, Marzani A, Palermo A, et al. Current developments in elastic and acoustic metamaterials science. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2024; 382(2278): 20230369. doi: 10.1098/rsta.2023.0369

[4]Jiao P, Mueller J, Raney JR, et al. Mechanical metamaterials and beyond. Nature Communications. 2023; 14(1): 6004. doi: 10.1038/s41467-023-41679-8

[5]Kushwaha MS, Halevi P, Dobrzynski L, et al. Acoustic band structure of periodic elastic composites. Physical Review Letters. 1993; 71(13): 2022–2025. doi: 10.1103/PhysRevLett.71.2022

[6]Ding W, Hollkamp JP, Patnaik S, et al. On the fractional homogenization of one-dimensional elastic metamaterials with viscoelastic foundation. Archive of Applied Mechanics. 2023; 93(1): 261–286. doi: 10.1007/s00419-022-02170-w

[7]Wang G, Wan S, Hong J, et al. Enhancement of the vibration attenuation characteristics in local resonance metamaterial beams: Theory and experiment. Mechanical Systems and Signal Processing. 2023; 188: 110036. doi: 10.1016/j.ymssp.2022.110036

[8]Ke Y, Yin J, He Y, et al. Locally multi-resonant meta-shells for broadband vibration suppression. International Journal of Mechanical Sciences. 2024; 278: 109452. doi: 10.1016/j.ijmecsci.2024.109452

[9]Frazier MJ, Hussein MI. Generalized Bloch’s theorem for viscous metamaterials: Dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex. Comptes Rendus Physique. 2016; 17(5): 565–577. doi: 10.1016/j.crhy.2016.02.009

[10]Zhao L, Lu Z, Ding H, et al. A viscoelastic metamaterial beam for integrated vibration isolation and energy harvesting. Applied Mathematics and Mechanics (English Edition). 2024; 45(7): 1243–1260. doi: 10.1007/s10483-024-3159-7

[11]Adham AI, Sorokin V, Mace B, et al. Broadband vibration isolation in a finite elastic rod with tailored resonators. Wave Motion. 2025; 136: 103528. doi: 10.1016/j.wavemoti.2025.103528

[12]Lemkalli B, Dudek KK, Kadic M, et al. W-shaped broadband attenuation of longitudinal waves through composite elastic metamaterial. Composites Part B: Engineering. 2025; 297: 112250. doi: 10.1016/j.compositesb.2025.112250

[13]Bagley RL, Torvik PJ. Fractional calculus—A different approach to the analysis of viscoelastically damped structures. AIAA Journal. 1983; 21(5): 741–748. doi: 10.2514/3.8142

[14]Bagley RL, Torvik PJ. On the fractional calculus model of viscoelastic behavior. Journal of Rheology. 1986; 30(1): 133–155. doi: 10.1122/1.549887

[15]Sepehri S, Mashhadi MM, Fakhrabadi MMS. Wave propagation in fractionally damped nonlinear phononic crystals. Nonlinear Dynamics. 2022; 110(2): 1683–1708. doi: 10.1007/s11071-022-07704-z

[16]Mahata S, Herencsar N, Kubanek D. On the design of power law filters and their inverse counterparts. Fractal and Fractional. 2021; 5(4): 197. doi: 10.3390/fractalfract5040197

[17]Kaczmarek MB, HosseinNia SH. Elastic metamaterials with fractional-order resonators. Fractional Calculus and Applied Analysis. 2023; 26(6): 2522–2539. doi: 10.1007/s13540-023-00220-w

[18]Mohajerin Esfahani P, Kuhn D. Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming. 2018; 171(1–2): 115–166. doi: 10.1007/s10107-017-1172-1

[19]Gao R, Kleywegt A. Distributionally robust stochastic optimization with Wasserstein distance. Mathematics of Operations Research. 2023; 48(2): 603–655. doi: 10.1287/moor.2022.1275

[20]Rahimian H, Mehrotra S. Frameworks and results in distributionally robust optimization. Open Journal of Mathematical Optimization. 2022; 3: 1–85. doi: 10.5802/ojmo.15

[21]Rockafellar RT, Uryasev S. Optimization of conditional value-at-risk. Journal of Risk. 2000; 2(3): 21–41. doi: 10.21314/JOR.2000.038

[22]Bartl D, Drapeau S, Obłój J, et al. Sensitivity analysis of Wasserstein distributionally robust optimization problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2021; 477(2256): 20210176. doi: 10.1098/rspa.2021.0176

[23]Storn R, Price K. Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization. 1997; 11(4): 341–359. doi: 10.1023/A:1008202821328

[24]Sobol IM. On quasi-Monte Carlo integrations. Mathematics and Computers in Simulation. 1998; 47(2–5): 103–112. doi: 10.1016/S0378-4754(98)00096-2

[25]Owen AB. Scrambled net variance for integrals of smooth functions. The Annals of Statistics. 1997; 25(4): 1541–1562. doi: 10.1214/aos/1031594731

[26]Matlack KH, Bauhofer A, Krödel S, et al. Composite 3D-printed metastructures for low-frequency and broadband vibration absorption. Proceedings of the National Academy of Sciences of the USA. 2016; 113(30): 8386–8390. doi: 10.1073/pnas.1600171113

[27]Wang X, Zhang C, Rui S, et al. Multi-scale material/structure integrated elastic metamaterial for broadband vibration absorbing. Materials & Design. 2024; 238: 112705. doi: 10.1016/j.matdes.2024.112705

[28]Palermo A, Krödel S, Marzani A, et al. Engineered metabarrier as shield from seismic surface waves. Scientific Reports. 2016; 6(1): 39356. doi: 10.1038/srep39356

[29]Blanchet J, Murthy K. Quantifying distributional model risk via optimal transport. Mathematics of Operations Research. 2019; 44(2): 565–600. doi: 10.1287/moor.2018.0936

[30]Blanchet J, Murthy K, Zhang F. Optimal transport-based distributionally robust optimization: Structural properties and iterative schemes. Mathematics of Operations Research. 2022; 47(2): 1500–1529. doi: 10.1287/moor.2021.1178

[31]Beyer HG, Sendhoff B. Robust optimization—A comprehensive survey. Computer Methods in Applied Mechanics and Engineering. 2007; 196(33–34): 3190–3218. doi: 10.1016/j.cma.2007.03.003

[32]Lion A. On the thermodynamics of fractional damping elements. Continuum Mechanics and Thermodynamics. 1997; 9(2): 83–96. doi: 10.1007/s001610050057

[33]Rossikhin YA, Shitikova MV. Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems. Acta Mechanica. 1997; 120(1–4): 109–125. doi: 10.1007/BF01174319

[34]Rossikhin YA, Shitikova MV. Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results. Applied Mechanics Reviews. 2010; 63(1): 010801. doi: 10.1115/1.4000563

[35]Mainardi F. An historical perspective on fractional calculus in linear viscoelasticity. Fractional Calculus and Applied Analysis. 2012; 15(4): 712–717. doi: 10.2478/s13540-012-0048-6

[36]Huang HH, Sun CT, Huang GL. On the negative effective mass density in acoustic metamaterials. International Journal of Engineering Science. 2009; 47(4): 610–617. doi: 10.1016/j.ijengsci.2008.12.007

[37]Mead DM. Wave propagation in continuous periodic structures: Research contributions from Southampton, 1964–1995. Journal of Sound and Vibration. 1996; 190(3): 495–524. doi: 10.1006/jsvi.1996.0076

[38]Sugino C, Xia Y, Leadenham S, et al. A general theory for bandgap estimation in locally resonant metastructures. Journal of Sound and Vibration. 2017; 406: 104–123. doi: 10.1016/j.jsv.2017.06.004

[39]Fournier N, Guillin A. On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields. 2015; 162(3–4): 707–738. doi: 10.1007/s00440-014-0583-7

[40]Rockafellar RT, Uryasev S. Conditional value-at-risk for general loss distributions. Journal of Banking & Finance. 2002; 26(7): 1443–1471. doi: 10.1016/S0378-4266(02)00271-6

[41]Saltelli A, Annoni P, Azzini I, et al. Variance based sensitivity analysis of model output: Design and estimator for the total sensitivity index. Computer Physics Communications. 2010; 181(2): 259–270. doi: 10.1016/j.cpc.2009.09.018

[42]Sobol IM. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation. 2001; 55(1–3): 271–280. doi: 10.1016/S0378-4754(00)00270-6

[43]Campolongo F, Cariboni J, Saltelli A. An effective screening design for sensitivity analysis of large models. Environmental Modelling & Software. 2007; 22(10): 1509–1518. doi: 10.1016/j.envsoft.2006.10.004

[44]Deb K, Pratap A, Agarwal S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation. 2002; 6(2): 182–197. doi: 10.1109/4235.996017

[45]Liu J, Zeng T, Ni HT, et al. A new local resonance metamaterial for flat and cylindrical structures depended on elastic chiral spiral beams. Science China Technological Sciences. 2024; 67(11): 3597–3611. doi: 10.1007/s11431-024-2739-x

[46]Wan S, Li L, Wang G, et al. A novel locally resonance metamaterial cylindrical shell with tower-shaped lattice for broadband vibration suppression. Mechanical Systems and Signal Processing. 2024; 216: 111510. doi: 10.1016/j.ymssp.2024.111510

[47]Hao S, Hui S, Lyu X, et al. A novel locally resonant metastructure with soft-material rings for broadband and low frequency vibration attenuation. Engineering Structures. 2022; 272: 114978. doi: 10.1016/j.engstruct.2022.114978