Computational Mechanics

, Volume 62, Issue 3, pp 285–307 | Cite as

An isogeometric approach for analysis of phononic crystals and elastic metamaterials with complex geometries

  • Ryan Alberdi
  • Guodong Zhang
  • Kapil Khandelwal
Original Paper


An isogeometric analysis (IGA) framework is presented to construct and solve dispersion relations for generating the band structure of periodic materials with complicated geometries representing phononic crystals and elastic metamaterials. As the dispersive properties depend on the microstructural geometry, an accurate representation of microstructural geometrical features is paramount. To this end, the ability of isogeometric analysis to exactly model complex curved geometries is exploited, and wave propagation in infinitely periodic solids is combined with isogeometric analysis. The benefits of IGA are demonstrated by comparing the results to those obtained using standard finite element analysis (FEA). It is shown that the IGA solutions can reach the same level of accuracy as FEA while using significantly fewer degrees of freedom. IGA is applied to phononic crystals and elastic metamaterials and the band structure for a variety of unit cells with complex microstructural geometries is investigated to illustrate the desirable dispersive effects in these metamaterials.


Isogeometric analysis Bloch waves Phononic crystals Elastic metamaterials Bandgaps and local resonance 



The presented work is supported in part by the US National Science Foundation through Grant CMMI-1055314. Any opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.


  1. 1.
    Lee J-H, Singer JP, Thomas EL (2012) Micro-/nanostructured mechanical metamaterials. Adv Mater 24:4782–4810. CrossRefGoogle Scholar
  2. 2.
    Pierre AD (2013) Acoustic metamaterials and phononic crystals. Springer, BerlinGoogle Scholar
  3. 3.
    Hussein MI, Leamy MJ, Ruzzene M (2014) Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl Mech Rev 66:040802-040802-38Google Scholar
  4. 4.
    Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51–78. MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sigalas MM, Economou EN (1992) Elastic and acoustic wave band structure. J Sound Vib 158:377–382. CrossRefGoogle Scholar
  6. 6.
    Kushwaha MS, Halevi P, Dobrzynski L, Djafarirouhani B (1993) Acoustic band-structure of periodic elastic composites. Phys Rev Lett 71:2022–2025.
  7. 7.
    Brillouin L (1946) Wave propagation in periodic structures, 1st edn. McGraw-Hill, LondonzbMATHGoogle Scholar
  8. 8.
    Liu Z, Zhang X, Mao Y, Zhu YY, Yang Z, Chan CT, Sheng P (2000) Locally resonant sonic materials. Science 289:1734–1736CrossRefGoogle Scholar
  9. 9.
    Wang G, Wen X, Wen J, Shao L, Liu Y (2004) Two-dimensional locally resonant phononic crystals with binary structures. Phys Rev Lett 93:154302CrossRefGoogle Scholar
  10. 10.
    Hirsekorn M, Delsanto PP, Batra NK, Matic P (2004) Modelling and simulation of acoustic wave propagation in locally resonant sonic materials. Ultrasonics 42:231–235. CrossRefGoogle Scholar
  11. 11.
    Wang P, Casadei F, Kang SH, Bertoldi K (2015) Locally resonant band gaps in periodic beam lattices by tuning connectivity. Phys Rev B 91:020103CrossRefGoogle Scholar
  12. 12.
    Sridhar A, Kouznetsova VG, Geers MGD (2016) Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum. Comput Mech 57:423–435. MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ruzzene M, Scarpa F, Soranna F (2003) Wave beaming effects in two-dimensional cellular structures. Smart Mater Struct 12:363CrossRefGoogle Scholar
  14. 14.
    Casadei F, Rimoli JJ (2013) Anisotropy-induced broadband stress wave steering in periodic lattices. Int J Solids Struct 50:1402–1414. CrossRefGoogle Scholar
  15. 15.
    Celli P, Gonella S (2014) Low-frequency spatial wave manipulation via phononic crystals with relaxed cell symmetry. J Appl Phys 115:103502. CrossRefGoogle Scholar
  16. 16.
    Liu Z, Chan CT, Sheng P (2002) Three-component elastic wave band-gap material. Phys Rev B 65:165116CrossRefGoogle Scholar
  17. 17.
    Sigalas MM, Garcia N (2000) Theoretical study of three dimensional elastic band gaps with the finite-difference time-domain method. J Appl Phys 87:3122–3125. CrossRefGoogle Scholar
  18. 18.
    Biwa S, Yamamoto S, Kobayashi F, Ohno N (2004) Computational multiple scattering analysis for shear wave propagation in unidirectional composites. Int J Solids Struct 41:435–457. CrossRefzbMATHGoogle Scholar
  19. 19.
    Phani AS, Woodhouse J, Fleck NA (2006) Wave propagation in two-dimensional periodic lattices. J Acoust Soc Am 119:1995–2005. CrossRefGoogle Scholar
  20. 20.
    Gonella S, Ruzzene M (2008) Analysis of in-plane wave propagation in hexagonal and re-entrant lattices. J Sound Vib 312:125–139. CrossRefzbMATHGoogle Scholar
  21. 21.
    Zhao J, Li Y, Liu WK (2015) Predicting band structure of 3D mechanical metamaterials with complex geometry via XFEM. Comput Mech 55:659–672. MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–95. MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cottrell JA, Hughes TJ, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, HobokenCrossRefzbMATHGoogle Scholar
  24. 24.
    Verhoosel CV, Scott MA, Hughes TJ, De Borst R (2011) An isogeometric analysis approach to gradient damage models. Int J Numer Meth Eng 86:115–134MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Akkerman I, Bazilevs Y, Calo VM, Hughes TJR, Hulshoff S (2008) The role of continuity in residual-based variational multiscale modeling of turbulence. Comput Mech 41:371–378. MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang G, Alberdi R, Khandelwal K (2016) Analysis of three-dimensional curved beams using isogeometric approach. Eng Struct 117:560–574. CrossRefGoogle Scholar
  27. 27.
    Zhang G, Khandelwal K (2016) Modeling of nonlocal damage-plasticity in beams using isogeometric analysis. Comput & Struct 165:76–95.
  28. 28.
    Cottrell J, Reali A, Bazilevs Y, Hughes T (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195:5257–5296MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Weeger O, Wever U, Simeon B (2013) Isogeometric analysis of nonlinear Euler-Bernoulli beam vibrations. Nonlinear Dyn 72:813–835MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43:3–37. MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wall WA, Frenzel MA, Cyron C (2008) Isogeometric structural shape optimization. Comput Methods Appl Mech Eng 197:2976–2988MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hughes TJR, Reali A, Sangalli G (2008) Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS. Comput Methods Appl Mech Eng 197:4104–4124. MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hughes TJR, Evans JA, Reali A (2014) Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput Methods Appl Mech Eng 272:290–320. MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Graff KF (1975) Wave motion in elastic solids. Oxford University Press, LondonzbMATHGoogle Scholar
  35. 35.
    Kittel C (1986) Introduction to solid state physics, 6th edn. Wiley, New YorkzbMATHGoogle Scholar
  36. 36.
    Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis, New YorkGoogle Scholar
  37. 37.
    Piegl L, Tiller W (1997) The NURBS book. 1997. Monographs in Visual CommunicationGoogle Scholar
  38. 38.
    Rogers DF (2000) An introduction to NURBS: with historical perspective. Elsevier, AmsterdamGoogle Scholar
  39. 39.
    Nguyen VP, Kerfriden P, Brino M, Bordas SPA, Bonisoli E (2014) Nitsche’s method for two and three dimensional NURBS patch coupling. Comput Mech 53:1163–1182. MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Cliffe KA, Garratt TJ, Spence A (1994) Eigenvalues of block matrices arising from problems in fluid mechanics. SIAM J Matrix Anal Appl 15:1310–1318. MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Langley RS (1994) On the modal density and energy flow characteristics of periodic structures. J Sound Vib 172:491–511. CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Civil and Environmental Engineering and Earth SciencesUniversity of Notre DameNotre DameUSA

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