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

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Abstract

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, IN, 46556, USA

    Ryan Alberdi, Guodong Zhang & Kapil Khandelwal

Authors
  1. Ryan Alberdi
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  2. Guodong Zhang
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  3. Kapil Khandelwal
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Corresponding author

Correspondence to Kapil Khandelwal.

Appendices

Appendix A: Control data for NURBS shapes

1.1 A.1 Geometry from Sect. 7.1

Fig. 17
figure 17

Patch discretization of the domain shown in Sect. 7.2

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Fig. 18
figure 18

Control point polygon of two patches in Fig. 17. a Patch 1, b Patch 2

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Table 6 Control points and weights for Patch 1 in Fig. 18a
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Table 7 Control points and weights for Patch 2 in Fig. 18b
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The domain in this example is divided into 12 patches as shown in Fig. 15. Only two patches, patch-1 and patch-2, are unique due to symmetry. The control point polygons for the two patches are shown in Fig. 16. Control point data and the corresponding weights for the two patches are provided in Tables 4 and 5, respectively.

Fig. 19
figure 19

Patch discretization of the domain shown in Sect. 7.3

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Fig. 20
figure 20

Control point polygon of two patches in Fig. 19.a Patch 1, b Patch 2

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Table 8 Control points and weights for Patch 1 in Fig. 20a
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A.2 Geometry from Sect. 7.2

The domain in this example is divided into 16 patches as shown in Fig. 17. Only two patches, Patch 1 and Patch 2, are unique due to symmetry. The control point polygons for the two patches are shown in Fig. 18. Control point data and the corresponding weights for the two patches are provided in Tables 6 and 7, respectively.

1.1 A.3 Geometry from Sect. 7.3

The domain with asymmetric hole in Example 7.3 is divided into 4 patches as shown in Fig. 19. Only two patches, Patch 1 and Patch 2, are unique due to symmetry. The control point polygons for the two patches are shown in Fig. 20. Control point data and the corresponding weights for the two patches are provided in Tables 8 and 9, respectively.

Table 9 Control points and weights for Patch 2 in Fig. 20b
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Alberdi, R., Zhang, G. & Khandelwal, K. An isogeometric approach for analysis of phononic crystals and elastic metamaterials with complex geometries. Comput Mech 62, 285–307 (2018). https://doi.org/10.1007/s00466-017-1497-x

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