Skip to main content
SpringerLink
Log in

Designing meta material slabs exhibiting negative refraction using topology optimization

  • RESEARCH PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

This paper proposes a topology optimization based approach for designing meta materials exhibiting a desired negative refraction with high transmission at a given angle of incidence and frequency. The approach considers a finite slab of meta material consisting of axis-symmetric designable unit cells subjected to an exterior field. The unit cell is designed to achieve the desired properties based on tailoring the response of the meta material slab under the exterior field. The approach is directly applicable to physical problems modeled by the Helmholtz equation, such as acoustic, elastic and electromagnetic wave problems. Acoustic meta materials with unit cell size on the order of half the wave length are considered as examples. Optimized designs are presented and their performance under varying frequency and angle of incidence is investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Notes

  1. 1 Note that the proposed formulation does not allow 𝜃 1=0o due to the formulation relying on (21).

References

  • Andkjær J, Sigmund O (2013) Topology optimized cloak for airborn sound. J Vib Acoust 135:041,011–1–041,001–5

    Article  Google Scholar 

  • Bendsøe MP, Sigmund O (2003) Topology optimization. Springer

  • Christiansen R E, Fernandez-Grande E, Sigmund O (2015a) Experimental validation of topology optimized acoustic cavity. J Acoust Soc Am 138(6):3470–3474. doi:10.1121/1.4936905

  • Christiansen RE, Lazarov BS, Jensen JS, Sigmund O (2015b) Creating geometrically robust designs for highly sensitive problems using topology optimization - acoustic cavity design. Structural and Multidiciplinary Optimization 52:737–754. doi:10.1007/s00,158--015--1265--5

  • Craster R V, Guenneau S (2013) Acoustic metamaterials - negative refraction, Imaging, Lensing and Cloaking. Springer Science+Business Media Dordrecht

  • Desmet W (1998) A wave based prediction technique for coupled vibro-acoustic analysis. PhD thesis, Katholieke Universiteit Leuven

  • Dühring MB, Jensen J S, Sigmund O (2008) Acoustic design by topology optimization. J Sound Vib 317:557–575

    Article  Google Scholar 

  • Diaz A R, Sigmund O (2009) A toplogy optimization method for design of negative permeability metamaterials. Structural and Multidiciplinary Optimization 41:163–177

    Article  MathSciNet  MATH  Google Scholar 

  • Goo S, Kook J, Koo K, Hyun J, Wang S (2014) Acoustic topology optimization for interior acoustic problem using the hybrid finite element - wave based method. In: The Eighth China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems

  • Guest J K, JH Prévost, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61:238–254

    Article  MathSciNet  MATH  Google Scholar 

  • Jansen M, Lazarov B, Schevenels M, Sigmund O (2013) On the similarities between micro/nano lithography and topology optimization projection methods. Struct Multidiscip Optim 48:717–730

    Article  MathSciNet  Google Scholar 

  • Krowne CM, Zhang Y (2007) Physics of Negative Refraction and Negative Index Material - Optical and Electronic Aspects and Diversified Approaches. Springer, Berlin Heidelberg

    Book  Google Scholar 

  • Lu L, Yamamoto T, Otomori M, Yamada T, Kazuhiro NS (2013) Topology optimization of an acoustic metamaterial with negative bulk modulus using local resonance. Finite Elem Anal Des 72:1–12

    Article  MathSciNet  MATH  Google Scholar 

  • Mills EM, Banks ML, Sprague JE, Finkel T (2003) Imaging by flat lens using negative refraction. Nature 426:404

    Article  Google Scholar 

  • Mortensen NA, Yan M, Sigmund O, Breinbjerg O (2010) On the unambiguous determination of effeffect optical properties of periodic metamaterials: a one-dimensional case study. Journal of the European Optical Society 5

  • Park J H, Ma P S, Kim Y Y (2015) Design of phononic crystals for self-colimation of elastic waves using topology optimization method. Structural and Multidiciplinary Optimization 51:1199– 1209

    Article  MathSciNet  Google Scholar 

  • Pendry J B (2000) Negative refraction makes a perfect lens. Phys Rev Lett 85:3966–3969

    Article  Google Scholar 

  • Philippe F D, Murray T W, Prada C (2015) Focusing on plates: controlling guided waves using negative refraction. Scientific Report 5:11,112–1–11,112–4

    Article  Google Scholar 

  • Pluymers B (2006) Wave based modelling methods for steady-state vibro-acoustics. PhD thesis, Katholieke Universiteit Leuven

  • Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24:359– 373

    Article  MathSciNet  MATH  Google Scholar 

  • Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex seperable approximations. SIAM Journal of Optimization 12:555–573

    Article  MathSciNet  MATH  Google Scholar 

  • Veselago V G (1968) The electrodynamics of substances with negative values of 𝜖 and μ. Soviet Physics Uspekhi 10:509– 514

    Article  Google Scholar 

  • Wang F, Lazarov B S, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Structural Multidiciplinary Optimization 43:767–784

    Article  MATH  Google Scholar 

  • Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside projections. Structural and Multidiciplinary Optimization 41:495–505

    Article  MathSciNet  MATH  Google Scholar 

  • Xu T, Agrawal A, Abashin M, Chau K J, Lezec H J (2013) All-angle negative refraction and active flat lensing of ultraviolet light. Nature 497:470–474

    Article  Google Scholar 

  • Zhang S, Fan W, Panoiu NC, Malloy KJ, Osgood RM, Brueck SRJ (2005) Experimental demonstration of near-infrared negative-index metamaterials. Phys Rev Lett 95:137,404–1–137, 404–4

    Article  Google Scholar 

  • Zhang S, Yin L, Fang N (2009) Focusing ultrasound with an acoustic metamaterial network. Phys Rev Lett 102:194,301–1–194,301–4

    Article  Google Scholar 

  • Zhang X, Liu Z (2004) Negative refraction of acoustic waves in two-dimensionl phononic crystals. Appl Phys Lett 85:341–343

    Article  Google Scholar 

  • Zhou M, Lazarov b, Sigmund O (2016) Topology optimization for optical microlithography with partially coherent illumination. Submitted

  • Zhou S, Li W, Sun G, Li Q (2010) A level-set procedure for the design of electromagnetic metamaterials. Optical Express 18

Download references

Acknowledgments

The work was financially supported by Villum Fonden through the research project Topology Optimization - the Next Generation (NextOpt).

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Nils Koppels Allé, B. 404, DK-2800, Lyngby, Denmark

    Rasmus E. Christiansen & Ole Sigmund

Authors
  1. Rasmus E. Christiansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ole Sigmund
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rasmus E. Christiansen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Christiansen, R.E., Sigmund, O. Designing meta material slabs exhibiting negative refraction using topology optimization. Struct Multidisc Optim 54, 469–482 (2016). https://doi.org/10.1007/s00158-016-1411-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-016-1411-8

Keywords

Search

Navigation