Abstract
We propose a numerical method for a topology optimisation of composite elastic metamaterial slabs. We aim to realise some anomalous functionalities such as perfect absorption, wave-mode conversion, and negative refraction by designing the shape and topology of (visco-)elastic inclusions. Instead of manipulating effective material constants, we propose to utilise the far-field characteristics of scattered waves. This allows us to achieve novel functionalities for waves in not only low- but also high-frequency ranges. The design sensitivity corresponding to the far-field characteristics is rigorously derived using the adjoint variable method and incorporated into a level-set–based topology optimisation algorithm. The design sensitivity is computed by the boundary element method with periodic Green’s function instead of the standard finite element method to rigorously deal with the radiation of scattered waves without absorbing boundaries. We show some numerical examples to demonstrate the effectiveness of the proposed method.
Similar content being viewed by others
References
Ambati M, Fang N, Sun C, Zhang X (2007) Surface resonant states and superlensing in acoustic metamaterials. Phys Rev B 75(19):195447. https://doi.org/10.1103/PhysRevB.75.195447
Amstutz S (2011) Analysis of a level set method for topology optimization. Optimization Methods and Software 26(4-5):555–573. https://doi.org/10.1080/10556788.2010.521557
Amstutz S, Andrä H (2006) A new algorithm for topology optimization using a level-set method. J Comput Phys 216(2):573–588. https://doi.org/10.1016/j.jcp.2005.12.015
Bebendorf M (2008) Hierarchical matrices. Springer, Berlin
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Bonnet M, Delgado G (2013) The topological derivative in anisotropic elasticity. The Quarterly Journal of Mechanics and Applied Mathematics 66(4):557. https://doi.org/10.1093/qjmam/hbt018
Burton AJ, Miller GF (1971) The application of integral equation methods to the numerical solution of some exterior boundary-value problems. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, vol 323, pp 201–210
Cadman JE, Zhou S, Chen Y, Li Q (2013) On design of multi-functional microstructural materials. J Mater Sci 48(1):51–66
Caloz C, Itoh T (2005) Electromagnetic metamaterials: transmission line theory and microwave applications. John Wiley & Sons
Christiansen RE, Sigmund O (2016) Designing meta material slabs exhibiting negative refraction using topology optimization. Struct Multidiscip Optim 54(3):469–482. https://doi.org/10.1007/s00158-016-1411-8
Diaz AR, Haddow AG, Ma L (2005) Design of band-gap grid structures. Struct Multidiscip Optim 29(6):418–431
Dong HW, Su XX, Wang YS, Zhang C (2014) Topological optimization of two-dimensional phononic crystals based on the finite element method and genetic algorithm. Struct Multidiscip Optim 50(4):593–604
Dong HW, Zhao SD, Wang YS, Zhang C (2017) Topology optimization of anisotropic broadband double-negative elastic metamaterials. Journal of the Mechanics and Physics of Solids 105:54–80. https://doi.org/10.1016/j.jmps.2017.04.009
Dong HW, Zhao SD, Wang YS, Zhang C (2018) Broadband single-phase hyperbolic elastic metamaterials for super-resolution imaging. Scientific Reports 8(1):2247
Dong HW, Zhao SD, Wei P, Cheng L, Wang YS, Zhang C (2019) Systematic design and realization of double-negative acoustic metamaterials by topology optimization. Acta Mater 172:102– 120
Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London a: Mathematical, Phys Eng Sci 241 (1226):376–396. https://doi.org/10.1098/rspa.1957.0133
Gazonas GA, Weile DS, Wildman R, Mohan A (2006) Genetic algorithm optimization of phononic bandgap structures. Int J Solids Struct 43(18):5851–5866. https://doi.org/10.1016/j.ijsolstr.2005.12.002
Isakari H, Takahashi T, Toshiro M (2017) A topology optimisation with level-sets of B-spline surface (in Japanese). Transactions of the Japan Society for Computational Methods in Engineering 17:125–130
Jensen JS (2007) Topology optimization problems for reflection and dissipation of elastic waves. Journal of Sound and Vibration 301(1):319–340
Kitahara M, Nakagawa K, Achenbach JD (1989) Boundary-integral equation method for elastodynamic scattering by a compact inhomogeneity. Comput Mech 5(2):129–144. https://doi.org/10.1007/BF01046482
Kweun JM, Lee HJ, Oh JH, Seung HM, Kim YY (2017) Transmodal fabry-pérot resonance: theory and realization with elastic metamaterials. Phys Rev Lett 118(20):205901. https://doi.org/10.1103/PhysRevLett.118.205901
Liu Z, Zhang X, Mao Y, Zhu YY, Yang Z, Chan CT, Sheng P (2000) Locally resonant sonic materials. Science 289(5485):1734–1736. https://doi.org/10.1126/science.289.5485.1734
Matsushima K, Isakari H, Takahashi T, Matsumoto T (2018a) A boundary element method for two-dimensional elastic periodic scattering and its application to topology optimisation (in Japanese). Transactions of the Japan Society for Computational Methods in Engineering (18):35–40
Matsushima K, Isakari H, Takahashi T, Matsumoto T (2018b) An investigation of eigenfrequencies of boundary integral equations and the Burton-Miller formulation in two-dimensional elastodynamics. International Journal of Computational Methods and Experimental Measurements 6(6):1037–1127
Noguchi Y, Yamada T, Otomori M, Izui K, Nishiwaki S (2015) An acoustic metasurface design for wave motion conversion of longitudinal waves to transverse waves using topology optimization. Appl Phys Lett 107(22):221909
Sigmund O, Jensen JS (2003) Systematic design of phononic band–gap materials and structures by topology optimization. Philosophical Transactions of the Royal Society of London Series a: Mathematical. Phys Eng Sci 361(1806):1001–1019
Sokolowski J, Zochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272. https://doi.org/10.1137/S0363012997323230
Yablonovitch E (1993) Photonic band-gap structures. J Opt Soc Am B 10(2):283–295. https://doi.org/10.1364/JOSAB.10.000283
Yang X, Kim YY (2018) Topology optimization for the design of perfect mode-converting anisotropic elastic metamaterials. Compos Struct 201:161–177. https://doi.org/10.1016/j.compstruct.2018.06.022
Yi G, Youn BD (2016) A comprehensive survey on topology optimization of phononic crystals. Struct Multidiscip Optim 54(5):1315–1344
Zhang L, Mei S, Huang K, Qiu CW (2016) Advances in full control of electromagnetic waves with metasurfaces. Advanced Optical Materials 4(6):818–833. https://doi.org/10.1002/adom.201500690
Zhou X, Hu G (2007) Acoustic wave transparency for a multilayered sphere with acoustic metamaterials. Phys Rev E 75(4):46606. https://doi.org/10.1103/PhysRevE.75.046606
Zhou X, Hu G, Lu T (2008) Elastic wave transparency of a solid sphere coated with metamaterials. Phys Rev B 77(2):24101
Zhou X, Liu X, Hu G (2012) Elastic metamaterials with local resonances: an overview. Theor Appl Mech Lett 2(4):41001
Funding
This work was supported by JSPS KAKENHI Grant Numbers JP19J21766, JP19H00740, JP17K14146.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Responsible Editor: YoonYoung Kim
Replication of results
Source codes used to produce the results in Section 4 are available upon request to the corresponding author.
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: Periodic Green’s function and far-field characteristics
1.1 A.1 Periodic Green’s function
To analyse the periodic scattering problem (9)–(15), we first consider Green’s function \(G^{\mathrm {p}}_{ij}\) satisfying:
and the radiation condition, where δij is the Kronecker delta, and δ(x) is the Dirac delta function. This Green’s function \(G^{\mathrm {p}}_{ij}\) is called periodic Green’s function and known to have the following representation:
where Gij is the fundamental solution for the two-dimensional elastodynamics, expressed by:
with the Hankel functions \(H^{(1)}_{n}\) of the first kind and order n and wavenumbers:
The lattice sum (70) would be the simplest expression of \(G^{\mathrm {p}}_{ij}\) but has computational limitations. From (71), we see that Gij asymptotically behaves as G(x,y) = O(|x − y|− 1/2) when |x − y| tends to the infinity if Im[λ] = Im[μ] = 0. This implies that the convergence speed of the lattice sum (70) is extremely slow; thus, we require another representation of \(G^{\mathrm {p}}_{ij}\) whose convergence is guaranteed and rapid.
For now, we assume that x2 − y2 ≠ 0 and consider the following Fourier transform of the fundamental solution Gij(x,y) with respect to x1:
where pL±(ξ), dL±(ξ), pT±(ξ), and dT±(ξ) are defined as follows:
Using Poisson’s summation formula, the lattice sum (70) can be converted into the following series:
where \( {p}^{\text {L}\pm }_{m}= {p}^{\text {L}\pm }(\xi _{m})\), \( {d}^{\text {L}\pm }_{m}= {d}^{\text {L}\pm }(\xi _{m})\), \( {p}^{\text {T}\pm }_{m}= {p}^{\text {T}\pm }(\xi _{m})\), \( {d}^{\text {T}\pm }_{m}= {d}^{\text {T}\pm }(\xi _{m})\), and
The series (79) converges rapidly because of the exponential functions unless |x2 − y2| becomes zero; otherwise, the summands become O(|m|− 1) as \(|m|\to \infty \). We can improve this convergence rate by using Kummer’s transformation and obtain:
where the vectors and functions with the tilde symbol are defined by replacing (λ, μ) with (−λ/qs,−μ/qs) (correspondingly (kL,kT) with (\(\mathrm {i}\sqrt {q_{s}} k_{\mathrm {L}}, \mathrm {i}\sqrt {q_{s}} k_{\mathrm {T}}\))). We can easily show that the summands of the first series in (83) become at worst (i.e. when |x2 − y2| = 0) \(O(|m|^{-2N_{K}-1})\) as \(|m|\to \infty \) when qs > 0 and cs solve the following linear system:
Thus, we determine cs by solving (86); qs > 0 are regarded as parameters. Note that (85) would suffer from a cancellation of siginificant digits in this case and thus require some transformations such as \(e^{z}-1=-2\mathrm {i}\mathrm {e}^{z/2}\sin \limits (iz/2)\). On the other hand, the first series in (83) always converges rapidly since \(\tilde {G}_{ij}( {x}, {y})=O(\mathrm {e}^{-k| {x}- {y}|})~(k>0)\) as \(| {x}- {y}|\to \infty \). Note that the representation (83)–(85) holds even if x2 − y2 = 0 though we assumed otherwise. For more details, refer to Matsushima et al. (2018a).
1.2 A.2 Far-field characteristics
Periodic Green’s function expressed by (79) implies that a scattered field can be expanded into a sum of plane P- and S-waves. This can be shown by substituting (79) into the representation formula (Kitahara et al. 1989):
which yields the plane-wave expansion (17) and formulae (21) and (22).
Appendix B: Boundary element method
We describe the numerical solution of the periodic scattering problem (9)–(15). We first convert it into the Burton-Miller-type boundary integral equations (Burton and Miller 1971):
where \(\mathcal {I}\) is the identity operator, and \(\mathcal {S}^{}\), \(\mathcal {D}^{}\), \(\mathcal {D}^{*}\), and \(\mathcal {N}^{}\) are the integral operators defined by:
and \(\mathcal {S}^{\prime }\) and \(\mathcal {D}^{\prime }\) are defined by replacing (ρ,λ,μ) in \(\mathcal {S}\) and \(\mathcal {D}\) with \((\rho ^{\prime }, \lambda ^{\prime }, \mu ^{\prime })\), respectively. Furthermore, ‘v.p.’ and ‘p.f.’ stand for Cauchy’s principal value and the finite part of divergent integrals, respectively. The coupling parameter \(\alpha \in \mathbb {C}\) is arbitrary if it has a non-zero imaginary part, but the best condition number of the boundary integrals (88) and 89 is often achieved when α = −i/(μkT) (Matsushima et al. 2018b).
The boundary integral equations (88) and (89) are numerically solved after being discretised by a collocation method with piecewise constant elements, which results in a system of linear equations with a fully populated coefficient matrix. To reduce the computational cost of the linear algebraic operations, we apply the \({\mathscr{H}}\)-matrix method (Bebendorf 2008) to the coefficient matrix and solve the linear system by an accelerated LU factorisation.
Rights and permissions
About this article
Cite this article
Matsushima, K., Isakari, H., Takahashi, T. et al. A topology optimisation of composite elastic metamaterial slabs based on the manipulation of far-field behaviours. Struct Multidisc Optim 63, 231–243 (2021). https://doi.org/10.1007/s00158-020-02689-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-020-02689-y