pith. sign in

arxiv: 2605.26435 · v2 · pith:5ZCOW4BPnew · submitted 2026-05-26 · ❄️ cond-mat.mtrl-sci · cs.NA· math.NA· physics.comp-ph

Gradient-Based Topology Optimization of Localized Defect Modes with Bandgap Preservation in Phononic Crystals

Xinlin Xu , Junji Kato This is my paper

Pith reviewed 2026-06-29 17:40 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cs.NAmath.NAphysics.comp-ph
keywords phononic crystalstopology optimizationdefect modesbandgap engineeringelastic wave localizationgradient-based optimizationlocalized resonances
0
0 comments X

The pith

A two-stage gradient optimization first opens a bandgap in a phononic host then tunes only the defect region to place one localized mode at a chosen frequency while pushing competing modes away.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a method to design periodic elastic media that trap waves at exact frequencies inside a bandgap. It splits the task into two stages: first shape the repeating host cell to create the gap, then adjust only the defect cell so one resonance lands on the target while others are driven out of the gap center. A single objective function handles both attraction of the desired mode and repulsion of the others, allowing the optimizer to track the correct branches automatically. Because the defect branches stay nearly flat, the calculation uses only the center point of the Brillouin zone and checks full dispersion afterward. Examples with different materials and cell sizes show the target mode lands accurately, stays isolated, and leaves most of the original gap intact.

Core claim

The central claim is that a gradient-based two-stage topology optimization framework can place a selected localized defect mode at a prescribed frequency inside a phononic bandgap while repelling non-target in-gap modes and largely preserving the host bandgap. The first stage optimizes the host unit cell to form the gap; the second stage modifies only the defect cell using a smooth mode-selection function that unifies attraction and repulsion in one objective. Optimization occurs at the Γ-point eigenspectrum because the branches of interest are nearly flat, with full dispersion relations verified afterward.

What carries the argument

Two-stage gradient-based topology optimization with a unified mode-selection objective that attracts one defect branch and repels others, evaluated at the Γ point.

If this is right

  • The target localized resonance appears at the chosen frequency with clear frequency separation from other in-gap modes.
  • The host bandgap remains largely open after defect optimization.
  • The final designs produce strong spatial localization of elastic waves at the defect.
  • The same procedure works for at least two different material combinations and two supercell sizes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested on three-dimensional or non-periodic host geometries to check whether the flat-branch assumption still holds.
  • If the repulsion term is removed, competing modes might drift into the center of the gap and reduce isolation.
  • The method supplies a concrete route to inverse design of vibration-trapping devices once fabrication constraints are added to the topology variables.

Load-bearing premise

The defect branches stay nearly flat, so optimizing only at the single Γ point captures the frequencies that matter across the full dispersion relation.

What would settle it

A computation on the optimized structures in which the target mode frequency moves by more than a few percent or merges with another in-gap mode once the full dispersion curve over the reduced Brillouin zone is calculated instead of the Γ point alone.

Figures

Figures reproduced from arXiv: 2605.26435 by Junji Kato, Xinlin Xu.

Figure 2
Figure 2. Figure 2: Supercell configuration (3×3 tiling of the unit cell) and its reduced IBZ. The dashed box highlights the central unit cell. where K and M are the global stiffness and mass matrices, and U is the global displacement vector. The element matrices are assembled from Ke = ∫ Ve BTDB dV, Me = ∫ Ve ρNTN dV, (3) with B denoting the strain–displacement matrix, N the shape-function matrix, and Ve the element volume. … view at source ↗
Figure 3
Figure 3. Figure 3: Point defect supercell. Because the supercell remains periodic, the same Bloch reduction applies. The corresponding eigenvalue problem is [ Ksc R (k sc) − ω 2Msc R (k sc) ] ϕ sc = 0, (6) where the superscript “sc” denotes quantities assembled over the supercell. For a perfect supercell, the dispersion consists of folded bulk bands originating from the underlying unit cell. When a defect is introduced, addi… view at source ↗
Figure 5
Figure 5. Figure 5: Sign convention of the quadratic exclusion function QEj . 8 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Attraction component fattract(ωj ) and repulsion component frepel(ωj ) of the Stage 2 objective function. the defect mode frequencies, yielding the discrete set of eigenfrequencies {ωj} and corresponding eigenvectors {ϕj} required for mode identification. A mode at frequency ωj with eigenvector ϕj is classified as a defect mode if it satisfies two criteria. First, its frequency lies within the base cell ba… view at source ↗
Figure 7
Figure 7. Figure 7: illustrates the selection function behavior for three shape exponents: β = 1 (blue), β = 2 (orange), and β = 5 (red). The vertical dashed lines represent defect mode frequencies present during optimization. For β = 1, the gradual transition means that modes in an intermediate frequency range receive comparable weights from both attraction and repulsion terms, leading to conflicting gradients. As β increase… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of analytical and FDA sensitivities 0 20 40 60 80 100 Element ID 0 2.0×10-6 4.0×10-6 6.0×10-6 8.0×10-6 1.0×10-5 1.2×10-5 1.4×10-5 1.6×10-5 Error [%] [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Initial unit cell configuration for Case 1. [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Unit cell designs from Stage 1 optimization. [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Defect mode frequency evolution for Case 1 ( [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Stage 2 optimization results for Case 1. [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Unit cell designs from Stage 1 optimization. [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Defect mode frequency evolution for Case 2 ( [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Stage 2 optimization results for Case 2. [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Defect mode frequency evolution for Case 3 ( [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Stage 2 optimization results for Case 3 ( [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
read the original abstract

Phononic crystals can confine elastic waves through localized defect states within bandgaps, offering promising opportunities for vibration control and energy localization. However, designing defect states at prescribed frequencies while maintaining adequate separation from other in-gap modes remains a significant challenge. Existing optimization approaches generally treat the target mode indirectly and provide limited control over competing localized modes. This study presents a gradient-based two-stage topology optimization framework for the frequency placement of localized defect modes in periodic elastic media. First, a host unit cell is optimized to create a bandgap around a prescribed frequency. Subsequently, only the defect cell is modified to attract a selected localized mode toward the target frequency while repelling non-target modes away from the central region of the bandgap. The formulation incorporates a smooth mode-selection function that combines mode attraction and repulsion within a unified objective, enabling automatic tracking of the relevant modes throughout the optimization process. Because the localized defect branches of interest are nearly flat, the optimization is performed using only the $\Gamma$-point eigenspectrum, while the corresponding dispersion relations over a reduced irreducible Brillouin zone are evaluated afterwards for verification. Numerical examples involving two material systems and two supercell sizes demonstrate accurate placement of localized resonances, clear separation from competing in-gap modes, and substantial preservation of the host bandgap. The resulting structures exhibit strong elastic-wave localization, highlighting the potential of the proposed approach for the design of phononic devices for vibration confinement and energy trapping.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents a gradient-based two-stage topology optimization framework for placing localized defect modes at prescribed frequencies in phononic crystals while preserving the host bandgap and separating target modes from competing in-gap modes. Stage one optimizes the host unit cell to open a bandgap around a target frequency; stage two modifies only the defect cell using a smooth mode-selection function in the objective that attracts the target mode and repels others. Optimization uses solely the Γ-point eigenspectrum on the basis that defect branches are nearly flat, with full dispersion relations evaluated post-optimization for verification. Numerical examples on two material systems and two supercell sizes are reported to achieve accurate frequency placement, mode separation, and bandgap preservation, with resulting structures showing strong wave localization.

Significance. If the results hold under the stated assumptions, the work provides a systematic, gradient-based method with explicit control over both target-mode attraction and competing-mode repulsion via a unified objective, which addresses limitations in prior indirect treatments of defect modes. The two-stage separation of bandgap creation from defect tuning and the use of automatic mode tracking are strengths that could facilitate design of phononic devices for vibration confinement. The numerical demonstrations across multiple systems add practical value, though the post-hoc nature of dispersion verification limits claims of robustness.

major comments (2)
  1. [Method section (two-stage framework)] Method section (two-stage framework, after bandgap stage): The optimization is performed exclusively at the Γ-point because 'the localized defect branches of interest are nearly flat,' yet no a-priori bound on dispersion, sensitivity analysis, or penalty term for non-flatness is supplied. This assumption is load-bearing for the central claim of accurate frequency placement and separation, as any k-dependent shift or hybridization would invalidate the Γ-only designs; post-optimization verification alone does not protect the gradient updates themselves.
  2. [Numerical examples section] Numerical examples section: While examples with two material systems and two supercell sizes are presented as demonstrating 'accurate placement' and 'clear separation,' the manuscript supplies no quantitative metrics (e.g., maximum frequency deviation across the reduced Brillouin zone or minimum separation distance) that directly test the flat-branch assumption to within the tolerance needed for the claimed performance.
minor comments (2)
  1. [Abstract] Abstract and method: The phrase 'substantial preservation of the host bandgap' is used without a precise definition (e.g., percentage retention of gap width or edge shift tolerance); a quantitative criterion should be stated.
  2. [Method section] The description of the smooth mode-selection function would benefit from an explicit equation reference or pseudocode to clarify how attraction and repulsion are combined into a single differentiable objective.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback on our manuscript. We address each major comment below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [Method section (two-stage framework)] The optimization is performed exclusively at the Γ-point because 'the localized defect branches of interest are nearly flat,' yet no a-priori bound on dispersion, sensitivity analysis, or penalty term for non-flatness is supplied. This assumption is load-bearing for the central claim of accurate frequency placement and separation, as any k-dependent shift or hybridization would invalidate the Γ-only designs; post-optimization verification alone does not protect the gradient updates themselves.

    Authors: We agree that an a-priori justification or bound would strengthen the methodological foundation. The near-flatness assumption stems from the strong localization of defect modes, which typically exhibit minimal dispersion in phononic crystals. While post-optimization verification is provided, we acknowledge that it does not safeguard the optimization process itself. In the revised manuscript, we will include a sensitivity analysis demonstrating the dispersion characteristics of the optimized defect modes across the Brillouin zone and discuss the validity of the Γ-point approximation for the reported designs. revision: yes

  2. Referee: [Numerical examples section] While examples with two material systems and two supercell sizes are presented as demonstrating 'accurate placement' and 'clear separation,' the manuscript supplies no quantitative metrics (e.g., maximum frequency deviation across the reduced Brillouin zone or minimum separation distance) that directly test the flat-branch assumption to within the tolerance needed for the claimed performance.

    Authors: We concur that explicit quantitative metrics would better substantiate the claims. The current manuscript relies on visual inspection of the dispersion relations for verification. To address this, we will add quantitative measures in the revised numerical examples section, including the maximum frequency deviation of the target mode over the reduced Brillouin zone and the minimum frequency separation to competing modes for each case study. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard optimization with post-hoc verification

full rationale

The paper applies standard gradient-based topology optimization to a custom objective combining mode attraction and repulsion. The Γ-point restriction is justified by the explicit modeling assumption that target defect branches are nearly flat, with dispersion checked only after optimization; this is an a-priori modeling choice rather than a definitional reduction or fitted input renamed as prediction. No self-citations appear as load-bearing uniqueness theorems, no ansatz is smuggled via prior work, and no result is shown to equal its inputs by construction. The numerical examples serve as external validation of the designs rather than tautological confirmation. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are identifiable. The framework relies on standard assumptions in topology optimization such as differentiable material models and finite-element discretization, which are not detailed here.

pith-pipeline@v0.9.1-grok · 5797 in / 1085 out tokens · 25384 ms · 2026-06-29T17:40:02.527541+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

54 extracted references · 41 canonical work pages

  1. [1]

    M. S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Physical Review Letters 71 (13) (1993) 2022–2025. doi:10.1103/ PhysRevLett.71.2022

    1993

  2. [2]

    Sigalas, E

    M. Sigalas, E. Economou, Elastic and acoustic wave band structure, Journal of Sound and Vibration 158 (2) (1992) 377–382. doi:10.1016/0022-460X(92)90059-7

    work page doi:10.1016/0022-460x(92)90059-7 1992

  3. [3]

    Martínez-Sala, J

    R. Martínez-Sala, J. Sancho, J. V. Sánchez, V. Gómez, J. Llinares, F. Meseguer, Sound atten- uation by sculpture, Nature 378 (6554) (1995) 241–241. doi:10.1038/378241a0. 27

    work page doi:10.1038/378241a0 1995

  4. [4]

    Muhammad, C. W. Lim, From Photonic Crystals to Seismic Metamaterials: A Review via Phononic Crystals and Acoustic Metamaterials, Archives of Computational Methods in Engi- neering 29 (2) (2022) 1137–1198. doi:10.1007/s11831-021-09612-8

    work page doi:10.1007/s11831-021-09612-8 2022

  5. [5]

    Y. Bai, Y. Chen, J. Zheng, Design of phononic crystal for enhancing low-frequency sound absorption in mufflers, Scientific Reports 14 (1) (2024) 28921. doi:10.1038/ s41598-024-79762-9

    2024

  6. [6]

    G. Lee, D. Lee, J. Park, Y. Jang, M. Kim, J. Rho, Piezoelectric energy harvesting using mechanical metamaterials and phononic crystals, Communications Physics 5 (1) (2022) 94. doi:10.1038/s42005-022-00869-4

    work page doi:10.1038/s42005-022-00869-4 2022

  7. [7]

    Laude, Principles and properties of phononic crystal waveguides, APL Materials 9 (8) (2021) 080701

    V. Laude, Principles and properties of phononic crystal waveguides, APL Materials 9 (8) (2021) 080701. doi:10.1063/5.0059035

    work page doi:10.1063/5.0059035 2021

  8. [8]

    Torres, F

    M. Torres, F. R. Montero De Espinosa, D. García-Pablos, N. García, Sonic Band Gaps in Finite Elastic Media: Surface States and Localization Phenomena in Linear and Point Defects, Physical Review Letters 82 (15) (1999) 3054–3057. doi:10.1103/PhysRevLett.82.3054

    work page doi:10.1103/physrevlett.82.3054 1999

  9. [9]

    Kafesaki, M

    M. Kafesaki, M. M. Sigalas, N. García, Frequency Modulation in the Transmittivity of Wave Guides in Elastic-Wave Band-Gap Materials, Physical Review Letters 85 (19) (2000) 4044–

    2000

  10. [10]

    doi:10.1103/PhysRevLett.85.4044

    work page doi:10.1103/physrevlett.85.4044

  11. [11]

    J. D. Joannopoulos, J. N. Winn, S. G. Johnson, Photonic Crystals: Molding the Flow of Light - Second Edition, Princeton University Press, Princeton, NJ, 2011. doi:10.1515/ 9781400828241

    2011

  12. [12]

    S.-H. Jo, H. Yoon, Y. C. Shin, B. D. Youn, Revealing defect-mode-enabled energy localiza- tion mechanisms of a one-dimensional phononic crystal, International Journal of Mechanical Sciences 215 (2022) 106950. doi:10.1016/j.ijmecsci.2021.106950

    work page doi:10.1016/j.ijmecsci.2021.106950 2022

  13. [13]

    J. O. Vasseur, P. A. Deymier, B. Djafari-Rouhani, Y. Pennec, A.-C. Hladky-Hennion, Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates, Physical Review B 77 (8) (2008) 085415. doi:10.1103/PhysRevB.77.085415

    work page doi:10.1103/physrevb.77.085415 2008

  14. [14]

    Khelif, B

    A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier, Ph. Lambin, L. Dobrzynski, Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal, Physical Review B 65 (17) (2002) 174308. doi:10.1103/PhysRevB.65.174308

    work page doi:10.1103/physrevb.65.174308 2002

  15. [15]

    A. H. M. Almawgani, H. Makhlouf Fathy, H. E. Alfassam, A. M. El-Sherbeeny, A. Hajjiah, H. A. Elsayed, M. R. Abukhadra, W. Al Zoubi, R. Semeda, M. Ismail Fathy, A. A. H. Al- Athwary, A. Mehaney, Fano resonance in one-dimensional quasiperiodic topological phononic crystals towards a stable and high-performance sensing tool, Scientific Reports 14 (1) (2024) 1...

    work page doi:10.1038/s41598-024-62268-9 2024

  16. [16]

    Jo, Electrically controllable behaviors in defective phononic crystals with inductive- resistive circuits, International Journal of Mechanical Sciences 278 (2024) 109485

    S.-H. Jo, Electrically controllable behaviors in defective phononic crystals with inductive- resistive circuits, International Journal of Mechanical Sciences 278 (2024) 109485. doi: 10.1016/j.ijmecsci.2024.109485

    work page doi:10.1016/j.ijmecsci.2024.109485 2024

  17. [17]

    G. Lee, J. Park, W. Choi, B. Ji, M. Kim, J. Rho, Multiband elastic wave energy localization for highly amplified piezoelectric energy harvesting using trampoline metamaterials, Mechanical Systems and Signal Processing 200 (2023) 110593. doi:10.1016/j.ymssp.2023.110593. 28

    work page doi:10.1016/j.ymssp.2023.110593 2023

  18. [18]

    M. P. Bendsøe, O. Sigmund, Topology Optimization: Theory, Methods, and Applications, Springer, Berlin ; New York, 2003

    2003

  19. [19]

    S. J. Van Den Boom, R. Abedi, F. Van Keulen, A. M. Aragón, A level set-based interface- enriched topology optimization for the design of phononic crystals with smooth boundaries, Computer Methods in Applied Mechanics and Engineering 408 (2023) 115888. doi:10.1016/ j.cma.2023.115888

    work page arXiv 2023

  20. [20]

    Huang, Y

    X. Huang, Y. M. Xie, EVOLUTIONARY TOPOLOGY OPTIMIZATION OF CONTINUUM STRUCTURES : METHODS AND APPLICATIONS

  21. [21]

    Y. F. Li, X. Huang, F. Meng, S. Zhou, Evolutionary topological design for phononic band gap crystals, Structural and Multidisciplinary Optimization 54 (3) (2016) 595–617. doi:10.1007/ s00158-016-1424-3

    2016

  22. [22]

    Sigmund, J

    O. Sigmund, J. S. Jensen, Systematic Design of Phononic Band-Gap Materials and Structures by Topology Optimization, Philosophical Transactions: Mathematical, Physical and Engineer- ing Sciences 361 (1806,) (2003) 1001–1019. arXiv:3559206

    2003

  23. [23]

    Y. Lu, Y. Yang, J. K. Guest, A. Srivastava, 3-D phononic crystals with ultra-wide band gaps, Scientific Reports 7 (1) (2017) 43407. doi:10.1038/srep43407

    work page doi:10.1038/srep43407 2017

  24. [24]

    W. Li, F. Meng, Y. F. Li, X. Huang, Topological design of 3D phononic crystals for ultra- wide omnidirectional bandgaps, Structural and Multidisciplinary Optimization 60 (6) (2019) 2405–2415. doi:10.1007/s00158-019-02329-0

    work page doi:10.1007/s00158-019-02329-0 2019

  25. [25]

    Q. Wu, J. He, W. Chen, Q. Li, S. Liu, Topology optimization of phononic crystal with pre- scribed band gaps, Computer Methods in Applied Mechanics and Engineering 412 (2023) 116071. doi:10.1016/j.cma.2023.116071

    work page doi:10.1016/j.cma.2023.116071 2023

  26. [26]

    Q. Wu, S. Liu, Q. Li, Y. He, Targeted-frequency bandgap maximization for phononic crystals via topology optimization, International Journal of Mechanical Sciences 302 (2025) 110553. doi:10.1016/j.ijmecsci.2025.110553

    work page doi:10.1016/j.ijmecsci.2025.110553 2025

  27. [27]

    Dong, X.-X

    H.-W. Dong, X.-X. Su, Y.-S. Wang, Multi-objective optimization of two-dimensional porous phononic crystals, Journal of Physics D: Applied Physics 47 (15) (2014) 155301. doi:10.1088/ 0022-3727/47/15/155301

    2014

  28. [28]

    H. Chen, Y. Fu, Y. Hu, L. Ling, A novel single variable based topology optimization method for band gaps of multi-material phononic crystals, Structural and Multidisciplinary Optimization 65 (9) (2022) 253. doi:10.1007/s00158-022-03355-1

    work page doi:10.1007/s00158-022-03355-1 2022

  29. [29]

    Liang, J

    X. Liang, J. Du, Design of phononic-like structures and band gap tuning by concurrent two- scale topology optimization, Structural and Multidisciplinary Optimization 61 (3) (2020) 943–

    2020

  30. [30]

    doi:10.1007/s00158-020-02489-4

    work page doi:10.1007/s00158-020-02489-4

  31. [31]

    J. Gao, Z. Luo, H. Li, P. Li, L. Gao, Dynamic multiscale topology optimization for multi- regional micro-structured cellular composites, Composite Structures 211 (2019) 401–417. doi: 10.1016/j.compstruct.2018.12.031

    work page doi:10.1016/j.compstruct.2018.12.031 2019

  32. [32]

    H. Chen, Y. Fu, L. Ling, Y. Hu, L. Li, Multiscale topology optimization for designing lo- cally resonant acoustic metamaterials with specified low-frequency bandgaps, Materials Today Communications 41 (2024) 110900. doi:10.1016/j.mtcomm.2024.110900. 29

    work page doi:10.1016/j.mtcomm.2024.110900 2024

  33. [33]

    Liu, G.-L

    C.-X. Liu, G.-L. Yu, Deep learning for the design of phononic crystals and elastic metamaterials, Journal of Computational Design and Engineering 10 (2) (2023) 602–614. doi:10.1093/jcde/ qwad013

    work page doi:10.1093/jcde/ 2023

  34. [34]

    D. Lee, B. D. Youn, S.-H. Jo, Deep-learning-based framework for inverse design of a defective phononic crystal for narrowband filtering, International Journal of Mechanical Sciences 255 (2023) 108474. doi:10.1016/j.ijmecsci.2023.108474

    work page doi:10.1016/j.ijmecsci.2023.108474 2023

  35. [35]

    W. W. Chen, R. Sun, D. Lee, C. M. Portela, W. Chen, Generative Inverse Design of Meta- materials with Functional Responses by Interpretable Learning, Advanced Intelligent Systems 7 (6) (2025) 2400611. doi:10.1002/aisy.202400611

    work page doi:10.1002/aisy.202400611 2025

  36. [36]

    Jensen, O

    J. Jensen, O. Sigmund, Topology optimization for nano-photonics, Laser & Photonics Reviews 5 (2) (2011) 308–321. doi:10.1002/lpor.201000014

    work page doi:10.1002/lpor.201000014 2011

  37. [37]

    Reyes, D

    D. Reyes, D. Martínez, M. Mayorga, H. Heo, E. Walker, A. Neogi, Optimization of the Spatial Configuration of Local Defects in Phononic Crystals for High Q Cavity, Frontiers in Mechanical Engineering 6 (2020) 592787. doi:10.3389/fmech.2020.592787

    work page doi:10.3389/fmech.2020.592787 2020

  38. [38]

    T. S. Kim, Y. Y. Kim, Mac-based mode-tracking in structural topology optimization, Com- puters & Structures 74 (3) (2000) 375–383. doi:10.1016/S0045-7949(99)00056-5

    work page doi:10.1016/s0045-7949(99)00056-5 2000

  39. [39]

    T. D. Tsai, C. C. Cheng, Structural design for desired eigenfrequencies and mode shapes using topology optimization, Structural and Multidisciplinary Optimization 47 (5) (2013) 673–686. doi:10.1007/s00158-012-0840-2

    work page doi:10.1007/s00158-012-0840-2 2013

  40. [40]

    J. S. Jensen, O. Sigmund, Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends, Applied Physics Letters 84 (12) (2004) 2022–2024. doi:10.1063/1.1688450

    work page doi:10.1063/1.1688450 2004

  41. [41]

    Pennec, J

    Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, P. A. Deymier, Two-dimensional phononic crystals: Examples and applications, Surface Science Reports 65 (8) (2010) 229–291. doi:10.1016/j.surfrep.2010.08.002

    work page doi:10.1016/j.surfrep.2010.08.002 2010

  42. [42]

    G. Yi, B. D. Youn, A comprehensive survey on topology optimization of phononic crys- tals, Structural and Multidisciplinary Optimization 54 (5) (2016) 1315–1344. doi:10.1007/ s00158-016-1520-4

    2016

  43. [43]

    H.-W. Dong, C. Shen, Z. Liu, S.-D. Zhao, Z. Ren, C.-X. Liu, X. He, S. A. Cummer, Y.-S. Wang, D. Fang, L. Cheng, Inverse design of phononic meta-structured materials, Materials Today 80 (2024) 824–855. doi:10.1016/j.mattod.2024.09.012

    work page doi:10.1016/j.mattod.2024.09.012 2024

  44. [44]

    The impact of individual head-related transfer function augmen- tation on spatial release from masking,

    H.-W. Dong, Y.-S. Wang, C. Zhang, Inverse design of high- Q wave filters in two-dimensional phononic crystals by topology optimization, Ultrasonics 76 (2017) 109–124. doi:10.1016/j. ultras.2016.12.018

    work page doi:10.1016/j 2017

  45. [45]

    P. Min, Z. Song, L. Yang, V. G. Ralchenko, J. Zhu, Optically Transparent Flexible Broadband Metamaterial Absorber Based on Topology Optimization Design, Micromachines 12 (11) (2021)

    2021

  46. [46]

    doi:10.3390/mi12111419

    work page doi:10.3390/mi12111419

  47. [47]

    Zhang, Y

    X. Zhang, Y. Li, Y. Wang, Z. Jia, Y. Luo, Narrow-band filter design of phononic crystals with periodic point defects via topology optimization, International Journal of Mechanical Sciences 212 (2021) 106829. doi:10.1016/j.ijmecsci.2021.106829. 30

    work page doi:10.1016/j.ijmecsci.2021.106829 2021

  48. [48]

    Ma, X.-L

    T.-X. Ma, X.-L. Tang, D. Li, Y.-S. Wang, Topological interface states in solid/liquid phononic crystal waveguides and sensing applications, Physics Letters A 530 (2025) 130109. doi:10. 1016/j.physleta.2024.130109

    work page arXiv 2025

  49. [49]

    C. Lan, G. Hu, L. Tang, Y. Yang, Energy localization and topological protection of a locally resonant topological metamaterial for robust vibration energy harvesting, Journal of Applied Physics 129 (18) (2021) 184502. doi:10.1063/5.0047965

    work page doi:10.1063/5.0047965 2021

  50. [50]

    T. J. Clark, S. Bernard, J. Ma, V. Dumont, J. C. Sankey, Optically Defined Phononic Crystal Defect, Physical Review Letters 133 (22) (2024) 226904. doi:10.1103/PhysRevLett.133. 226904

    work page doi:10.1103/physrevlett.133 2024

  51. [51]

    Brillouin, Wave propagation in periodic structures (1946)

    L. Brillouin, Wave propagation in periodic structures (1946)

    1946

  52. [52]

    M. I. Hussein, M. J. Leamy, M. Ruzzene, Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook, Applied Mechanics Reviews 66 (4) (2014) 040802. doi:10.1115/1.4026911

    work page doi:10.1115/1.4026911 2014

  53. [53]

    X. Xu, K. Nishiguchi, J. Kato, A two-stage topology optimization framework for designing multi-scale elastic metamaterials with cross-scale dual bandgaps, SSRN preprint, under review. A vailable at SSRN: https://ssrn.com/abstract=6625038 (Mar. 2026). doi:10.2139/ssrn. 6625038

    work page doi:10.2139/ssrn 2026

  54. [54]

    A. P. Seyranian, E. Lund, N. Olhoff, Multiple eigenvalues in structural optimization problems, Structural Optimization 8 (4) (1994) 207–227. doi:10.1007/BF01742705. 31

    work page doi:10.1007/bf01742705 1994