Pith. sign in

REVIEW 3 major objections 4 minor 40 references

Hybrid TPMS-based Designs for Anisotropic Acoustic Metamaterials: Numerical Simulation and Topology Optimization

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that hybridizing two sheet-based TPMS unit cells of unequal size produces acoustic metamaterials whose bandgaps and passbands differ sharply between the two periodic directions, with optimized designs reaching up to…

desk verdict Competent numerical screening study for hybrid TPMS acoustic metamaterials; the main risk is that passband-rich y-direction coverage may be inflated by non-transmitting flat modes, and the paper's own APR check does not fully close that gap. read the letter →

arxiv 2506.04378 v1 pith:MOUXPVIQ submitted 2025-06-04 physics.app-ph

Mohamed Shendy , Nima Maftoon , Armaghan Salehian This is my paper
classification physics.app-ph
keywords anisotropicacousticmetamaterialshybridTPMSunitcellstriplyperiodicminimalsurfacesbandgapandpassbandoptimizationexhaustivesearchdirectionalwaveguidingfiniteelementBlochanalysisaudiblefrequencyrange
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that acoustic metamaterials can have genuinely different band behavior along two perpendicular periodic directions if the unit cell is made by hybridizing two sheet-based triply periodic minimal surface (TPMS) shapes of different sizes. The key move is the $C_2/C_1<1$ scenario: the smaller second subunit cells are blocked along the $y$-periodicity, so the structure becomes bandgap-rich along $x$ and passband-rich along $y$. An exhaustive search over 978 normalized designs, expanded into 392,178 actual designs by a frequency-scaling rule, selects optimal cells for four audible frequency ranges; the best low-frequency design covers 62.90% of 20 Hz to 5 kHz and the best high-frequency design covers 93.53% of 15 kHz to 20 kHz. If the central claim is right, directional waveguiding can be achieved in a single periodic structure without separate demultiplexing channels.

What carries the argument

The carrying object is the hybrid unit cell: two sheet-based TPMS subunit cells, each described by a relative density and a size, assembled with design variables $X=[\rho_1^*, \rho_2^*, C_2/C_1, C_1]^T$. TPMS here means a minimal surface that repeats periodically in all three directions, used as a solid sheet that shapes the air cavity. The mechanism that generates anisotropy is the $C_2/C_1<1$ arrangement: the second subunit cells are not connected along $y$, so their $y$-faces act as rigid barriers and the only $y$-pathway runs through the first subunit cells; this yields bandgaps along $x$ and passbands along $y$. Band behavior is computed with a finite-element Bloch-theorem eigenvalue problem under periodic boundary conditions, stored in normalized form $n=fC_1/c$, and expanded to actual designs with $f=nc/C_1$; the search then maximizes the objective $\min[BC_x, PC_y]$, the smaller of the $x$-bandgap and $y$-passband coverages of the target range.

What would settle it

Fabricate the optimal Nevious-Nevious unit cell ($C_2/C_1=0.7$, $C_1=50$ mm, densities 10% and 30%) and measure its transmission spectrum along both axes in an impedance tube over 20 Hz to 5 kHz. If the measured band edges along $x$ deviate noticeably from the scaled predictions, or if the $y$-direction does not show the predicted passband, then the uniform-scaling assumption that generates all 392,178 designs is wrong.

Watch

Extended reading notes

Core claim

The central claim is that hybridizing two sheet-based TPMS subunit cells with different relative densities and sizes produces anisotropic acoustic metamaterials whose bandgaps and passbands differ sharply between the two periodic axes. The paper reports that when $C_2/C_1<1$, only the first subunit cell connects to neighboring unit cells along the $y$-direction while the second subunit cell's $y$-faces are sealed by solid barriers, giving the metamaterial bandgap-rich transmission along $x$ and passband-rich transmission along $y$. The optimized designs for the four considered ranges are a Nevious-Nevious cell ($C_2/C_1=0.7$, $\rho_1^*=10\%$, $\rho_2^*=30\%$, $C_1=50$ mm, objective 62.90%) for 20 Hz-5 kHz, a Gyroid-Gyroid cell (76.71%) for 5-10 kHz, and Diamond-Diamond cells (83.35% and 93.53%) for 10-15 kHz and 15-20 kHz, where the objective is the minimum of the bandgap coverage along $x$ and the passband coverage along $y$. Acoustic pressure responses of a 3-by-3 repeat of the optimal Nevious-Nevious design show attenuation bands that match the predicted bandgap structure for each actuation direction.

Load-bearing premise

The results rest on a scaling premise: each normalized band structure transfers to any physical size through $f=nc/C_1$, so 392,178 actual designs can be ranked without re-solving the wave equation, and the only size filter is that air cavities stay 20 times thicker than the entropy boundary layer. If viscous or thermal losses, thin-wall effects, or deviations in the curve-fitted relative densities break that simple scaling at small $C_1$ values or at hybrid junctions, the identified optimal designs would not be the true optima.

Editorial extensions

If this is right

  • Hybrid unit cells with $C_2/C_1<1$ give a single-structure route to directional waveguiding: sound is filtered along one periodic axis and transmitted along the other, with no demultiplexing channels required.
  • The exhaustive search identifies optimized designs for each of the four audible sub-ranges, with the minimum of $x$-bandgap and $y$-passband coverage rising from 62.90% (20 Hz-5 kHz) to 93.53% (15-20 kHz).
  • Acoustic pressure-response simulations of a 3-by-3 metamaterial made from the optimal Nevious-Nevious cell confirm that attenuation regions align with the predicted bandgaps for actuation along $x$ and along $y$.
  • Because bandgap data are stored in normalized form and converted by a one-line scaling relation, the same 978 normalized designs cover every cell-size choice without additional finite-element solves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The normalized bandgap database could be reused for other $C_1$ sets or other objectives (for example, requiring bandgaps in both directions at different frequencies), making follow-up optimizations nearly free once the database exists.
  • Because the directionality is embedded in the unit cell, rotating or stacking optimal cells could produce more complex routing networks, such as turning a wave from $x$ to $y$, without external channel structures.
  • A direct experimental check should also probe the scaling assumption: measuring the optimal Nevious-Nevious cell in an impedance tube would reveal whether viscous and thermal boundary-layer losses shift the band edges in the narrow cavities, which the paper does not test.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes hybrid TPMS-based acoustic metamaterial unit cells, each formed by two subunit cells of the same TPMS family with different relative densities and size ratio. For the case C2/C1 < 1, the second subunit cells are isolated along the y-periodicity by rigid barriers, which the authors argue produces bandgap-rich behavior along the x-periodicity and passband-rich behavior along the y-periodicity. Bandgaps and passbands are computed with an FE-Bloch method (Eqs. 2-4), stored in normalized form via f = n c / C1 (Eq. 5), and converted to 392,178 actual designs by sweeping 401 values of C1. An exhaustive search maximizes the objective function O.F. = min[B.Cx, P.Cy] (Eq. 6) for four audible frequency ranges. Optimal designs are reported for Nevious-Nevious, Gyroid-Gyroid, and Diamond-Diamond families, with objective-function values from 62.90% to 93.53%. A finite-structure acoustic pressure response (APR) simulation of a 3x3 array of the low-frequency optimal Nevious-Nevious design is used to cross-check the predicted bandgaps under two actuation scenarios.

Significance. If the reported coverage values are meaningful, the paper offers a computationally efficient way to design anisotropic acoustic metamaterials with strongly direction-dependent bandgaps and passbands, which is a useful step toward directional waveguiding devices. The exhaustive-search methodology over nearly four hundred thousand designs is a strength, as is the finite-structure APR check that at least confirms attenuation in predicted bandgap regions. The main caveat is that the central claim of 'passband-rich' behavior along y rests on a spectral classification that may count non-transmissive localized modes as passbands; the APR verification does not currently demonstrate energy transmission in the passbands. This needs to be addressed before the quantitative coverage claims can be accepted.

major comments (3)
  1. [2.2, 2.3, Eq. (6), Table 2] The objective function and the reported coverage values count any frequency covered by a Bloch eigenfrequency along Gamma-Y as a passband. For the C2/C1 < 1 scenario, Section 2.2 states that the second subunit cells are isolated along y by rigid barriers (Figures 3-B and 4). The eigenmodes of these isolated cavities are standing-wave modes with near-zero group velocity; they appear as flat bands in the Gamma-Y band structure but do not transmit acoustic energy. Including them in P.Cy inflates the objective function, and the flagship low-frequency Nevious-Nevious optimal design (C2/C1 = 0.7, O.F. = 62.90%) may be optimal only under this classification. The paper needs to demonstrate that the frequencies counted as passbands along y actually support energy transport, for example by computing group velocities and excluding flat bands, or by direct transmission simulation through a finite array.
  2. [3.2, Figure 14] The APR verification in Section 3.2 only checks that attenuation regions in the finite-structure response coincide with the predicted bandgaps; it does not verify that the reported passband frequencies transmit along the y-periodicity. In particular, for Actuation Scenario II, the response at sensing location Y could be dominated by cavity resonances rather than by a propagating wave through the first-subunit pathway. The authors should quantify transmission in passband regions, e.g., by computing a transmission coefficient or by comparing the transmitted SPL with a reference empty waveguide, for both actuation scenarios. Without this, the claim that the hybrid metamaterial 'directs waves along both periodicities' is not fully supported.
  3. [2.4, Eq. (5)] The conversion of all 978 normalized designs into 392,178 actual designs assumes that the eigenfrequency scaling f = n c / C1 is exact for every combination of rho1*, rho2*, and C2/C1 across the full C1 range of 10-50 mm. This is valid only if the generated air geometry scales uniformly with C1. Since the unit cell geometry is built from curve-fitted relative-density functions (Section 2.1), it is not obvious that the geometry, wall thicknesses, and subunit-cell connection faces scale exactly. The 20x boundary-layer filter is a necessary but not sufficient check. The authors should spot-check a few normalized designs by running direct FE-Bloch simulations at multiple C1 values and comparing the resulting bandgap edges with the scaled predictions, especially for small C1 where thin-wall and meshing issues are most likely.
minor comments (4)
  1. [2.3, Eqs. (3.1)-(3.2)] The Bloch phase factors are written with opposite signs in Eqs. (3.1) and (3.2). This may be a convention choice, but it should be justified or made consistent to avoid confusion.
  2. [2.2, Figure 3] The term 'semi-periodic pattern' for the x-axis is not defined; the description of the first and second design scenarios would be clearer if the periodicity along each axis were stated explicitly for both C2/C1 = 1 and C2/C1 < 1.
  3. [3.1, Table 2 and Figure 12] The text says the anisotropic effect becomes more pronounced at higher frequency ranges and cites the Diamond-Diamond design with C2/C1 = 1 as the best example; however, for C2/C1 = 1 the two subunit cells are connected along y, which is the less anisotropic scenario. The discussion should clarify why this design is nonetheless classified as anisotropic.
  4. [Data Availability] The data availability statement says data are available from the corresponding author upon reasonable request. Given that the paper reports a search over 392,178 designs, providing the design-generation scripts or at least the bandgap data for all normalized designs would substantially improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: exhaustive search over FE-Bloch spectra with exact uniform-scaling conversion; APR is an independent cross-check.

full rationale

The paper's derivation chain is a direct exhaustive search: normalized eigenfrequencies are computed by FE-Bloch for 978 geometries; actual frequencies are obtained by the uniform-scaling relation f = n c / C1 (Eq. 5), which is an exact identity for the lossless Helmholtz eigenproblem under proportional scaling, not a fitted constant. The objective function (Eq. 6) is the optimization target itself, not a prediction derived from inputs. The APR finite-structure simulation (Section 3.2) is an independent cross-check that does not feed back into the search. Self-citations to [21,22,38] provide curve-fitted relative-density functions and the boundary-layer filter; these are geometric/physical inputs from prior work, not the bandgap/passband result, and are not used to define the target quantities. The definition of passbands as eigenfrequency coverage along the Gamma-Y path is a modeling convention; any concern that isolated-cavity modes inflate passband counts is a physical validity limitation, not a circular derivation. No step reduces by construction to its own inputs, and no fitted parameter is renamed as a prediction. Therefore, no circularity is identified.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

Most of the scientific content rests on standard periodic-media theory plus three domain assumptions specific to this design study: impedance-mismatch air-only acoustics, uniform scaling with C1, and the validity of curve-fitted relative-density relations. No new physical entities are introduced; the hybrid cells are geometric designs. The exhaustive search uses no fitted constants except inherited curve-fit coefficients and the 20x boundary-layer threshold.

free parameters (2)
  • TPMS relative-density curve-fit coefficients = not reported (inherited from Refs [22,38])
    rho1* and rho2* are mapped to actual TPMS geometry through curve-fitted functions from prior work; errors in these coefficients affect every bandgap computation.
  • Cavity-to-entropy-boundary-layer ratio threshold = 20
    Designs are kept only when cavities are at least 20 times the entropy boundary layer thickness; this hand-chosen threshold, adopted from Ref [22], controls which C1 values survive the search.
assumptions (5)
  • domain assumption The impedance mismatch between solid TPMS walls and air is large enough that sound propagates only in the air domain, with rigid boundary conditions at solid-air interfaces.
    Invoked in Section 2.3 to justify modeling only the air unit cell; this neglects solid-wall vibration and viscothermal losses.
  • standard math Bloch periodic boundary conditions on a single unit cell represent an infinite periodic metamaterial.
    Equations 3.1 and 3.2 apply periodic boundary conditions; the paper then retains only the Gamma-X and Gamma-Y paths of the first Brillouin zone for directional bandgaps.
  • domain assumption Normalized eigenfrequencies scale exactly as f = n c / C1 when the unit cell is uniformly scaled.
    Section 2.4 converts all 978 normalized designs into 392,178 actual designs using Equation 5; no correction is made for finite-size, boundary-layer, or thin-wall effects at small C1.
  • domain assumption Curve-fitted relative-density relations from Refs [22,38] remain accurate for the hybrid two-TPMS geometry.
    Section 2.1 defines rho1* and rho2* through these fits; the paper does not directly measure or verify the resulting hybrid geometry's volume fraction.
  • domain assumption Bandgaps along the Gamma-X and Gamma-Y paths are sufficient to characterize directional waveguiding behavior.
    Section 2.3 limits the Brillouin-zone sampling to these two paths; oblique propagation and finite-array edge effects are not part of the optimization objective.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hybrid TPMS-based Designs for Anisotropic Acoustic Metamaterials: Numerical Simulation and Topology Optimization." pith.science (2026). https://pith.science/paper/MOUXPVIQ

@misc{pith2026250604378,
  author       = {Pith},
  title        = {Pith review of: Hybrid TPMS-based Designs for Anisotropic Acoustic Metamaterials: Numerical Simulation and Topology Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MOUXPVIQ}},
  note         = {Machine review of arXiv:2506.04378}
}
read the original abstract

Anisotropic acoustic metamaterials have received significant scholarly attention in recent years due to their capacity to manipulate wave propagation across various directions. This property is integral to applications involving directional wave guiding. Nonetheless, previously proposed anisotropic acoustic metamaterials exhibited commonalities in bandgaps along x and y periodicities, limiting their efficacy for such applications. Therefore, this work introduces hybrid Triply Periodic Minimal Surfaces (TPMSs)-based anisotropic acoustic metamaterials that manifest abundant bandgap and passband characteristics along the x and y periodicities. Four design families were considered: Primitive-Primitive sheet-based, Gyroid-Gyroid sheet-based, Diamond-Diamond sheet-based, and Nevious-Nevious sheet-based. A computationally efficient exhaustive search was employed to identify optimal hybrid metamaterial designs for various desired audible frequency ranges (i.e., ranging from 20 Hz to 20 kHz). In total, 392,178 designs were assessed in this search. The optimal designs demonstrated pronounced bandgap and passband characteristics across both periodicities, thereby positioning them as promising candidates for directional wave guiding applications. For instance, the optimal Nevious-Nevious sheet-based unit cell design achieved a minimum coverage of 62.90% for bandgaps and passbands within the frequency range of 20 Hz to 5 kHz. Following the identification of optimal unit cell designs, the acoustic pressure responses of hybrid anisotropic acoustic metamaterials, constructed from repeated optimal unit cells, were computed. This involved considering two actuation scenarios, which pertain to exciting the system along both x and y periodicities, revealing responses in alignment with optimized band characteristics.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

40 extracted references · 37 canonical work pages

  1. [1]

    Metamaterial-based real-time communication with high information density by multipath twisting of acoustic wave

    Wu K, Liu JJ, Ding Y jiang, et al. Metamaterial-based real-time communication with high information density by multipath twisting of acoustic wave. Nat Commun 2022; 13. doi:10.1038/s41467-022-32778-z

    work page doi:10.1038/s41467-022-32778-z 2022

  2. [2]

    Exploring the fascinating world of combinational photonic devices: A review

    Noonepalle H, Palacharla V , Dubey SK, et al. Exploring the fascinating world of combinational photonic devices: A review. Optik (Stuttg) 2024; 311. doi:10.1016/j.ijleo.2024.171923

    work page arXiv 2024

  3. [3]

    preprint_Signal_processing

    Ammari H, Davies B. preprint_Signal_processing. ArXiv 2022

    work page 2022

  4. [4]

    Architected acoustic metamaterials: An integrated design perspective

    Comandini G, Ouisse M, Ting VP, et al. Architected acoustic metamaterials: An integrated design perspective. Appl Phys Rev 2025; 12

    work page 2025

  5. [5]

    Acoustic Metamaterials for Noise Reduction: A Review

    Gao N, Zhang Z, Deng J, et al. Acoustic Metamaterials for Noise Reduction: A Review. Adv Mater Technol 2022; 7

    work page 2022

  6. [6]

    Emerging topics in nanophononics and elastic, acoustic, and mechanical metamaterials: An overview

    Krushynska AO, Torrent D, Aragón AM, et al. Emerging topics in nanophononics and elastic, acoustic, and mechanical metamaterials: An overview. Nanophotonics 2023; 12: 659–686

    work page 2023

  7. [7]

    Adaptive noise cancellation with directional microphones

    Elko GW. Adaptive noise cancellation with directional microphones. J Acoust Soc Am 1996; 100: 2696. doi:10.1121/1.417064

    work page doi:10.1121/1.417064 1996

  8. [8]

    Solid State Lasers XXIX : Technology and Devices

    Clarkson WA, Shori RK. Solid State Lasers XXIX : Technology and Devices. Bellingham, Washington: SPIE; 2020

    work page 2020

Show all 40 references
  1. [9]

    Architected Lattices for Simultaneous Broadband Attenuation of Airborne Sound and Mechanical Vibrations in All Directions

    Bilal OR, Ballagi D, Daraio C. Architected Lattices for Simultaneous Broadband Attenuation of Airborne Sound and Mechanical Vibrations in All Directions. Phys Rev Appl 2018; 10. doi:10.1103/PhysRevApplied.10.054060

    2018 doi

  2. [10]

    Metamaterials for simultaneous acoustic and elastic bandgaps

    Elmadih W, Chronopoulos D, Zhu J. Metamaterials for simultaneous acoustic and elastic bandgaps. Sci Rep 2021; 11. doi:10.1038/s41598-021-94053-3

    2021 doi

  3. [11]

    Lightweight architected hollow sphere foams for simultaneous noise and vibration control

    Jiang H, Chen Y . Lightweight architected hollow sphere foams for simultaneous noise and vibration control. J Phys D Appl Phys 2019; 52. doi:10.1088/1361-6463/ab22ee

    2019 doi

  4. [12]

    Acoustoelastic phononic metamaterial for isolation of sound and vibrations

    Aravantinos-Zafiris N, Kanistras N, Sigalas MM. Acoustoelastic phononic metamaterial for isolation of sound and vibrations. J Appl Phys 2021; 129. doi:10.1063/5.0041970

    2021 doi

  5. [13]

    Local resonance – Helmholtz lattices with simultaneous solid-borne elastic waves and air-borne sound waves attenuation performance

    Li G, Chen Y , Chen W, et al. Local resonance – Helmholtz lattices with simultaneous solid-borne elastic waves and air-borne sound waves attenuation performance. Applied Acoustics 2022; 186. doi:10.1016/j.apacoust.2021.108450

    2022

  6. [14]

    Acoustic Metamaterial with Fractal Coiling Up Space for Sound Blocking in a Deep Subwavelength Scale

    Xia B, Li L, Liu J, et al. Acoustic Metamaterial with Fractal Coiling Up Space for Sound Blocking in a Deep Subwavelength Scale. J Vib Acoust 2018; 140. doi:10.1115/1.4037514 25

    2018 doi

  7. [15]

    Tuning of acoustic bandgaps in phononic crystals with helmholtz resonators

    Li JB, Wang YS, Zhang C. Tuning of acoustic bandgaps in phononic crystals with helmholtz resonators. J Vib Acoust 2013; 135. doi:10.1115/1.4023812

    2013 doi

  8. [16]

    Acoustic band gaps of three-dimensional periodic polymer cellular solids with cubic symmetry

    Chen Y , Yao H, Wang L. Acoustic band gaps of three-dimensional periodic polymer cellular solids with cubic symmetry. J Appl Phys 2013; 114. doi:10.1063/1.4817168

    2013 doi

  9. [17]

    Acoustic band gaps and elastic stiffness of PMMA cellular solids based on triply periodic minimal surfaces

    Abueidda DW, Jasiuk I, Sobh NA. Acoustic band gaps and elastic stiffness of PMMA cellular solids based on triply periodic minimal surfaces. Mater Des 2018; 145: 20–27. doi:10.1016/j.matdes.2018.02.032

    2018 doi

  10. [18]

    Insights into acoustic properties of seven selected triply periodic minimal surfaces-based structures: A numerical study

    Lu JY , Silva T, Alzaabi F, et al. Insights into acoustic properties of seven selected triply periodic minimal surfaces-based structures: A numerical study. Journal of Low Frequency Noise Vibration and Active Control 2024; 43: 208–219. doi:10.1177/14613484231190986

    2024 doi

  11. [19]

    Gyroid-based acoustic materials and devices

    Guo Y , Ruzzene M, Rosa M. Gyroid-based acoustic materials and devices. 2024

    2024

  12. [20]

    Topological Surface States in a Gyroid Acoustic Crystal

    Guo Y , Rosa MIN, Ruzzene M. Topological Surface States in a Gyroid Acoustic Crystal. Advanced Science 2023; 10. doi:10.1002/advs.202205723

    2023 doi

  13. [21]

    An Extensive Parametric Analysis and Optimization to Design Unidimensional Periodic Acoustic Metamaterials for Noise Attenuation

    Shendy M, Oluyemi M, Maftoon N, et al. An Extensive Parametric Analysis and Optimization to Design Unidimensional Periodic Acoustic Metamaterials for Noise Attenuation. Applied Sciences (Switzerland) 2024; 14. doi:10.3390/app14167272

    2024 doi

  14. [22]

    TOPOLOGY OPTIMIZATION OF ONE- DIMENSIONAL PERIODIC ACOUSTIC METAMATERIAL FOR AIRBORNE NOISE ATTENUATION

    Shendy M, Maftoon N, Salehian A. TOPOLOGY OPTIMIZATION OF ONE- DIMENSIONAL PERIODIC ACOUSTIC METAMATERIAL FOR AIRBORNE NOISE ATTENUATION. In: Proceedings of the ASME 2025 44 International Conference on Ocean, Offshore and Arctic Engineering. 2025

    2025

  15. [23]

    Induced transparency based subwavelength acoustic demultiplexers

    Gu T, Cheng Y , Wen Z, et al. Induced transparency based subwavelength acoustic demultiplexers. J Phys D Appl Phys 2021; 54. doi:10.1088/1361-6463/abe07c

    2021 doi

  16. [24]

    Multi-functional resonant acoustic wave router

    Richoux O, Achilleos V , Theocharis G, et al. Multi-functional resonant acoustic wave router. J Phys D Appl Phys 2020; 53. doi:10.1088/1361-6463/ab78d9

    2020 doi

  17. [25]

    Phononic crystal-based acoustic demultiplexer design via bandgap-passband topology optimization

    Bao Y , Jia Z, Tian Q, et al. Phononic crystal-based acoustic demultiplexer design via bandgap-passband topology optimization. Compos Struct 2025; 351. doi:10.1016/j.compstruct.2024.118622

    2025

  18. [26]

    Wave propagation in periodic structures : electric filters and crystal lattices

    Brillouin L. Wave propagation in periodic structures : electric filters and crystal lattices. 1953

    1953

  19. [27]

    Acoustic frequency filter based on anisotropic topological phononic crystals

    Chen ZG, Zhao J, Mei J, et al. Acoustic frequency filter based on anisotropic topological phononic crystals. Sci Rep 2017; 7. doi:10.1038/s41598-017-15409-2

    2017 doi

  20. [28]

    Tunable bandgaps and acoustic characteristics of perforated Miura-ori phononic structures

    Zhang X, Huang X, Lu G. Tunable bandgaps and acoustic characteristics of perforated Miura-ori phononic structures. Int J Mech Sci 2023; 253: 108389. doi:10.1016/j.ijmecsci.2023.108389 26

    2023

  21. [29]

    Tunable auxetic metamaterials for simultaneous attenuation of airborne sound and elastic vibrations in all directions

    Kheybari M, Daraio C, Bilal OR. Tunable auxetic metamaterials for simultaneous attenuation of airborne sound and elastic vibrations in all directions. Appl Phys Lett 2022

    2022

  22. [30]

    Tunable Asymmetric Transmissions via Anisotropic Acoustic Metamaterials

    Zhang Z, Bai L, Xu L, et al. Tunable Asymmetric Transmissions via Anisotropic Acoustic Metamaterials. Physica Status Solidi - Rapid Research Letters 2021; 15. doi:10.1002/pssr.202100163

    2021 doi

  23. [31]

    Anisotropic metamaterials for full control of acoustic waves

    Christensen J, García De Abajo FJ. Anisotropic metamaterials for full control of acoustic waves. Phys Rev Lett 2012; 108. doi:10.1103/PhysRevLett.108.124301

    2012 doi

  24. [32]

    Novel hybrid minimal surface-based lattice materials

    Ejeh CJ, Barsoum I, Abu Al-Rub RK. Novel hybrid minimal surface-based lattice materials. Mater Des 2025; 253. doi:10.1016/j.matdes.2025.113959

    2025

  25. [33]

    Protective performance of hybrid triply periodic minimal surface lattice structure

    Zhang Y , Chen Y , Li J, et al. Protective performance of hybrid triply periodic minimal surface lattice structure. Thin-Walled Structures 2024; 194. doi:10.1016/j.tws.2023.111288

    2024

  26. [34]

    A study of multi-stage energy absorption characteristics of hybrid sheet TPMS lattices

    Zhang J, Xie S, Li T, et al. A study of multi-stage energy absorption characteristics of hybrid sheet TPMS lattices. Thin-Walled Structures 2023; 190. doi:10.1016/j.tws.2023.110989

    2023

  27. [35]

    Mechanical and thermal property analysis and optimization design of hybrid lattice structure based on triply periodic minimal surfaces

    Ye H, Tian F, He W, et al. Mechanical and thermal property analysis and optimization design of hybrid lattice structure based on triply periodic minimal surfaces. Thin-Walled Structures 2024; 203. doi:10.1016/j.tws.2024.112203

    2024

  28. [36]

    Enhancing mechanical properties and damage tolerance of additive manufactured ceramic TPMS lattices by hybrid design

    Lu C, Ding J, Jiang X, et al. Enhancing mechanical properties and damage tolerance of additive manufactured ceramic TPMS lattices by hybrid design. Journal of the American Ceramic Society 2024; 107: 6524–6531. doi:10.1111/jace.19978

    2024 doi

  29. [37]

    On hybrid cellular materials based on triply periodic minimal surfaces with extreme mechanical properties

    Chen Z, Xie YM, Wu X, et al. On hybrid cellular materials based on triply periodic minimal surfaces with extreme mechanical properties. Mater Des 2019; 183. doi:10.1016/j.matdes.2019.108109

    2019

  30. [38]

    Topology Optimization Via Spatially-Varying TPMS

    Xu W, Zhang P, Yu M, et al. Topology Optimization Via Spatially-Varying TPMS. IEEE Trans Vis Comput Graph 2023; doi:10.1109/TVCG.2023.3268068

    2023

  31. [39]

    Acoustics

    Pierce AD. Acoustics. Third Edition. Springer; 2019

    2019

  32. [121]

    doi:10.1063/5.0104266

    doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.