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REVIEW 3 major objections 5 minor 1 cited by

Graded phononic metamaterials: Scalable design meets scalable microfabrication

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Graded phononic waveguides with hundreds of thousands of unit cells can now be designed and fabricated on a silicon wafer.

desk verdict A solid experimental demonstration of wave guiding in a microfabricated silicon phononic film, but the headline 'hundreds of thousands of unit cells' claim is not supported by the data in this preprint. read the letter →

arxiv 2507.01874 v1 pith:4OMKWFYL submitted 2025-07-02 physics.app-ph

Charles Dorn , Vignesh Kannan , Ute Drechsler , Dennis M. Kochmann This is my paper
classification physics.app-ph
keywords phononicmetamaterialsspatiallygradedraytracinginversedesignmicrofabricationsilicon-on-insulatorwaveguidingbroadband
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

The paper aims to remove the scalability bottleneck that has kept phononic metamaterials at toy scales: it pairs an inverse design method that handles hundreds of thousands of unit cells with a silicon-wafer fabrication route that can realize them as free-standing films. The design side uses ray tracing through locally periodic Bloch dispersion relations, so optimization cost scales with the number of rays rather than unit cells, and a modular tile library lets large waveguides be assembled from small designed building blocks. The fabrication side adapts photolithography and deep reactive ion etching of silicon-on-insulator wafers to produce free-standing architected films, demonstrated here with roughly 600,000 unit cells on a 100 mm wafer. The central experimental claim is that a designed figure-eight waveguide, targeted at 750 kHz, actually guides broadband elastic waves along the intended path in the fabricated sample, with measured displacement fields matching finite element simulations.

What carries the argument

Ray tracing in graded metamaterials, governed by the Hamiltonian system $\dot{x} = \partial \omega/\partial k$ and $\dot{k} = -\partial \omega/\partial x$, uses the local dispersion relation of the beam unit cell as a Hamiltonian to compute trajectories; an adjoint-state optimization shapes these rays to prescribed exit positions and wave vectors. The modular tile library then assembles individually designed tiles, with matching boundary values of $\theta$, into larger waveguides without recomputation.

What would settle it

Fabricate a graded silicon film with a sharp gradient across just a few unit cells, launch the same broadband laser pulse, and compare the measured displacement along a designed ray path with finite element simulation; a large deviation or loss of guiding would indicate where the local-periodicity assumption breaks down.

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Extended reading notes

Core claim

The paper demonstrates an end-to-end path from computer-aided design to physical sample for spatially graded phononic metamaterials at a scale that was previously impractical. Its central discovery is that a ray tracing model, built on locally computed Bloch dispersion surfaces for a beam-lattice unit cell parameterized by one angle $\theta$, is accurate and fast enough to serve as the forward model for inverse design of waveguides spanning tens of thousands of unit cells, and that the resulting designs can be fabricated by standard semiconductor processing into free-standing silicon films. Two tile types are designed—one converting a point excitation into four outgoing plane waves, the other rotating an incident plane wave by $90^\circ$—and assembled into figure-eight and cross waveguides. Transient finite element simulations reproduce the ray-predicted paths, and laser-excited interferometric measurements on the fabricated figure-eight wafer confirm wave guiding over a band from about 250 to 800 kHz, well beyond the 750 kHz design frequency.

Load-bearing premise

The ray tracing model is based on local Bloch analysis, so it assumes the spatial grading is smooth enough that neighboring unit cells are almost periodic; if the grading is too sharp, the predicted rays will not match the actual wave field.

Editorial extensions

If this is right

  • Waveguide designs spanning hundreds of thousands to millions of unit cells become computationally tractable, since the optimization cost scales with the number of rays rather than the number of cells.
  • Semiconductor wafer processing turns metamaterial fabrication into a batch, chip-style process, enabling many samples per wafer and direct integration with MEMS and on-chip signal-processing devices.
  • The demonstrated broadband guiding suggests that target-frequency designs may inherit a wide operational band, easing practical deployment.
  • Free-standing architected silicon films open a route to studying wave attenuation in the architecture alone, in the absence of intrinsic material damping and substrate losses.
  • The modular tile approach allows new functions to be added by designing new tiles, so functionality can grow incrementally without redesigning the whole device.

Reading between the lines

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

  • The same design pipeline could optimize higher dispersion branches and out-of-plane or in-plane targets, suggesting extensions to multi-mode or topology-switching waveguides.
  • The ray tracing forward model could be tested more aggressively at sharper-than-demonstrated gradings and at higher frequencies, where corners and amplitude effects are more pronounced.
  • Because the ray-optimized paths are computed at a single frequency, one could test whether multi-frequency or broadband-weighted cost functions produce even wider operational bands.
  • The combination of a general ray model with a standardized tile library points toward reusable 'metamaterial standard cells' analogous to logic gates in electronics, though the paper does not pursue this analogy.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript presents a framework for inverse design of graded phononic metamaterials based on ray tracing and modular tile assembly, combined with a silicon-on-insulator wafer microfabrication process to create free-standing architected films. The authors design two tile types by optimizing ray trajectories, assemble them into figure-eight (192×192 unit cells) and cross (256×256 unit cells) patterns, and validate the figure-eight design experimentally using a pump-probe laser/interferometer setup. Experimental line-scan and frequency-resolved data show good agreement with finite-element simulations over a broad band, and the authors claim that the design/fabrication framework enables waveguides with hundreds of thousands of unit cells and the potential to scale to millions.

Significance. If its claims hold, this work is a significant advance because it combines an efficient ray tracing inverse-design method with a wafer-scale microfabrication route for free-standing elastic metamaterials, a combination that is rare and of broad interest for MEMS-scale wave manipulation. The experimental validation is a genuine strength: the agreement between the measured line scans (Fig. 4) and the independent finite-element simulations, including the frequency-resolved comparisons in SI Sec. 5, provides strong evidence that the fabricated 36,864-cell waveguide behaves as designed, with no apparent parameter fitting in the experimental channel. The public code/data repository supports reproducibility. The central caveats are that the headline scale claim (hundreds of thousands of realized unit cells) is not supported by data in this manuscript, and the ray tracing smoothness assumption is not established for large tile assemblies.

major comments (3)
  1. [Abstract; Microfabrication (main text, p. 8); SI Sec. 1.4] The abstract states that the framework 'enables the design and realization of complex waveguides including hundreds of thousands of unit cells,' but the experimental realization in this manuscript is a 192×192 (36,864-cell) figure-eight (Fig. 2c, Fig. 4). The 256×256 cross (~66,000 cells) is validated only by finite-element simulation (SI Fig. S4), and the ~143,000-cell assembly in SI Fig. S2 is presented only as a θ distribution and ray paths, with no transient wavefield simulation. The ~600,000-cell wafer mentioned in the Microfabrication section is attributed to companion reference [32], which is not part of this submission and is not available for assessment. The 'realization' half of the headline claim is therefore not supported by evidence in this manuscript; the text should clearly separate what is demonstrated here (a ~37,000-cell prototype with experimental validation) from what is claimed via the companion paper or via ray-only designs.
  2. [Introduction (p. 3); SI Sec. 1.2 (Eqs. S1-S2); SI Sec. 1.4] The ray tracing model assumes 'smooth spatial gradings and hence locally an approximately periodic medium' (Introduction), and the ray equations (S1)-(S2) involve ∂ω/∂x = (∂ω/∂θ)(∂θ/∂x). The tile assembly enforces continuity of θ at tile boundaries (θ = 0.4L on each tile perimeter) but does not enforce continuity of ∇θ, so the normal derivative of θ can jump at tile boundaries. Such a jump produces a discontinuity in k̇, which is inconsistent with the smooth-grading assumption on which ray tracing is based. The authors do not quantify the gradient jumps at tile boundaries in the 3×3 and 4×4 assemblies, nor do they validate the ~143,000-cell assembly (SI Fig. S2) with a wavefield simulation. The good FE/experiment agreement for the 3×3 figure-eight mitigates the concern for that particular design, but it does not establish that much larger assemblies with many tile boundaries are accurately predicted by ray tracing. The authors should either enforce or verify C1 continuity of θ across tile boundaries, or validate a large assembly with transient FE.
  3. [SI Sec. 1.3] The description of the design-variable interpolation is incomplete. The text states that the design grid is refined from a spacing of 16L to L and that unit cell designs are 'interpolated' from the design grid, but it does not specify the interpolation scheme (e.g., piecewise linear, spline) or how the spatial derivative ∂θ/∂x is evaluated in the ray tracing system (S1)-(S2) and in the adjoint gradient (S11). If the final design is piecewise constant per unit cell, then ∂θ/∂x is a sum of delta functions and the ray tracing equations are not well-defined; if a smooth interpolation is used, the manuscript should state it. This detail is essential for reproducing the optimization and for assessing the smoothness assumption raised in the preceding comment.
minor comments (5)
  1. [Introduction, Conclusion] The text says the structures span 'three orders of magnitude in length scales'; the fabricated figure-eight spans 3 cm with L = 100 µm (ratio 300), while the 100-mm wafer with 100-µm cells (ratio 1000) comes from companion [32]. Please specify which structure is meant and avoid overstating the demonstrated range.
  2. [References] Reference [32] is listed as a companion manuscript without a preprint identifier or status note; since the 600,000-cell claim rests on it, readers need a link or a clear statement of its availability.
  3. [SI Sec. 1.3] The manuscript does not report the values of w1 and w2 used in Eq. (S3) for the two tile designs; please provide them for reproducibility.
  4. [Fig. 4b] The text says two measurement points have poor signal-to-noise ratio 'marked by the black arrow for line scan L2,' but the figure appears to show only one arrow; please clarify the notation.
  5. [SI Sec. 2.2] The transient FE simulation uses a 1 µs half-sine excitation and states that it 'spans approximately the same frequency range' as the experimental 1 ns laser pulse. Please quantify the spectral content of the experimental acoustic pulse or justify why the approximation is adequate for the comparisons in Fig. 4 and SI Sec. 5.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation is present: the experimental wave-guiding evidence is independent of the ray-tracing design objective, and the 600,000-cell realization claim rests on companion data that constitutes an evidence gap rather than a circular reduction.

full rationale

The paper's derivation chain is an inverse-design loop: beam-FE Bloch analysis produces local dispersion relations, the ray system (S1)-(S2) is used as the forward model, and the cost function (S3) is optimized to match prescribed ray exit states. The optimized ray paths therefore match the design objective by construction, but the paper's load-bearing claims are not those ray paths; they are the transient FE wavefields (Figs. S3-S4), the measured displacement line scans (Fig. 4), and the broadband spectral agreement (SI Sec. 5), none of which are inputs to the optimization. No parameter is fitted to the experimental data, and the experiment provides an independent check of both the beam-FE dispersion model and the ray-tracing approximation. The transient FE validation shares the beam-FE modeling framework used to build the dispersion table, so the numerical portion is an internal-consistency check rather than a fully independent model, but the experiment breaks the self-reference. Flag for verification rather than circularity: the abstract's 'hundreds of thousands' realization claim is not evidenced in this preprint, because the fabricated device is the 192x192 figure-eight, the 256x256 cross is simulation-only, SI Fig. S2 is a 143k ray/theta design without a transient wavefield, and the 600k wafer is attributed to companion [32]; additionally, the smooth-grading assumption in SI Sec. 1.2 is a stated modeling assumption, not a hidden fit. Hence no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a chain of modeling assumptions: local periodicity and smooth grading for ray tracing, out-of-plane mode decoupling, thin-beam finite element modeling, and interpolation of dispersion surfaces. No new physical entities are introduced. The most consequential free parameters are the design frequency, the boundary theta_0 value, and the unreported cost weights in the optimization objective.

free parameters (3)
  • Design target frequency omega_0 = 750 kHz
    Chosen for all ray-tracing optimization examples; not fitted to experimental data, but the central wave guiding claim is demonstrated at and around this frequency.
  • Boundary and excitation design value theta_0 = 0.4L
    Chosen 'through numerical experiments' (SI Sec 1.3.1) to be suitable; it fixes initial ray conditions and ensures tile compatibility.
  • Cost function weights w1 and w2 = not reported
    Eq. S3 weights position error and wave-vector error in the optimization objective; the relative weighting is not stated, so the exact objective used to generate the tile designs cannot be reconstructed from the text.
assumptions (5)
  • domain assumption The grading is smooth enough that ray tracing with local Bloch dispersion relations is valid
    Main text and SI Sec 1.2: 'assuming smooth spatial gradings and hence locally an approximately periodic medium'. All inverse designs rely on this.
  • domain assumption Out-of-plane and in-plane wave motions decouple, and only the lowest out-of-plane dispersion surface matters
    SI Sec 2.1: 'in- and out-of-planes are decoupled... we only consider the lowest out-of-plane dispersion surface'. The experiments excite and measure out-of-plane motion.
  • domain assumption Timoshenko beam finite element model with nominal geometry and literature silicon moduli represents the fabricated beams
    SI Sec 2; experimental agreement supports this after the fact, but it is assumed in both design and validation.
  • standard math Frequency is conserved along each ray because the local dispersion relation is time-independent
    SI Sec 1.3; used to fix each ray to the design frequency of 750 kHz.
  • domain assumption Unit cell dispersion can be interpolated across the theta design space from a 50 by 50 k-space grid and 100 theta samples
    SI Sec 2.1; finite differences and interpolation supply the first and second derivatives needed by the ray equations and adjoint method.

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Cite this review

Pith. "Pith review of Graded phononic metamaterials: Scalable design meets scalable microfabrication." pith.science (2026). https://pith.science/paper/4OMKWFYL

@misc{pith2026250701874,
  author       = {Pith},
  title        = {Pith review of: Graded phononic metamaterials: Scalable design meets scalable microfabrication},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4OMKWFYL}},
  note         = {Machine review of arXiv:2507.01874}
}
read the original abstract

Metamaterials are a new generation of advanced materials, exhibiting engineered microstructures that enable customized material properties not found in nature. The dynamics of metamaterials are particularly fascinating, promising the capability to guide, attenuate, and focus waves at will. Phononic metamaterials aim to manipulate mechanical waves with broad applications in acoustics, elastodynamics, and structural vibrations. A key bottleneck in the advancement of phononic metamaterials is scalability -- in design, simulation, and especially fabrication (e.g., beyond tens of unit cells per spatial dimension). We present a framework for scalable inverse design of spatially graded metamaterials for elastic wave guiding, together with a scalable microfabrication method. This framework enables the design and realization of complex waveguides including hundreds of thousands of unit cells, with the potential to extend to millions with no change in protocol. Scalable design is achieved via optimization with a ray tracing model for waves in spatially graded beam lattices. Designs are fabricated by photolithography and etching of silicon wafers to create free-standing microarchitected films. Wave guiding is demonstrated experimentally, using pulsed laser excitation and an interferometer for displacement measurements. Broadband wave guiding is demonstrated, indicating the promise of our scalable design and fabrication methods for on-chip elastic wave manipulation.

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Forward citations

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Reference graph

Works this paper leans on

47 extracted references · 46 canonical work pages · cited by 1 Pith paper

  1. [32]

    , author Dorn, C

    author Kannan, V. , author Dorn, C. , author Drechsler, U. & author Kochmann, D. title Microscale architected materials for elastic wave guiding: Fabrication and dynamic characterization across length and time scales ( year 2025 )

    work page 2025

  2. [1]

    title Wave propagation in periodic structures

    author Brillouin, L. title Wave propagation in periodic structures . journal McGraw-Hill volume 2 ( year 1946 )

    work page 1946

  3. [2]

    author Hussein, M. I. , author Leamy, M. J. & author Ruzzene, M. title Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook . journal Applied Mechanics Reviews volume 66 , pages 040802 ( year 2014 )

    work page 2014

  4. [3]

    , author Javelaud, E

    author Br \^u l \'e , S. , author Javelaud, E. , author Enoch, S. & author Guenneau, S. title Experiments on seismic metamaterials: molding surface waves . journal Physical Review Letters volume 112 , pages 133901 ( year 2014 )

    work page 2014

  5. [4]

    author Manushyna, D. et al. title Application of vibroacoustic metamaterials for structural vibration reduction in space structures . journal Mechanics Research Communications volume 129 , pages 104090 ( year 2023 )

    work page 2023

  6. [5]

    author Dub c ek, T. et al. title In-sensor passive speech classification with phononic metamaterials . journal Advanced Functional Materials volume 34 , pages 2311877 ( year 2024 )

    work page 2024

  7. [6]

    , author Guo, B

    author Chen, Z. , author Guo, B. , author Yang, Y. & author Cheng, C. title Metamaterials-based enhanced energy harvesting: A review . journal Physica B: Condensed Matter volume 438 , pages 1--8 ( year 2014 )

    work page 2014

  8. [7]

    , author Degertekin, F

    author Tol, S. , author Degertekin, F. L. & author Erturk, A. title Phononic crystal luneburg lens for omnidirectional elastic wave focusing and energy harvesting . journal Applied Physics Letters volume 111 ( year 2017 )

    work page 2017

Show all 47 references
  1. [8]

    author Zhu, J. et al. title Acoustic rainbow trapping . journal Scientific Reports volume 3 , pages 1728 ( year 2013 )

    2013

  2. [9]

    & author Kochmann, D

    author Dorn, C. & author Kochmann, D. M. title Conformally graded metamaterials for elastic wave guidance . journal Extreme Mechanics Letters volume 65 , pages 102091 ( year 2023 )

    2023

  3. [10]

    , author Rimoli, J

    author Trainiti, G. , author Rimoli, J. J. & author Ruzzene, M. title Wave propagation in undulated structural lattices . journal International Journal of Solids and Structures volume 97 , pages 431--444 ( year 2016 )

    2016

  4. [11]

    author Lin, S.-C. S. , author Huang, T. J. , author Sun, J.-H. & author Wu, T.-T. title Gradient-index phononic crystals . journal Physical Review B—Condensed Matter and Materials Physics volume 79 , pages 094302 ( year 2009 )

    2009

  5. [12]

    & author Chan, C

    author Chen, H. & author Chan, C. T. title Acoustic cloaking and transformation acoustics . journal Journal of Physics D: Applied Physics volume 43 , pages 113001 ( year 2010 )

    2010

  6. [13]

    , author Chen, Y

    author Nassar, H. , author Chen, Y. & author Huang, G. title Polar metamaterials: a new outlook on resonance for cloaking applications . journal Physical Review Letters volume 124 , pages 084301 ( year 2020 )

    2020

  7. [14]

    , author Cui, H

    author Shaikeea, A. , author Cui, H. , author O’Masta, M. , author Zheng, X. R. & author Deshpande, V. S. title The toughness of mechanical metamaterials . journal Nature Materials volume 21 , pages 297--304 ( year 2022 )

    2022

  8. [15]

    , author Singer, J

    author Lee, J.-H. , author Singer, J. P. & author Thomas, E. L. title Micro-/nanostructured mechanical metamaterials . journal Advanced Materials volume 24 , pages 4782--4810 ( year 2012 )

    2012

  9. [16]

    , author Thiel, M

    author B \"u ckmann, T. , author Thiel, M. , author Kadic, M. , author Schittny, R. & author Wegener, M. title An elasto-mechanical unfeelability cloak made of pentamode metamaterials . journal Nature Communications volume 5 , pages 4130 ( year 2014 )

    2014

  10. [17]

    author Bauer, J. et al. title Nanolattices: an emerging class of mechanical metamaterials . journal Advanced Materials volume 29 , pages 1701850 ( year 2017 )

    2017

  11. [18]

    author Meza, L. R. et al. title Reexamining the mechanical property space of three-dimensional lattice architectures . journal Acta Materialia volume 140 , pages 424--432 ( year 2017 )

    2017

  12. [19]

    & author Shin, Y

    author Harinarayana, V. & author Shin, Y. title Two-photon lithography for three-dimensional fabrication in micro/nanoscale regime: A comprehensive review . journal Optics & Laser Technology volume 142 , pages 107180 ( year 2021 )

    2021

  13. [20]

    & author Daraio, C

    author Kr \"o del, S. & author Daraio, C. title Microlattice metamaterials for tailoring ultrasonic transmission with elastoacoustic hybridization . journal Physical Review Applied volume 6 , pages 064005 ( year 2016 )

    2016

  14. [21]

    author Kiefer, P. et al. title A multi-photon (7 7)-focus 3d laser printer based on a 3d-printed diffractive optical element and a 3d-printed multi-lens array . journal Light: Advanced Manufacturing volume 4 , pages 28--41 ( year 2024 )

    2024

  15. [22]

    author Li, Z. et al. title Inverse design enables large-scale high-performance meta-optics reshaping virtual reality . journal Nature Communications volume 13 , pages 1--11 ( year 2022 )

    2022

  16. [23]

    , author Pestourie, R

    author Li, Z. , author Pestourie, R. , author Lin, Z. , author Johnson, S. G. & author Capasso, F. title Empowering metasurfaces with inverse design: principles and applications . journal ACS Photonics volume 9 , pages 2178--2192 ( year 2022 )

    2022

  17. [24]

    author Moore, D. T. title Gradient-index optics: a review . journal Applied Optics volume 19 , pages 1035--1038 ( year 1980 )

    1980

  18. [25]

    , author Djafari-Rouhani, B

    author Jin, Y. , author Djafari-Rouhani, B. & author Torrent, D. title Gradient index phononic crystals and metamaterials . journal Nanophotonics volume 8 , pages 685--701 ( year 2019 )

    2019

  19. [26]

    title Seismic ray theory ( publisher Cambridge University Press Cambridge , year 2001 )

    author Cerven \`y , V. title Seismic ray theory ( publisher Cambridge University Press Cambridge , year 2001 )

    2001

  20. [27]

    & author Wolf, E

    author Born, M. & author Wolf, E. title Principles of optics: E lectromagnetic theory of propagation, interference and diffraction of light ( publisher Elsevier , year 2013 )

    2013

  21. [28]

    & author Kochmann, D

    author Dorn, C. & author Kochmann, D. M. title Ray theory for elastic wave propagation in graded metamaterials . journal Journal of the Mechanics and Physics of Solids volume 168 , pages 105049 ( year 2022 )

    2022

  22. [29]

    & author Kochmann, D

    author Dorn, C. & author Kochmann, D. M. title Inverse design of graded phononic materials via ray tracing . journal Journal of Applied Physics volume 134 ( year 2023 )

    2023

  23. [30]

    author Zelhofer, A. J. & author Kochmann, D. M. title On acoustic wave beaming in two-dimensional structural lattices . journal International Journal of Solids and Structures volume 115 , pages 248--269 ( year 2017 )

    2017

  24. [31]

    author Telgen, B. et al. title Rainbow trapping of out-of-plane mechanical waves in spatially variant beam lattices . journal Journal of the Mechanics and Physics of Solids volume 191 , pages 105762 ( year 2024 )

    2024

  25. [33]

    author Zega, V. et al. title Microstructured phononic crystal isolates from ultrasonic mechanical vibrations . journal Applied Sciences volume 12 , pages 2499 ( year 2022 )

    2022

  26. [34]

    , author Tang, L

    author Hu, G. , author Tang, L. , author Liang, J. , author Lan, C. & author Das, R. title Acoustic-elastic metamaterials and phononic crystals for energy harvesting: A review . journal Smart Materials and Structures volume 30 , pages 085025 ( year 2021 )

    2021

  27. [35]

    , author Kujala, Z

    author Charara, M. , author Kujala, Z. , author Lee, S. & author Gonella, S. title Spatially selective drop-motion programming using metamaterials . journal Proceedings of the Royal Society A volume 481 , pages 20240429 ( year 2025 )

    2025

  28. [36]

    Rainbow trapping of out-of-plane mechanical waves in spatially variant beam lattices

    Bastian Telgen, Vignesh Kannan, Jean-Charles Bail, Charles Dorn, Hannah Niese, and Dennis M Kochmann. Rainbow trapping of out-of-plane mechanical waves in spatially variant beam lattices. Journal of the Mechanics and Physics of Solids, 191: 0 105762, 2024

    2024

  29. [37]

    Inverse design of graded phononic materials via ray tracing

    Charles Dorn and Dennis M Kochmann. Inverse design of graded phononic materials via ray tracing. Journal of Applied Physics, 134 0 (19), 2023 a

    2023

  30. [38]

    Conformally graded metamaterials for elastic wave guidance

    Charles Dorn and Dennis M Kochmann. Conformally graded metamaterials for elastic wave guidance. Extreme Mechanics Letters, 65: 0 102091, 2023 b

    2023

  31. [39]

    Ray theory for elastic wave propagation in graded metamaterials

    Charles Dorn and Dennis M Kochmann. Ray theory for elastic wave propagation in graded metamaterials. Journal of the Mechanics and Physics of Solids, 168: 0 105049, 2022

    2022

  32. [40]

    Adjoint nonlinear ray tracing

    Arjun Teh, Matthew O'Toole, and Ioannis Gkioulekas. Adjoint nonlinear ray tracing. ACM Transactions on Graphics, 41 0 (4): 0 1--13, 2022

    2022

  33. [41]

    On the limited memory BFGS method for large scale optimization

    Dong C Liu and Jorge Nocedal. On the limited memory BFGS method for large scale optimization. Mathematical Programming, 45 0 (1): 0 503--528, 1989

    1989

  34. [42]

    Steven G. Johnson. The NLopt nonlinear-optimization package. https://github.com/stevengj/nlopt, 2007

    2007

  35. [43]

    ae108, 2020

    Mechanics and Materials Lab. ae108, 2020

    2020

  36. [44]

    What is the Young's modulus of silicon? Journal of Microelectromechanical Systems, 19 0 (2): 0 229--238, 2010

    Matthew A Hopcroft, William D Nix, and Thomas W Kenny. What is the Young's modulus of silicon? Journal of Microelectromechanical Systems, 19 0 (2): 0 229--238, 2010

    2010

  37. [45]

    Wave propagation in two-dimensional periodic lattices

    A Srikantha Phani, J Woodhouse, and NA Fleck. Wave propagation in two-dimensional periodic lattices. The Journal of the Acoustical Society of America, 119 0 (4): 0 1995--2005, 2006

    1995

  38. [46]

    On acoustic wave beaming in two-dimensional structural lattices

    Alex J Zelhofer and Dennis M Kochmann. On acoustic wave beaming in two-dimensional structural lattices. International Journal of Solids and Structures, 115: 0 248--269, 2017

    2017

  39. [47]

    https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.butter.html

    butter 2014; S ci P y v1.15.2 M anual --- docs.scipy.org. https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.butter.html. [Accessed 17-04-2025]

    2014

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