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arxiv: 1907.09297 · v1 · pith:YBHU44HZnew · submitted 2019-07-18 · ⚛️ physics.class-ph · physics.app-ph

Graded metasurface for enhanced sensing and energy harvesting

Jacopo M. De Ponti , Andrea Colombi , Raffaele Ardito , Francesco Braghin , Alberto Corigliano , Richard V. Craster This is my paper

Pith reviewed 2026-05-24 19:18 UTC · model grok-4.3

classification ⚛️ physics.class-ph physics.app-ph
keywords metasurfacerainbow trappingenergy harvestingelastic wavespiezoelectricgraded structuresresonant rodswave focusing
0
0 comments X

The pith

Graded resonant rod arrays on elastic plates trap waves by frequency to create large local displacements for piezoelectric energy harvesting.

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

The paper examines how spatial grading in arrays of resonant rods attached to elastic plates produces rainbow trapping of the A0 wave mode. Different frequency components accumulate at separate locations along the array, each generating substantially larger displacement amplitudes than would occur in a uniform array. These amplified motions are then coupled to piezoelectric elements to improve sensing and energy conversion. Numerical simulations on plates and extensions to half-spaces quantify the gains in displacement and harvested power. A reader would care because the approach offers a subwavelength route to broadband, frequency-selective wave focusing that could raise the output of vibration-powered devices.

Core claim

Graded resonant arrays of rods on elastic plates enable rainbow trapping that produces large displacement amplitudes usable for enhanced piezoelectric sensing and energy harvesting, with advantages over uniform arrays shown via numerical simulation. The work concentrates on elastic plate models where the A0 mode dominates, develops accurate models of the phenomena, and extends the analysis to Rayleigh waves on an elastic half-space.

What carries the argument

Rainbow trapping produced by spatial grading of resonant rod arrays, which spatially separates frequency components and accumulates large displacement amplitudes at specific resonator positions on the plate.

If this is right

  • Graded arrays produce broadband focusing and convert surface waves to bulk waves at chosen locations.
  • The spatially separated large amplitudes allow multiple piezoelectric harvesters to operate at different frequencies simultaneously.
  • Numerical models predict higher harvested power per unit area than uniform resonant arrays.
  • The same grading principle applies to Rayleigh waves on half-spaces.
  • Subwavelength rod spacing still achieves the required frequency-dependent trapping.

Where Pith is reading between the lines

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

  • The same graded layout could be used to improve structural-health-monitoring sensors that detect specific vibration frequencies.
  • Varying the grading slope would allow designers to target particular ambient vibration spectra without changing the overall device footprint.
  • Combining the metasurface with electrical tuning circuits might compensate for any small detuning caused by the piezoelectric load itself.
  • Experimental prototypes on thin plates would provide the clearest test of whether fabrication tolerances preserve the predicted amplitude gains.

Load-bearing premise

The amplified displacements from rainbow trapping can be efficiently converted to electrical energy via piezoelectric coupling without significant detuning, damping, or fabrication imperfections that would reduce the predicted gains.

What would settle it

A direct numerical or laboratory comparison in which the electrical power extracted from a graded array is not materially higher than from an otherwise identical uniform array would falsify the claimed harvesting advantage.

Figures

Figures reproduced from arXiv: 1907.09297 by Alberto Corigliano, Andrea Colombi, Francesco Braghin, Jacopo M. De Ponti, Raffaele Ardito, Richard V. Craster.

Figure 1
Figure 1. Figure 1: FIG. 1. The elastic metasurface used for energy harvesting (a) and conventional dispersion curves [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Harvester shown with the piezoelectric patches (a), the harvesting fundamental mode (b) [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison between the analytical and numerical transmission spectrum in short circuit [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Power output vs. electrical conductance and frequency (a). Dashed white line shows the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Harvester on a plate strip with (a), and without (b), the metasurface at time t = 10 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Six harvesters on a plate strip with (a) and without (b) the metasurface at time t = 10 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Six harvesters on an half-space strip with (a) and without (b) the metasurface (equal to [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
read the original abstract

In elastic wave systems, combining the powerful concepts of resonance and spatial grading within structured surface arrays enable resonant metasurfaces to exhibit broadband wave focusing, mode conversion from surface (Rayleigh) waves to bulk (shear) waves, and spatial frequency selection. Devices built around these concepts allow for precise control of surface waves, often with structures that are subwavelength, and utilise rainbow trapping that separates the signal spatially by frequency. Rainbow trapping yields large amplifications of displacement at the resonator positions where each frequency component accumulates. We investigate whether this amplification, and the associated control, can be used to create energy harvesting devices; the potential advantages and disadvantages of using graded resonant devices as energy harvesters is considered. We concentrate upon elastic plate models for which the A0 mode dominates, and take advantage of the large displacement amplitudes in graded resonant arrays of rods, to design innovative metasurfaces that focus waves for enhanced piezoelectric sensing and energy harvesting. Numerical simulation allows us to identify the advantages of such graded metasurface devices and quantify its efficiency, we also develop accurate models of the phenomena and extend our analysis to that of an elastic half-space and Rayleigh surface waves.

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 claims that graded resonant rod arrays on elastic plates can achieve rainbow trapping of A0-mode waves, producing spatially separated large displacement amplifications that enable enhanced piezoelectric sensing and energy harvesting compared to uniform arrays. Numerical FEM simulations are used to identify and quantify efficiency advantages, with extensions to Rayleigh-wave models on elastic half-spaces; the work positions these metasurfaces as broadband focusing devices for energy conversion applications.

Significance. If the mechanical amplification results hold under realistic transduction, the graded metasurface concept could provide a route to broadband elastic-wave harvesters that spatially segregate frequencies for improved capture efficiency, with relevance to structural monitoring and vibration scavenging. The numerical demonstration of rainbow trapping in subwavelength rod arrays on plates is a clear modeling strength, though the absence of electromechanical coupling limits direct applicability to harvesting performance.

major comments (2)
  1. [Numerical simulations / results] Numerical simulations (abstract and results sections): the FEM models treat only the elastic plate + rod system (A0 mode propagation and graded resonances) and do not incorporate piezoelectric patches, the electromechanical coupling matrix, or electrical boundary conditions. This omission is load-bearing for the central efficiency claims, as displacement amplitudes and any quantified harvesting gains cannot be taken as evidence that mechanical amplification survives realistic piezoelectric loading, detuning, or damping.
  2. [Abstract / efficiency claims] Abstract and efficiency quantification: the text states that numerical simulation 'quantify its efficiency' and identifies 'advantages' over uniform arrays, yet no specific numerical values, comparison metrics, error bars, or power-output figures are supplied in the available description. Without these, the advantage claims rest on unshown data and cannot be evaluated for robustness.
minor comments (2)
  1. [Abstract] Abstract grammar: 'combining the powerful concepts of resonance and spatial grading within structured surface arrays enable' should read 'enables'; 'we concentrate upon' is acceptable but 'on' is more standard.
  2. [Abstract / introduction] Notation and terminology: 'rainbow trapping' and 'mode conversion from surface (Rayleigh) waves to bulk (shear) waves' are used without initial definition or reference to prior literature on the same phenomena in elastic metasurfaces.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments and recommendation. We address the two major comments point-by-point below, agreeing on the need for greater clarity regarding model scope and quantitative presentation. Revisions will be made to the abstract, results, and discussion sections to improve the manuscript.

read point-by-point responses

  1. Referee: [Numerical simulations / results] Numerical simulations (abstract and results sections): the FEM models treat only the elastic plate + rod system (A0 mode propagation and graded resonances) and do not incorporate piezoelectric patches, the electromechanical coupling matrix, or electrical boundary conditions. This omission is load-bearing for the central efficiency claims, as displacement amplitudes and any quantified harvesting gains cannot be taken as evidence that mechanical amplification survives realistic piezoelectric loading, detuning, or damping.

    Authors: We agree that the simulations are limited to the elastic mechanical system (plate + graded rods) and do not include piezoelectric patches or electromechanical coupling. The quantified efficiency and advantages refer to mechanical displacement amplification via rainbow trapping relative to uniform arrays. We will revise the manuscript to explicitly state this scope, add discussion of how piezoelectric loading could introduce detuning or damping, and note that coupled simulations represent a logical next step. This clarifies the claims without overstating current results. revision: yes

  2. Referee: [Abstract / efficiency claims] Abstract and efficiency quantification: the text states that numerical simulation 'quantify its efficiency' and identifies 'advantages' over uniform arrays, yet no specific numerical values, comparison metrics, error bars, or power-output figures are supplied in the available description. Without these, the advantage claims rest on unshown data and cannot be evaluated for robustness.

    Authors: The efficiency advantages are quantified in the results section via FEM comparisons of displacement fields and frequency responses between graded and uniform cases. To allow direct evaluation, we will revise the abstract and main text to report explicit metrics such as peak amplification ratios at trapped frequencies. Error bars do not apply to these deterministic simulations; power-output figures are absent because the models are mechanical only (addressed in the first response). revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on direct numerical simulation of standard elastic wave equations

full rationale

The paper derives its claims about rainbow trapping amplification and efficiency advantages through numerical FEM simulation of the A0 mode in graded rod arrays on elastic plates, extending to half-space Rayleigh waves. No load-bearing step reduces a prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness result, or renames an input as an output. The modeling follows conventional wave propagation without self-definitional loops or ansatz smuggling. The analysis is therefore self-contained against external benchmarks of elastic metasurface theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5747 in / 985 out tokens · 17644 ms · 2026-05-24T19:18:17.754285+00:00 · methodology

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Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    in acoustics and Fano resonances [19], enabling the creation of low-frequency band-gaps in structures relatively small compared to the wavelength; these ideas can be augmented by tuning or spatially grading the resonators to broaden or decrease band-gaps. At a smaller scale, and consequently higher frequency, wave redirection and protection [20–23] promis...

    work page 2018

  2. [2]

    J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Transactions on Microwave Theory and Techniques , 47(11):2075–2084, 1999

    work page 2075

  3. [3]

    D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire. Metamaterials and negative refractive index. Science, 305:788–792, 2004

    work page 2004

  4. [4]

    S. A. Ramakrishna and T. M. Grzegorczyk. Physics and applications of negative refractive index materials. CRC Press, 2008

    work page 2008

  5. [5]

    D. H. Werner. Broadband Metamaterials in Electromagnetics: Technology and Applications . Pan Stanford Publishing, 2016

    work page 2016

  6. [6]

    Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, and P. Sheng. Locally resonant sonic materials. Science, 289(5485):1734–1736, 2000

    work page 2000

  7. [7]

    R. V. Craster and S. Guenneau. Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking. London: Springer, 2012

    work page 2012

  8. [8]

    World Scientific Handbook of Metamaterials and Plasmonics: Volume 2: Elastic, Acoustic and Seismic Metamaterials

    R Craster and S Guenneau. World Scientific Handbook of Metamaterials and Plasmonics: Volume 2: Elastic, Acoustic and Seismic Metamaterials . World Scientific, 2017

    work page 2017

  9. [9]

    Brˆ ul´ e, E

    S. Brˆ ul´ e, E. H. Javelaud, S. Enoch, and S. Guenneau. Experiments on seismic metamaterials: Molding surface waves. Phys. Rev. Lett., 112:133901, 2014

    work page 2014

  10. [10]

    Antonakakis, R

    T. Antonakakis, R. V. Craster, and S. Guenneau. Moulding and shielding flexural waves in elastic plates. Euro. Phys. Lett., 105:54004, 2014

    work page 2014

  11. [11]

    Torrent, Y

    D. Torrent, Y. Pennec, and B. Djafari-Rouhani. Omnidirectional refractive devices for flexural waves based on graded phononic crystals. J. Appl. Phys. , 116(22):224902, 2014

    work page 2014

  12. [12]

    Achaoui, T

    Y. Achaoui, T. Antonakakis, S. Brule, R. Craster, S. Enoch, and S. Guenneau. Clamped seismic metamaterials: Ultra-low frequency stop bands. New J. Phys. , 19:063022, 2017

    work page 2017

  13. [13]

    Miniaci, A

    M. Miniaci, A. Krushynska, F. Bosia, and N. M Pugno. Large scale mechanical metamaterials as seismic shields. New J. Phys. , 18(8):083041, 2016

    work page 2016

  14. [14]

    D’Alessandro, R

    L. D’Alessandro, R. Ardito, F. Braghin, A. Corigliano, Low frequency 3D ultra-wide vibration attenuation via elastic metamaterial. Sci. Rep., 9:8039, 2019

    work page 2019

  15. [15]

    D’Alessandro, B

    L. D’Alessandro, B. Bahr, L. Daniel, D. Weinstein, R. Ardito Shape optimization of solidair porous phononic crystal slabs with widest full 3D bandgap for in-plane acoustic waves.Journal 18 of Computational Physics , Vol. 344, 2017

    work page 2017

  16. [16]

    Dertimanis, I.A

    V.K. Dertimanis, I.A. Antoniadis, and E.N. Chatzi. Feasibility analysis on the attenuation of strong ground motions using finite periodic lattices of mass-in-mass barriers. J. Engng Mech., 142(9):04016060, 2016

    work page 2016

  17. [17]

    Finocchio, O

    G. Finocchio, O. Casablanca, G. Ricciardi, U. Alibrandi, F. Garesc, M. Chiappini, and B. Azzerboni. Seismic metamaterials based on isochronous mechanical oscillators. Appl. Phys. Lett., 104(19):191903, 2014

    work page 2014

  18. [18]

    Carta, I.S

    G. Carta, I.S. Jones, N. V. Movchan, A. B. Movchan, and M. J. Nieves. Gyro-elastic beams for the vibration reduction of long flexural systems. Proc. R. Soc. Lond. A, 473:2017.0136, 07 2017

    work page arXiv 2017

  19. [19]

    Lemoult, M

    F. Lemoult, M. Fink, and G. Lerosey. Acoustic resonators for far-field control of sound on a subwavelength scale. Phys. Rev. Lett., 107:064301, 2011

    work page 2011

  20. [20]

    A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar. Fano resonances in nanoscale structures. Rev. Mod. Phys., 82:2257–2298, 2010

    work page 2010

  21. [21]

    Colombi, V

    A. Colombi, V. Ageeva, R. Smith, R. Patel, M. Clark, R. Craster, A. Clare, S. Guenneau, and P. Roux. Enhanced sensing and conversion of ultrasonic rayleigh waves by elastic metasurfaces. Sci. Rep., 7:6750, 02 2017

    work page 2017

  22. [22]

    A. Colombi. Resonant metalenses for flexural waves. J. Acoust. Soc. Am., 140(5):EL423, 2016

    work page 2016

  23. [23]

    Matlack, A

    K.H. Matlack, A. Bauhofer, S. Kr¨ odel, A. Palermo and C. Daraio, Composite 3D-printed metastructures for low-frequency and broadband vibration absorption. Proc. Natl. Acad. Sci., 113(30):8386–8390, 2016

    work page 2016

  24. [24]

    Cardella, P

    D. Cardella, P. Celli, and S. Gonella. Manipulating waves by distilling frequencies: a tunable shunt-enabled rainbow trap. Smart Materials and Structures , 25(8):085017, 2016

    work page 2016

  25. [25]

    Erturk and N

    A. Erturk and N. Elvin. Advances in Energy Harvesting Methods . Springer, 2013

    work page 2013

  26. [26]

    Erturk and D

    A. Erturk and D. J. Inman. Piezoelectric Energy Harvesting. John Wiley and Sons, Ltd, 2011

    work page 2011

  27. [27]

    S. R. Anton and H. A. Sodano. A review of power harvesting using piezoelectric materials (2003-2006). Smart Materials and Structures , 16(3):R1–R21, May 2007

    work page 2003

  28. [28]

    Ahmed, F

    R. Ahmed, F. Mir, and S. Banerjee. A review on energy harvesting approaches for renewable energies from ambient vibrations and acoustic waves using piezoelectricity. Smart Materials and Structures, 26(8):085031, July 2017. 19

    work page 2017

  29. [29]

    Carrara, M

    M. Carrara, M. R. Cacan, J. Toussaint, M. Leamy, M. Ruzzene, and A. Erturk. Metamaterial- inspired structures and concepts for elastoacoustic wave energy harvesting. Smart Materials and Structures, 22:065004, 04 2013

    work page 2013

  30. [30]

    H. Lv, X. Tian, M. Y. Wang, and D. Li. Vibration energy harvesting using a phononic crystal with point defect states. Appl. Phys. Lett. , 102:034103, January 2013

    work page 2013

  31. [31]

    Wu, L.-W

    L.-Y. Wu, L.-W. Chen, and C.-M. Liu. Acoustic energy harvesting using resonant cavity of a sonic crystal. Appl. Phys. Lett. , 95:013506, June 2009

    work page 2009

  32. [32]

    Gonella, A

    S. Gonella, A. C. To, and W. K. Liu. Interplay between phononic bandgaps and piezoelectric microstructures for energy harvesting. J. Mech. Phys. Solids , 57:621633, November 2009

    work page 2009

  33. [33]

    S. Tol, F. L. Degertekin, and A. Erturk. Phononic crystal luneburg lens for omnidirectional elastic wave focusing and energy harvesting. Appl. Phys. Lett. , 111:013503, July 2017

    work page 2017

  34. [34]

    S. Tol, F. L. Degertekin, and A. Erturk. Gradient-index phononic crystal lens based enhance- ment of elastic wave energy harvesting. Appl. Phys. Lett. , 109:064902, May 2016

    work page 2016

  35. [35]

    Zareei, A

    A. Zareei, A. Darabi, M. J. Leamy, and M.-R. Alam. Continuous profile flexural GRIN lens: Focusing and harvesting flexural waves. Appl. Phys. Lett. , 112:023901, January 2018

    work page 2018

  36. [36]

    Analysis of multifunctional piezoelectric metastructures for low- frequency bandgap formation and energy harvesting

    C Sugino and A Erturk. Analysis of multifunctional piezoelectric metastructures for low- frequency bandgap formation and energy harvesting. J. Phys. D: Appl. Phys , 51:215103, May 2018

    work page 2018

  37. [37]

    Cveticanin and M

    L. Cveticanin and M. Zukovic. Negative effective mass in acoustic metamaterial with nonlinear mass-in-mass subsystems. Communications in Nonlinear Science and Numerical Simulation , 51:89 – 104, 2017

    work page 2017

  38. [38]

    Casalotti, S

    A. Casalotti, S. El-Borgi, and W. Lacarbonara. Metamaterial beam with embedded nonlinear vibration absorbers. International Journal of Non-Linear Mechanics , 98:32–42, 2018

    work page 2018

  39. [39]

    Colombi, D

    A. Colombi, D. Colquitt, P. Roux, S. Guenneau, and R. V. Craster. A seismic metamaterial: The resonant metawedge. Sci. Rep., 6:27717, 2016

    work page 2016

  40. [40]

    Wu, Z.-G

    T-T. Wu, Z.-G. Huang, T.-C. Tsai, and T.-C. Wu. Evidence of complete band gap and resonances in a plate with periodic stubbed surface. Appl. Phys. Lett. , 93(11):111902, 2008

    work page 2008

  41. [41]

    Pennec, B

    Y. Pennec, B. Djafari-Rouhani, H. Larabi, J. O. Vasseur, and A. C. Hladky-Hennion. Low- frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin ho- mogeneous plate. Phys. Rev. B , 78:104105, Sep 2008. 20

    work page 2008

  42. [42]

    Achaoui, A

    Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude. Experimental observation of locally-resonant and bragg band gaps for surface guided waves in a phononic crystal of pillars. Phys. Rev. B , 83(10):10401, 2011

    work page 2011

  43. [43]

    Rupin, F

    M. Rupin, F. Lemoult, G. Lerosey, and P. Roux. Experimental demonstration of ordered and disordered multi-resonant metamaterials for Lamb waves. Phys. Rev. Lett., 112:234301, 2014

    work page 2014

  44. [44]

    N. C. Perkins and C. D. Mote. Comments on curve veering in eigenvalue problems. J. Sound Vib., 106:451–463, 1986

    work page 1986

  45. [45]

    L. D. Landau and E. M. Lifshitz. Quantum Mechanics non-relativistic theory . Pergamon Press, 1958

    work page 1958

  46. [46]

    Colombi, P

    A. Colombi, P. Roux, S. Guenneau, P. Gueguen, and R.cV. Craster. Forests as a natural seis- mic metamaterial: Rayleigh wave bandgaps induced by local resonances.Sci. Rep., 5(5):19238, 2016

    work page 2016

  47. [47]

    Kaina, F

    N. Kaina, F. Lemoult, M. Fink, and G. Lerosey. Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials. Nature, 525:77–81, 2015

    work page 2015

  48. [48]

    Colombi, R

    A. Colombi, R. Craster, D. Colquitt, Y. Achaoui, S. Guenneau, P. Roux, and M. Rupin. Elastic wave control beyond band-gaps: Shaping the flow of waves in plates and half-spaces with subwavelength resonant rods. Frontiers in Mechanical Engineering, 3, 05 2017

    work page 2017

  49. [49]

    C. Kittel. Introduction to Solid State Physics . Wiley, Hoboken, NJ, 2005

    work page 2005

  50. [50]

    www.comsol.com, 2012

    COMSOL. www.comsol.com, 2012

    work page 2012

  51. [51]

    E. G. Williams, P. Roux, M. Rupin, and W. A. Kuperman. Theory of multiresonant meta- materials for A0 lamb waves. Phys. Rev. B , 91:104307, 2015

    work page 2015

  52. [52]

    K. L. Tsakmakidis, A. D. Boardman, and O. Hess. Trapped rainbow storage of light in metamaterials. Nature, 450:397–401, 2007

    work page 2007

  53. [53]

    Gafforelli, R

    G. Gafforelli, R. Ardito, and A. Corigliano. Improved one-dimensional model of piezoelectric laminates for energy harvesters including three dimensional effects. Composite Structures, 127:369–381, 2015

    work page 2015

  54. [54]

    Ardito, A

    R. Ardito, A. Corigliano, G. Gafforelli, C. Valzasina, F. Procopio, R. Zafalon. Advanced model for fast assessment of piezoelectric micro energy harvesters. Frontiers in Materials , Vol. 3, 2016

    work page 2016

  55. [55]

    Rajagopal, M

    P. Rajagopal, M. Drozdz, E.A Skelton, Lowe M. J S, and R.V. Craster. On the use of absorbing layers to simulate the propagation of elastic waves in unbounded isotropic media 21 using commercially available finite element packages. NDT E International , 51:30 – 40, 2012. 22

    work page 2012