pith. sign in

arxiv: 2509.23290 · v1 · submitted 2025-09-27 · ⚛️ physics.flu-dyn · physics.app-ph

A Framework to Systematically Study the Nonlinear Fluid-Structure Interaction of Phononic Materials with Aerodynamic Flows

Vinod Ramakrishnan , Arturo Machado Burgos , Sangwon Park , Kathryn H. Matlack , Andres Goza This is my paper

Pith reviewed 2026-05-18 13:04 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.app-ph
keywords phononic materialsfluid-structure interactionvortex sheddingbehavioral parametersaerodynamic flowsflat platelift coefficientnonlinear dynamics
0
0 comments X

The pith

Phononic materials in aerodynamic flows can be described by four behavioral parameters that determine their effects on vortex shedding.

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

The paper introduces a systematic framework to study the nonlinear fluid-structure interactions of phononic materials with fluid flows. It identifies four key behavioral parameters that are separate from but map to the material's structural properties. These parameters are effective stiffness, truncation resonance frequency, dynamic displacement amplitude, and unit cell mass. Using high-fidelity simulations of laminar flow over a flat plate, the work demonstrates how each parameter influences distinct aspects of the lift coefficient during vortex shedding. This framework is presented as essential for investigating and designing phononic materials in broader fluid flow applications.

Core claim

Our study proposes four critical PM behavioral parameters -- effective stiffness, truncation resonance frequency, a quantity representing the dynamic displacement amplitude, and unit cell mass -- that influence the spectral characteristics of the vortex-shedding process inherent to the flat plate system. Results show connections between each parameter and distinct behavior in the lift coefficient in FSI.

What carries the argument

Four behavioral parameters for phononic materials in FSI (effective stiffness, truncation resonance frequency, dynamic displacement amplitude, and unit cell mass) that map to structural parameters and shape flow spectra.

If this is right

  • Each of the four behavioral parameters produces a distinct signature in the lift coefficient during fluid-structure interaction.
  • The parameters allow systematic quantification of how phononic materials modulate flow unsteadiness and vortex shedding.
  • Identifying behavioral parameters rather than only structural ones simplifies the study of nonlinear FSI in aerodynamic flows.
  • The approach supports passive flow control design using phononic materials in flat-plate configurations.

Where Pith is reading between the lines

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

  • The same behavioral-parameter approach could be tested in turbulent or compressible flow regimes to address transition delay or shock-boundary-layer control.
  • Experimental measurements on fabricated phononic plates could validate the simulation-derived mappings to structural design variables.
  • Designers might target specific behavioral parameters to achieve desired changes in lift or drag without exhaustive structural optimization.

Load-bearing premise

High-fidelity strongly coupled simulations can reliably quantify the influence of these behavioral parameters on FSI dynamics and that a clear mapping exists from behavioral parameters back to the underlying structural parameters of the phononic material.

What would settle it

A simulation or experiment in which independently varying one behavioral parameter produces no distinct predicted change in the vortex-shedding spectrum or lift coefficient would challenge the framework.

Figures

Figures reproduced from arXiv: 2509.23290 by Andres Goza, Arturo Machado Burgos, Kathryn H. Matlack, Sangwon Park, Vinod Ramakrishnan.

Figure 1
Figure 1. Figure 1: Problem setup and phononic subsystem. (a) Two-dimensional flat plate of chord [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diatomic Phononic Material. (a) Mass-spring models for an ungrounded and grounded diatomic PMs. (b) Dispersion [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: PM behavioral parameters. (a) Dispersion curves of three distinct diatomic PMs engineered with the same truncation [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Resonant dynamics and behavioral parameters in different mass-spring models. (a.i) A single mass-spring oscillator [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Compliant-section reparametrization and marker redistribution. (a) Undeformed arc-length locations [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Vortex-shedding process as a function of different PM truncation resonance frequencies. (a.i) The surface [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Fourier frequency spectra in the post-transient state ( [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Steady and Dynamic PM-FSI parameters. (a) Time-average (mean), and (b) dynamic amplitude of (i) [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
read the original abstract

Phononic materials (PMs) are periodic media that exhibit novel elastodynamic responses. While PMs have made progress in vibration-mitigation applications, recent studies have demonstrated the potential of PMs to passively and adaptively modulate flow behavior through fluid-structure interaction (FSI). For example, PMs have been shown to delay laminar-to-turbulent transition and mitigate unsteadiness in shock-boundary layer interactions. However, a systematic framework to relate the effect of specific PM behaviors to the FSI dynamics is lacking. Such a framework is essential to systematically investigate the complex and nonlinear coupled dynamics of the FSI. Further, parameters that are not typically considered in PM models become critical, such as the vibration amplitude. This article addresses this gap by proposing FSI-relevant ``behavioral'' parameters, distinct from the structural parameters of the PM, but with a clear mapping provided to them. We use high-fidelity, strongly coupled simulations to quantify the FSI between a novel configuration of laminar flow past a flat plate, equipped with a PM. Our study proposes four critical PM behavioral parameters -- effective stiffness, truncation resonance frequency, a quantity representing the dynamic displacement amplitude, and unit cell mass -- that influence the spectral characteristics of the vortex-shedding process inherent to the flat plate system. Results show connections between each parameter and distinct behavior in the lift coefficient in FSI. While the focus of this work is on the PM-FSI dynamics in an aerodynamic flow, we argue that identifying these behavioral parameters is key to unlocking scientific study and design with phononic materials in fluid flows more broadly.

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

1 major / 2 minor

Summary. The manuscript proposes a systematic framework for investigating nonlinear fluid-structure interaction (FSI) between phononic materials (PMs) and aerodynamic flows. It introduces four behavioral parameters—effective stiffness, truncation resonance frequency, dynamic displacement amplitude, and unit cell mass—that are distinct from but mappable to the PM structural parameters. High-fidelity, strongly coupled simulations of laminar flow over a flat plate equipped with a PM are used to demonstrate connections between each behavioral parameter and distinct features in the spectral content of the lift coefficient arising from vortex shedding.

Significance. If the behavioral parameters can be shown to be independently controllable with an invertible mapping to structural design variables, the framework would provide a valuable tool for extending PM applications from vibration control to passive flow modulation in aerodynamics. The use of strongly coupled simulations to link parameter variations to lift-coefficient spectra is a positive step toward falsifiable design rules, though the overall impact depends on resolving questions of parameter orthogonality.

major comments (1)
  1. [Definition of behavioral parameters and simulation methodology] The central claim that the four behavioral parameters can be systematically varied to map distinct effects onto vortex-shedding spectra requires explicit demonstration that dynamic displacement amplitude can be adjusted independently while holding effective stiffness, truncation resonance frequency, and unit cell mass fixed. In a strongly coupled nonlinear FSI system, amplitude emerges from the instantaneous force balance; without shown constraints (e.g., external forcing or damping) that preserve invariance of the other three parameters, the set is not orthogonal and the framework reduces to post-hoc correlation rather than a controllable design space. This issue is load-bearing for the proposed framework and must be addressed with concrete simulation protocols or additional constraints.
minor comments (2)
  1. Clarify the precise definition and units of the 'quantity representing the dynamic displacement amplitude' to avoid ambiguity with emergent response quantities.
  2. Provide explicit mapping equations or tables showing how each behavioral parameter translates back to the underlying PM structural parameters (e.g., lattice geometry, material moduli).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive and detailed review. The emphasis on demonstrating orthogonality among the behavioral parameters is well taken, as it directly affects the framework's utility for systematic design. We address the major comment below and commit to revisions that strengthen this aspect of the work.

read point-by-point responses

  1. Referee: [Definition of behavioral parameters and simulation methodology] The central claim that the four behavioral parameters can be systematically varied to map distinct effects onto vortex-shedding spectra requires explicit demonstration that dynamic displacement amplitude can be adjusted independently while holding effective stiffness, truncation resonance frequency, and unit cell mass fixed. In a strongly coupled nonlinear FSI system, amplitude emerges from the instantaneous force balance; without shown constraints (e.g., external forcing or damping) that preserve invariance of the other three parameters, the set is not orthogonal and the framework reduces to post-hoc correlation rather than a controllable design space. This issue is load-bearing for the proposed framework and must be addressed with concrete simulation protocols or additional constraints.

    Authors: We agree that independent controllability is essential for the framework to move beyond observed correlations toward a true design space. The dynamic displacement amplitude is indeed an emergent quantity in the coupled FSI problem. In the current manuscript the four parameters are obtained by mapping from distinct structural features of the phononic unit cells (stiffness from spring constants, resonance from band-gap edges, mass from density/geometry, and amplitude from the resulting modal participation under load). To address the referee's point directly, the revised manuscript will add a new subsection with targeted simulation protocols: we will introduce controlled variations in local geometry and internal damping that primarily affect displacement amplitude while re-computing and confirming that effective stiffness, truncation resonance frequency, and unit cell mass remain invariant to within numerical tolerance. These additional cases will be presented alongside the existing results to demonstrate the required orthogonality. revision: yes

Circularity Check

0 steps flagged

No circularity: behavioral parameters proposed and validated via independent simulations

full rationale

The paper proposes four behavioral parameters (effective stiffness, truncation resonance frequency, dynamic displacement amplitude, unit cell mass) with an asserted mapping to underlying PM structural parameters, then quantifies their influence on vortex-shedding spectra using high-fidelity strongly coupled FSI simulations. No equations, derivations, or self-citations are shown that reduce any claimed prediction or parameter to a fitted quantity defined from the same data or to a prior result by the same authors. The framework is self-contained against external simulation benchmarks rather than self-referential definitions, so the central claim does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central contribution rests on introducing four new behavioral parameters and assuming that high-fidelity simulations can isolate their effects on vortex shedding. No free parameters are explicitly fitted in the abstract; the parameters themselves function as the primary invented concepts.

axioms (1)
  • domain assumption High-fidelity, strongly coupled simulations accurately capture the nonlinear FSI dynamics between the phononic material and the laminar flow.
    Invoked to quantify the influence of the behavioral parameters on the lift coefficient and vortex-shedding spectra.
invented entities (1)
  • Behavioral parameters (effective stiffness, truncation resonance frequency, dynamic displacement amplitude, unit cell mass) no independent evidence
    purpose: To provide a clear mapping from phononic material behavior to FSI outcomes distinct from conventional structural parameters.
    These four quantities are proposed as the key levers for studying PM-FSI; independent evidence outside the simulations is not described.

pith-pipeline@v0.9.0 · 5842 in / 1448 out tokens · 46063 ms · 2026-05-18T13:04:47.319167+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    D. M. Kochmann, K. Bertoldi, Exploiting microstructural instabilities in solids and structures: from metamaterials to structural transitions, Appl. Mech. Rev. 69 (2017) 050801

    work page 2017

  2. [2]

    G. U. Patil, K. H. Matlack, Review of exploiting nonlinearity in phononic materials to enable nonlinear wave responses, Acta Mech. 233 (2022) 1–46

    work page 2022

  3. [3]

    P. Jiao, J. Mueller, J. R. Raney, X. Zheng, A. H. Alavi, Mechanical metamaterials and beyond, Nat. Commun. 14 (2023) 6004

    work page 2023

  4. [4]

    B. Wu, W. Jiang, J. Jiang, Z. Zhao, Y. Tang, W. Zhou, W. Chen, Wave manipulation in intelligent metamaterials: recent progress and prospects, Adv. Funct. Mater. 34 (2024) 2316745

    work page 2024

  5. [5]

    M. S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (1993) 2022

    work page 1993

  6. [6]

    M. S. Kushwaha, Classical band structure of periodic elastic composites, Int. J. Mod. Phys. B 10 (1996) 977–1094. 27

    work page 1996

  7. [7]

    J. Lee, Y. Y. Kim, Elastic metamaterials for guided waves: from fundamentals to applications, Smart Mater. Struct. 32 (2023) 123001

    work page 2023

  8. [8]

    Vishwakarma, M

    U. Vishwakarma, M. Irfan, G. Theocharis, R. Chaunsali, Edge states with hidden topology in spinner lattices, Commun. Phys. 8 (2025) 83

    work page 2025

  9. [9]

    Zhang, W

    Y. Zhang, W. Z. Jiang, W. Jiang, X. Y. Zhang, J. Dong, Y. M. Xie, K. E. Evans, X. Ren, Recent advances of auxetic metamaterials in smart materials and structural systems, Adv. Funct. Mater. (2025) 2421746

    work page 2025

  10. [10]

    Wu, K.-W

    Z. Wu, K.-W. Wang, On the wave propagation analysis and supratransmission prediction of a metastable modular metastructure for non-reciprocal energy transmission, J. Sound Vib. 458 (2019)

    work page 2019

  11. [11]

    Akbari-Farahani, S

    F. Akbari-Farahani, S. Ebrahimi-Nejad, From defect mode to topological metamaterials: A state-of-the-art review of phononic crystals & acoustic metamaterials for energy harvesting, Sens. Actuator A Phys. (2023) 114871

    work page 2023

  12. [12]

    Martinez, M

    F. Martinez, M. Maldovan, Metamaterials: Optical, acoustic, elastic, heat, mass, electric, magnetic, and hydrodynamic cloaking, Mater. Today Phys. 27 (2022) 100819

    work page 2022

  13. [13]

    Aksoy, H

    B. Aksoy, H. Shea, Multistable shape programming of variable-stiffness electromagnetic devices, Sci. Adv. 8 (2022) eabk0543

    work page 2022

  14. [14]

    M. V. Bastawrous, M. I. Hussein, Closed-form existence conditions for bandgap resonances in a finite periodic chain under general boundary conditions, J. Acoust. Soc. Am. 151 (2022) 286–298

    work page 2022

  15. [15]

    Al Ba’ba’a, M

    H. Al Ba’ba’a, M. Nouh, T. Singh, Dispersion and topological characteristics of permutative polyatomic phononic crystals, Proc. R. Soc. A 475 (2019) 20190022

    work page 2019

  16. [16]

    H. B. Al Ba’ba’a, H. Yousef, M. Nouh, A blueprint for truncation resonance placement in elastic diatomic lattices with unit cell asymmetry, JASA Express Lett. 4 (2024) 077501

    work page 2024

  17. [17]

    Ramakrishnan, K

    V. Ramakrishnan, K. H. Matlack, A quantitative study of energy localization characteristics in defect-embedded monoatomic phononic crystals, J. Sound Vib. 614 (2025) 119164

    work page 2025

  18. [18]

    M. I. Hussein, S. Biringen, O. R. Bilal, A. Kucala, Flow stabilization by subsurface phonons, Proc. R. Soc. A 471 (2015) 20140928

    work page 2015

  19. [19]

    C. L. Willey, C. J. Barnes, V. W. Chen, K. Rosenberg, A. Medina, A. T. Juhl, Multi-input multi-output phononic subsurfaces for passive boundary layer transition delay, J. Fluids Struct. 121 (2023) 103936

    work page 2023

  20. [20]

    Michelis, A

    T. Michelis, A. Putranto, M. Kotsonis, Attenuation of tollmien–schlichting waves using resonating surface-embedded phononic crystals, Phys. Fluids 35 (2023)

    work page 2023

  21. [21]

    C. J. Barnes, C. L. Willey, K. Rosenberg, A. Medina, A. T. Juhl, Initial computational investigation toward passive transition delay using a phononic subsurface, in: AIAA Scitech 2021 Forum, 2021, p. 1454

    work page 2021

  22. [22]

    Schmidt, H

    R. Schmidt, H. Yousef, I. Roy, C. Scalo, M. Nouh, Perturbation energy extraction from a fluid via a subsurface acoustic diode with sustained downstream attenuation, J. Appl. Phys. 137 (2025)

    work page 2025

  23. [23]

    Keogh, J

    M. Keogh, J. McTighe, J. Dahl, O. R. Bilal, Experimental observation of flow instability control by metamaterial subsurfaces, arXiv preprint arXiv:2504.02053 (2025)

    work page arXiv 2025

  24. [24]

    J. D. Navarro, D. Balderas, E. J. LaLonde, J. C. Velasquez-Gonzalez, E. N. Hoffman, C. S. Combs, D. Restrepo, Stabilization of hypersonic shockwave/boundary-layer interactions with phononic metamaterials, Matter (2025)

    work page 2025

  25. [25]

    Compliant coatings, Annu. Rev. Fluid Mech. 20 (1988) 393–420

    work page 1988

  26. [26]

    Gad-el Hak, Compliant coatings: A decade of progress, Appl

    M. Gad-el Hak, Compliant coatings: A decade of progress, Appl. Mech. Rev. 49 (1996) S147–S157

    work page 1996

  27. [27]

    C. H. Williamson, R. Govardhan, Vortex-induced vibrations, Annu. Rev. Fluid Mech. 36 (2004) 413–455

    work page 2004

  28. [28]

    Sarpkaya, A critical review of the intrinsic nature of vortex-induced vibrations, J

    T. Sarpkaya, A critical review of the intrinsic nature of vortex-induced vibrations, J. Fluids Struct. 19 (2004) 389–447

    work page 2004

  29. [29]

    M. J. Shelley, J. Zhang, Flapping and bending bodies interacting with fluid flows, Annu. Rev. Fluid Mech. 43 (2011) 449–465

    work page 2011

  30. [30]

    T. B. Benjamin, Effects of a flexible boundary on hydrodynamic stability, J. Fluid Mech. 9 (1960) 513–532

    work page 1960

  31. [31]

    M. T. Landahl, On the stability of a laminar incompressible boundary layer over a flexible surface, J. Fluid Mech. 13 (1962) 609–632

    work page 1962

  32. [32]

    Carpenter, A

    P. Carpenter, A. Garrad, The hydrodynamic stability of flow over kramer-type compliant surfaces. part 1. tollmien-schlichting instabilities, J. Fluid Mech. 155 (1985) 465–510

    work page 1985

  33. [33]

    Shelley, N

    M. Shelley, N. Vandenberghe, J. Zhang, Heavy flags undergo spontaneous oscillations in flowing water, Phys. Rev. Lett. 94 (2005) 094302

    work page 2005

  34. [34]

    Doar´ e, S

    O. Doar´ e, S. Michelin, Piezoelectric coupling in energy-harvesting fluttering flexible plates: linear stability analysis and conversion efficiency, J. Fluids Struct. 27 (2011) 1357–1375

    work page 2011

  35. [35]

    Luhar, A

    M. Luhar, A. S. Sharma, B. McKeon, A framework for studying the effect of compliant surfaces on wall turbulence, J. Fluid Mech. 768 (2015) 415–441

    work page 2015

  36. [36]

    A. Goza, T. Colonius, J. E. Sader, Global modes and nonlinear analysis of inverted-flag flapping, J. Fluid Mech. 857 (2018) 312–344

    work page 2018

  37. [37]

    Pfister, N

    J.-L. Pfister, N. Fabbiane, O. Marquet, Global stability and resolvent analyses of laminar boundary-layer flow interacting with viscoelastic patches, J. Fluid Mech. 937 (2022) A1

    work page 2022

  38. [38]

    Fredsoe, B

    J. Fredsoe, B. M. Sumer, Hydrodynamics around cylindrical structures, volume 12, World Scientific, 1997

    work page 1997

  39. [39]

    Shiels, A

    D. Shiels, A. Leonard, A. Roshko, Flow-induced vibration of a circular cylinder at limiting structural parameters, J. Fluids Struct. 15 (2001) 3–21

    work page 2001

  40. [40]

    B. S. Connell, D. K. Yue, Flapping dynamics of a flag in a uniform stream, J. Fluid Mech. 581 (2007) 33–67

    work page 2007

  41. [41]

    T. Khan, W. Li, J. Zhang, T. I.-P. Shih, Local vibrations and lift performance of low reynolds number airfoil, Propuls. Power Res. 6 (2017) 79–90. Volume 6 Issue 2 (Special Issue) 2017

    work page 2017

  42. [42]

    W. Kang, M. Xu, W. Yao, J. Zhang, Lock-in mechanism of flow over a low-reynolds-number airfoil with morphing surface, Aerosp. Sci. Technol. 97 (2020) 105647. 28

    work page 2020

  43. [43]

    Thompson, A

    E. Thompson, A. Goza, Surface morphing for aerodynamic flows at low and stalled angles of attack, Phys. Rev. Fluids 7 (2022) 024703

    work page 2022

  44. [44]

    Khelif, A

    A. Khelif, A. Adibi, Phononic Crystals: Fundamentals and Applications, Springer, 2015

    work page 2015

  45. [45]

    M. H. Bae, S. H. Kim, H. M. Seung, J. H. Oh, Opening bandgap in monoatomic-diatomic convertible metamaterial with nonlinearity, Int. J. Mech. Sci. 289 (2025) 110067

    work page 2025

  46. [46]

    Machado Burgos, S

    A. Machado Burgos, S. Park, N. O’Brien, K. H. Matlack, A. Goza, Fluid-structure interaction between an unsteady vortex-driven aerodynamic flow and a diatomic phononic subsurface, in: AIAA SCITECH 2024 Forum, 2024, p. 1353

    work page 2024

  47. [47]

    Bloch, ¨Uber die quantenmechanik der elektronen in kristallgittern, Zeitschrift f¨ ur Physik 52 (1929) 555–600

    F. Bloch, ¨Uber die quantenmechanik der elektronen in kristallgittern, Zeitschrift f¨ ur Physik 52 (1929) 555–600

    work page 1929

  48. [48]

    P. A. Deymier, Acoustic metamaterials and phononic crystals, volume 173, Springer science & Business media, 2013

    work page 2013

  49. [49]

    M. I. Hussein, M. J. Frazier, Metadamping: An emergent phenomenon in dissipative metamaterials, J. Sound Vib. 332 (2013) 4767–4774

    work page 2013

  50. [50]

    R. K. Narisetti, M. J. Leamy, M. Ruzzene, A perturbation approach for predicting wave propagation in one-dimensional nonlinear periodic structures, J. Vib. Acoust. 132 (2010)

    work page 2010

  51. [51]

    Arretche, K

    I. Arretche, K. H. Matlack, Centrifugal forces enable band gaps that self-adapt to synchronous vibrations in rotating elastic metamaterial, Mech. Syst. Signal Process. 202 (2023) 110689

    work page 2023

  52. [52]

    Ramakrishnan, M

    V. Ramakrishnan, M. J. Frazier, Architected material with independently tunable mass, damping, and stiffness via multi-stability and kinematic amplification, J. Acoust. Soc. Am. 153 (2023) 1283–1292

    work page 2023

  53. [53]

    Nemat-Nasser, A

    S. Nemat-Nasser, A. Srivastava, Negative effective dynamics mass-density and stiffness: Micro-architecture and phononic transport in periodic composites, AIP Adv. 1 (2011) 041502

    work page 2011

  54. [54]

    C. S. Wojnar, D. M. Kochmann, A negative-stiffness phase in elastic composites can produce stable extreme effective dynamic but not static stiffness, Philos. Mag. 94 (2014) 532–555

    work page 2014

  55. [55]

    C.-T. Lin, A. Goza, H. J. Bae, Active control for turbulent drag reduction by periodic blowing and suction, in: AIAA AVIATION FORUM AND ASCEND 2024, 2024, p. 3636

    work page 2024

  56. [56]

    A. Goza, T. Colonius, A strongly-coupled immersed-boundary formulation for thin elastic structures, J. Comp. Phys. 336 (2017) 401–411

    work page 2017

  57. [57]

    A. Goza, S. Liska, B. Morley, T. Colonius, Accurate computation of surface stresses and forces with immersed boundary methods, J. Comp. Phys. 321 (2016) 860–873

    work page 2016

  58. [58]

    Colonius, K

    T. Colonius, K. Taira, A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions, Comput. Methods Appl. Mech. Eng. 197 (2008) 2131–2146

    work page 2008

  59. [59]

    Taira, T

    K. Taira, T. Colonius, The immersed boundary method: a projection approach, J. Comp. Phys. 225 (2007) 2118–2137. 29

    work page 2007