WP2 has been in charge to identify and develop the theoretical and numerical methods needed to extend to aeroacoustic and aeronautical applications the applicability of metamaterials starting from the state of the art prior to AERIALIST.

An initial step in the definition of a general method for the analysis of the response of a generic metamaterial has been a review of the classic acoustic analogy to include unconventional acoustic behaviours within the aeroacoustic theory. Revisiting the classic Lighthill acoustic analogy, the formulation describing the propagation of an acoustic perturbation within a quiescent metafluid had been generalized for a metamaterial in an arbitrary state including the general effects of the metafluid properties in presence of a non-zero velocity fieldThe equations governing the propagation of an acoustic disturbance in a moving fluid have been revisited within the framework of the Lorentzian differential geometry, considering the spacetime continuum as a whole and exploiting the analytical framework of special and general relativity. This allowed for the development of spacetime corrections based on spacetime coordinate transformations, to be applied on static-designed metacontinua to make them effective in a moving hosting fluid, i.e. in an aeroacoustic environment. With this basis, a Space Time Transformation Acoustics (STTA) has been developed as a framework for the design of metamaterial devices based on coordinate transformation including the presence of background aerodynamics. As the Standard TA, also STTA is a model of a continuum, which feedback is the Hooke’s tensor required from the metamaterial to obtain the desired metabehviour.

Pictorial representation of the matching of the propagation pattern at the interface between the moving hosting medium and the metafluid at rest (left), Effect of the Taylor correction on the cloaking efficiency of a cylindrical device at M = 0.35 for three different frequencies (right).

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WP2 also studied the solid mechanics of metamaterial microstructures. The objectives of this part of the work were

  • Explore the influence of anisotropy in vibro-acoustic unconventional meta-behavior and provide potential applications in aeronautics.
  • Design interesting metamaterials with advanced meta-behavior in both mechanics and acoustics for further experimental validation of the most promising concepts.
  • Develop numerical tools for the design and optimization of the tailored metamaterials.
  • Develop a numerical method to characterize the anisotropic material properties based on the design model.
  • Design acoustic metamaterials based on Kelvin cell structures.

Starting from a defined micro-structural geometry, that is parametrised in a systematic and controlled way, i.e. the tetrakaidecahedron well known as Kelvin Cell, a micro-structural numerical model is used to predict the deformation field response to a known load and then used to extract a homogenised elastic and acoustic equivalent model of the structure. The estimation of the equivalent continuum properties is conducted via optimization, matching the displacement field of the micro-structural model, retrieving the complete set of 21 independent components of the elastic Hooke’s tensor needed to characterize a fully anisotropic elasticity response. The methodology proved to be able to capture orthotropicity and anisotropicity of micro-structures, producing coherent homogenized continua. The acoustic response is also evaluated and allows to design metamaterials based on Kelvin cell structures.

AERIALIST efforts have been dedicated also to the modelling of metasurfaces for extraordinary reflection. WP2 developed analytical and numerical models to predict and design surface treatments based on the so called Generalized Snell’s Law of aeronautical interest. Since D2.2, this concept has been included in the concept of interest in AERIALIST since allows large modification of the acoustic field by a thin surface treatment (on the order of few centimeters thickness at the frequencies of interest) which could be feasible in aeronautical applications.

Starting from a well assessed design found during the literature review activity of WP1 (based on coiled channels), BEM and FEM models have been validated and then the analysis has been extended to an AERIALIST original design (based on off–resonance cavities). 

An optimization–based design technique has been developed, involving the construction of a metamodel of the elementary unit of the metasurfaces, built from FEM simulations, to cut the computational cost of the design optimization aimed at extending the bandwidth of the metabehaviour. The designed metasurfaces have been experimentally tested and the whole procedure hence validated.

The modelling activity also allowed to bridge the gap between the GSL-based and the TA design frameworks (see D5.5), linking the two models through the equivalence of the effects of a tailored coordinate transformation and a phase-delayer cell. This widens the set of designs usable to effectively realize the modeled effects, and allows for efficient simulations and optimization of metasurfaces with reduced order models, linking with the other WP2 activities.

The PGMS has been the first example of closure of the AERIALIST design loop, from the theoretical principle, the generalized Snell’s law, to the realization of a demonstrator and a validation of the numerical results against experiments. AERIALIST had the opportunity to show the PGMS demonstrator rig at the 23rd CEAS-ASC Workshop held in Rome. The impressive effect of the metasurfaces, audible in the video below, on the emitted sound from the rig was theoretically and numerically predicted by the models developed by WP2 and WP3 gave the fundamental contribution to the manufacturing of the rig. 

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WP2 activities on reduced order modelling have been focused also on visco-thermal acoustic losses effects. The lossy acoustics in a fine absorptive medium can be modelled by linearised visco-thermal fluid mechanical models. Solution of the related system of equations is extremely expensive and precludes the solutions for structures of conjoined cell models. Clearly a model reduction strategy is needed such as for example the Low Reduced Frequency approaches whereby a simple Helmholtz model is formulated enabling the simulation of much more extensive structures. The losses themselves are modelled approximately using equivalent cylindrical duct losses. The link between more complex structures such as stacked spheres is done through the use of an hydraulic radius r, Using the corrections for averaged density and bulk modulus, both a function of frequency, in cylindrical bores.

Once that the equivalent Helmholtz system has been validated against thermo-viscous acoustic simulations of the fully micromodelled single cell, macro models of multiple cell structures have then be prepared using the reduced Helmholtz models for comparison with an impedance tube test program, developed in WP3, under various loading scenarios, giving consistent results and further validating the method.

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Another model reduction strategy leading to a macro model has been studied. A transfer matrix approach provides a key to building up full depth absorbent models from individual modelled test data.

The individual elements of the matrix can be determined by for example driving the system at inlet (a plane wave forcing was used in the present case) and then imposing an open followed by a closed boundary condition.

The overall multicell system response matrix for n cells can then be quickly calculated. 

There is promising agreement between the experimental measurement and the efficient micro-macro numerical simulation procedure.

Notwithstanding experimental error, clear uncertainties in the modelling process also include those associated with dimensional estimates of narrower sections and the modelling of surface finish.

Although the numerical predictions underestimate the loss, they are positioned well on the frequency axis. Tentative predictions with a doubled viscosity, to account for increases surface roughness, give very encouraging results.

The use of efficient viscothermal models at a cell level provides a direct route via the transfer matrix approach to modelling built up cellular acoustically absorptive structures.

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Finally, WP2 analyzed the potential of metamaterials for the realization of device concepts, proposed in the project, of aeronautical interest in term of numerical simulations, such as VSI and Enhanced Shielding for innovative configurations.

The concept of a VSI can take advantage of metamaterials and metasurface treatments in modifying the directivity of the nacelle.

The Enhanced Shielding concept is of interest especially for innovative configurations of aircraft, with engines mounted on top of the main body, also with distributed propulsion systems. The shadowing effect is augmented by the extraordinary reflection achieved with the PGMS

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Studies were conducted via numerical optimization, tailoring the metasurfaces for this particular applications, achieving the desired metabehaviour. The optimization process can be integrated in a multidisciplinary context, and uncertainties taken into account.

Preliminary simulations with simplified geometries and acoustic sources provided encouraging results and deserve to be further developed in the future.