Ultrasonic bandgaps in viscoelastic 1D-periodic media: Mechanical modeling and experimental validation.

Multi-material additive manufacturing is receiving increasing attention in the field of acoustics, in particular towards the design of micro-architectured periodic media used to achieve programmable ultrasonic responses. To unravel the effect of the material properties and spatial arrangement of the printed constituents, there is an unmet need in developing wave propagation models for prediction and optimization purposes. In this study, we propose to investigate the transmission of longitudinal ultrasound waves through 1D-periodic biphasic media, whose constituent materials are viscoelastic. To this end, Bloch-Floquet analysis is applied in the frame of viscoelasticity, with the aim of disentangling the relative contributions of viscoelasticity and periodicity on ultrasound signatures, such as dispersion, attenuation, and bandgaps localization. The impact of the finite size nature of these structures is then assessed by using a modeling approach based on the transfer matrix formalism. Finally, the modeling outcomes, i.e., frequency-dependent phase velocity and attenuation, are confronted with experiments conducted on 3D-printed samples, which exhibit a 1D periodicity at length-scales of a few hundreds of micrometers. Altogether, the obtained results shed light on the modeling characteristics to be considered when predicting the complex acoustic behavior of periodic media in the ultrasonic regime.

Precise-Integration Time-Domain Formulation for Optical Periodic Media.

A numerical formulation based on the precise-integration time-domain (PITD) method for simulating periodic media is extended for overcoming the Courant-Friedrich-Levy (CFL) limit on the time-step size in a finite-difference time-domain (FDTD) simulation. In this new method, the periodic boundary conditions are implemented, permitting the simulation of a wide range of periodic optical media, i.e., gratings, or thin-film filters. Furthermore, the complete tensorial derivation for the permittivity also allows simulating anisotropic periodic media. Numerical results demonstrate that PITD is reliable and even considering anisotropic media can be competitive compared to traditional FDTD solutions. Furthermore, the maximum allowable time-step size has been demonstrated to be much larger than that of the CFL limit of the FDTD method, being a valuable tool in cases in which the steady-state requires a large number of time-steps.

High-frequency homogenization in periodic media with imperfect interfaces.

In this work, the concept of high-frequency homogenization is extended to the case of one-dimensional periodic media with imperfect interfaces of the spring-mass type. In other words, when considering the propagation of elastic waves in such media, displacement and stress discontinuities are allowed across the borders of the periodic cell. As is customary in high-frequency homogenization, the homogenization is carried out about the periodic and antiperiodic solutions corresponding to the edges of the Brillouin zone. Asymptotic approximations are provided for both the higher branches of the dispersion diagram (second-order) and the resulting wave field (leading-order). The special case of two branches of the dispersion diagram intersecting with a non-zero slope at an edge of the Brillouin zone (occurrence of a so-called Dirac point) is also considered in detail, resulting in an approximation of the dispersion diagram (first-order) and the wave field (zeroth-order) near these points. Finally, a uniform approximation valid for both Dirac and non-Dirac points is provided. Numerical comparisons are made with the exact solutions obtained by the Bloch-Floquet approach for the particular examples of monolayered and bilayered materials. In these two cases, convergence measurements are carried out to validate the approach, and we show that the uniform approximation remains a very good approximation even far from the edges of the Brillouin zone.

A rational framework for dynamic homogenization at finite wavelengths and frequencies.

In this study, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media (including the source term) at finite frequencies and wavenumbers spanning the first Brillouin zone. We take the eigenvalue problem for the unit cell of periodicity as a point of departure, and we consider the projection of germane Bloch wave function onto a suitable eigenfunction as descriptor of effective wave motion. For generality the finite wavenumber, finite frequency homogenization is pursued in R d via second-order asymptotic expansion about the apexes of 'wavenumber quadrants' comprising the first Brillouin zone, at frequencies near given (acoustic or optical) dispersion branch. We also consider the junctures of dispersion branches and 'dense' clusters thereof, where the asymptotic analysis reveals several distinct regimes driven by the parity and symmetries of the germane eigenfunction basis. In the case of junctures, one of these asymptotic regimes is shown to describe the so-called Dirac points that are relevant to the phenomenon of topological insulation. On the other hand, the effective model for nearby solution branches is found to invariably entail a Dirac-like system of equations that describes the interacting dispersion surfaces as 'blunted cones'. For all cases considered, the effective description turns out to admit the same general framework, with differences largely being limited to (i) the eigenfunction basis, (ii) the reference cell of medium periodicity, and (iii) the wavenumber-frequency scaling law underpinning the asymptotic expansion. We illustrate the analytical developments by several examples, including Green's function near the edge of a band gap and clusters of nearby dispersion surfaces.

Nonlinear Bloch waves and balance between hardening and softening dispersion.

The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unit-cell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size.

On the dynamic homogenization of periodic media: Willis' approach versus two-scale paradigm.

When considering an effective, i.e. homogenized description of waves in periodic media that transcends the usual quasi-static approximation, there are generally two schools of thought: (i) the two-scale approach that is prevalent in mathematics and (ii) the Willis' homogenization framework that has been gaining popularity in engineering and physical sciences. Notwithstanding a mounting body of literature on the two competing paradigms, a clear understanding of their relationship is still lacking. In this study, we deploy an effective impedance of the scalar wave equation as a lens for comparison and establish a low-frequency, long-wavelength dispersive expansion of the Willis' effective model, including terms up to the second order. Despite the intuitive expectation that such obtained effective impedance coincides with its two-scale counterpart, we find that the two descriptions differ by a modulation factor which is, up to the second order, expressible as a polynomial in frequency and wavenumber. We track down this inconsistency to the fact that the two-scale expansion is commonly restricted to the free-wave solutions and thus fails to account for the body source term which, as it turns out, must also be homogenized-by the reciprocal of the featured modulation factor. In the analysis, we also (i) reformulate for generality the Willis' effective description in terms of the eigenfunction approach, and (ii) obtain the corresponding modulation factor for dipole body sources, which may be relevant to some recent efforts to manipulate waves in metamaterials.

Comparative study of periodicity estimation methods using ultrasonic signals

Abstract Introduction Various signal-processing techniques have been proposed to extract quantitative information about internal structures of tissues from the original radio frequency (RF) signals instead of an ultrasound image. The quantifiable parameter called the mean scatterer spacing (MSS) can be useful to detect changes in the quasi-periodic microstructure of tissues such as the liver or the spleen, using ultrasonic signals. Methods We evaluate and compare the performance of three classic methods of spectral estimation to calculate the MSS without operator intervention: Tufts-Kumaresan, SAC (Spectral Autocorrelation) and MUSIC (MUltiple SIgnal Classification). Initially the evaluations were performed with 10,000 signals simulated from a model in which the variables of interest are controlled, and then, real signals from sponge phantoms were used. Results For the simulated signals, the performance of all three methods decreased with increasing Ad or jitter levels. For the sponges, none of the methods accurately estimated the pore size. Conclusion For the simulated signals, Tufts-Kumaresan had the lowest performance, whereas SAC and MUSIC had similar results. For sponges, only Tufts-Kumaresan was able to detect the increase in the size of the pores of the sponge, although most often, it estimated sizes larger than expected.

Detecção automática do espaçamento médio de meios periódicos por sinais ultrassônicos retroespalhados / Automatic detection algorithms for mean scatterer space estimation using ultrasound backscattered signals

A interpretação da imagem ultrassônica, por ocorrer de modo visual e qualitativa, traz uma variação inter e intra-observador importante. A adoção de métodos quantitativos é uma forma de diminuir esta dependência. Entre tais métodos está a quantificação do espaçamento médio entre espalhadores (Mean Scatterer Spacing - MSS), que pode ser útil para detectar mudanças na microestrutura quasi-periódica de tecidos como o hepático ou o esplênico. Neste trabalho foram avaliados três métodos clássicos de estimação espectral para cálculo do MSS (sem intervenção do operador): BURG, WIENER e MUSIC. O intuito é comparar suas potencialidades para a estimação automática de espaçamento médio de espalhadores ultrassônicos. Inicialmente as avaliações foram realizadas com 10.000 sinais simulados a partir de um modelo em que se tem controle das variáveis de interesse, e em seguida foram utilizados sinais reais de phantoms de fios de nylon imersos em água. O método de BURG não conseguiu estimar adequadamente o espaçamento em sinais de phantom, tendo apresentado resultados equivalentes aos outros métodos deste trabalho somente para sinais simulados. O método de WIENER para os sinais simulados apresentou resultados de menor percentual de acerto, ficando em segundo lugar, para os sinais dos phantoms. O método de subespaço MUSIC apresentou melhor desempenho global em relação a BURG e WIENER, com resultados de 100% de acerto para o phantom de fio de nylon de 1,2 mm e 91,45% para 0,8 mm considerando uma janela de acerto de 10%.

The interpretation of ultrasound imaging is essentially visual and qualitative, so there are important inter and intra-observer variations. Quantification methods aim at decreasing this dependency. Among those, the quantification of the Mean Scatterer Spacing (MSS) can be useful to detect changes in the microstructure of quasi-periodic tissues, such as liver or spleen. This study evaluated the following methods of spectral estimation for calculating the MSS (without requiring operator intervention): BURG, WIENER and MUSIC. The aim is to compare their potential for automatic estimation of MSS from ultrasonic scattering signals. Initially, the evaluation has been carried out using 10,000 simulated signals, with the aim of studying the behavior of the methods using a model in which the variables of interest can be controlled. Then, the methods have been applied to real signals of nylon phantoms immersed in water. The BURG method could not estimate the spacing of US phantom signals, presenting results similar to the other methods only for simulated signals. The WIENER method for the simulated signals was in second place in terms of percentage of success, when considering signals from the phantoms. The subspace method MUSIC had the best performance from all three methods.