Bloch Floquet Research Articles

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

Machine learning-based optimal design of an acoustic black hole metaplate for enhanced bandgap and load-bearing capacity

This paper introduces a novel machine learning-based optimization strategy for multi-functional acoustic black hole (ABH) metaplates. The primary objective is to achieve a multi-functional metaplate with excellent performance in elastic wave attenuation and load-bearing capacity simultaneously. The paper begins by describing the design of nanocomposite ABH metaplates, presenting a new pathway to realize multi-functional metaplates. Then, a semi-analytical method, based on plate theory and the Bloch–Floquet theorem, is introduced to consider the band structure of the nanocomposite metaplates. Through systematic analysis, the impacts of the ABH effect, nanocomposite reinforcements, and the viscoelastic damping layer on the bandgaps and strain energy compliance are highlighted. Meanwhile, two optimization objectives representing bandgap characteristics and in-plane stiffness are derived respectively. Subsequently, a deep learning surrogate model is employed to establish a relationship involving significant parameters with the optimization objectives. The performance evaluation confirms accuracy and computational speed of the surrogate model. Finally, an optimization strategy based on deep reinforcement learning is proposed to obtain multi-functional metaplates with superior bandgaps, enhanced in-plane stiffness, or both. The robustness and efficiency of the strategy are demonstrated under different tests. The results show that the proposed strategy can achieve identical results as the genetic algorithm and nondominated sorting genetic algorithm-II, while surpassing them in computational efficiency and balancing multiple objectives. The findings of this study serve as valuable references for the future development and application of multi-functional advanced metamaterials.

Ultrasound monitoring of multiphase architectured media: Bandgap tracking via the measurement of the reflection coefficient

Characterizing the mechanics of periodic scaffolds used for bone tissue repair, while maintaining their structural integrity, is a challenging problem. By leveraging concepts arising from the bulk phononic crystal community, here we investigate the reflection of elastic waves propagating through a water-immersed biphasic architectured medium in the ultrasonic regime. Towards this goal, Bloch–Floquet analysis is applied on a 2D unit cell made of a soft inclusion embedded in a hard matrix, to recover its corresponding phononic band structure. Exploring the modal conversion at the boundary between the homogeneous incident medium and the architectured one allows identifying a bandgap within the considered frequency range, which exhibits a significant sensitivity to varying volume fraction of the soft phase. Conducting further numerical analyzes, which account for the viscoelasticity of the two constituent phases, along with the finite-size and bounded nature of the architectured medium, shows that a sudden amplitude rise of the reflection coefficient takes place at a frequency that is closely related to the upper limit of this bandgap. This hypothesis is experimentally verified on 3D-printed bio-mimicking samples, which exhibit an in-plane periodicity at a length scale of a few hundred micrometers. Altogether, the reported results suggest that tracking prohibited frequency bands via the measurement of the reflection coefficient allows for the monitoring of micro-architectured media like scaffolds.

Wave propagation analysis for three typical box girders: Dominant wave modes and their applications

Because of the prevalent application of bridges in transportation, problems such as vibration and noise of bridges, especially in densely populated urban areas, are increasingly arousing concerns. The vibro-acoustic behaviors of bridges are important for vibration and noise control. Although these vibro-acoustic behaviors have been reported and discussed in several studies in terms of modal analysis, this study focuses on the probe into the vibration properties of three typical box girders (BGs) from the wave motion perspective: concrete, steel–concrete composite, and steel BGs. The dispersion characteristics and mode shapes of waves propagating in BGs are identified using the wave finite element method (WFEM). This calculation method combines the finite element method (FEM) with the Bloch–Floquet theorem, and it is validated by the existing literature and FEM. The wave propagation characteristics of the three typical BGs are then compared to analyze their similarities and differences in vibration. Furthermore, the contribution of each wave mode to the whole vibration and the local vibration of BGs is determined; consequently, the important vibration mechanism can be identified and ranked in different frequency bands. Lastly, some feasible application of vibration control for BGs is given in the light of the vibration mechanism. The study results provide theoretical and practicable foundation of vibration control for BG bridges in urban rail trafficsystems

Effects of corrugated boundaries on Rayleigh waves in a piezoelectric semiconductor substrate covered with a metal layer

In the present work, analytical solutions for Rayleigh waves propagating in a metal covered layer over a piezoelectric semiconductor (PSC) substrate with corrugated surface and corrugated interface are obtained. The solutions of the various physical fields can be expressed as a summation of infinite harmonic waves by using the Bloch–Floquet theorem due to the existence of the periodic corrugated surface and corrugated interface. The effects of the height and contour shape of the corrugated surface and corrugated interface, the doping density of the PSC substrate and the thickness of the metal covered layer on the dispersion and attenuation curves and mode shapes of Rayleigh waves are discussed via numerical example. The results show that the existence of the corrugated surface will lead to the decrease of the phase velocity and non-monotonous change with wavenumber, besides, and significantly increases the attenuation near k0=π/Λ(k0 and Λ are the wavenumber and period, respectively). In addition, the effects of periodic corrugated surface are much greater than those of periodic corrugated interface and are closely related with the ratio of the wavelength to the structure period. Moreover, the sinusoidal corrugated surface can be used to better control the propagation characteristics of Rayleigh waves by comparison.

Shear horizontal wave propagation on a piezoelectric semiconductor substrate under slight ridge or thin metal strip gratings

In the present work, analytical solutions for shear horizontal (SH) waves propagating in an n-type piezoelectric semiconductor (PSC) substrate under periodic slight ridge or thin metal strip gratings are obtained. The solutions of the various physical fields can be expressed as a summation of infinite harmonic waves by using the Bloch–Floquet theorem due to the existence of the periodic ridge or metal grating. Based on the numerical calculations, the effects of the width, height and shape of the ridge, the doping density of the PSC substrate and the material properties of the metal grating on the dispersion and attenuation curves of the SH waves are investigated. The results show that both the periodic ridge and metal grating have evident effects on the propagation characteristics and displacement mode shapes of the SH waves and the effects are closely related with the ratio of the wavelength to the structure period. In addition, the metal gratings can better control the propagation characteristics of SH waves than the ridges, namely, the effects of the metal gratings are more obvious and flexible than that of the ridges.

Bending–torsion coupled wave in thin-walled mono-symmetric metabeam: A non-dimensional analysis

The influence of the periodic resonators on wave dispersion of a thin-walled mono-symmetric beam is studied in this paper. The primary focus is to develop a non-dimensional framework to investigate the coupled bending–torsion wave propagation analytically. The resonators are placed eccentrically and are capable to form a subwavelength attenuation band in the beam due to its periodicity. The governing equations corresponding to the bending and torsional motion of a metabeam are presented in a dimensionless form, including the warping effect. The transfer matrix approach is adopted to obtain the dispersive phenomena using the Bloch–Floquet principle. The modal characteristics of a mono-symmetric beam without a resonator are computed from the frequency response function and are validated with finite element results. Subsequently, the impact of various non-dimensional parameters on the attenuation band and band merging, conversion, and bifurcation phenomena are investigated through a comprehensive parametric study. It is observed that the occurrence of the band gap primarily depends on the resonator properties and its location. A substantial band gap is achieved in the low-frequency range, which will be useful for the application in the field of vibration control and acoustics.

Geometric Aspects of Nonlinear and Nonequilibrium Phenomena

We review recent developments in the research of nonlinear and nonequilibrium phenomena in solids focusing on their geometrical aspects. We start with introducing the basic concepts of geometrical phases of Bloch electrons and Floquet theory for periodically driven systems. Then we review recent attempts to engineer topological phases in nonequilibrium matters such as graphene, magnets, and superconductors irradiated with circularly polarized light. We next review a bulk photovoltaic effect of inversion broken materials focusing on the shift current response. The shift current is described with Berry connections and has a close relationship to the modern theory of polarization. We further review recent extensions of the shift current to correlated electron systems. Finally, we explain the geometric diabatic time evolution in the Zener tunneling process and its consequences on nonreciprocal transport.

Non-negative intensity for a heavy fluid-loaded stiffened plate

Localisation of sound sources on vibrating structures is a critical part of the design in many engineering applications. In structures with stiffeners, so-called Bloch–Floquet waves are generated due to the interaction between the flexural waves in the host structure and the flexural/torsional waves of the stiffeners. It is known that the Bloch–Floquet waves have a significant contribution to the radiated sound. However, it is not understood which area of the vibrating stiffened structures contributes significantly to the radiation in the far-field. Non-negative intensity (NNI) is a powerful tool developed recently to locate the surface regions on structures that can contribute to the radiated sound power. Although NNI has been used for several distinct structures under different excitations, it has not been considered for analysing structures with stiffeners. In this work, NNI is evaluated for an infinite fluid-loaded stiffened plate subjected to a point force to localise the sources of sound and to shed light on the mechanism involved in the far-field radiation. An analytical model formulated in the wavenumber domain is presented to carry out fast calculations of the plate vibroacoustic responses and the NNI maps. A parametric study is then performed by comparing the vibroacoustic responses with the NNI maps to highlight the capability of NNI for sound source localisation. This is achieved by analysing the results for stiffened/unstiffened structures with excitation on/between the stiffeners, and at frequencies either in a passband or in a stopband. Moreover, the NNI maps are further explained and interpreted using identified Bloch–Floquet radiating bands.

Cascaded parametric amplification based on spatiotemporal modulations

Active devices have drawn considerable attention owing to their powerful capabilities to manipulate electromagnetic waves. Fast and periodic modulation of material properties is one of the key obstacles to the practical implementation of active metamaterials and metasurfaces. In this study, to circumvent this limitation, we employ a cascaded phase-matching mechanism to amplify signals through spatiotemporal modulation of permittivity. Our results show that the energy of the amplified fundamental mode can be efficiently transferred to that of the high harmonic components if the spatiotemporal modulation travels at the same speed as the signals. This outstanding benefit enables a low-frequency pump to excite parametric amplification. The realization of cascaded parametric amplification is demonstrated by finite-difference time-domain (FDTD) simulations and analytical calculations based on the Bloch–Floquet theory. We find that the same lasing state can always be excited by an incidence at different harmonic frequencies. The spectral and temporal responses of the space-time modulated slab strongly depend on the modulation length, modulation strength, and modulation velocity. Furthermore, the cascaded parametric oscillators composed of a cavity formed by photonic crystals are presented. The lasing threshold is significantly reduced by the cavity resonance. Finally, the excitation of cascaded parametric amplification relying on the Si-waveguide platform is demonstrated. We believed that the proposed mechanism provides a promising opportunity for the practical implementation of intense amplification and coherent radiation based on active metamaterials.

Broad-Angle Multichannel Metagrating Diffusers

We present a semianalytical scheme for the design of broad-angle multichannel metagratings (MGs), sparse periodic arrangements of loaded conducting strips (meta-atoms), embedded in a multilayer printed circuit board (PCB) configuration. By judicious choice of periodicity and angles of incidence, scattering off such an MG can be described via a multiport network, where the input and output ports correspond to different illumination and reflection directions associated with the same set of propagating Floquet–Bloch (FB) modes. Since each of these possible scattering scenarios can be modeled analytically, constraints can be conveniently applied on the modal reflection coefficients (scattering matrix entries) to yield a diffusive response, which, when resolved, produce the required MG geometry. We show that by demanding a symmetric MG configuration, the number of independent S parameters can be dramatically reduced, enabling satisfaction of multiple such constraints using a single sparse MG. Without any full-wave optimization, this procedure results in a fabrication-ready layout of a multichannel MG, enabling retroreflection suppression and diffusive scattering from numerous angles of incidence simultaneously. This concept, verified experimentally via a five-channel MG prototype, offers an innovative solution to both monostatic and bistatic radar cross Section (RCS) reduction, avoiding design and implementation challenges associated with dense metasurfaces used for this purpose.

Spectral-Element Spectral-Integral (SESI) Method for Doubly Periodic Problems With 3-D Scatterers Embedded in Multiple Regions of Layered Media

In this work, a hybrid 3-D spectral-element spectral-integral (SESI) method is developed by combining the spectral element method (SEM) and spectral integral method (SIM) for the fast evaluation of electromagnetic (EM) scattering from double Bloch (Floquet) periodic problems with 3-D scatterers embedded in multiple regions of layered media. This 3-D SESI method is a further extension of the previously developed 2-D SESI method. In the proposed method, the computational domain is divided into several SEM and SIM subdomains, where the subdomains are coupled by Riemann transmission conditions (RTCs) to allow the use of nonconforming meshes for these subdomains. Meanwhile, the 3-D periodic layered medium Green’s function (PLMGF) is derived and employed to simulate 3-D structures in layered media without their volumetric discretization. Four numerical experiments show that the developed SIM method is highly efficient and accurate; the SESI method is more efficient than the traditional finite element method (FEM), thus is a promising solver for designing and optimizing frequency selective surfaces (FSSs), metamaterials, and metasurfaces.

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.

Alternative Finite Element Eigenvalue Formulations for the Simulation of Arbitrarily Bianisotropic Media

We derive two alternative formulations for the implementation of arbitrarily bianisotropic materials for the finite element analysis of eigenmode problems. The e-formulation solves for the Bloch–Floquet periodic envelope of the electric field, and the e–b formulation computes the periodic envelopes of the electric field and magnetic induction. Implementation is carried out in in-house codes and in the Weak Form Module of COMSOL Multiphysics, exhibiting identical results. Extracted dispersion diagrams are compared to the existing solutions in the literature showing perfect agreement. Convergence rates between the proposed formulations are illustrated, with the scaled version of e–b formulation exhibiting faster convergence, due to its linear matrix system and its well-conditioned matrices.

Nonlinear Ultrasonic Detection Enhanced by 3-D Printed Phononic Crystals

Nonlinear ultrasonic methods are a novel non-destructive testing and evaluation technology due to their high sensitivity to micro-defects. However, the inherent nonlinearity of ultrasonic equipment hinders the feature extraction with micro-defects information and limits the full-scale use of nonlinear ultrasonic NDT techniques. In this work, we propose a material-enabled filter using phononic crystals (PCs) as functional units to eliminate the unapplicable inherent nonlinearity component. The PCs composed of mass blocks and connective layers symmetrical about the host plate, are simultaneously processed by melting deposition of PLA. The finite element method with Bloch–Floquet boundary condition is implemented to obtain the bandgap at 100 kHz. Based on the mass-spring model, the mechanism of bandgap formation is proved to be attributed to the negative effective mass density band induced by the local resonance of mass block. The PCs successfully opens a propagation wave transmission gap and eliminates the inherent nonlinearity to improve the nonlinear detection of micro-defects in the aluminum plate. This work provides a useful methodology for the wide application of PC-based nonlinear detection methods in the non-destructive testing field.

Semiclassical dynamics of electrons in a space-time crystal: Magnetization, polarization, and current response

A space-time crystal is defined as a quantum-mechanical system with both spatial and temporal periodicity. Such a system can be described by the Floquet-Bloch (FB) theory. We first formulate a semiclassical theory by constructing a wave packet through the superposition of the FB wave functions, and we derive the equations of motion of FB electrons subjected to slowly varying external fields (not to be confused with the fast-changing Floquet drive), revealing behaviors similar to ordinary Bloch electrons but with quantities modified in the Floquet context. Specifically, we study the local magnetic moment due to the self-rotation of the wave packet, a contribution to total magnetization from the Berry curvature in $\mathbf{k}$-space, and the polarization of a fully occupied FB band. Based on the semiclassical theory, we can also show the fingerprint of the energy flow in such an energy-nonconserved system. We then discuss the density matrix of a FB system attached to a thermal bath, which allows us to investigate quantities involving many electrons in the noninteracting limit. As an application, we calculate the intrinsic current response in an oblique space-time metal showing the nonequilibrium nature of the FB system. The current response can also be related to the acoustoelectric effect. Overall, we develop a systematic approach for studying space-time crystals, and we provide a powerful tool to explore the electronic properties of this exotic system with coupled space and time.

Negative refraction in a single-phase flexural metamaterial with hyperbolic dispersion

We analyze the band structure of a single-phase metamaterial involving low-frequency flexural resonances by combining asymptotic homogenization and Bloch–Floquet analysis. We provide the closed-form expression of the dispersion relation in the whole Brillouin zone. The dispersion relation involves two effective, frequency-dependent, mass densities associated with symmetric and antisymmetric flexural resonances of the beams at the microscopic scale. We demonstrate that our simple locally-resonant structure produces at low-frequency band-gaps and, in the hyperbolic regions of the dispersion diagram, negative refraction. Our findings are validated by direct numerical calculations.

Sound transmission of active elastic wave metamaterial with double locally resonant substructures surrounded by external mean flow

The elastic wave metamaterial with active feedback control is presented to discuss its dynamic effective parameter and sound transmission properties. Both studies are initiated from the local resonance with the active feedback control and dynamic structural response. The acoustic-structure coupling of the metamaterial with resonators and orthogonal stiffeners are analyzed with effects of the external mean flow. Different analytical methods are used to derive the effective mass density and sound transmission loss (STL) of the elastic wave metamaterial. The effective density is obtained by the dynamic effective medium method firstly. And the second one applies the spatial harmonic expansion method with the principle of virtual work for the STL based on the Bloch–Floquet theorem and Poisson summations. The effective mass density shows frequency-dependent properties and appears positive and negative values in a certain frequency region simultaneously. This research also illustrates both characteristics can be tuned by the active feedback control with the acceleration and displacement feedback control actions. Numerical results of coupled structures show that the changes of transmission characteristics in the certain frequency region are consistent with the effective mass density properties of uncoupled unit cell structure. In addition, influences of the lattice constants, incident angles and Mach number on the STL are illustrated and discussed.

Flexural wave propagation in periodic Micropolar-Cosserat panels: Spectral Element Formulation

In this research, a perspective of the dispersion relation of flexural waves on periodic structures is proposed. The rotational theory of micro-elements is stem to explore the favorable impact of size effect on structures through the localization of the strain energy. The panels are modelled as a one-dimensional (1-D) Micropolar–Cosserat (MC) continuum to explore the effect of length-scale or micro-rotational waves of solids. The spectral element formulation of a panel and the transfer matrix of the unit cell is presented within the framework of the state-space method. The propagation constant in the eigenvalue domain is established based on the Bloch–Floquet theorem to account for the periodicity of the panel. The MC theory in analyzing the band-gap (stop-band) and pass-band characteristics of the unit cell panels is validated by a commercial finite element software package-COMSOL Multiphysics. The mode shape of unit cell for the start and end point of band-gap is further analyzed within the realms of finite element method (FEM) analysis. The appropriate limiting condition reverts MC spectral element into Timoshenko (TM) beam model. The comparison of Timoshenko beam model with the spectral element of 1-D plane-stress (PS) analysis is also investigated. It is observed that the shear wave of the MC model is coupled with a micro-rotational wave of micro-elements, unlike the classical continuum framework. The response of the finite structure subjected to various frequencies is presented too. Presence of vibration attenuation in the specified band-gap (BG) is justified through the elemental dynamic stiffness (DS) matrix of unit cells. This reduced the level of vibration at the specific frequency range within the attenuation band or disintegrates it into two bands. The attenuation bandwidth and its location is significantly affected by the modulation of both geometry and material properties of the beam. Investigation of flexural wave propagation based on the proposed 1-D MC beam theory shows good agreement with the two-dimensional (2-D) plane-stress FEM analysis.