In this article, defect state of photonic crystal regarding parity time known as PT-symmetric is explored thoroughly. For this purpose, transfer matrix technique in setting Bloch theorem is employed ...to characterize the dispersion relation regarding PT-symmetric defect state. The study uncovers the norteworthy wave phenomenon entitle “non-reciprocity” is exhibited. Moreover, an increasing the in-plane component of wavevector contributes two or more branches in the short wavelength boundary of the stop-band which is not observed in conventional structures. The reflectance of structure is conducted across the optical regime leads to minimum point that aligns with the wavelength of the eigen mode for wavevector value (kx = 0). These findings of research may have significant applications in various areas of photonics and optics likes filters, modern optical photonic cavity resonators and thermo-photovoltaic devices etc.
Wider bandgap and band merging phenomena in simultaneous negative mass and stiffness metamaterial were reported and studied previously. In this letter, an additional feature, namely double ...attenuation peak, of simultaneous negative mass and stiffness metamaterial has been identified. The generation of double attenuation peaks is hinged upon the resonance coupling of longitudinal and transverse resonator. Double attenuation peaks ensure a significant level of spatial attenuation throughout the attenuation band.
•Elucidates the double peak attenuation phenomenon in dual negative metamaterial.•Double peak attenuation band provides significant level of spatial attenuation.•Resonance coupling is the main cause of the double attenuation peaks in a single band.•Criterion for this phenomenon is also derived.
Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional (2D) hyperbolic space, a non-Euclidean space of uniform ...negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; whether a formal Bloch theorem can be rigorously established; and whether hyperbolic band theory can describe finite lattices realized in an experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps correspond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only 1D irreps appear, and the hyperbolic band theory previously developed becomes exact.
After recalling the first Brillouin zone (BZ) and the first irreducible Brillouin zone (IBZ) of a lattice in terms of its plane crystallographic group, we investigate the danger of restricting a ...band-gap detection to the contour of the IBZ, instead of its full IBZ. Based on hundreds of porous phononic crystal simulations, we provide for the 17 plane crystallographic groups (i) statistics of the band-gap localizations, (ii) probabilities to get non-full band-gaps, and (iii) averages of the bandwidth error made when only the IBZ contour is considered. It is found that for phononic crystals, the IBZ contour provides accurate results only for highly symmetric lattices.
Periodic structures attenuate wave propagation in a specified frequency range, such that a desired bandgap behaviour can be obtained. Most periodic structures are produced by additive manufacturing. ...However, it is recently found that the variability in the manufacturing processes can lead to a significant deviation from the desired behaviour. This paper investigates the elastic wave propagation of stochastic hexagonal periodic lattice structures considering micro-structural variability. Thus, the effect of uncertainties in the material and geometrical parameters of the unit cell is quantified on the wave propagation in hexagonal lattices. Based on Bloch’s theorem and the finite element method, the band structures are determined as a function of the frequency and wave vector at the unit cell level and later scaled-up via full-scale simulations of finite metamaterials with a prescribed number of cells. State of the practice machine learning techniques, namely the Gaussian process, multi-layer perceptron, radial basis neural network and support vector machine, are employed as grey-box meta-models to capture the stochastic wave propagation response. The results demonstrate good accuracy by validation with Monte Carlo simulations. The study illustrates that considering the effect of uncertainties on the wave propagation behaviour of hexagonal periodic lattices is critical for their practical applicability and desirable performance. Based on the results, the manufacturing tolerances of the hexagonal lattices can be obtained to attain a bandgap within a certain frequency band.
The extended plane wave expansion (EPWE) method is formulated in order to obtain the complex dispersion diagram of bulk waves in 1-D viscoelastic phononic crystal solids considering the standard ...linear solid model (SLSM). This new formulation is important to handle evanescent Bloch waves in 1-D viscoelastic periodic structures for mechanical wave attenuation. The viscoelastic effect can increase the unit cell wave attenuation in 1-D periodic structures. In addition, the increase of relaxation time (up to ωa/2πct=7) and the decrease of final state of shear modulus (considering ωa/2πct≤1.9) enhance the unit cell wave attenuation.
A new model for determining band gaps for elastic wave propagation in three-dimensional (3-D) periodic two-phase composites is developed using a modified couple stress theory that accounts for ...microstructure effects. Three types of composites, each containing a different kind of inclusion – spherical, cubic, and cube with square-rod connections, are considered, with the third one representing a co-continuous composite. The plane wave expansion method and the Bloch theorem for periodic media are employed to solve the elastic wave equations in each case, which are converted to an eigenvalue problem. The band gaps are determined from solving the characteristic equation and plotting the resulting eigen-frequencies. The new non-classical model reduces to the classical elasticity-based model when microstructure effects are suppressed. To quantitatively illustrate the newly developed model, a parametric study is conducted for 3-D periodic composites with the three kinds of inclusions. The numerical results reveal that the first band gap values predicted by the current non-classical model are smaller than those predicted by the classical elasticity-based model, and the difference between the two sets of band gap values is large when the unit cell size is very small. Also, it is seen that the volume fraction and inclusion shape have significant effects on the band gap size. These indicate that large band gaps can be attained by tailoring microstructural parameters including the unit cell size, volume fraction and inclusion shape.
Flexural wave propagation through a sandwich beam, composed of two parallel Euler–Bernoulli beams connected with the translational and rotational springs at a periodic interval (acronym ...metasandwich), is analytically investigated in this paper. Dimensionless governing parameters are obtained by non-dimensionalizing the governing differential equations. Band structure of the metasandwich has been obtained by implementing Bloch’s theorem in conjunction with the transfer matrix. The two parallel beams are modeled separately, which demonstrated the key novelty of the paper. A new band categorization criteria of the wave phenomena in metasandwich is proposed within the scope of this paper. Furthermore, an exploration of the parametric space has also been conducted to observe the variation of the band structure. The underlying physics of the wave propagation or attenuation in the different bands are conceptualized from its respective mode shapes and energy plots. The similarity between the mode shapes of the parallel beams has been analyzed by using cross recurrence plots.
•Analytical solution of sandwich metastructure.•New bandgap categorization criterion.•Insights to band attenuation using mode shape, total energy and recurrence plots.
This article presents an efficient reduced formulation of the Bloch Operator Finite Element method to calculate complex band structures of periodic waveguides. The use of a Bloch operator formulation ...allows building and solving a Bloch eigenvalue problem along a generic wave direction, thus being not limited to the unit cell Irreducible Brillouin Zone (IBZ) edges, so that band gap directionality and material absorption in elastic and damped waveguides can be fully disclosed. The proposed Reduced-Order Modeling (ROM) exploits a small set of Bloch modes, extracted at relevant frequency locations along one or more wave directions and post-processed with a Singular Value Decomposition, to reduce the dimensions of the eigenvalue problem. The performances of the proposed numerical technique are evaluated in terms of accuracy and computational saving by analyzing a linear elastic and a damped bi-periodic stubbed plate. Results demonstrate that the reduced formulation yields accurate predictions of propagative, evanescent and complex wave solutions with a reduction in computational time of more than one order of magnitude with respect to the full model calculations. Complex band structures can thus be efficiently computed over the whole IBZ.
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