Homogenization of periodic structures via Bloch decomposition

In this paper, the classical problem of homogenization of elliptic operators in arbitrary domains with periodically oscillating coefficients is considered. Using Bloch wave decomposition, a new proof of convergence is furnished. It sheds new light and offers an alternate way to view the classical results. In a natural way, this method leads us to work in the Fourier space and thus in a framework dual to the one used by L. Tartar [Problemes d'Homogeneisation dans les Equations aux: Derivees Partielles, Cours Peccot au College de Prance, 1977] in his method of homogenization. Further, this technique offers a nontraditional way of calculating the homogenized coefficients which is easy to implement in the computer.

Triply-periodic balance surfaces

The list of possible G-H symmetries for triply-periodic balance surfaces, given by Koch and Fischer, is studied with a view to finding new types. It is found that balance surfaces exist for all the cases that were labelled as undecided in the work of Koch and Fischer. For some of these, new minimal surfaces have been found, with the aid of Brakke's 'Surface Evolver'. (C) 1997 Elsevier Science B.V.

Detection of a periodic structure embedded in surface roughness, for various correlation functions

This paper deals with surface profilometry, where we try to detect a periodic structure, hidden in randomness using the matched filter method of analysing the intensity of light, scattered from the surface. From the direct problem of light scattering from a composite rough surface of the above type, we find that the detectability of the periodic structure can be hindered by the randomness, being dependent on the correlation function of the random part. In our earlier works, we had concentrated mainly on the Cauchy-type correlation function for the rough part. In the present work, we show that this technique can determine the periodic structure of different kinds of correlation functions of the roughness, including Cauchy, Gaussian etc. We study the detection by the matched filter method as the nature of the correlation function is varied.

Mode-coupling glass transition in a fluid confined by a periodic potential

We show that a fluid under strong spatially periodic confinement displays a glass transition within mode-coupling theory at a much lower density than the corresponding bulk system. We use fluctuating hydrodynamics, with confinement imposed through a periodic potential whose wavelength plays an important role in our treatment. To make the calculation tractable we implement a detailed calculation in one dimension. Although we do not expect simple 1d fluids to show a glass transition, our results are indicative of the behavior expected in higher dimensions. In a certain region of parameter space we observe a three-step relaxation reported recently in computer simulations [S. H. Krishnan, Ph.D. thesis, Indian Institute of Science (2005); Kim et al., Eur. Phys. J. Special Topics 189, 135 (2010)] and a glass-glass transition. We compare our results to those of Krakoviack [Phys. Rev. E 75, 031503 (2007)] and Lang et al. [Phys. Rev. Lett. 105, 125701 (2010)].

Quenching across quantum critical points in periodic systems: Dependence of scaling laws on periodicity

We study the quenching dynamics of a many-body system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin-1/2 chain with equal xx and yy couplings and subject to a periodically varying magnetic field in the (z) over cap direction or, equivalently, a tight-binding model of spinless fermions with a periodic local chemical potential, having period 2q, where q is a positive integer. For a linear quench of the strength of the magnetic field (or chemical potential) at a rate 1/tau across a quantum critical point, we find that the density of defects thereby produced scales as 1/tau(q/(q+1)), deviating from the 1/root tau scaling that is ubiquitous in a range of systems. We analyze this behavior by mapping the low-energy physics of the system to a set of fermionic two-level systems labeled by the lattice momentum k undergoing a nonlinear quench as well as by performing numerical simulations. We also show that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of tau, although it may exhibit a crossover at intermediate values of tau. Finally, for the case where a zz coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of q and the interaction strength.

A unified model for the dynamics of driven ribbon with strain and magnetic order parameters

We develop a unified model to explain the dynamics of driven one dimensional ribbon for materials with strain and magnetic order parameters. We show that the model equations in their most general form explain several results on driven magnetostrictive metallic glass ribbons such as the period doubling route to chaos as a function of a dc magnetic field in the presence of a sinusoidal field, the quasiperiodic route to chaos as a function of the sinusoidal field for a fixed dc field, and induced and suppressed chaos in the presence of an additional low amplitude near resonant sinusoidal field. We also investigate the influence of a low amplitude near resonant field on the period doubling route. The model equations also exhibit symmetry restoring crisis with an exponent close to unity. The model can be adopted to explain certain results on magnetoelastic beam and martensitic ribbon under sinusoidal driving conditions. In the latter case, we find interesting dynamics of a periodic one orbit switching between two equivalent wells as a function of an ac magnetic field that eventually makes a direct transition to chaos under resonant driving condition. The model is also applicable to magnetomartensites and materials with two order parameters. (C) 2013 American Institute of Physics. http://dx.doi.org/10.1063/1.4790845]

Majorana fermions in superconducting 1D systems having periodic, quasiperiodic, and disordered potentials

We present a unified study of the effect of periodic, quasiperiodic, and disordered potentials on topological phases that are characterized by Majorana end modes in one-dimensional p-wave superconducting systems. We define a topological invariant derived from the equations of motion for Majorana modes and, as our first application, employ it to characterize the phase diagram for simple periodic structures. Our general result is a relation between the topological invariant and the normal state localization length. This link allows us to leverage the considerable literature on localization physics and obtain the topological phase diagrams and their salient features for quasiperiodic and disordered systems for the entire region of parameter space. DOI: 10.1103/PhysRevLett.110.146404

On the zeta function of a periodic-finite-type shift

Periodic-finite-type shifts (PFT's) are sofic shifts which forbid the appearance of finitely many pre-specified words in a periodic manner. The class of PFT's strictly includes the class of shifts of finite type (SFT's). The zeta function of a PET is a generating function for the number of periodic sequences in the shift. For a general sofic shift, there exists a formula, attributed to Manning and Bowen, which computes the zeta function of the shift from certain auxiliary graphs constructed from a presentation of the shift. In this paper, we derive an interesting alternative formula computable from certain ``word-based graphs'' constructed from the periodically-forbidden word description of the PET. The advantages of our formula over the Manning-Bowen formula are discussed.

Periodic architecture for high performance shock absorbing composites

A novel composite architecture consisting of a periodic arrangement of closely-spaced spheres of a stiff material embedded in a soft matrix is proposed for extremely high damping and shock absorption capacity. Efficacy of this architecture is demonstrated by compression loading a composite, where multiple steel balls were stacked upon each other in a polydimethylsiloxane (PDMS) matrix, at a low strain-rate of 0.05 s(-1) and a very high strain-rate of >2400 s(-1). The balls slide over each other upon loading, and revert to their original position when the load is removed. Because of imposition of additional strains into the matrix via this reversible, constrained movement of the balls, the composite absorbs significantly larger energy and endures much lesser permanent damage than the monolithic PDMS during both quasi-static and impact loadings. During the impact loading, energy absorbed per unit weight for the composite was, 8 times larger than the monolithic PDMS.

A Guided Wave approach for the detection of damage in a structure having elements with periodic damage

A wave propagation based approach for the detection of damage in components of structures having periodic damage has been proposed. Periodic damage pattern may arise in a structure due to periodicity in geometry and in loading. The method exploits the Block-Floquet band formation mechanism, a feature specific to structures with periodicity, to identify propagation bands (pass bands) and attenuation bands (stop bands) at different frequency ranges. The presence of damage modifies the wave propagation behaviour forming these bands. With proper positioning of sensors a damage force indicator (DFI) method can be used to locate the defect at an accuracy level of sensor to sensor distance. A wide range of transducer frequency may be used to obtain further information about the shape and size of the damage. The methodology is demonstrated using a few 1-D structures with different kinds of periodicity and damage. For this purpose, dynamic stiffness matrix is formed for the periodic elements to obtain the dispersion relationship using frequency domain spectral element and spectral super element method. The sensitivity of the damage force indicator for different types of periodic damages is also analysed.

On existence of periodic solutions for stable interval plants with odd, sector type nonlinearities

In several systems, the physical parameters of the system vary over time or operating points. A popular way of representing such plants with structured or parametric uncertainties is by means of interval polynomials. However, ensuring the stability of such systems is a robust control problem. Fortunately, Kharitonov's theorem enables the analysis of such interval plants and also provides tools for design of robust controllers in such cases. The present paper considers one such case, where the interval plant is connected with a timeinvariant, static, odd, sector type nonlinearity in its feedback path. This paper provides necessary conditions for the existence of self sustaining periodic oscillations in such interval plants, and indicates a possible design algorithm to avoid such periodic solutions or limit cycles. The describing function technique is used to approximate the nonlinearity and subsequently arrive at the results. Furthermore, the value set approach, along with Mikhailov conditions, are resorted to in providing graphical techniques for the derivation of the conditions and subsequent design algorithm of the controller.

Zero miss distance guidance using feedforward and periodic control

There have been attempts at obtaining robust guidance laws to ensure zero miss distance (ZMD) for interceptors with parametric uncertainties. All these laws require the plant to be of minimum phase type to enable the overall guidance loop transfer function to satisfy strict positive realness (SPR). The SPR property implies absolute stability of the closed loop system, and has been shown in the literature to lead to ZMD because it avoids saturation of lateral acceleration. In these works higher order interceptors are reduced to lower order equivalent models for which control laws are designed to ensure ZMD. However, it has also been shown that when the original system with right half plane (RHP) zeros is considered, the resulting miss distances, using such strategies, can be quite high. In this paper, an alternative approach using the circle criterion establishes the conditions for absolute stability of the guidance loop and relaxes the conservative nature of some earlier results arising from assumption of in�nite engagement time. Further, a feedforward scheme in conjunction with a lead-lag compensator is used as one control strategy while a generalized sampled hold function is used as a second strategy, to shift the RHP transmission zeros, thereby achieving ZMD. It is observed that merely shifting the RHP zero(s) to the left half plane reduces miss distances signi�cantly even when no additional controllers are used to ensure SPR conditions.