Infinitely Periodic Research Articles

Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains

The Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set P~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{P}$$\\end{document} of lifted points in the hyperbolic plane. Since P~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{P}$$\\end{document} is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than 12g-6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$12g-6$$\\end{document} with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.

Sparsity of Radiating Characteristic Modes on Infinite Periodic Structures

Characteristic modes on infinite periodic structures are studied using spectral dyadic Green's functions. This formulation demonstrates that, in contrast to the modal analysis of finite structures, the number of radiating characteristic modes is limited by unit cell size and incident wavevector (i.e., scan angle or phase shift per unit cell). The reflection tensor is decomposed into modal contributions from radiating modes, indicating that characteristic modes are a predictably sparse basis in which to study reflection phenomena.

The Comparative Review of Wave Properties of Finite and Infinite Periodic Crack Arrays

The Comparative Review of Wave Properties of Finite and Infinite Periodic Crack Arrays

Development of a Cyclic Periodic Wave Function Approach for the Study of Infinitely Periodic Solid-State Systems.

The ab initio cyclic periodic wave function (CPWF) approach is developed for the treatment of infinitely periodic systems. Using the full infinite Hamiltonian operator, as well as symmetrically identical basis set wave functions that preserve the translational symmetry of the electron density of the system, this approach can be applied at the Hartree–Fock level, or correlation can be directly included by the usual modes. In this approach, all many-body interactions are included, and no edge effects occur. Initial test calculations of the CPWF method at the ab initio Hartree–Fock level are performed on the chains of hydrogen fluoride molecules.

Equivalent Surface Impedance of an Infinite Periodic Array of Slot Impedance Loads Based on an Equilateral Triangle Cross-Section Cavity with a Dielectric Coating

In this paper, we consider a two-dimensional problem of determining the equivalent surface impedance of an infinite array of slot impedance loads with a dielectric layer. Each element of the array contains three regions: the first region occupies the entire half-space above the interface and the primary field is excited in this region by a plane wave; the second region does not contain excitation sources and is a dielectric layer; and the third region also does not contain exciting sources and is bounded by the walls of the cavity with a equilateral triangle cross section. The second and third regions are connected through one or several slots in a perfectly conducting screen located at their border. A strip conductor is located in the opening of each slot. The problem was solved by the method of integral equations, for the numerical implementation of which the Krylov–Bogolyubov method was used. Numerical results are presented as the dependences of the equivalent surface impedance on the geometric dimensions of the structure at fixed values ​​of the array period, the incidence angle of the electromagnetic wave, and the thickness of the dielectric layer. A comparative analysis of the obtained dependences with the characteristics of a similar array without a dielectric layer is performed.

A Cyclic Periodic Wave Function Approach for the Study of Infinitely Periodic Solid-State Systems. I. Application to the C-H···π(C≡C) Hydrogen Bonding Systems.

The cyclic periodic wave function (CPWF) approach is applied at the AM1 and PM3 semiempirical levels of approximation to infinitely periodic solid-state systems stabilized by weak CH–pi (C–H···π) interactions between repeat units. The reliability of the AM1 and PM3 methods for modeling C–H···π bonding is first demonstrated using two representative dimer systems: the T-shaped ethyne dimer and the T-shaped propyne dimer. The CPWF method is then applied to two different crystal systems that are stabilized by C–H···π interactions: (1) pent-4-ynoic acid solid and (2) a series of three infinite crystal systems—tetrakis(4-ethynylphenyl)methane solid, tetraethynylmethane solid, and tetrabutadiynylmethane solid. A comparison of our results with available data demonstrates that the use of the CPWF approach at the AM1 and PM3 levels of approximation provides a convenient and reliable method for the study of infinitely periodic systems containing very weak C–H···π bonding.

A Cyclic Periodic Wave Function Approach for the Study of Infinitely Periodic Solid-State Systems: II. Application to Helical Polysaccharides.

The cyclic periodic wave function (CPWF) approach is applied at the AM1 and PM3 semiempirical levels of approximation to two infinitely periodic polymer systems in the solid state. The two polysaccharides of interest here are (1→3)-β-d-glucan and (1→3)-β-d-xylan. Our calculated results show excellent agreement with the available data for the two polysaccharides and demonstrate that the use of the CPWF approach at the AM1 and PM3 levels of approximation provides a convenient and reliable method for the study of infinitely periodic bonds of two different types: moderately strong O–H···O hydrogen bonding and strong C–O–C covalent bonding.

Sagittal Plane Waves in Infinitely Periodic Multilayered Composites Composed of Alternating Viscoelastic and Elastic Solids

In order to design phononic crystals whose band-gaps are located in low-frequency ranges, researchers commonly adopt low stiffness polymeric materials as key constituents and exploit the high impedance mismatch between metals and polymers. However, there has been very little research on wave propagation at arbitrary angles in the sagittal plane of viscoelastic-elastic multilayered composites because there exist the intricate wave attenuation characteristics at the layer interfaces. The objective of our investigation is to obtain analytical dispersion relation for oblique wave motion in the sagittal plane of infinitely periodic multilayered composite composed of alternating viscoelastic and elastic solids, where the attenuation of harmonic plane waves is found to occur only in the direction perpendicular to the layers. By using this wave propagation characteristic, we directly apply the semi-analytical approach employed in elastic multilayered composites to calculate the dispersion relation of sagittal plane waves in alternating viscoelastic-elastic multilayered composites. Specifically, we consider a bilayered composite composed of alternating aluminum and polyurethane elastomer, whose complex-valued viscoelastic moduli are experimentally determined by performing dynamic mechanical analysis (DMA). The analysis shows that the alternating viscoelastic-elastic layered composite does not possess a phononic band-gap regardless of incident angles. In addition, wave motions at oblique angles are found to travel with a wide range of frequency contents compared to wave motions perpendicular to the layers. The presented analysis demonstrates that wave dispersion relation in viscoelastic-elastic layered composites is distinctly different from the corresponding elastic counterpart, and highlights the importance of the viscoelastic modeling of polymeric materials in wave dispersion analysis.

Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method

The finite element (FE) method offers an efficient framework to investigate the evolution of phononic crystals which possess materials or geometric nonlinearity subject to external loading. Despite its superior efficiency, the FE method suffers from spectral distortions in the dispersion analysis of waves perpendicular to the layers in infinitely periodic multilayered composites. In this study, the analytical dispersion relation for sagittal elastic waves is reformulated in a substantially concise form, and it is employed to reproduce spatial aliasing-induced spectral distortions in FE dispersion relations. Furthermore, through an anti-aliasing condition and the effective elastic modulus theory, an FE modeling general guideline is provided to overcome the observed spectral distortions in FE dispersion relations of infinitely periodic multilayered composites, and its validity is also demonstrated.

Induction of an Infinite Periodic Minimal Surface by Endowing An Amphiphilic Zwitterion with Halogen-Bond Ability.

We have designed an amphiphilic zwitterion with an iodine‐substituted imidazolium cation. Although it forms a layered assembly with flat interfaces, the addition of an equimolar amount of bis(trifluoromethane)sulfonimide results in the formation of a bicontinuous cubic liquid‐crystalline assembly with a primitive‐type infinite periodic minimal surface, where its zwitterionic headgroup sits regularly. IR measurements revealed that halogen bond between the iodine atoms on the imidazolium cation and the anions is involved in its molecular‐assembly behavior. The present results clearly indicate the potential utility of halogen bonding to control the dimensionality and continuity of the ionic/nonionic interface of amphiphiles in bulk and consequent mesophase patterns, which may be a significant new molecular technology for precisely arranging functional molecules on a 3D continuous interfaces.

On the complexity of aperiodic Fourier modal methods for finite periodic structures

The Fourier modal method (FMM) is based on Fourier expansions of the electromagnetic field and is inherently built for infinitely periodic structures. When the infinite periodicity assumption is not realistic, the finiteness of the structure has to be incorporated into the model. In this paper we discuss the recent extensions of the FMM for finite periodic structures and analyze their complexity both with respect to the main discretization parameter Nˆ as well as with respect to the number of periods R. We show that among the three FMM-based approaches able to represent finiteness, the aperiodic Fourier modal method with alternative discretization has the lowest computational cost given by either O(Nˆ3log2R) or O(Nˆ2R) depending on the values of Nˆ and R. This result demonstrates that the method is highly suited for rigorous modeling of scattering from large periodic structures. For instance, for Nˆ=100 and R<1000 the complexity of the aperiodic Fourier modal method with alternative discretization is comparable to the complexity of the standard FMM.

Transparent Boundary Conditions for Evolution Equations in Infinite Periodic Strips

We consider the solution of a generic equation $\gamma\rho(\mathbf{x})\partial^p_tu(\mathbf{x},t)-\Delta u(\mathbf{x},t) +V(\mathbf{x})u(\mathbf{x},t) = f(\mathbf{x},t)$, $\mathbf{x} = (x,y)$, for $t>0$, $p=1,2$ in a domain $\Omega$ which is infinite in $x$ and bounded in $y$. We assume that $f(\cdot,t)$ is supported for all $t>0$ in $\Omega_0 = \{\mathbf{x} \in \Omega \; | \; -a_- < x < a_+\}$ and that $\rho(\mathbf{x})$ and $V(\mathbf{x})$ are x-periodic in $\Omega \setminus \Omega_0$. We consider the associated $\theta$-scheme in time, to obtain a semidiscretized problem. We then show how to obtain for each time step exact boundary conditions on the vertical segments, $\Gamma_0^- = \{\mathbf{x}\in \Omega\; | \; x=-a_-\}$ and $\Gamma_0^+ = \{\mathbf{x}\in \Omega \;| \; x=a_+\}$, that will enable us to find the solution on $\Omega_0 \cup \Gamma_0^+ \cup \Gamma_0^-$. Then the solution can be extended in $\Omega$ in a straightforward manner from the values on $\Gamma_0^-$ and $\Gamma_0^+$. The method is ba...

New Formula for Evaluating the Number of Unit Cell of a Finite Periodic Structure Considered as Infinite Periodic One

This paper presents a modeling and an analysis of one-dimensional periodic structure composed of a cascade connection of N cells considered as infinite. The ABCD matrix representations with the Floquet analysis have been used to derive the dispersion relation and input impedance of infinite periodic structure. The transmission matrix for the N identical cascaded cells has been successfully used to obtain an efficient and easy-to-use formula giving the necessary number of cells such that they can be considered infinite. As an illustrative example, the formula is applied and verified to finite size TL periodically loaded with obstacles. Scattering parameters and the input impedance of the structure are expressed and plotted.

Method of Moments Simulation of Infinitely Periodic Structures Combining Metal With Connected Dielectric Objects

A method of moments (MoM) technique is described for the simulation of infinitely periodic structures made of complex elements, involving both metal and dielectric parts. A numerical approach based on the MoM was developed, using the EFIE formulation for metallic objects and a modified PMCHWT formulation for the air-dielectric interfaces. The method is also based on an efficient algorithm for the computation of the singly and doubly periodic Green's functions and their gradients. The use of such functions for both inside and outside problems enables the analysis of. structures where the dielectric volumes of successive cells are connected. The accuracy of the MoM approach is assessed in several cases, based on energy conservation checks and on comparisons with analytical models and data available in the literature. Examples are shown for arrays of disconnected and connected dielectric objects, dielectric electromagnetic band-gap (EBG) superstrates and dielectric slabs. Examples with metallic parts embedded in connected dielectric material are given in the case of frequency selective surfaces (FSS) and tapered-slot antennas.

Band gaps and elastic waves in disordered stacks: normal incidence

We present a development of previous work where we studied the reflection and transmission properties of an array of circular cylindrical voids in an elastic matrix. The voids were taken to be infinitely long, equally spaced and of the same radius. In this paper we improve the method to deal with the problems of inclusions as well as voids and study various forms of disorder within the array, as well as stack designs showing good low–pass and band–pass characteristics in transmission.

Electron density modeling and reconstruction of infinite periodic minimal surfaces (IPMS) based phases in lipid-water systems. II. Reconstruction of D surface based phases

This is the second of two papers dealing with the structure of lipid-water phases based on Infinite Periodic Minimal Surfaces (IPMS). The first paper describes mathematical modeling of such phases. In this paper, a new reconstruction method, called the methyl trough search, is described and used to solve the structures based on powder pattern X-ray diffraction data. Structures are derived for both a single chain lipid-water system (mono-olein) and a diacyl phospholipid-water system (2-2 methyl butyl 16:0 phosphatidylcholine). The methyl trough search uses the low electron density of the lipid methyl tails to determine the correct phasing for the electron density reconstruction. The data are consistent with a structure based on the IPMS D surface. The results are compared to other methods used to solve the mono-olein structure; the structure of the diacyl lipid has never before been solved. We discuss the subtleties involved in reconstruction of D surface based phases and the substantial artifacts that arise in low-resolution reconstructions of hydrocarbon lipids lacking heavy-atom sites.

The Micellar Cubic Phases of Lipid-Containing Systems: Analogies with Foams, Relations with the Infinite Periodic Minimal Surfaces, Sharpness of the Polar/Apolar Partition

Des 7 phases cubiques clairement identifiees dans les systemes lipidiques, 2 sont bicontinues, 4 micellaires. Parmi ces dernieres, 3 sont du type I: une (Q 223 ) comporte deux types de micelles, deux autres sont formees de micelles identiques quasi-spheriques, empaquetees selon le mode face-centre (Q 225 ) ou centre (Q 229 ). Tout comme les mousses, ces structures peuvent etre decrites en termes d'assemblages denses de polyhedres: 12-hedres et 14-hedres deformes dans Q 223 , dodecahedres rhombiques dans Q 225 , octahedres tronques dans Q 229 . Dans les mousses la geometrie des faces et de leurs jonctions est censee obeir aux conditions de Plateau, du moins aux faibles teneurs en eau: ces conditions sont satisfaites dans Q 223 , peuvent etre satisfaites dans Q 229 au prix d'une petite distortion des faces hexagonales, mais il est impossible de les satisfaire dans Q 225 . On peut egalement representer ces structures en termes de surfaces minimales periodiques et infinies (IPMS): on trouve, en effet, que deux types de IPMS, F-RD pour Q 225 et I-WP pour Q 229 , coincident avec l'une des surfaces d'egale densite electronique des cartes 3D. Ces IPMS partagent l'espace en deux labyrinthes non congruents: dans les phases micellaires un de ces labyrinthes contient les micelles hydratees, l'autre l'eau. Si on associe a ces surfaces des interactions interfaciales, on peut alors penser que ces interactions sont minimales puisque les surfaces sont minimales. Une autre caracteristique des phases micellaires est que le rayon de leur noyau apolaire, determine en faisant l'hypothese que les chaines paraffiniques sont totalement immiscibles avec les tetes polaires et l'eau, depasse souvent la longueur des chaines etirees. On peut eviter ce paradoxe en admettant que chaines et tetes polaires sont partiellement miscibles entre elles.

Free Wave Propagation In Rotationally Restrained Infinite Periodic Beams

A suitably chosen deflection function is used to analyze the free vibration of rotationally restrained infinite periodic beams on transversely rigid supports by the wave approach. The assumed complex modes of wave motion which satisfy the wave boundary conditions are used in a Galerkin type of analysis to predict the vibration frequencies in any desired pair of adjacent propagation bands for a wide range of rotational constraint parameter. Since the function satisfies the geometric and force boundary conditions of the beam, it yields better results than those that can be obtained by using functions satisfying only the geometric boundary conditions. The function can be used to formulate deflection functions for plates for analyzing two-dimensional wave propagation in periodic plates, rotationally restrained at the transversely rigid supports.

New surface patches for minimal balance surfaces. IV. Catenoids with spout-like attachments

The way in which Schoen [Infinite Periodic Minimal Surfaces Without Self-intersections (1970), NASA Tech. Note No. D-5541] derived a simply connected surface patch for a C(H) surface cannot be generalized. One may, however, subdivide a C(H) surface into larger patches that are not simply connected. Surface patches of analogous shape give rise to five families of minimal balance surfaces unknown so far: tetragonally and orthorhombically distorted C(P) surfaces, surfaces complementary to Schoen's R2 and R3 surfaces with genus 25 and 37, respectively, and orthorhombic surfaces of a fifth family with genus 5 that are also complementary to oP surfaces.

AB Initio Calculation of the Polarizability and Hyperpolarizabilrties of Infinite Periodic Polymers by the Sum-Overstates approach

ABSTRACTWe investigate the applicability of the Genkin-Mednis perturbative approach to the ab initio calculation of the static (hyper)polarizabilities of infinite periodic polymers. Test calculations on molecular hydrogen and polyacetylene chains are presented to assess some of the computational problems inherent to the method.