Triangular Domain Research Articles

Adhesive Microstructures: Anisotropic Adhesion Properties of Triangular‐Tip‐Shaped Micropillars (Small 16/2011)

The cover image illustrates the anisotropic adhesion property of triangular-tip-shaped micropillars, which are proposed as a new design for bio-inspired adhesive structures. These unique micropillars show directional shear adhesion with respect to the change of peeling direction (8.8–17.2 J m−2) as well as relatively high normal adhesion (8.2–10 N m−2). The directional adhesion is ascribed to the propagation line of cracks under triangular contact domains. When cracks initiate from the vertex, they can easily propagate to the inner region, resulting in relatively lower shear adhesion. In contrast, when peeled from the side of the triangle, the cracks form a pseudoline contact, and thus the crack propagation is retarded. For more information, please read the Communication “Anisotropic Adhesion Properties of Triangular-Tip-Shaped Micropillars” by K. Y. Suh and co-workers, beginning on page 2296.

Inverted intracontinental basin and vertical tectonics: The Saharan Atlas in Tunisia

The present-day architecture of the Saharan Atlas in Tunisia can be defined by two principal models: (1) The first model emphasizes a general SW–NE geological structure in the North forming successive and parallel bands (the Tellian zone, the diapir zone) and the central Atlas, which are cut by the southern Atlas ranges located within a NW–SE corridor. These zones are bordered to the East by the “North–South Axis”. (2) The second model defines the Tunisian Atlas in terms of an E–W strike-slip corridor, which initially controls the sedimentary facies distribution during the Meso-Cenozoïc, and which then generates elongate en echelon folds in the sedimentary cover by dextral shearing. In this study, we aim to show that the Saharan Atlas in Tunisia appears today as a triangular megablock, that we call the Tunisian Block (TB), bounded by three structural trends (N–S, SW–NE and NW–SE) belonging to the African strike-slip fault network: (1) The eastern boundary appears as a complex faulted and folded corridor limiting the folded zone of the central Atlas in the West and the depressed zone of the Sahel in the East: it corresponds to the “North–South Axis” as defined classically in the literature. (2) The southern boundary also corresponds to a faulted belt (Gafsa–Negrine-Tozeur corridor), which cuts off the continuation of the North–South axis southward into the Gabès region; it corresponds to the Southern Saharan Atlas, delimited by the Gafsa fault in the North and the Negrine-Tozeur fault in the South. (3) The northern boundary, trending SW–NE, appears rather in the form of a reverse tectonic bundle, facing SE or S (oblique convergence), whose major feature corresponds to the El Alia-Téboursouk fault. This northern boundary cuts across and delimits the N–S corridor towards the North, in such a way that its extension is limited at both extremities. Finally, the inner part of the TB actually corresponds to a mosaic of second-order blocks, each of which contains an arrangement of widely spaced SW–NE trending anticlines forming the main relief separated by vast plains very often occupied by sebkhas. The paleogeographic and structural evolution of this region during the Mesozoic and Palaeogene shows that the TB, along with its limits as defined here, developed an increasingly distinct identity at a very early stage, being characterized by an extensive and/or transtensive tectonic regime. Finally, the Tunisian Atlas Chain defines a triangular domain that owes its origin and particular character precisely because of the paleogeographic and structural history of this paleoblock. The boundaries of this paleoblock remain mobile, thus tectonically controlling the geometry and morphology of a typical intracontinental basin. The extension directions and the frequent changes of stress regime (or rotations) are related to the existence of two active basins: the strike-slip margin of the western branch of Tethys and the Mesogea oceanic basin, with tectonic activity becoming alternately dominant in one or other of the basins at different times. In this context, the Tunisian basin is characterized by rhythmic sedimentation, composed of a succession of filling sequences linked to the continuing tectonic instability of the sedimentary floor associated with two major crises: one at the end of the Aptian and the other at the end of the Ypresian. The vertical movements related to the extension and/or transtension of the blocks is accentuated by Triassic salt tectonics, giving rise to linear (salt axes) or point (salt domes) structures that lead to the formation of shoal zones during development of the basin, thus enhancing the vertical tectonics. The diapirism developed slowly and gradually from late Triassic through to Langhian times, leading to numerous sedimentary wedges on the flanks of the structures. The uprise of the diapirs exhibits three pauses corresponding to the end-Aptian, end-Ypresian and pre-Burdigalian. The vertical tectonics is characterized by abundant drape folds giving rise to an extensional fault-related folding and strike-slip/dip-slip faults creating frequent unconformities that are nevertheless always localized. Finally, the folded chain results from the structural inversion of this paleoblock from Tortonian times onward. We can only account for the various folds-axis directions in the context of an intracontinental chain where the pre-existing major vertical faults are able to develop on the surface as draped-folds in a transpressive regime by the local reorientation of stresses in crustal-scale faults. In detail, the structures produced by this vertical tectonic activity, which are profoundly controlled by inheritance, display a highly original style at very shallow levels in the crust.

Triangular domain extension of algebraic trigonometric Bézier-like basis

In computer aided geometric design (CAGD), Bezier-like bases receive more and more considerations as new modeling tools in recent years. But those existing Bezier-like bases are all defined over the rectangular domain. In this paper, we extend the algebraic trigonometric Bezier-like basis of order 4 to the triangular domain. The new basis functions defined over the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry, boundary representation, linear independence and so on. We also prove some properties of the corresponding Bezier-like surfaces. Finally, some applications of the proposed basis are shown.

Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions

For the Goursat problem, we consider a triangular domain with mixed Dirichlet and impedance boundary conditions imposed on it. We develop an algorithm for its numerical solution mainly based on Runge–Kutta method and trapezoidal formula. Iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition. Error estimates are derived. Examples are presented to illustrate the effectiveness of the method.

Microstructural and magnetic properties of (La0.7Sr0.3MnO3)0.7:(Mn3O4)0.3 nanocomposite thin films

Epitaxial (La0.7Sr0.3MnO3)0.7:(Mn3O4)0.3 (LSMO:Mn3O4) nanocomposite thin films were grown on SrTiO3 (001) substrate by a pulsed laser deposition technique. The nanocomposite structures vary from triangular domains, to vertically aligned columns, and finally to smaller spherical domains as the deposition frequency varies from 1, 5, to 10 Hz, respectively. The strain in LSMO is systematically tuned, but that of the Mn3O4 phase is relatively stable as the deposition frequency increases. The tunable strain is found directly related to the different domain and grain boundary (GB) structures. Physical properties including saturation magnetization, Curie temperature (TC), magnetoresistance and metal–insulator transition temperature (TMI), all show systematic trends as the deposition frequency varies. This study reveals that the domain/GBs tunability achieved in nanocomposite thin films can affect the lattice strain and further tune their ferromagnetic properties.

Shape of ferroelectric domains in LiNbO3 and LiTaO3 from defect/domain-wall interactions

The observed orientations and shapes of ferroelectric domains in stoichiometric and congruent LiTaO3 and LiNbO3 are explained in terms of the differences in the energetics of domain walls, and the energetics of defect/domain-wall interactions. Density functional theory calculations show that the energy of a Y-wall, parallel to (112¯0), is lower than that of the X-walls, parallel to (101¯0), in both stoichiometric LiNbO3 and LiTaO3. As a result the domains are hexagonal and delineated by Y-walls. In congruent LiTaO3, the interaction of antisite defects with domains walls can reverse this order of stability, leading to triangular domains delineated by one type of X-wall. In congruent LiNbO3, the interactions of the antisite defects with the domain walls are too weak to reverse the order of stability of the domains walls; as a result the domains remain hexagonal.

Bivariate polynomial and continued fraction interpolation over ortho-triples

By means of the barycentric coordinates expression of the interpolating polynomial over each ortho-triple, some properties are obtained. Moreover, the explicit coefficients in terms of B-net for one ortho-triple, and two ortho-triples are worked out, respectively. Thus the computation of multiple integrals can be converted into the sum of the coefficients in terms of the B-net over triangular domain much effectively and conveniently. Based on a new symmetrical algorithm of partial inverse differences, a novel continued fractions interpolation scheme is presented over arbitrary ortho-triples in R 2, which is a bivariate osculatory interpolation formula with one-order partial derivatives at all corner points in the ortho-triples. Furthermore, its characterization theorem is presented by three-term recurrence relations. The new scheme is advantageous over the polynomial one with some numerical examples.

Constructing C1 triangular patch of degree six by the Boolean sum of an approximation operator and an interpolation operator

A new method to construct C 1 triangular patches which satisfy the given boundary curves and cross-boundary slopes is presented. The Boolean sum of an approximation operator and an interpolation operator is employed to construct the triangular patch. The approximation operator is used to construct a polynomial patch of degree six. The polynomial of degree six affords more freedoms, which makes the approximation operator not only approximate the given boundary interpolation conditions but also have a better approximation precision in the interior of the triangle, so that the triangular patch has a better precision on both the boundary and the interior of the triangular domain. The interpolation operator is utilized to build an interpolation patch which satisfies the given boundary conditions. The Boolean sum of the approximation and interpolation patches forms the triangular patch. Comparison results of the new method with other three methods are given.

Charge-driven structural transformation and valence versatility of boron sheets in magnesium borides

Based on density-functional theory simulations, we have predicted a series of stable magnesium borides, MgB${}_{x}$, with a broad range of stoichiometries, $2<x\ensuremath{\leqslant}16$, by removing magnesium atoms from MgB${}_{2}$. The layered boron structures are preserved through an in-plane topological transformation of the hexagonal lattice domains to triangular lattice domains. The process can be reversibly switched as the charge transfer changes with Mg insertion and extraction. The mechanism of such a charge-driven transformation originates from the versatile valence state of boron in its planar form. The discovery of these new physical phenomena suggests the design of a high-capacity magnesium boron battery.

Effects of electron irradiation on the ferroelectric 180° in-plane nanostripe domain structure in a thin film prepared from a bulk single crystal of BaTiO3 by focused ion beam

Effects of electron irradiation on the ferroelectric 180° in-plane nanostripe domain structure in a thin film prepared from a bulk single crystal of BaTiO3 by focused ion beam were studied. The domain structure transformed into a characteristic 90° in-plane nanostripe domain structure under intense electron irradiation. In particular, an unconventional triangular 90° in-plane nanostripe domain structure was observed. Polarization analysis suggests the existence of an incomplete or half tetradomain vortex at the boundary of the triangular domain structure. Together with the help of phase-field simulations using time-dependent Ginzburg–Landau equations, it is suggested that such a domain structure is created by an anisotropic in-plane electric field, which is plausibly induced by an anisotropic interaction of the incident electron beam with the ferroelectric material.

Green’s Function Technique and Global Optimization in Reconstruction of Elliptic Objects in the Regular Triangle

The reconstruction problem for elliptic voids located in the regular (equilateral) triangle is studied. A known point source is applied to the boundary of the domain, and it is assumed that the input data is obtained from the free-surface input data over a certain finite-length interval of the outer boundary. In the case when the boundary contour of the internal object is unknown, we propose a new algorithm to reconstruct its position and size on the basis of the input data. The key specific character of the proposed method is the construction of a special explicit-form Green's function satisfying the boundary condition over the outer boundary of the triangular domain. Some numerical examples demonstrate good stability of the proposed algorithm.

Improved Haar and Walsh functions over triangular domains

Abstract Piecewise constant orthogonal functions over triangular domains play an important role in many applications. In the present paper, some Haar and Walsh functions over triangular domains are constructed. Compared with the previously proposed Haar functions in [5] , the new Haar functions take only integer. For any continuous function, the uniform convergence of the new Haar–Fourier series is proved. Moreover, based on the relation between the new Haar and Walsh functions, the uniform convergence of the Walsh–Fourier series is studied. Additionally, we obtain the relation between the Walsh functions in the two different orders, Paley order and Hadamard order, which have not been discussed previously.

An analytical solution for groundwater flow to a vertical well in a triangle-shaped aquifer

The problem of drawdown and determination of velocity field near a fully penetrating pumping well which is located in a confined aquifer is investigated. An isosceles triangular domain defining the geometry of the aquifer is considered in this paper. The combination of streams with/without impervious layer forms the boundaries of the domain of interest. The concept of image well is employed to formulate the problem. The method of Green's Function together with two dimensional Fourier series expansion is used to obtain the closed-form solution for the problem. The isopotential lines and corresponding velocity field are presented for a particular position of pumping well. Also, the water depletion from interfacing streams is analytically calculated for different positions of pumping well, and the results are plotted. The plots can be effectively used in water resources management and environmental engineering.

Evaluation of Weakly Singular Integrals Via Generalized Cartesian Product Rules Based on the Double Exponential Formula

Various weakly singular integrals over triangular and quadrangular domains, arising in the mixed potential integral equation formulations, are computed with the help of novel generalized Cartesian product rules. The proposed integration schemes utilize the so-called double exponential quadrature rule, originally developed for the integration of functions with singularities at the endpoints of the associated integration interval. The final formulas can easily be incorporated in the context of singularity subtraction, singularity cancellation and fully-numerical methods, often used for the evaluation of multidimensional singular integrals. The performed numerical experiments clearly reveal the superior overall performance of the proposed method over the existing numerical integration methods.

Structure of chlorine on Ag(111): Evidence of the(3×3)reconstruction

The structure of the chlorine induced reconstruction of Ag(111) has been studied by a combination of low-temperature scanning tunneling microscopy (STM), low-energy electron diffraction (LEED), and density-functional theory (DFT). We demonstrate that previously observed mysterious LEED pattern arises as a result of diffraction from a system of small $(15--30\text{ }\text{\AA{}})$ triangular antiphase domains with a new $(3\ifmmode\times\else\texttimes\fi{}3)$ superstructure. In our model supported by DFT calculations, within a $(3\ifmmode\times\else\texttimes\fi{}3)$ unit cell the upper silver layer reconstructs forming a couple of three-atom triangles placed in fcc and hcp sites of the substrate. Chlorine atoms occupy fourfold hollow sites between these triangles. The corner holes, which look like depressions in the STM images, are also occupied by chlorine atoms.

Heatline analysis of heat recovery and thermal transport in materials confined within triangular cavities

Heat recovery from hot fluids in material processing industries is important for environmental and thermal management. Present work involves numerical visualization of heat flow in entrapped cavities filled with hot materials. The concept of heatline is used to visualize the heat energy trajectory. The system involves entrapped triangular cavities filled with hot fluid ( Pr = 0.015, 0.026, 0.7 and 1000). At low Rayleigh number ( Ra), it is found that the heatlines are smooth and perfectly normal to the isotherms indicating the dominance of conduction for both the triangles. As Ra increases, flow slowly becomes convection dominant. Multiple heat flow circulations with high intensity are formed within the lower triangular domain especially for low Pr numbers, whereas, less intense convective heat flow circulations are observed for the upper triangle. Multiple circulations are absent for both the triangular domains involving fluids with higher Pr numbers. It is observed that the heat transfer rates are monotonic for the upper triangle whereas a few local maxima in heat transfer rates occur for smaller Pr within lower triangular domain. Overall, fluid with any Pr may be useful for enhanced heat transfer within the upper triangle but fluid with high Pr may be preferred for the lower triangle.

Triangular domain extension of linear Bernstein-like trigonometric polynomial basis

In computer aided geometric design (CAGD), the Bernstein-Bezier system for polynomial space including the triangular domain is an important tool for modeling free form shapes. The Bernstein-like bases for other spaces (trigonometric polynomial, hyperbolic polynomial, or blended space) has also been studied. However, none of them was extended to the triangular domain. In this paper, we extend the linear trigonometric polynomial basis to the triangular domain and obtain a new Bernstein-like basis, which is linearly independent and satisfies positivity, partition of unity, symmetry, and boundary representation. We prove some properties of the corresponding surfaces, including differentiation, subdivision, convex hull, and so forth. Some applications are shown.

Atomic-Scale Study of Ge-Induced Incommensurate Phases on Si(111)

Two Ge-induced incommensurate phases, γ and β, on Si(111) are observed and studied by in situ scanning tunneling microscopy. The γ phase consists of aligned triangular domains whose stacking sequence is faulted with respect to the Si(11)-1 × 1 surface. The β phase consists of two kinds of triangular domains whose stacking sequences are faulted and unfaulted with respect to the Si(111)-1 × 1 surface, respectively. In the β phase, two types of domain walls, “zigzag" and “face-to-face", form to release the strain. The triangular domains all exhibit a quasi-1 × 1 hexagonal close-packed structure. By studying the structural evolution from magic clusters to incommensurate structures, the structure models for γ and β phases are proposed.

Patterned Hydrogenation of Graphene: Magnetic Quantum Dot Array

We design a new class of chemically modified graphenes through selective hydrogenation, which contain an array of triangular carbon domains separated by hydrogenated carbon stripes. By using a density-functional theory method within the generalized gradient approximation in the Perdew-Burke-Ernzerhof functional form, we show that each carbon triangle possesses notable magnetic moment on the edges. The chemically patterned graphene can be also viewed as an array of semiconducting quantum dots, where the hydrogenated carbon stripes play the role of quantum confinement. By adjusting the width of hydrogenated carbon stripes or the length scale of uncoated carbon triangles, the magnetic moment, band gap, and optical properties of the selectively hydrogenated graphene can be tuned.

Maximizing Neumann fundamental tones of triangles

We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proven for the harmonic and arithmetic means of the first two nonzero eigenvalues.