Two-dimensional Singularities Research Articles

The slow viscous flow around a general rectangular doubly-periodic arrays of infinite slender cylinders

Abstract The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media and microfluidic arrays, this solution is important for many real-world systems. We asymptotically determine the flow around a general rectangular doubly-periodic array of infinite slender cylinders, extending the existing asymptotic solution for square arrays. The flow in the cell is represented by a collection of doubly-periodic, rapidly-convergent two-dimensional singularity solutions, and the boundary condition on the surface of the cylinder is solved asymptotically in powers of the cylinder radius. The asymptotic solution provides an easily computed closed-form estimate for the flow and forces as a function of the radius and the dimensions of the cell. The force is compared to results from lattice-Boltzmann simulations of low-Reynolds-number flows in the same geometry, and the accuracy of the no-slip condition on the surface of the cylinder, predicted by the asymptotic theory, is assessed. Finally, the behaviour of the flow, flux, force and effective permeability of the cell is investigated as a function of the geometric parameters. The structure of the asymptotic permeability is consistent with previous single-geometry predictions but provides a closed-form estimate for how the aspect ratio of the cell changes the leading-order behaviour. These models could be used to help understand the flows within porous systems composed of fibres and systems involving periodic arrays such as systems based on deterministic lateral displacement .

Application of two dimensional Morlet wavelet transform in damage detection for composite laminates

Wavelet transform as a novel signal processing method for singularity detection has been introduced into vibration-based nondestructive testing (NDT) methods. However, for plate-like structures, the mode-shape-based NDT methods depend on the powerful two-dimensional (2D) singularity detection method. The 2D continuous wavelet transform (CWT) based on 2D Morlet wavelet function has high sensitivity to singularity in certain direction, but its anisotropic behavior strongly limits its application in damage detection for plate-like structures. In this paper, a modified 2D Morlet wavelet function is proposed to make the 2D CWT behave like a quasi-isotropic transform and show high sensitivity to damage in all directions. After a series of numerical simulations are conducted to investigate the performance of the modified 2D Morlet wavelet function and verify its effectivity in damage detection, a damage identification procedure for composite laminates is experimentally conducted and damage index (DI) is also determined by the modified wavelet coefficient. Results indicate that the 2D CWT based on the modified 2D Morlet wavelet can provide a baseline-free NDT method based on MS, and its effectivity and robustness are experimentally confirmed.

Engineering phase and polarization singularity sheets

Optical phase singularities are zeros of a scalar light field. The most systematically studied class of singular fields is vortices: beams with helical wavefronts and a linear (1D) singularity along the optical axis. Beyond these common and stable 1D topologies, we show that a broader family of zero-dimensional (point) and two-dimensional (sheet) singularities can be engineered. We realize sheet singularities by maximizing the field phase gradient at the desired positions. These sheets, owning to their precise alignment requirements, would otherwise only be observed in rare scenarios with high symmetry. Furthermore, by applying an analogous procedure to the full vectorial electric field, we can engineer paraxial transverse polarization singularity sheets. As validation, we experimentally realize phase and polarization singularity sheets with heart-shaped cross-sections using metasurfaces. Singularity engineering of the dark enables new degrees of freedom for light-matter interaction and can inspire similar field topologies beyond optics, from electron beams to acoustics.

Three-dimensional singularity index at interface corner in prismatic butt joint under arbitrary material combination

In this paper, the three-dimensional (3D) singularity index at the interface corner λcorner is newly analyzed for the prismatic butt joint in comparison with the two-dimensional (2D) singularity index along the interface edge λ2D previously studied. The finite element eigenanalysis reveals that the ratio of the singularity index is in the range of 0.85≤λcorner/λ2D≤1.15 for the all range of material combination. Under 2D bad pair condition expressed by Dundurs’ parameters α,β as αα-2β>0, the ratio is in the range 0.85≤λcorner/λ2D<1. The 3D bad pair condition when the singular stress appears at the corner is clarified on the α,β map in comparison with 2D well-known expression αα-2β>0. Simple and convenient discriminant formula at the 3D interface corner is also indicated for the appearance of the stress singularity.

Singularity-Exponent-Domain Image Feature Transform.

Combining the generalized fractal theory and the time-frequency distribution, the image feature decomposition in the singularity exponent domain is studied in this paper. With the theoretical derivation and quantitative analysis, the singularity-exponent-domain image feature transform (SIFT) method is proposed to analyze and process images from new feature dimensions. If one derives from the generalized fractal characteristics of the image, the two-dimensional frequency variables of the 2D time-frequency transform of the image can be used to estimate the two-dimensional singularity power spectrum (SPS) in the space dimension. As a consequence, it leads to the SPS distribution of the original image in the spatial domain, i.e., SIFT images. Based on the SIFT, the feature transform images with different singularity exponent and feature curves of singularity power spectrum with respect to different physical regions can thus be obtained. The SIFT is rigorously derived from the 2D-SPS and the Pseudo Wigner-Ville distribution (PWVD). In addition, the feature images based on the SIFT is proved to be the SNR independence in the GWN background. In order to validate the effectiveness of feature extraction, the proposed methodology is tested on the breast ultrasound images, the visual images, and the synthetic aperture radar (SAR) images. Furthermore, the SAR target detection method based on the SIFT images is proposed, and the experiment results indicate that the proposed algorithm is superior in performance to the traditional CFAR or 2D-SPS method. In fact, this new SIFT is promising to provide a technical approach for image feature extraction, target detection, and recognition.

Description of Degenerate Two-Dimensional Singularities with Single Critical Point

Description of Degenerate Two-Dimensional Singularities with Single Critical Point

Effect of sudden change in free stream velocity on the wind turbine airfoil performance based on boundary element method

In this study, the aerodynamic effects of sudden change in magnitude of incident velocity are investigated in detail for wind turbine airfoils. For this purpose, the unsteady panel method which is based on two-dimensional singularity approach has been developed. The model is first validated via using both the analytical functions and experimental data. The jump in free-stream velocity has been supposed to be applied in two different cases. The first is the airfoil at initially steady state and the second is the airfoil experiencing sinusoidal free-stream velocity. For both cases, the parametric study has been performed consequently investigating the effects of velocity difference magnitude, phase angle, airfoil characteristic and relative velocity oscillation amplitude. The response pattern of lift coefficient is shown to be a combination of three distinct behaviors: an initial jump, a secondary fall and a damped like envelope. The results showed that when the sudden velocity change is being applied on steady case, the time constant of recovery is finite and limited, while for the oscillatory case, the recovery time scale is considerably increased. Moreover, the local change in the normalized lift coefficient seems to be less than 0.3 for the steady case, while it can rise to 0.9 in dynamic oscillatory case. The magnitude of overshoot and undershoot has meaningful correlation with both relative velocity change and the initial phase in which the change occurs.

Effects of sudden change in pitch angle on oscillating wind turbine airfoil performances

In this paper, the aerodynamic performances of a wind turbine airfoil in sinusoidal pitching motion was numerically predicted and analyzed using the two-dimensional singularity method. A parametric study based on the sudden pitch angle rate variation, the reduced frequency and the pitching amplitudes have been investigated.The results highlight the sudden change effect in pitch angle on the aerodynamic performances of the airfoil and the shedding wake. The aspect of lift and drag coefficient loops and hysteresis were customized regarding to the pure sinusoidal motion case. The dynamic stall phenomenon, which occurs during the rapid change in pitch angle, produces more drag forces influencing the wind turbines performance.

Cohomology of ideals in elliptic surface singularities

We introduce the the normal reduction number of two-dimensional normal singularities and prove that elliptic singularity has normal reduction number two. We also prove that for a two-dimensional normal singularity which is not rational, it is Gorenstein and its maximal ideal is a $p_{g}$-ideal if and only if it is a maximally elliptic singularity of degree $1$.

The symmetric signature

ABSTRACTWe define two related invariants for a d-dimensional local ring (R,𝔪,k) called syzygy and differential symmetric signature by looking at the maximal free splitting of reflexive symmetric powers of two modules: the top-dimensional syzygy module of the residue field and the module of Kähler differentials ΩR∕k of R over k. We compute these invariants for two-dimensional ADE singularities obtaining 1∕|G|, where |G| is the order of the acting group, and for cones over elliptic curves obtaining 0 for the differential symmetric signature. These values coincide with the F-signature of such rings in positive characteristic.

Good ideals and $${p_{g}}$$ p g -ideals in two-dimensional normal singularities

In this paper, we introduce the notion of $${p_{g}}$$ -ideals and $${p_{g}}$$ -cycles, which inherits nice properties of integrally closed ideals on rational singularities. As an application, we prove an existence of good ideals for two-dimensional Gorenstein normal local rings. Moreover, we classify all Ulrich ideals for two-dimensional simple elliptic singularities.

Detection by regular schemes in degree two

Using Lipman's results on resolution of two-dimensional singularities, we provide a form of resolution of singularities in codimension two for reduced quasi-excellent schemes. We deduce that operations of degree less than two on algebraic cycles are characterised by their values on classes of regular schemes. We provide several applications of this "detection principle", when the base is an arbitrary regular excellent scheme: integrality of the Chern character in codimension less than three, existence of weak forms of the second and third Steenrod squares, Adem relation for the first Steenrod square, commutativity and Poincar\'e duality for bivariant Chow groups in small degrees. We also provide an application to the possible values of the Witt indices of non-degenerate quadratic forms in characteristic two.

On the Parameterization of Germs of Two-Dimensional Singularities

We consider a germ of a two-dimensional complex singularity \((X,x_0)\), irreducible at \(x_0\) and \(F\) the exceptional divisor of a desingularization. We prove that if there exists a normal isolated singularity \((Z,z_0)\) with simply connected link and a surjective holomorphic map \(f:(Z,z_0)\rightarrow (X,x_0)\) then all irreducible components of \(F\) are rational, and if all irreducible components of \(F\) are rational then there exists a surjective holomorphic map \(f:(\mathbb {C}^2,0)\rightarrow (X,x_0)\).

Nonlinear lifting theory for unsteady WIG in proximity to incident water waves. Part 1: Two-dimension

A nonlinear analysis is made for determining the two-dimensional unsteady potential-flow characteristics about a wing subject to wing-in-ground effect (WIG) operating above progressive water waves. The dynamic boundary condition requiring the constant pressure and the kinematic boundary condition prescribing the continuity in the vertical velocity are satisfied on the undisturbed free surface. The boundary conditions imposed on the free surface are linear, but the kinematic boundary condition satisfied on the foil surface is nonlinear. Through the derivation and evaluation of the time-domain Green's functions for two-dimensional singularities above a free surface, the influence of water waves on the lift performance of the two-dimensional WIG is addressed using the discrete vortex method. Furthermore, the roll-up of the wake vorticity is considered. The comparison of present work with other numerical results available in the literature shows the validation of the present approach.

Interface Cracks in Kirchhoff Anisotropic Thin Plates of Dissimilar Materials

This paper investigates the problem of stretching and bending deformations of a Kirchhoff anisotropic thin plate composed of two dissimilar materials bonded along a straight interface containing a crack. Our analysis makes use of the Stroh octet formalism developed recently by Cheng and Reddy (Cheng and Reddy, 2002, “Octet Formalism for Kirchhoff Anisotropic Plates,” Proc. R. Soc. Lond., A458, pp. 1499–1517; Cheng and Reddy, 2003, “Green’s Functions for Infinite and Semi-Infinite Anisotropic Thin Plates,” ASME J. Appl. Mech., 70, pp. 260–267; Cheng and Reddy, 2004, “Laminated Anisotropic Thin Plate With an Elliptic Inhomogeneity,” Mech. Mater., 36, pp. 647–657; Cheng and Reddy, 2005, “Structure and Properties of the Fundamental Elastic Plate Matrix,” J. Appl. Math. Mech., 85, pp. 721–739) for Kirchhoff anisotropic plates. It is found that the interfacial crack-tip field consists of a pair of two-dimensional oscillatory singularities, which are explicitly determined. Two complex intensity factors are proposed to evaluate the two oscillatory singularities.

The electric and magnetic polarization singularities of Gaussian vortex beam in the free-space propagation

Using the Fresnel vector diffraction integral and taking the Gaussian vortex beam as an example, the analytical expressions for the electric and magnetic components of Gaussian vortex beam propagating in the free-space are derived, and used to study the variations of electric and magnetic polarization singularities in free space. It is found that there exist two-dimensional (2D) and three-dimensional (3D) electric and magnetic polarization singularities in the free-space, which do not coincide in general. By varying waist width ratio, amplitude ratio, and propagation distance, the motion of polarization singularities takes place. In the 2D and the 3D electric and magnetic fields, the V-point may appear under a certain condition.

Ripplet transform type II transform for feature extraction

Current image representation schemes have limited capability of representing two-dimensional (2D) singularities (e.g. edges in an image). Wavelet transform has better performance in representing one-dimensional (1D) singularities than Fourier transform. Recently invented ridgelet and curvelet transform achieve better performance in resolving 2D singularities than wavelet transform. To further improve the capability of representing 2D singularities, this study proposes a new transform called ripplet transform type II (ripplet-II). The new transform is able to capture 2D singularities along a family of curves in images. In fact, ridgelet transform is a special case of ripplet-II transform with degree 1. Ripplet-II transform provides the freedom in parameter settings, which can be optimised for specific problems. Ripplet-II transform can be used for feature extraction because of its efficiency in representing edges and textures. Experiments in texture classification and image retrieval demonstrate that the ripplet-II transform-based scheme outperforms wavelet and ridgelet transform-based approaches.

Existence conditions, phonon spectrum and vibrational chcracteristics of Van der Waals crystals with BCC structure

Conditions of elastic stability and lattice dynamics of a body centered cubic crystal (BCC) having Van der Waals interaction among atoms have been studied. It was shown that for potentials describing interaction within crystals, BCC structure exists as a metastable state in a narrow interval of lattice constant values. Character of stability loss of the BCC crystal structure on borders of the allowed interatomic distances interval is analyzed. It was carried out that melting takes place at the highest border while transition to another crystal structure is at the lowest one. In the last case phonon spectrum has two-dimensional singularities, and low temperature oscillatory characteristics are well described using two-dimensional Debue model.

A two-dimensional model of low-Reynolds number swimming beneath a free surface

Biological organisms swimming at low-Reynolds number are often influenced by the presence of rigid boundaries and soft interfaces. In this paper, we present an analysis of locomotion near a free surface with surface tension. Using a simplified two-dimensional singularity model and combining a complex variable approach with conformal mapping techniques, we demonstrate that the deformation of a free surface can be harnessed to produce steady locomotion parallel to the interface. The crucial physical ingredient lies in the nonlinear hydrodynamic coupling between the disturbance flow created by the swimmer and the free boundary problem at the fluid surface.

On Adjacent Ideals in Two-Dimensional Rational Singularities

Let (R, ℳ) be a two-dimensional rational singularity with algebraically closed residue field. We investigate, for complete ℳ-primary ideals J ⊂ I in R, when length =length for all natural numbers n ≥ 1. This enables us to obtain some new results concerning adjacent and almost adjacent ideals in R. For example, in the special case where the associated graded ring of R is an integrally closed domain, we are able to show that e(I) = e(ℳ) +1 where I is a complete ideal adjacent to ℳ and e denotes the multiplicity. This result suggests the more general question of how much the multiplicities can differ for two adjacent ℳ-primary ideals in case there are no restrictions on the associated graded ring of R. We are able to answer this question using the theory of degree functions developed by Rees and Sharp, thereby generalizing the work of Noh.