Singular Field Research Articles

Shaping optical spin flow topologies by the translation of tailored orbital phase flow

Beneath being a well-established tool in optical manipulation for spinning and eventually sorting trapped objects, transverse energy flow (TEF) of light represents a versatile, illustrative and instructive approach to understand singular light fields and their properties. Up to now, these transverse flows were mainly investigated in its orbital part, for example, as in optical vortices, whereas the study of complex spin flow density (SFD) topologies has not yet reached the same mature state. Especially, the defined customization of energy flow and respective SFD fields has not yet been initiated. By the application of higher-order Laguerre– as well as Ince–Gaussian modes, being well-studied tools in optical manipulation, we demonstrate for the first time to our knowledge the development of sophisticated SFD configurations. We show shaping the flow field and controlling the type, number, spatial location and order of formed singular points in these TEFs. Our approach facilitates on-demand sculpting energy flow fields and their complex topologies for future applications in, for example, optical manipulation based on tailored polarization structures.

The Action of a Plane Singular Holomorphic Flow on a Non-invariant Branch

AbstractWe study the dynamics of a singular holomorphic vector field at $(\mathbb{C}^{2},0)$. Using the associated flow and its pullback to the blow-up manifold, we provide invariants relating the vector field, a non-invariant analytic branch of curve, and the deformation of this branch by the flow. This leads us to study the conjugacy classes of singular branches under the action of holomorphic flows. In particular, we show that there exists an analytic class that is not complete, meaning that there are two elements of the class that are not analytically conjugated by a local biholomorphism embedded in a one-parameter flow. Our techniques are new and offer an approach dual to the one used classically to study singularities of holomorphic vector fields.

Enhanced discretization of surface integral equations for resonant scattering analysis of sharp-edged plasmonic nanoparticles

The surface integral equation (SIE) method, discretized with the method of moments, is a well-established methodology for the scattering analysis of subwavelength plasmonic nanoparticles. SIEs are usually discretized with low-order basis functions that preserve the normal continuity of the surface currents across the edges arising in the meshed boundary, such as Rao-Wilton-Glisson (RWG) functions. However, the plasmonic enhancement modeling on sharp-edged particles is an extremely challenging task, especially due to the singular fields exerted at sharp corners, exposing a slow (or no) convergence in the computation of the scattering and absorption spectra. In this paper, we propose an alternative discretization strategy based on a discontinuous basis function set in conjunction with a volumetric-tetrahedral testing scheme. We demonstrate the potential of the proposed discretization scheme by studying scattering and absorption spectra of three canonical plasmonic polyhedra, i.e., a hexahedral, an octahedral, and a tetrahedral silver inclusion. The results expose an improved accuracy and faster convergence in both far-field and near-field regions when compared to the standard RWG implementation. The proposed discretization scheme can offer faster and more accurate routes towards the exploration and design of the plasmonic resonant spectrum of sharp-edged nanoparticles and nanoantennas.

Finite element simulation of elastoplastic field near crack tips and results for a central cracked plate of LE-LHP material under tension

The elastoplastic field near crack tips is investigated through finite element simulation. A refined mesh model near the crack tip is proposed. In the mesh refining area, element size continuously varies from the nanometer scale to the micrometer scale and the millimeter scale. Graphics of the plastic zone, the crack tip blunting, and the deformed crack tip elements are given in the paper. Based on the curves of stress and plastic strain, closely near the crack tip, the stress singularity index and the stress intensity factor, as well as the plastic strain singularity index and the plastic strain intensity factor are determined. The stress and plastic strain singular index vary with the load, while the dimensions of the stress and the plastic strain intensity factors depend on the stress and the plastic strain singularity index, respectively. The singular field near the elastoplastic crack tip is characterized by the stress singularity index and the stress intensity factor, or alternatively the plastic strain singularity index and the plastic strain intensity factor. At the end of the paper, following Irwin’s concept of fracture mechanics, $$\sigma_{\delta K}$$ criterion and $$\varepsilon_{\delta Q}$$ criterion are proposed. Besides, crack tip angle criterion is also presented.

On stress singularity at crack tip in elasticity

Abstract In order to gain a better understanding of the stress singularity at the crack tip in elasticity, the relevant solutions are investigated. Based on the rigorous solution of a plate containing a crack and its corresponding asymptotic expression, the stress singularity at the crack tip was described in details. The stress singularity at the crack tip is identified as: the infinite stress exists at the crack tip, and the stress components at the crack tip are multivalued. Besides, the rigorous solution of the infinite plate containing a crack may be derived from the rigorous solution of the infinite plate containing an elliptical hole, which is solved from Mushelishvili’s approach. As half minor-axial length approaches zero, the stress concentration field of a plate with an elliptical hole evolves into the stress singularity field of a plate with a crack. Thus, the stress intensity factor characterizing the stress singularity field near the crack tip may also be read as characterizing the infinite stress concentration field near the crack tip. Therefore, the mechanical concept of stress intensity factor can be identified as follows: stress intensity factor is a parameter characterizing the strength of infinite stress concentration field at the crack tip.

Basic singular fields in the theory of impulsive supersonic leading-edge noise

This paper determines the impulsive sound fields produced by sharp-edged gusts striking the leading edge of a supersonic blade or aerofoil, for example in a turbofan aeroengine or a counter-rotating propeller system. A full three-dimensional theory is provided, so that the gust edges can be at any orientation relative to the blade. Complete details are given of the sound fields produced by gust edges in the spanwise and streamwise directions, and by many combinations of such edges, including corners. The mathematical theory depends on singular sound fields produced by gusts with a delta-function upwash; these are used to derive exact analytical formulae for impulsive sound fields of different three-dimensional shapes, and also a Green’s function representation of the field which is especially adapted to numerical evaluation. Gusts with top-hat profiles are given particular attention, and also the effect of Gaussian-function smoothing of both delta-function and top-hat profiles. The investigation is complementary to that in a companion paper (Powles and Chapman, 2019), which determines the smooth sound fields produced by single-frequency gusts. Fourier integration provides the relation between the two types of field.

Stress and Strain of Cracked Orthotropic Sheet

The analytic function of complex variable including a material parameter is analyzed fully. The new stress function has been formed by the complex function method. Typical shear mode II crack states are considered to orthotropic materials. The boundary shear loading problem of cracked orthotropic sheet is studied and the formulae for stress fields are derived. The singular strain fields are also discussed. And the expressions of the singular stress and strain in the crack-tip are determined with inclusion of the material elasticity coefficients.

A Modified Gravity Theory: Null Aether**Supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK)

General quantum gravity arguments predict that Lorentz symmetry might not hold exactly in nature. This has motivated much interest in Lorentz breaking gravity theories recently. Among such models are vector-tensor theories with preferred direction established at every point of spacetime by a fixed-norm vector field. The dynamical vector field defined in this way is referred to as the “aether”. In this paper, we put forward the idea of a null aether field and introduce, for the first time, the Null Aether Theory (NAT)—a vector-tensor theory. We first study the Newtonian limit of this theory and then construct exact spherically symmetric black hole solutions in the theory in four dimensions, which contain Vaidya-type non-static solutions and static Schwarzschild-(A)dS type solutions, Reissner-Nordström-(A)dS type solutions and solutions of conformal gravity as special cases. Afterwards, we study the cosmological solutions in NAT: We find some exact solutions with perfect fluid distribution for spatially flat FLRW metric and null aether propagating along the x direction. We observe that there are solutions in which the universe has big-bang singularity and null field diminishes asymptotically. We also study exact gravitational wave solutions—AdS-plane waves and pp-waves—in this theory in any dimension . Assuming the Kerr-Schild-Kundt class of metrics for such solutions, we show that the full field equations of the theory are reduced to two, in general coupled, differential equations when the background metric assumes the maximally symmetric form. The main conclusion of these computations is that the spin-0 aether field acquires a “mass” determined by the cosmological constant of the background spacetime and the Lagrange multiplier given in the theory.

Strain and stress mapping by mechanochemical activation of spiropyran in poly(methyl methacrylate)

AbstractThe relationship between fracture‐induced mechanophore activation and the strain and stress ahead of a propagating crack in poly(methyl methacrylate) (PMMA) is studied. The mechanophore spiropyran is used as a secondary cross‐linker in rubber toughened PMMA, and the spiropyran‐linked material is subjected to fracture testing. Mechanophore activation is detected and analysed by fluorescence imaging. Digital image correlation is used to measure the strain field ahead of the crack tip, whereas the corresponding stress field is calculated using the Hutchinson–Rice–Rosengren singularity field equations. Mechanophore activation follows a power law dependence on distance from the crack tip and provides both a qualitative and quantitative measure of the strain and stress fields ahead of the crack.

Calculation of 3D residual notch stress intensity factors by means of the peak stress method

In the recent literature new approaches were developed to include the residual stress effects in fatigue strength assessments of welded joints. Among these, the local approach based on the calculation of the Residual Notch Stress Intensity Factors (R-NSIFs) seems very promising. Unfortunately, its main drawback is linked to the difficulty to assess such parameters by means of numerical simulation of the welding process, because of the high mesh density required to capture the residual stress field singularity. In this scenario, new numerical methodologies are needed which use coarse meshes, such as the Peak Stress Method (PSM). In this work, an investigation is described aimed at analyzing the PSM applicability to estimate the R-NSIF values in a 3D butt-welded joint. It was shown that the R-NSIF values obtained ‘via’ the PSM by using 3D models are quite higher (about 45%) as compared to estimations obtained by means of a 2D model, under generalized plain strain condition. This was attributed to the limitation of 2D models to correctly assess the residual stress field induced by welding processes.

Regularization of proximal point algorithms in Hadamard manifolds

In this paper, we consider the regularization method for exact as well as for inexact proximal point algorithms for finding the singularities of maximal monotone set-valued vector fields. We prove that the sequences generated by these algorithms converge to an element of the set of singularities of a maximal monotone set-valued vector field. A numerical example is provided to illustrate the inexact proximal point algorithm with regularization. Applications of our results to minimization problems and saddle point problems are given in the setting of Hadamard manifolds.

Buckling and post-buckling analysis of cracked stiffened panels via an X-Ritz method

A multi-domain eXtended Ritz formulation, called X-Ritz, for the analysis of buckling and post-buckling of stiffened panels with cracks is presented. The theoretical framework is based on the First-order Shear Deformation Theory and accounts for von Kármán's geometric nonlinearities. The structure is modeled as assembly of plate elements. Penalty techniques are used to fulfill the continuity condition along the edges of contiguous elements and to satisfy essential boundary conditions requirements. The use of an extended set of approximating functions allows to model through-the-thickness cracks and to capture the crack opening and tip singular fields as well as the structural behavior within each single domain. Numerical results for buckling and post-buckling of cracked stiffened panels are compared with finite elements simulations and literature solutions, showing the accuracy and potential of the proposed approach.

Spectral Flow for Dirac Operators with Magnetic Links

This paper is devoted to the study of the spectral properties of Dirac operators on the three-sphere with singular magnetic fields supported on smooth, oriented links. As for Aharonov–Bohm solenoids in the Euclidean three-space, the flux carried by an oriented knot features a $$2\pi $$-periodicity of the associated operator. For a given link, one thus obtains a family of Dirac operators indexed by a torus of fluxes. We study the spectral flow of paths of such operators corresponding to loops in this torus. The spectral flow is in general nontrivial. In the special case of a link of unknots, we derive an explicit formula for the spectral flow of any loop on the torus of fluxes. It is given in terms of the linking numbers of the knots and their writhes.

Cracks as Limits of Eshelby Inclusions

As limiting behaviors of Eshelby ellipsoidal inclusions with transformation strain, crack solutions can be obtained both in statics and dynamics (for self-similarly expanding ones). Here is presented the detailed analysis of the static tension and shear cracks, as distributions of vertical centers of eigenstrains and centers of antisymmetric shear, respectively, inside the ellipse being flattened to a crack, so that the singular external field is obtained by the analysis, while the interior is zero. It is shown that a distribution of eigenstrains that produces a symmetric center of shear cannot produce a crack. A possible model for a Barenblatt type crack is proposed by the superposition of two elliptical inclusions by adjusting their small axis and strengths of eigenstrains so that the singularity cancels at the tip.

Propagation of Chaos for a Class of First Order Models with Singular Mean Field Interactions

Dynamical systems of $N$ particles in $\Bbb{R}^{D}$ interacting by a singular pair potential of mean field type are considered. The systems are assumed to be of gradient type and the existence of a macroscopic limit in the many particle limit is established for a large class of singular interaction potentials in stochastic as well as deterministic settings. The main assumption on the potentials is an appropriate notion of quasi-convexity. When $D=1$ the convergence result is sharp when applied to strongly singular repulsive interactions and for a general dimension $D$ the result applies to attractive interactions with Lipschitz singular interaction potentials, leading to stochastic particle solutions to the corresponding macroscopic aggregation equations. The proof uses the theory of gradient flows in Wasserstein spaces of Ambrosio, Gigli, and Savaré.

Memoria oficial y memorias colectivas. La creación y eliminación del recuerdo a Domiciano en los espacios de representación pública de las ciudades hispanas

We analyze the construction of the Roman official memory about Domitian into the publics spaces of imperial representation in the Hispanic cities. The condemnation of his memory is a singular field of study. Therefore, we intend to analyze how was the reception of the image and official propaganda in Hispanic cities, and how the damnatio memoriae was materialized, through the analysis of portraits and votive and honorific tributes preserved in the Iberian Peninsula.

ON THE SCATTERING OF SURFACE WAVES BY UNDERWATER OBSTACLES

The paper considers a 2D linear problem on scattering of the low-amplitude surface wave in an incompressible fluid that propagates over the bottom obstacle in form of a vertical rectangular wall of the finite of infinitesimal length. The bottoms of the channel and obstacle walls are absolutely rigid. The solution is found by the method of partial domains implying the subdivision of the complex geometry of the overall domain of wave field existence into the subdomains of canonical shapes. This approach allows the formal representing of the corresponding general solutions as the series with respect to eigenfunctions satisfying part of the boundary conditions in each subdomain. The unknown coefficients present in these general solutions may be found from matching conditions that postulate continuity of pressure and normal velocity fields at the boundaries of adjacent subdomains. Applying algebraization procedure to matching conditions for the corresponding general solutions leads to generation of infinite systems of linear equations with respect to the unknown coefficients. The obtained systems possess slow convergence that is determined by the presence of velocity singularities in angular points of the obstacle. To accelerate the convergence, the method of improved reduction is applied in analyzing the infinite systems. It is based on the a priori hypotheses about the asymptotic behavior of the unknown coefficients in general solutions that is controlled by the order of field singularity in the vicinity of the corresponding angular point. The dependence of calculated energy discrepancy on the amount of considered terms in the reduced system is assessed. The method of improved reduction is shown to provide the better convergence quality at moderate number of kept terms (50 to 70), in comparison with the ordinary reduction. The use of the improved reduction also leads to some widening of the region for which the numerical error remains within acceptable limits.

Static and dynamic fracture analysis in elastic solids using a multiscale extended isogeometric analysis

The main objective of this paper is to develop a novel multiscale approach for simulating static and stationary dynamic crack problems in two-dimensional elastic solids through an extended isogeometric analysis using the Non-Uniform Rational B-Splines (NURBS). This multiscale strategy consists of two techniques that the Nitsche’s method is applied for coupling different length-scales meshes on one hand, and on the other hand, a local enrichment technique is taken to describe the cracks independently of the mesh in the refined region by introducing the enrichment function into the isogeometric analysis displacement approximation. This method inherits the advantage of XIGA and simultaneously improves his limitation on the local mesh refinement. The proposed approach possesses several advantages in fracture modeling: (i) the singularities of the crack-tip fields can be captured precisely and conveniently by the enrichment techniques; (ii) higher-order convergence rates is achieved and (iii) stress smoothing results are achieved owing to higher continuity of NURBS basis functions. Numerical results of static and stationary dynamic crack problems validate its accuracy and efficiency.

Magnetized orbifolds and localized flux

Magnetized orbifolds play an important role in compactifications of string theories and higher-dimensional field theories to four dimensions. Magnetic flux leads to chiral fermions, it can be a source of supersymmetry breaking and it is an important ingredient of moduli stabilization. Flux quantization on orbifolds is subtle due to the orbifold singularities. Generically, Wilson line integrals around these singularities are non-trivial, which can be interpreted as localized flux. As a consequence, flux densities on orbifolds can take the same values as on tori. We determine the transition functions for the flux vector bundle on the orbifold T2∕Z2 and the related twisted boundary conditions of zero-mode wave functions. We also construct “untwisted” zero-mode functions that are obtained for singular vector fields related to the Green’s function on a torus, and we discuss the connection between zeros of the wave functions and localized flux. Twisted and untwisted zero-mode functions are related by a singular gauge transformation.

The optical signal analysis (OSA) method to process fringe patterns containing displacement information

In preceding publications, the authors have analyzed different aspects of the recovery of displacement information encoded in scalar fields of gray levels. The signals that contain displacement information can be generated by different optical methods or can be computer generated, for example, in the images of the magnetic resonance method (MRI). This paper merges ideas of the authors’ previous developments, included in the manuscript references, with new developments. These developments simplify fundamental basic steps of the process of retrieving displacement information. They also extend the applicability of basic algorithms to a very diverse variety of patterns, including patterns that contain displacement singularities. Singularities in displacement fields indicate departures of the standard continuity conditions assumed in the mathematical model of Continuum Mechanics, these discontinuities can be local or can divide the field into piecewise continuous regions. To illustrate the practical aspect of these derivations the proposed methodology is applied to very complex patterns. The analysis of these patterns with standard techniques will require complex and time-consuming pattern processing procedures. In the approach presented in this paper, the same task can be accomplished by a much simpler and general scheme.