Order Of Singularity Research Articles

Sharp focusing of a light field with polarization and phase singularities of an arbitrary order

Using the Richards-Wolf formalism, we obtain general expressions for all components of the electric and magnetic strength vectors near the sharp focus of an optical vortex with the topological charge m and nth-order azimuthal polarization. From these equations, simple consequences are derived for different values of m and n. If m=n>1, there is a non-zero intensity on the optical axis, like the one observed when focusing a vortex-free circularly polarized light field. If n=m+2, there is a reverse flux of light energy near the optical axis in the focal plane. The derived expressions can be used both for simulating the sharp focusing of optical fields with the double singularity (phase and polarization) and for a theoretical analysis of focal distributions of the intensity and the Poynting vector, allowing one to reveal the presence of rotational symmetry or the on-axis reverse energy flux, as well as the focal spot shape (a circle or a doughnut).

Reverse and toroidal flux of light fields with both phase and polarization higher-order singularities in the sharp focus area.

Based on the Richards-Wolf formalism, we obtain for the first time a set of explicit analytical expressions that completely describe a light field with a double higher-order singularity (phase and polarization), as well as distributions of its intensity and energy flux near the focus. A light field with the double singularity is an optical vortex with a topological charge m and with nth-order cylindrical polarization (azimuthal or radial). From the theory developed, rather general predictions follow. 1) For any singularity orders m and n, the intensity distribution near the focus has a symmetry of order 2(n - 1), while the longitudinal component of the Poynting vector has always an axially symmetric distribution. 2) If n = m + 2, there is a reverse energy flux on the optical axis near the focus, which is comparable in magnitude with the forward flux. 3) If m ≠0, forward and reverse energy fluxes rotate along a spiral around the optical axis, whereas at m = 0 the energy flux is irrotational. 4) For any values of m and n, there is a toroidal energy flux in the focal area near the dark rings in the distribution of the longitudinal component of the Poynting vector.

Vortex beams with high-order cylindrical polarization: features of focal distributions

A detailed analytical and numerical study of the effect of phase vortex singularity has been carried out when focusing high-order cylindrical polarized fields. The possibility of forming a light spot with reduced side lobes when focusing of ring vortex beams with a high-order cylindrical polarization in the paraxial mode. The formation of a compact focal spot is possible at sharp focusing of the Gaussian beam when the order of polarization and the order of the vortex phase singularity coincide. When the orders of polarization and the vortex phase do not coincide, focused intensity distributions resembling “chamomiles” and “cogwheels”. Such distributions can be used in laser printing of plasmonic nanotextures for chemo- and biosensing applications.

All order α′ corrections to fermionic couplings of D-brane-Anti-D-brane effective actions

We first point out to all order α′ corrections to lower order fermionic couplings and their effective actions of type II super string theory. In order to reveal all symmetries of the particular D-brane Anti-D-brane system, and to construct not only all the singularity structures but also all order α′ corrections of fermionic couplings to tachyons, we employ all the Conformal Field Theory (CFT) methods to a six point function. Basically, we construct all the correlation functions of four spin operators and two world sheet fermion fields. The S-matrix of two fermion fields, two real tachyons in the presence of a closed string Ramond-Ramond (RR) field in type II will be explored. Applying all the symmetries of the amplitude, a proper expansion for D-brane Anti D-brane is also found out. Having carried out all the Effective Field Theory (EFT) techniques, one would discover all order singularity structures of string theory. The entire algebraic computations of the amplitude of <VC−1(z,z̄) Vψ̄−1/2(x1) Vψ−1/2(x2) VT0(x3) VT0(x4)> have also been derived. Various new string couplings are revealed as well. Eventually, making comparison the S-matrix with all EFT parts, all order α′ higher derivative corrections to two fermions- two tachyons in the world volume of D-brane Anti D-brane of type II are discovered accordingly.

On the optimal choice of fibre orientation angle in off-axis tensile test using oblique end-tabs: Theoretical and experimental studies

Shear strength evaluation in composite materials still remains a not fully solved problem. One of most promising proposals to characterise a material in shear by means of a simple tensile test is the off-axis tension test with oblique end-tabs whose inclination coincides with that of the longitudinal isodisplacement lines. This end-tab configuration presents an essential difficulty since the tab angle depends on the material properties including the shear modulus G12, which is one of the values to be obtained from off-axis tension tests.In the present work, the study of the most suitable fibre orientation for the performance of off-axis tests under an oblique tabs configuration has been addressed. An analytical study on the dependence of the tab angle ϕ on both the fibre orientation θ and the material properties E11, E22, ν12 and G12 has been carried out, the quotient G12/E11 being identified as the parameter that controls the evolution of the tab angle.The influence of deviations in the tab angle on the stress state near the corners of specimen has been analysed by means of a novel study of the nominal singular stress state at these points. The results of this study show that negative deviations in the angle of the tabs contribute to decrease the severity of the stress state at the corner, whereas positive deviations contribute to increase the order of the stress singularity. Additionally, the range of fibre orientations in which the evaluation of shear strength S is feasible has been determined, i.e., the effect of σ12 in the failure is clearly dominant versus the effect of the rest of the stress-state components.In the experimental part of this work, four different fibre orientation angles (5°, 10°, 15° and 20°) have been considered to perform the off-axis tension test under oblique end-tabs configuration. The experimental results of these tests show that only for 10° fibre orientation angle the majority of specimens fail at the central zone under a uniform stress state. Finally, for a fibre orientation angle θ = 10°, specimens with induced positive deviations in tab angle of +3° and +7° have been tested, it being observed that all the failures appears at the end of specimens.On the basis of both theoretic and experimental results, the optimal fibre orientation angle to use in the tests is θ = 10°. The tab angle must be evaluated on the basis of an estimation of the material properties (quotient G12/E11), it being recommendable, in case of ignoring the value of G12, to use a tab angle lesser than the theoretical value to avoid positive deviations.

Les inventions partagées et les innovations concurrentielles : des approches nouvelles

Les concepts « d’invention partagée » et « d’innovation concurrentielle » ont été parfois utilisés dans des domaines comme l’économie et la gestion des inventions et des innovations, l’économie industrielle et l’histoire des techniques. Analysant la littérature disponible, cet article proposera une approche nouvelle de ces concepts et de leurs relations réciproques. Le concept d’invention partagée sera défini comme un ensemble de savoirs des acteurs rassemblés dans des intelligences partagées et des dynamiques sociales développant la cohérence de ces inventions et les réinventions de techniques anciennes. Le concept d’innovation concurrentielle sera défini par des associations d’expériences de certains acteurs motivés par des dynamiques sociales promouvant la concurrence et cherchant à complémenter des inventions existantes en construisant leurs forces concurrentielles. Pour être cohérentes, ces inventions partagées et ces innovations concurrentielles seront aussi définies par des trajectoires permettant de dépasser leurs singularités pour réaliser de manière efficace ces partages et ces concurrences. Codes JEL : O3, L, A10

Asymptotic analysis of out-of-plane strain and displacement fields at angular corners

This paper presents three-dimensional analytical solutions for displacement and strain fields near angular corners in plates subjected to in-plane loading. The approach is based on the first-order plate theory, which represents an elementary extension of the classical plane theory of linear elasticity. Utilising the Kontorovich–Lebedev transform, the asymptotic behaviour of the out-of-plane displacement field near the apex is investigated for both Mode I and Mode II loadings. The analytical solutions obtained in the present work correctly predict several three-dimensional effects, such as the order of stress singularity and the intensity of the coupled (local) out-of-plane singular mode under remote in-plane shear loading as well as the scale effect associated with the plate thickness. The developed solutions are more general than many previous analytical results obtained for crack and notch geometries and, essentially, generalise the classical William's solution for angular corners. The analytical predictions compare favourably with three-dimensional finite element modelling and experimental results.

Sheaf lines of Yang–Mills instanton sheaves

We calculate a sheaf line in [Formula: see text] which is the real line supporting sheaf points on [Formula: see text] of [Formula: see text] Yang–Mills (YM) instanton (or [Formula: see text] complex YM instanton) sheaves for some given ADHM data we obtained previously. We found that this sheaf line is indeed a special jumping line over [Formula: see text] spacetime. In addition, we calculate the singularity structure of the connection [Formula: see text] and the field strength [Formula: see text] at the corresponding singular point on [Formula: see text] of this sheaf line. We found that the order of singularity at the singular point on [Formula: see text] associated with the sheaf line in [Formula: see text] is higher than those of other singular points associated with normal jumping lines. We conjecture that this is a general feature for sheaf lines among jumping lines.

A singular element based on dual interpolation BFM for V-shaped notches

• A new singular element based on dual interpolation boundary face method is proposed. • The new element takes into account the variable singularity orders at the notch tip . • Special basis function is introduced to build the shape functions of the proposed element. • The dual interpolation method can provide much higher accuracy than traditional interpolation method. • The numerical results for stress intensity factors obtained by the new element are convergent. A singular element based on dual interpolation boundary face method (DiBFM) is presented for solving V-shaped notch problems in this paper. The stress field around sharp notches is singular, and the singularity orders vary with the notch angle. Thus an element with usual shape functions or traditional singular element cannot lead to high accurate results. To accurately model the distribution of displacement around the notch tip, a new displacement singular element based on DiBFM is proposed. The new element takes into account the variable singularity orders at the notch tip. The dual interpolation method which combines conventional polynomial element interpolation and moving least-squares approximation can provide much higher accuracy than traditional interpolation method. With the proposed singular element based on DiBFM, more accurate displacement results in the vicinity of the notch tip can be evaluated, thus more accurate stress intensity factor (SIF) of the V-shaped notches can be obtained. Numerical examples have demonstrated the validity and accuracy of our method.

Regularization of saddle-fold singularity for nonsmooth differential systems

This paper is concerned with the saddle-fold singularity that is a codimension two singularity for piecewise smooth planar differential systems. The approach consists in apply the Regularization's Method to the unfolding of the saddle-fold singularity in order to obtain a 2-parameter family of smooth vector fields. Our main result says that the regularized family undergoes thought a classical Bogdanov–Takens bifurcation.

Enrichment functions for weak singularities in 2D elastic problems with isotropic and orthotropic materials

In this paper a new enrichment technique is presented for weak singularities in isotropic and orthotropic 2D linear elasticity problems. With absolutely no information of the singularity order, in a novel numerical approach, the enriching bases are constructed through a weighted residual imposition of the partial differential equation. Conceptually, the bases can be categorized as equilibrated basis functions. As another advantage of the proposed technique, it will be shown that the enriched solution may be extracted for either isotropic or orthotropic materials in a similar manner, noting that singularity effect in orthotropic materials has not been as widely referred to as for isotropic materials in the literature. The presented examples selected from the well-known literature reveal the accuracy and applicability of the enrichment in the framework of a boundary type method. The method may also be found useful for the researchers focusing on the eXtended Finite Element Method or other similar mesh-based or mesh-less methods.

Singularity evaluation for single material and bi-material propagating V-notches

By introducing displacement asymptotic expansion around the vertex of a propagating V-notch into elasticity governing equation, radial boundary conditions and interfacial continuity conditions, the singularity eigen analysis for a propagating V-notch is transformed into solving a set of eigen ordinary differential equations. The singularity orders and corresponding eigen angular functions for a propagating V-notch can be obtained after applying interpolating matrix method in order to solve the established eigen equation. The singularities for single material and bi-material propagating V-notch are respectively analyzed under different material properties, propagating directions and propagating velocities. It is found that the relationship between stress singularity and propagating velocity depends on the propagating direction as well as the singularity variation with respect to propagating velocity is not monotonous. For a single material V-notch, the singularity will be the weakest when the propagating direction is along the bisector of V-notch, and it will be the strongest when the propagating direction is perpendicular to the notch bisector. For a bi-material V-notch, the strongest singularity for a bi-material V-notch occurs when the propagating direction is along the harder material.

Disturbance-observer-based fuzzy terminal sliding mode control for MIMO uncertain nonlinear systems

• Modelling and simulation of a novel five-rotor unmanned aerial vehicle with an extra rotor. • Proposing a disturbance-observer-based fuzzy terminal sliding mode controller . • Considering MIMO uncertain nonlinear systems with unmeasured external disturbance . • Unknown non-symmetric input saturation and control singularity are investigated. • Closed-loop stability and convergence of the state estimations in finite time are proved. This study is concerned with the design of a disturbance-observer-based fuzzy terminal sliding mode controller (FTSMC) for multi-input multi-output (MIMO) uncertain nonlinear systems by considering unknown non-symmetric input saturation and control singularity. The disturbance observer is proposed for the unmeasured external disturbance and guarantees the convergence of the disturbance estimation error to zero in a finite time. The terminal sliding mode controller (TSMC) is designed for MIMO uncertain nonlinear systems by utilizing the output of the proposed disturbance observer. This control scheme combines the disturbance-observer-based TSMC with a fuzzy logic system in the presence of unknown non-symmetric input saturation and control singularity in order to reduce chattering phenomena. Finite time asymptotic stability, convergence of the disturbance observer, and convergence of the closed-loop system are proved via Lyapunov stability theorem. In addition, a five-rotor unmanned aerial vehicle (UAV) is employed in the numerical simulations to demonstrate the effectiveness and performance of the proposed control scheme. Disturbance observer estimates the payload and flight endurance of the five-rotor UAV. Genetic algorithm (GA) optimization is used to specify the parameters of the disturbance-observer-based TSMC (GATSMC) to decrease chattering. Finally, the superior performance of FTSMC is investigated over TSMC and GATSMC.

A Crack Close to and Perpendicular to an Interface: Resolution of Anomalous Behavior With a Cohesive Zone

In this investigation, we consider a crack close to and perpendicular to a bimaterial interface. If the crack tip is at the interface then, depending on material properties, the order of the stress singularity will be equal to, less than, or greater than one-half. However, if the crack tip is located any finite distance away from the interface the stress field is square-root singular. Thus, as the crack tip approaches the interface, the stress intensity factor approaches zero (for cases corresponding to a singularity of order less than one-half) or infinity (for a singularity of order greater than one-half). The implication of this behavior is that for a finite applied pressure the crack will either never reach the interface or will reach the interface with vanishing small applied pressure. In this investigation, a cohesive zone model is used in order to model the crack behavior. It is found that the aforementioned anomalous behavior for the crack without a cohesive zone disappears and that the critical value of the applied pressure for the crack to reach the interface is finite and depends on the maximum stress of the cohesive zone model, as well as on the work of adhesion and the Dundurs' parameters.

Electric and elastic singularity eigenanalysis of the V-notch in a piezoelectric Reissner plate

The singular eigenstate with respect to the stress and electric displacement at the vertex of a V-notch in a piezoelectric plate is investigated. The displacement and electric potential near the vertex of a V-notch are expressed as series asymptotic expansions, and their typical terms are substituted into the governing equations to establish the singularity eigenequations, from which the singularity orders and angular eigenfunctions can be yielded by the interpolating matrix method. The coupling effect of elastic and electric singularities can be found by distinguishing the singularity orders from their angular eigenfunctions. There are most four singularity orders for the V-notch in a piezoelectric Reissner plate, which is one more than the one in a homogeneous plate. Two orders are corresponding to the singularity of bending moment and electric displacement in the thickness direction, and another two orders are with respect to the singularity of shear force and electric displacement in the mid-plane. The free and electrically opened, and clamped and electrically closed boundary conditions are respectively introduced to investigate the influence of boundary conditions on the singularity. It is found that the singularity orders with respect to bending moment and electric displacement in the thickness direction change with boundary conditions, while the singularity orders with respect to shear force and electric displacement in the mid-plane are identical under these two kinds of boundary conditions.

HYPERGRAPH GRAMMAR BASED MULTI-THREAD MULTI-FRONTAL DIRECT SOLVER WITH GALOIS SCHEDULER

In this paper we analyze two dimensional grids with point and edge singularities in order to develop an efficient graph grammar based multi-frontal direct solver algorithm. We express these grids by hypergraph models. For these meshes we define a sequence of graph grammar productions expressing the construction of frontal matrices, elimination of fully assembled nodes, merging of resulting Schur complements, and repeating the process of elimination and merging until a single frontal matrix remains. The dependency relation between graph grammar productions is analyzed, and the dependency graph is plot, which is equivalent to the elimination tree of the multi-frontal solver algorithm. We utilize classical multi-frontal solver algorithm, and the graph grammar productions allows us to construct an efficient elimination tree, based on the graph representation of the computational mesh, and not the global matrix itself. The graph grammar productions are assigned to nodes of the dependency graph, and they are implemented as tasks in the GALOIS system and scheduled according to the developed dependency graph over the shared memory parallel machine. We show that our graph grammar based solver outperforms parallel MUMPS solver.

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.

Identification analysis of order of singularity near vertex on interface in three-dimensional bonded structures

Identification analysis of order of singularity near vertex on interface in three-dimensional bonded structures

Analysis of plastic stress singularities of cracks and wedges under the plane stress conditions

The aim of this paper is to analyze the plastic stress singularities of the V-shaped notches and cracks under the plane stress conditions and mixed mode. Based on the formulation of assuming an asymptotic displacement field with respect to the radial coordinate centered at the notch tip, the stress components near the notch tip are expressed as the asymptotic series expansions in term of the radial coordinate by satisfying the plastic yield criterion and the nonlinear stress-strain relation. Then, the series expansions of the stress and displacement components are substituted into the governing differential equations of the plastic theory. Hence, the nonlinear second-order system of ordinary differential equations (ODEs) with the plastic stress singularity orders and the associated eigenfunctions are deduced. The interpolating matrix method is applied to solving the nonlinear eigenvalue problems of the ODEs by an iteration procedure. Thus, the leading plastic stress singularity orders and the associated eigenvectors of the displacement and stress fields in the notch tip region are achieved simultaneously. In addition, the numerical examples are given to demonstrate the accuracy and the effectiveness of the present approach to solve the plastic stress singularities of the plane V-shaped notches and cracks. Furthermore, some mechanical behaviors related to the plastic stress singularities in the notch tip region are investigated.

Stress analysis in transverse loading of soft core sandwich plates with various boundary conditions

In this paper, the transverse loading of sandwich plate is formulated to study the three-dimensional stress field in the sandwich plates for various edge conditions. The formulation is based on the weak formulation approach. A complete three-dimensional displacement field is considered and the weak formulation approach is employed to obtain the governing equations of the plate using the three dimensional equilibrium equations of elasticity. An analytical solution is presented for governing equations when two opposite edges of plate are simply supported. A one-step stress recovery scheme is used to compute the out-of-plane stresses in the sandwich plates. A comparison is made with the predictions of exact elasticity solutions in the open literature and very good agreements are achieved. The distribution of stresses is investigated for various boundary conditions and the log-linear procedure is employed to study the order of stress singularity at free and clamped edge of the plate. It is seen that the present approach accurately predicts the distribution of out-of-plane stresses and local concentration of stresses in the vicinity of free and clamped edges of sandwich structures.