Order Of Singularity Research Articles

Evaluation of heat flux intensity factor for a V-notched structure by the isogeometric boundary element method

The conventional boundary element method using piecewise polynomial interpolation cannot accurately simulate the singular heat flux field around the vertex of the V-notch. Herein, the singularity eigen-analysis combined with the isogeometric boundary element method is proposed to calculate the singular heat flux field. The V-notched structure is divided into two parts, in which one is the heat flux singularity sector near the vertex and the other is the remained structure without heat flux singularity. In the singularity sector, the asymptotic expansion of the heat flux is introduced to transform the heat conduction governing equation into ordinary differential eigen equation, from which the singularity orders and eigen angular functions can be determined, except for the amplitude coefficients in the asymptotic expansion. The boundary integral equations for the heat conduction analysis established on the remained structure are discretized by the non-uniform rational B-spline (NURBS) elements. The amplitude coefficients, which are corresponding to the heat flux intensity factors, can be yielded by coupling the isogeometric boundary integral equations with the singularity asymptotic expansion analysis. A coordinate system transformation method is then proposed to transform the heat conduction governing equations of orthotropic and anisotropic material into the one of the isotropic material, and the heat flux intensity factors are approved to be invariable before and after coordinate transformation. Since the NURBS elements are used, fewer elements are required to evaluate the heat flux intensity factors compared with the conventional boundary element method.

An Analytical Approach for the Near‐Tip Field Around V‐Notch in Orthotropic Materials

ABSTRACTThis study introduces a novel theoretical framework for determining the notch stress intensity factors (NSIFs) in composite materials. The approach involves utilizing Irwin's complex stress functions for Mode I and Mode II loading conditions. By investigating displacement and singular stress components around V‐notch tips, the research aims to provide a comprehensive understanding of NSIFs in composite materials. Three different composite materials were considered in the study to explore the relationship between stress and material properties. The stress singularity order, denoted as , is determined through eigenequations. Moreover, explicit solutions for displacement and singular stress fields are derived for a more thorough understanding of the behavior around notches in composite materials. To validate the proposed theoretical framework, a rigorous numerical study is conducted using FEM on finite‐size notches. The results obtained from the numerical simulations are compared with theoretical predictions to assess the accuracy and reliability of the developed approach.

Singularity analysis for V‐notches in a piezoelectric multimaterial and functionally graded piezoelectric materials

AbstractThis study develops a novel singularity analysis method that addresses the limitations posed by piezoelectric material quantity and the variation patterns of property functions in functionally graded piezoelectric materials (FGPMs). Given that piezoelectric composite materials consist of multiple materials, the material properties of FGPMs vary with respect to angle. By introducing the asymptotic assumption that governs the physical fields close to the notch vertex into the static equilibrium equations, a comprehensive set of characteristic equations is formulated. All singularity orders and corresponding characteristic angle functions of the notch are determined by numerically solving the established characteristic equations. The effects of the notch opening angle, piezoelectric material polarization direction, and boundary conditions on the electromechanical field singularity at the notch are assessed. The judicious selection of material variation patterns can alleviate notch singularity in FGPMs.

Cancelling the effect of sharp notches or cracks with graded elastic modulus materials

Recent technologies permit to build materials which have elastic spatially varying modulus which can also imitate solutions adopted in Nature to optimize some structures. It has been shown that for example the stress concentration due to a hole in an infinite plate can be cancelled with a radially varying modulus making it similar to load-bearing bones which seem to resist structural failures even in the presence of blood vessel holes (foramina). Here, we attempt to study the classical problem of a sharp wedge (which includes the important case of a crack) looking for stresses varying as power law of the distance from the notch tip, σ∼rα, with a modulus varying as E∼rβ. In the inhomogeneous case the order of singularity of the LEFM case decreases if β>0, as confirmed by FEM investigations. Hence, we can remove stress singularities, which suggests an interesting alternative to the “rounding” of the notch. More in general, since for many materials it has been found that both strength and modulus are power laws of the density, using the so called strength-modulus exponent ratio we can obtain optimal design by keeping the asymptotic stress constantly equal to the strength. The present investigation paves the way for a new optimization strategy in the problems which eliminates size-scale effects due to singular stress fields, with potentially very wide applications.

Evolution of topological singularities below the light line in momentum space.

Polarization singularities that exist in momentum space have brought new opportunities in various fields such as enhanced optical nonlinearity, structured laser sources, and light field manipulation. However, previous researches have predominantly focused on the polarization singularities above the light line, because they have no leakage and are referred to bound states in the continuum. Here, by extending the polarization fields to Fourier components of the evanescent field on a dielectric metasurface, polarization singularities of different Fourier orders are discovered below the light line. When continuously changing the geometrical parameters of the metasurface, a Fourier order transition process of the polarization singularity is observed through the bandgap closing at the boundary of the Brillouin zone, which finally leads to the annihilation of two singularities with opposite topological charges below the light line. These findings expand the understanding of polarization singularities in the near-field region and may find applications in light field manipulation and light-matter interaction.

Enrichment of the element free Galerkin method for cracks and notches without a priori knowledge of the analytical singularity order

We introduce a novel enrichment technique for the element free Galerkin (EFG) method to capture the infinite stress field around the crack or notch tip. The singular enrichment bases are built through a weighted residual application of the governing partial differential equation. A special mapping is used to adapt the bases with the singularities. These bases are used in a moving least squares approximation to enrich the smooth polynomial bases of the EFG within the singularity-dominated zone in an intrinsic approach. The novelty of the method is that no a priori knowledge of the analytical singularity order is required to reproduce the singular bases, but they are automatically built during the solution procedure. The diffraction technique has been used to model the discontinuity of the notch tip. The effect of various parameters, including the enriched region size, enrichment factors, nodal distribution and irregularities, has been examined. Validation of the method using available analytical solutions or finite element modeling, revealed that the proposed enrichment is quite effective and accurate for displacement and stress components.

Determining Topological Charge of Bessel-Gaussian Beams Using Modified Mach-Zehnder Interferometer

The orbital angular momentum (OAM) associated with structured singular beams carries vital information crucial for studying various properties and applications of light. Determining OAM through the interference of light is an efficient method. The interferogram serves as a valuable tool for analyzing the wavefront of structured beams, especially identifying the order of singularity. In this study, we propose a modified Mach–Zehnder interferometer architecture to effectively determine the topological charge of Bessel–Gaussian (BG) beams. Several numerically generated self-referenced interferograms have been used for analysis. Moreover, this study examines the propagation property and phase distribution within BG beams after they are obstructed by an aperture in the interferometer setup.

The linear term of the Poincaré map at singularities of planar vector fields

The aim of this work is to give conditions on the parameters of a family of analytic planar vector fields with a fixed Newton diagram and a monodromic singularity in order to guarantee that the coefficient (generalized first Lyapunov constant) of the linear part of the Poincaré map has an explicit formula that depends only on the Newton diagram. As far as we know, all the monodromic classes appearing in the literature satisfy that formula.

Evaluation of mixed-mode stress intensity factors for anisotropic elastic solids with interface singularity

Abstract When a structure is composed of many different parts and each part is made of different materials, it is very possible that many interface corners appear in several local fields of the structure. Due to the mismatch of elastic properties, stress singularity may occur at the interface corners. The singular orders of stresses and their associated stress intensity factors (SIFs) are two important parameters for understanding failure initiation. Since the singular orders of general interface corners may be real or complex, distinct or repeated, several different definitions of SIFs have been proposed for different types of interfaces. In order to build a direct connection among all general interface corners (including cracks and interface cracks), a unified definition of SIFs was proposed in our previous studies. To calculate the uni-defined SIF accurately and efficiently, a special boundary element using the fundamental solution satisfying interface continuity conditions is established. The singular integrals involved in the associated boundary integral equations are solved by the setting of proper rigid body motions and the approximation through interpolation of nearby non-singular solutions. To avoid the unstable and inaccurate region near the corners, in this study we employ the path-independent H-integral, and derive the relations between H-integral and SIFs. Moreover, the auxiliary solutions of displacements and tractions required in the H-integral are provided for different integral paths. Several numerical examples, such as interface cracks, interface corners, single lap joints and delamination, are presented to demonstrate the methodology employed in this paper.

The blow-up method applied to monodromic singularities

The blow-up method proves its effectiveness to characterize the integrability of the resonant saddles giving the necessary conditions to have formal integrability and the sufficiency doing the resolution of the associated recurrence differential equation using induction. In this work we apply the blow-up method to monodromic singularities in order to solve the center-focus problem. The case of nondegenerate monodromic singularities is straightforward since any real nondegenerate monodromy singularity can be embedded into a complex system with a resonant saddle. Here we apply the method to nilpotent and degenerate monodromic singularities solving the center problem when the center conditions are algebraic.

Admissible H$_\infty$ Control of Fuzzy Singular Fractional Order Multi-Agent Systems With External Disturbances

This brief consider the problem about admissible consensus of fuzzy singular fractional order multi agent systems (FSFOMAS). By designing a new distributed fuzzy dynamic output control strategy, the coupled system achieves admissible while satisfying <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_\infty$</tex-math> </inline-formula> performance. Then, when the fuzzy models reduce to ordinary fractional order multi agent systems (FOMAS), a new sufficient condition for determining system consensus is given. The above results all depend only on the eigenvalues of the Laplace matrix and the state matrix of the system and consist of linear matrix inequalities (LMIs). Finally, a numerical example and a practical circuit example can demonstrate the feasibility and validity of the theorems. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Due to the unique memorability of fractional orders, their application to fuzzy multi-agent systems will bring system performance metrics closer to reality. This paper investigates the stabilization of singular fuzzy fractional-order multi-agent systems with the perturbation case, the pulse case. We know that in practical applications a single variable does not accurately describe the phenomenon. Therefore we have introduced the concepts of multi-agent, fuzzy, singular and fractional orders in order to describe the system model more accurately. The main objective is to bring the system into equilibrium in the closest possible approximation to reality. The performance metrics of the system are optimised to bring the system to an optimal state. Finally, two examples are used to illustrate the validity of our theorems.

Frictional weakening leads to unconventional singularities during dynamic rupture propagation

Earthquakes i.e. frictional ruptures, are commonly described by singular solutions of shear crack motions. These solutions assume a square root singularity order around the rupture tip and a constant shear stress value behind it, implying scale-independent edge-localized energy. However, recent observations of large-scale thermal weakening accompanied by decreasing shear stress potentially affecting the singularity order can challenge this assumption. In this study, we replicate earthquakes in a laboratory setting by conducting stick-slip experiments on PMMA samples under normal stress ranging from 1 to 4 MPa. Strain gauges rosettes, located near the frictional interface, are used to analyze each rupture event, enabling the investigation of shear stress evolution, slip velocity, and material displacement as a function of distance from the rupture tip. Our analysis of the rupture dynamics provides compelling experimental evidence of frictional rupture driven by enhanced thermal weakening. The observed rupture fronts exhibit unconventional singularity orders and display slip-dependent breakdown work (on-fault dissipated energy). Moreover, these findings elucidate the challenges associated with a priori estimating the energy budget controlling the velocity and final extent of a seismic rupture, when thermal weakening is activated during seismic slip.

Multiple-order singularity expansion method

Physical systems and signals are characterized by complex functions of the frequency in the harmonic domain. The extension of such functions to the complex frequency plane, and in particular expansions and factorized forms of the harmonic-domain functions in terms of their poles and zeros, is of high interest to describe the physical properties of a system, and study its response dynamics in the temporal and harmonic domains. In this work, we start from a general property of continuity and differentiability of the complex functions to derive the multiple-order singularity expansion method. We rigorously derive the common singularity and zero expansion and factorization expressions, and generalize them to the case of singularities of arbitrary order, while deducing the behavior of these complex frequencies from the simple hypothesis that we are dealing with physically realistic signals.

Global convergence in non-relativistic limits for Euler-Maxwell system near non-constant equilibrium

In this paper, we study the global-in-time convergence of non-relativistic limits from Euler-Maxwell systems to Euler-Poisson systems near non-constant equilibrium states by letting the reciprocal of the speed of light ν:=1/c→0. In previous studies, the dissipative estimates for the electric field E appear to be singular and the orders of singularities for divE and ∇×E are different, hence a div-curl decomposition should be considered in our case, and this makes it unclear the preservation of the anti-symmetric structure of the system and the global-in-time L2–estimate of the smooth solutions. To overcome these difficulties, we find a strictly convex entropy of the system that is valid for the case of non-constant equilibrium to obtain the global L2–estimate and use some induction arguments to close the estimates. It is worth mentioning that in our proof, very careful and accurate estimates of solutions are needed and we show that the electric field E is actually non-singular, which is vital and necessary for our proof.

Projection continuation for minimal coordinate set formulation and singularity detection of redundantly constrained system dynamics

The formulation of (possibly redundantly) constrained system dynamics using coordinate projection onto a subspace locally tangent to the constraint manifold is revisited using the QR factorization of the constraint Jacobian matrix, using column pivoting to identify a suitable subspace, possibly detect any singular configurations that may arise, and extract it. The evolution of the QR factorization is integrated along with that of the constraint Jacobian matrix as the solution evolves, generalizing to redundant constraints a recently proposed true continuation algorithm that tracks the evolution of the subspace of independent coordinates. The resulting subspace does not visibly affect the quality of the solution, as it is merely a recombination of that resulting from the blind application of the QR factorization but avoids the artificial algorithmic irregularities or discontinuities in the generalized velocities that could otherwise result from arbitrary reparameterizations of the coordinate set, and identifies and discriminates any further possible motions that arise at singular configurations. The characteristics of the proposed subspace evolution approach are exemplified by solving simple problems with incremental levels of redundancy and singularity orders.

Determination of singular and higher order non-singular stress for angularly heterogeneous material notch

The complete stress distribution around the tip of angularly heterogeneous material V-notch includes singular stress term and higher order non-singular stress term. The latter has important role on the prediction of the fracture, while it is usually neglected due to the evaluation difficulty. The idea of combing the singularity analysis near the notch vertex with the finite element analysis is established to deal with this difficulty. Firstly, a set of variable coefficient differential equations for the singular analysis of angularly heterogeneous material notch are built. The interpolating matrix method is introduced to solve the singular characteristic equations to prepare the singularity order and characteristic angular function. The V-notched structure is then analyzed by the coarse finite element meshes. The displacements from the finite element analysis are substituted into the singularity asymptotic expression to form a set of over-determined equation. The least square method is applied to yield the amplitude coefficients in the series asymptotic expansion. The complete stress field near the angularly heterogeneous material V-notch tip is then constructed according to the obtained amplitude coefficients, singularity orders, characteristic angular functions, together with their first order derivatives. The singular and non-singular stress term can be respectively calculated according to the truncated series terms. The reconstructed stresses have the same precision of the displacement from the view point of the finite element analysis. The role of the non-singular stress on the stress intensity factors of angularly heterogeneous material notch is then investigated.

High-order spectral singularity

Exceptional point and spectral singularity are two types of singularity that are unique to non-Hermitian systems. Here, we report the high-order spectral singularity as a high-order pole of the scattering matrix for a non-Hermitian scattering system, and the high-order spectral singularity is a unification of the exceptional point and spectral singularity. At the high-order spectral singularity, the scattering coefficients have high-order divergence and the scattering system stimulates high-order lasing. The wave emission intensity is polynomially enhanced, and the order of the growth in the polynomial intensity linearly scales with the order of the spectral singularity. Furthermore, the coherent input controls and alters the order of the spectral singularity. Our findings provide profound insights into the fundamentals and applications of high-order spectral singularities.

Rolling bearing faults identification based on multiscale singular value

To solve the vulnerability of singular value denoising to the effective number of singular value orders and influence of Hankel matrix structure, the paper has proposed a multiscale singular value denoising algorithm (M-SVD). Meanwhile, in concerning that when a fault of bearing occurs, the fault feature information contained in vibration signal tends to be weak and complex, the M-SVD method is blended with variational mode decomposition (VMD) algorithm in identification. Firstly, when bearing faults occur, raw vibration acceleration signals are decomposed using the superiority of VMD algorithm for non-stationary signal; in addition, considering that with the larger difference between singular values of signal, feature information will be more obvious, the method takes advantage of sensibility of variance to signal intensity and proposes to constitute Hankel matrix based on variance feature of signal. During the construction of Hankel matrix, the method to determine effective length of variance sequence (optimal scale) has been brought forward by sample entropy which is able to precisely measure the regularity of signal and detect the weak change of signal. Furthermore, component signals are denoised in line with reconstructed Hankel matrix and the spectrum of signals denoised are selected to describe fault characteristic information and judge the failure types. For the sake of verification of the accuracy and effectiveness of proposed method, the paper has analyzed and verified the data of rolling bearing in 3 different failure types and 2 different installation positions of accelerometers. The comparative analysis with classical singular value difference spectrum (SVDS) and singular value median decomposition (SVMD) denoising has further proved that presented VMD combined M-SVD method can effectively restrain noise while retaining fault characteristic information of signal. Therefore, the identification for rolling bearing fault types can be implemented, accurately.

Stress singularity of one-dimensional hexagonal piezoelectric quasicrystal composites due to thermal effect

In the framework of thermo-electro-elasticity, the present paper investigates the singular behaviors of interface corners, interface cracks, composite wedges and spaces for one-dimensional hexagonal quasicrystal. The stress function and temperature variation can be described as the exponential form with a view to stress and heat flux singularities. Based on the Stroh formalism, the analytical expressions of singular orders of stress and heat flux are easily established by simple multiplication of the crucial matrix. Numerical examples of the singular orders are given for some general cases including single, bi-material, and tri-material wedges and spaces under different boundary conditions. Numerical results show that the geometry structures, material properties, boundary conditions, and heat conduction coefficients have great influences on singularities, but thermal moduli have no effect on singularities.

Spatially segmented SVD clutter filtering in cardiac blood flow imaging with diverging waves

Ultrafast ultrasound imaging enables the visualization of rapidly changing blood flow dynamics in the chambers of the heart. Singular value decomposition (SVD) filters outperform conventional high pass clutter rejection filters for ultrafast blood flow imaging of small and shallow fields of view (e.g., functional imaging of brain activity). However, implementing SVD filters can be challenging in cardiac imaging due to the complex spatially and temporally varying tissue characteristics. To address this challenge, we describe a method that involves excluding the proximal portion of the image (near the chest wall) and divides the reduced field of view into overlapped segments, within which tissue signals are expected to be spatially and temporally coherent. SVD filtering with automatic selection of cut-off singular vector orders to remove tissue and noise signals is implemented for each segment. Auto-thresholding is based on the coherence of spatial singular vectors, delineating tissue, blood, and noise subspaces within a spatial similarity matrix calculated for each segment. Filtered blood flow signals from the segments are reconstructed and then combined and Doppler processing is used to form a set of blood flow images. Preliminary experimental results suggest that the spatially segmented approach improves the separation of the tissue and blood subsets in the spatial similarity matrix so that automatic thresholding is significantly improved, and tissue clutter can then be rejected more effectively in cardiac ultrafast imaging, compared to using the full field of view. In the case studied, spatially segmented SVD improved the rate of correct automatic selection of thresholds from 78% to 98.7% for the investigated cases and improved the post-filter power of blood signals by an average of more than 10 dB during a cardiac cycle.