Order Of Singularity Research Articles

Numerical determination of the singularity order of a system of differential equations

We consider moving singular points of systems of ordinary differential equations. A review of Painlevé’s results on the algebraicity of these points and their relation to the Marchuk problem of determining the position and order of moving singularities by means of finite difference method is carried out. We present an implementation of a numerical method for solving this problem, proposed by N. N. Kalitkin and A. Al’shina (2005) based on the Rosenbrock complex scheme in the Sage computer algebra system, the package CROS for Sage. The main functions of this package are described and numerical examples of usage are presented for each of them. To verify the method, computer experiments are executed (1) with equations possessing the Painlevé property, for which the orders are expected to be integer; (2) dynamic Calogero system. This system, well-known as a nontrivial example of a completely integrable Hamiltonian system, in the present context is interesting due to the fact that coordinates and momenta are algebraic functions of time, and the orders of moving branching points can be calculated explicitly. Numerical experiments revealed that the applicability conditions of the method require additional stipulations related to the elimination of superconvergence points.

Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area 4π the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomon formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.

P.831Trajectories of suicidal ideation after a first-episode psychosis: a growth mixture modeling approach

Singular mode shapes around the sharp frictional contact interfaces of different materials with linear elastic behavior are derived semi-analytically using the asymptotic eigen-function expansion method. The characteristic equation which is not proposed in a short explicit expression in the literature is extracted in symbolic terms; then solved numerically and the singularity order is derived using Newton's iterative method for the first time in the literature.The extracted eigen-shapes of the high gradient stress fields and the corresponding displacement fields satisfy the compatibility of deformations and tractions on the contact interface with arbitrary frictional interface equations. These asymptotic mode shapes are then implemented into the finite element interpolation using Partition of Unity Finite Element Method (PUFEM). Since the additional enrichment degrees of freedom on the singular contact nodes have the physical meaning of generalized stress intensity factor (GSIF), the direct GSIFs obtained are more accurate and the contour integral manipulations need not be maintained in the post processing phase of the contact analysis.Finally, some engineering problems are solved which demonstrate the capacities of the approach concerning the accuracy of GSIF obtained from the direct-PUFEM proposed compared with that derived by reciprocal work contour integral method used in traditional finite element analyses.

Crack tip fields and fracture resistance parameters based on strain gradient plasticity

The crack tip mechanics of strain gradient plasticity solids is investigated analytically and numerically. A first-order mechanism-based strain gradient (MSG) plasticity theory based on Taylor’s dislocation model is adopted and implemented in the commercial finite element package ANSYS by means of a user subroutine. Two boundary value problems are considered, a single edge tension specimen and a biaxially loaded plate. First, crack tip fields are characterized. Strain gradient effects associated with dislocation hardening mechanisms elevate crack tip stresses relative to conventional plasticity. A parametric study is conducted and differences with conventional plasticity predictions are quantified. Moreover, the asymptotic nature of the crack tip solution is investigated. The numerical results reveal that the singularity order predicted by the first-order MSG theory is equal or higher to that of linear elastic solids. Also, the crack tip field appears not to have a separable solution. Moreover, contrarily to what has been shown in the higher order version of MSG plasticity, the singularity order exhibits sensitivity to the plastic material properties. Secondly, analytical and numerical approaches are employed to formulate novel amplitude factors for strain gradient plasticity. A generalized J-integral is derived and used to characterize a nonlinear amplitude factor. A closed-form equation for the analytical stress intensity factor is obtained. Amplitude factors are also derived by decomposing the numerical solution for the crack tip stress field. Nonlinear amplitude factor solutions are determined across a wide range of values for the material length scale l and the strain hardening exponent N. The domains of strain gradient relevance are identified, setting the basis for the application of first-order MSG plasticity for fracture and damage assessment.

Well-posedness and approximation of some one-dimensional Lévy-driven non-linear SDEs

In this article, we are interested in the strong well-posedness together with the numerical approximation of some one-dimensional stochastic differential equations with a non-linear drift, in the sense of McKean–Vlasov, driven by a spectrally-positive Lévy process and a Brownian motion. We provide criteria for the existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type without non-degeneracy assumption following the approach developed by Li and Mytnik (2011). The strong convergence rate of the propagation of chaos for the associated particle system and of the corresponding Euler–Maruyama scheme are also investigated. In particular, the strong convergence rate of the Euler–Maruyama scheme exhibits an interplay between the regularity of the coefficients and the order of singularity of the Lévy measure around zero.

An enriched dual boundary element method formulation for linear elastic crack propagation

This study presents an eXtended Boundary Element Method (XBEM) formulation for simulating the linear elastic crack growth in two-dimensional domains. A displacement approximation enrichment based on the Williams first-order expansions is used to represent the near-tip square root behavior predicted by the linear elastic fracture mechanics. This strategy also enables the direct evaluation of the Stress Intensity Factors (SIFs) after a crack tip tying constraint is enforced at the crack tip to accommodate the additional enrichment parameters. Additionally, discontinuous functions are embedded into the displacement approximation of elements intercepted by cracks to avoid remeshing of these elements. For the discontinuous enrichment, new equations are provided by imposing displacement continuity conditions at the crack mouth. Shifted enrichment functions are used to preserve the physical meaning of the nodal parameters and to reduce the singularity order of the integral kernels containing the enrichment terms in the XBEM formulation. Furthermore, an enriched traction approximation is proposed to apply concentrated forces and support points along the boundary. When considering the XBEM formulation, changes in the conventional BEM are minimal, which facilitates the implementation of the extended approach into existing codes. Three numerical examples demonstrate the application of the proposed XBEM formulation in fracture modeling.

The calculation of singular orders for composite material anti-plane propagating V-notches

Because of geometric or material discontinuities, stress singularities can occur around the vertex of a V-notch. The singular order is an important parameter for characterizing the degree of the stress singularity. The present paper focuses on the calculation of the singular order for the anti-plane propagating V-notch in a composite material structure. Starting from the governing equation of elastodynamics and the displacement asymptotic expansion, an ordinary differential eigen equation with respect to the singular order is proposed. The interpolating matrix method is then employed to solve the established eigen equation to conduct the singular orders. The effects of the material principal axis direction, shear modulus, and propagation velocity and acceleration on the singular orders of the V-notch are respectively investigated, and some conclusions are drawn. The singular orders of the V-notches decrease with an increase in the material principal axis direction angle, except for the crack, whose singular orders do not change with the principal axis direction. The singular orders increase with an increase in the shear modulus $$G_{13} $$ , while they decrease with an increase in the shear modulus $$G_{23} $$ . The singular orders increase as the magnitude of the propagation velocity increases, while they decrease as the direction of the propagation velocity increases. The singular orders increase with an increase in the value of the propagation acceleration, while they decrease with an increase in the direction of the propagation acceleration. The singular orders become larger when the ratio of $$G_{13}^{(2)} /G_{13}^{(1)} $$ increases, while they become smaller when the ratio of $$G_{23}^{(2)} /G_{23}^{(1)} $$ increases, and they increase with an increase in the mass density ratio for the bi-material V-notch.

Local well-posedness of the topological Euler alignment models of collective behavior

In this paper we address the problem of well-posedness of multi-dimensional topological Euler-alignment models introduced by Roman and Tadmor (2018 arXiv:1806.01371). The main result demonstrates local existence and uniqueness of classical solutions in class (ρ, u) ∈ Hm+α × Hm+1 on the periodic domain , where 0 < α < 2 is the order of singularity of the topological communication kernel ϕ(x, y), and m = m(n, α) is large. Our approach is based on new sharp coercivity estimates for the topological alignment operator which render proper a priori estimates and help stabilize viscous approximation of the system.

Singular Stresses at a Vertex and Along a Singular Line in Three-‎dimensional Piezoelectric Bonded Joints

Singular stress fields in three-dimensional piezoelectric bonded joints are investigated at a vertex and along a free edge (the singular line) of an interface. Two perfectly bonded joints, which are different in the side surface shapes, are considered. The joints consist of multi-terms of singularity. Two-major terms of singularity are investigated in details. The orders of singularity at the vertex and along the singular line are calculated using three-dimensional finite element eigen-analysis. The intensities of singularity are calculated using the conservative integral. The intensities of singularity at several points located on the singular line are examined. The relationships between the intensities of singularity and the distances from the vertex are plotted to determine how the vertex singularity affects the singularity along the singular line. Finally, the relationships between singular stress fields at the vertex and along the singular line are considered.

Discontinuous Galerkin methods using poly-sinc approximation

In this paper, we develop a discontinuous Galerkin method using poly-Sinc approximation, in which non-equidistant points are generated by conformal mappings, such as Sinc points. Sinc points are characterized by their dense distribution near the endpoints of an interval, which render them effective in dealing with jumps introduced by the partitioning process of the interval in a discontinuous Galerkin scheme. Sinc methods are characterized by their fast convergence rate of exponential order and effective handling of singularities. We use equidistant, Chebyshev, and Sinc partitions to handle the singularities. We demonstrate that, using poly-Sinc-based discontinuous Galerkin approximation, Sinc partitions outperforms those with equidistant and Chebyshev partitions as we near the singularity. We also demonstrate that for Sinc partitions, poly-Sinc-based discontinuous Galerkin approximation outperforms those with Chebyshev points. We used the stability condition of the discontinuous Galerkin method to demonstrate the stability of our approach. It was shown that the approximation error between a smooth function and its poly-Sinc approximation over the global partition has a convergence rate of exponential order. For a function with a singularity at an endpoint, we use the weighted L2 norm over the Sinc partitions. Using the weighted L2 norm, we show that, by computing the ℓ2 norm over all Sinc partitions constituting the global partition except the last partition, the approximation error between the exact solution of the ordinary differential equation and its poly-Sinc-based discontinuous Galerkin approximation has a convergence rate of exponential order.

Singularity modal analysis of frictional interfaces in orthotropic composites

Shear crack or sliding contact within or between orthotropic composites contains innovative local solutions in terms of stress singularity, which are concentrated in this study. For this purpose, a finite element asymptotic expansion is proposed, which extracts the singular point solution for different frictional crack or contact problems in orthotropic composite design.The investigations illustrate the fact that many situations in orthotropic fracture and contact mechanics deal with stress singularities of order higher than the conventional cracks in isotropic materials and even the free crack conditions in the same orthotropic cases.In this paper, the significance of different situations for shear crack and various configuration of frictional contact between orthotropic composites are discussed in terms of the order of singularity. The consistent singular enrichment function derived may be utilized in numerical simulations of fracture and contact phenomena. Direct eXtended Finite Element Method (Direct-XFEM) proposed facilitates an efficient algorithm for implementation of the singularity modal analysis results.

An example of harmonic map into the spheres with the singularity of order 4

The radial map w(x)=x||x|| is well known as harmonic map into the sphere with a point singularity. Misawa and Nakauchi (0000) gave two examples of harmonic maps with the singularities xixj||x||2 and xixjxk||x||3 at x=0 respectively. In this paper we give a new example of harmonic maps into the spheres. It has the singularity xixjxkxl||x||4 at x=0.

Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations

We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today, hypersingular integral equations of this type are widely used in physics and technology. The convergence of the proposed method is based on the Lyapunov stability theory of solutions of ordinary differential equation systems. The advantage of the method for linear equations is in simplicity of unique solvability verification for the approximate equations system in terms of the operator logarithmic norm. This makes it possible to estimate the norm of the inverse matrix for an approximating system. The advantage of the method for nonlinear equations is that neither the existence or reversibility of the nonlinear operator derivative is required. Examples are given illustrating the effectiveness of the proposed method.

Mechanics of crack path selection in microtransfer printing: Challenges and opportunities for process control

Microtransfer printing is a manufacturing technique that relies on controlled selective delamination between two interfaces to transfer thin solid films and chips. Stamps in these processes are often designed to leverage strategies that modify the interfacial stress distribution at the stamp interface to achieve a specific adhesion response, however the effect of a modified stress distribution at the stamp interface on crack path selection between the two interfaces in the system is unclear. This paper investigates how the stress distribution beneath the stamp can affect the delamination path between the two interfaces in a microtransfer printing process using mechanics modeling. In general, altering the stress distribution at the stamp/chip interface also alters the stress distribution at the chip/substrate interface. For a sufficiently thin chip with no or small initial defects at the interfaces, the common approach of tuning adhesion by reducing the strength of stress singularity near the edge of the stamp does not provide a robust route to control the delamination path. An alternative stamp design strategy, guided by the singularity order of the stress distribution, is proposed and analyzed. In the proposed stamp design, the delamination path can be controlled through the stamp thickness and the application of a shear displacement. This work provides a fundamental understanding of the mechanics of the microtransfer printing process as well as guidance for designing successful microtransfer printing processes.

The Cauchy Problem for Parabolic Equations with Degeneration

Annotation. For a second-order parabolic equation, the multipoint in time Cauchy problem is considered. The coefficients of the equation and the boundary condition have power singularities of arbitrary order in time and space variables on a certain set of points. Conditions for the existence and uniqueness of the solution of the problem in Hölder spaces with power weight are found.

Limit cycles bifurcating from periodic orbits near a centre and a homoclinic loop with a nilpotent singularity of Hamiltonian systems

For a planar analytic near-Hamiltonian system, whose unperturbed system has a family of periodic orbits filling a period annulus with the inner boundary an elementary centre and the outer boundary a homoclinic loop through a nilpotent singularity of arbitrary order, we characterize the coefficients of the terms with degree greater than or equal to 2 in the expansion of the first order Melnikov function near the homoclinic loop. Based on these expression of the coefficients, we discuss the limit cycle bifurcations and obtain more number of limit cycles which bifurcate from the family of periodic orbits near the homoclinic loop and the centre. Finally, as an application of our main results we study limit cycle bifurcation of a (m + 1)th order Liénard system with an elliptic Hamiltonian function of degree 4, and improve the lower bound of the maximal number of the isolated zeros of the related Abelian integral for any m ≥ 4.

Singularity of Electromechanical Coupling Field in Piezoelectric Composite Junctions

The double singularities including singular stress field and singular electric displacement field, in the tips of piezoelectric composite junctions, are analyzed by the interpolation matrix method (IMM). The double singularity analysis problem of piezoelectric composite junctions is converted into eigenvalue solution problem of ordinary differential equations with variable coefficients under corresponding boundary conditions. In numerical examples, the first couple of singularity orders and the corresponding characteristic angular functions of displacement and electric potential for the electromechanical coupling field are obtained and comparisons are presented to validate the accuracy of the proposed method. The singularity of the electromechanical coupling field at the tip of piezoelectric composite material junctions is closely related to the bonding angle and fiber direction. According to the numerical results, the best scheme can be configured for the combination of dissimilar materials.

A Generation of Special Triangular Boundary Element Shape Functions for 3D Crack Problems

This paper focuses on tackling the two drawbacks of the dual boundary element method (DBEM) when solving crack problems with a discontinuous triangular element: low accuracy of the calculation of integrals with singularity and crack front element must be utilized to model the square-root property of displacement. In order to calculate the integrals with higher order singularity, the triangular elements are segmented into several subregions which consist of subtriangles and subpolygons. The singular integrals in those subtriangles are handled by the singularity subtraction technique in the integration space and can be regularized and accurately calculated. For the nearly singular integrals in those subpolygons, the element subdivision technique is employed to improve the calculation accuracy. In addition, considering the location of the crack front in the element, special crack front elements are constructed based on a 6-node discontinuous triangular element, in which the displacement extrapolation method is introduced to obtain the stress intensity factors (SIFs) without consideration of orthogonalization of the crack front mesh. Several numerical results are investigated to fully verify the validation of the presented approach.

Partition of unity finite element for analysis of contact interface stress singularity

Singular mode shapes around the sharp frictional contact interfaces of different materials with linear elastic behavior are derived semi-analytically using the asymptotic eigen-function expansion method. The characteristic equation which is not proposed in a short explicit expression in the literature is extracted in symbolic terms; then solved numerically and the singularity order is derived using Newton's iterative method for the first time in the literature.The extracted eigen-shapes of the high gradient stress fields and the corresponding displacement fields satisfy the compatibility of deformations and tractions on the contact interface with arbitrary frictional interface equations. These asymptotic mode shapes are then implemented into the finite element interpolation using Partition of Unity Finite Element Method (PUFEM). Since the additional enrichment degrees of freedom on the singular contact nodes have the physical meaning of generalized stress intensity factor (GSIF), the direct GSIFs obtained are more accurate and the contour integral manipulations need not be maintained in the post processing phase of the contact analysis.Finally, some engineering problems are solved which demonstrate the capacities of the approach concerning the accuracy of GSIF obtained from the direct-PUFEM proposed compared with that derived by reciprocal work contour integral method used in traditional finite element analyses.

A note on Pageau et al. (1994) results for the order of stress singularities in multimaterial junctions

In 1994, Pageau, Joseph and Biggers Jr. evaluated “The order of stress singularities for bonded and disbonded three-material junctions”. Their work represented an important finding on multimaterial corners, since their formulation can be used to study the stress state on these junctions. Despite the relevance, the graphs of some results presented in their paper might lead to an improper interpretation, and consequently to a wrong asymptotic stress representation. This is evidenced and discussed in the present work, in which an analytical algorithm was implemented, allowing to reproduce, to complete and to clarify the original results. The physical interpretation of the divergences indicates, among other observations, that the failure mode associated with the highest stress singularity order in a multimaterial junction may change by only varying the Young’s modulus of one of the materials.