Showing papers on "Singularity published in 1986"
1,096 citations
TL;DR: In this article, a variational formulation and computational aspects of a three-dimensional finite-strain rod model, considered in Part I, are presented, which bypasses the singularity typically associated with the use of Euler angles.
Abstract: The variational formulation and computational aspects of a three-dimensional finite-strain rod model, considered in Part I, are presented. A particular parametrization is employed that bypasses the singularity typically associated with the use of Euler angles. As in the classical Kirchhoff-Love model, rotations have the standard interpretation of orthogonal, generally noncommutative, transformations. This is in contrast with alternative formulations proposed by Argyris et al. [5–8], based on the notion of semitangential rotation. Emphasis is placed on a geometric approach, which proves essential in the formulation of algorithms. In particular, the configuration update procedure becomes the algorithmic counterpart of the exponential map. The computational implementation relies on the formula for the exponential of a skew-symmetric matrix. Consistent linearization procedures are employed to obtain linearized weak forms of the balance equations. The geometric stiffness then becomes generally nonsymmetric as a result of the non-Euclidean character of the configuration space. However, complete symmetry is recovered at an equilibrium configuration, provided that the loading is conservative. An explicit condition for this to be the case is obtained. Numerical simulations including postbuckling behavior and nonconservative loading are also presented. Details pertaining to the implementation of the present formulation are also discussed.
986 citations
TL;DR: In this paper, the authors examined Rosenhead's point-vortex approximation and sought to understand better the source of this difficulty, using discrete Fourier analysis, it was shown that perturbations introduced spuriously by computer roundoff error are responsible for the irregular point vortex motion that occurs at a smaller time as the number of points is increased, this source of computational error is controlled here by using either higher precision arithmetic or a new filtering technique.
Abstract: The initial-value problem for perturbations of a flat, constant-strength vortex sheet is linearly ill posed in the sense of Hadamard, owing to Kelvin-Helmholtz instability. Previous numerical studies of this problem have experienced difficulty in converging when the mesh was refined. The present work examines Rosenhead’s point-vortex approximation and seeks to understand better the source of this difficulty. Using discrete Fourier analysis, it is shown that perturbations introduced spuriously by computer roundoff error are responsible for the irregular point-vortex motion that occurs at a smaller time as the number of points is increased. This source of computational error is controlled here by using either higher precision arithmetic or a new filtering technique. Computations are presented which use a linear-theory growing eigenfunction of small amplitude/wavelength ratio as the initial perturbation. The results indicate the formation of a singularity in the vortex sheet at a finite time as previously found for other initial data by Moore and Meiron, Baker & Orszag using different techniques of analysis. Numerical evidence suggests that the point vortex approximation converges up to but not beyond the time of singularity formation in the vortex sheet. For large enough initial amplitude, two singularities appear along the sheet at the critical time.
433 citations
TL;DR: A review of the wide variety of predictions that results from a Landau-type of description of the nematic-isotropic phase transition is given in this paper, which includes a discussion of the nature of the order parameter and of the various types of possible phases.
291 citations
TL;DR: In this paper, the global existence problem for regular solutions of the relativistic Vlasov-Maxwell equations was studied for the case where the plasma density vanishes a priori for velocities near the speed of light.
Abstract: The global existence problem is studied for regular solutions of the relativistic Vlasov-Maxwell equations. If it is assumed that the plasma density vanishes a priori for velocities near the speed of light, then regular solutions with arbitrary initial data exist in all of space and time. This assumption is either postulated for a solution or is arranged for all solutions through a modification of the equations themselves.
255 citations
TL;DR: In this paper, the main emphasis is placed on the analysis when the exact solution has singularity of xα-type, and the first part analyzes the p-version, the second the h-version and general h-p version and the final third part addresses the problems of the adaptive h-P version.
Abstract: This paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting The main emphasis is placed on the analysis when the (exact) solution has singularity of xα-type. The first part analyzes thep-version, the second theh-version and generalh-p version and the final third part addresses the problems of the adaptiveh-p version.
246 citations
TL;DR: In this paper, Maslowe et al. presented a new perturbation approach using a nonlinear critical layer (i.e., nonlinear terms are restored within a thin layer).
Abstract: The normal mode approach to investigating the stability of a parallel shear flow involves the superposition of a small wavelike perturbation on the basic flow. Its evolution in space and/or time is then determined. In the linear inviscid theory, if ū(y) is the basic velocity profile, then a singularity occurs at critical points yc, where ū = c, the perturbation phase speed. This is plausible intuitively because energy can be exchanged most efficiently where the wave and mean flow are travelling at the same speed. The problem is of the singular perturbation type; when viscosity or nonlinearity, for example, are restored to the governing equations, the singularity is removed. In this lecture, the classical viscous theory is first outlined before presenting a newer perturbation approach using a nonlinear critical layer (i.e., nonlinear terms are restored within a thin layer). The application to the case of a density stratified shear flow is discussed and, finally, the results are compared qualitatively with radar observations and also with recent numerical simulations of the full equations. ∗Address for correspondence: Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6, Canada. e-mail: maslowe@math.mcgill.ca
246 citations
TL;DR: In this paper, it was shown that the AR quiver of reflexive R-modules of a finite subgroup G of GL(2, C), where C is the complex numbers, is the same as the ar-quiver of the reflexive modules of the quotient singularity associated with G. The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the fimte-dimensional complex represen-tations of G, where G is a finite group, is isomorphic to the
Abstract: The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimen- sional representations of a finite subgroup G of GL(2, C), where C is the complex numbers, is isomorphic to the AR quiver of the reflexive modules of the quotient singularity associated with G. Over the past decade, almost split sequences have been playing an increasingly important role in the representation theories of finite-dimensional algebras and classical orders (see (5 and 3, 8) for basic existence theorems in these contexts). While they have been known for some time to exist in higher-dimensional situations (3), it has not been at all clear how they related to singularity theory, if at all. The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the fimte-dimensional complex represen- tations of a finite subgroup G of GL(2,C), where C is the complex numbers, is isomorphic to the AR quiver of the reflexive modules over the quotient singularity R associated with G. As in the case of finite-dimensional algebras, the AR quiver of reflexive R-modules is defined in terms of the almost split sequences of reflexive R-modules. In the case G c SL(2,C), McKay observed that the underlying graph of the McKay quiver, with the trivial module removed, is isomorphic to the desingulariza tion graph of the associated singularity. Various explanations of this phenomenon have been given by Knorrer, Gonzalez-Sprinberg-Verdier (6) and Artin-Verdier (1), which along the way have established, most explicitly in (1), a natural one-to-one correspondence between the indecomposable reflexive R-modules and the nodes of the desingularization graph. But why the almost split sequences describe the edge of the desingularization graph still remains to be explained. An effort has been made to make this paper as self-contained as possible. In particular, no prior knowledge of almost split sequences is required. Before describing the contents of the six sections of this paper we fix some notation. Throughout this paper G is a finite group, k an algebraically closed field of characteristic not dividing the order of G and V a two-dimensional k-representation of G. Setting S = k((X, Y)), the k-algebra of formal power series, the two-dimen- sional representation V gives a linear action of G on S as a group of k-algebra automorphisms. We denote by S(G) the skew group ring given by this action.
232 citations
TL;DR: In this paper, the properties of the three basic versions of the finite element method in the one-dimensional setting are analyzed in detail, and the main emphasis is placed on the analysis when the exact solution has singularity of x sub alpha type.
Abstract: : This paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting. The main emphasis is placed on the analysis when the (exact) solution has singularity of x sub alpha type. The first part analyzes the p-version, the second the h-version and general h-p version and the final third part addresses the problems of the adaptive h-p version. Additional keywords: Legendre polynomials; Legendre expansion; estimates; Asymptotic behavior; Approximation (mathematics); Analytic functions. (Author)
221 citations
TL;DR: In this paper, the structure and formation of singularities in solutions to nonlinear dispersive evolution equations were studied. But the singularities were not observed in the case of nonlinear evolution equations.
Abstract: (1986). On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations. Communications in Partial Differential Equations: Vol. 11, No. 5, pp. 545-565.
185 citations
TL;DR: This paper presents a new hybrid element approach based on displacement fields which fulfil a priori the non-homogeneous Lagrange equation (Trefftz method) and applies it to plate bending, showing the excellent accuracy and efficiency of the new elements.
Abstract: This paper presents a new hybrid element approach and applies it to plate bending. In contrast to more conventional models, the formulation is based on displacement fields which fulfil a priori the non-homogeneous Lagrange equation (Trefftz method). The interelement continuity is enforced by using a stationary principle together with an independent interelement displacement. The final unknowns are the nodal displacements and the elements may be implemented without any difficulty in finite element libraries of standard finite element programs. The formulation only calls for integration along the element boundaries which enables arbitrary polygonal or even curve-sided elements to be generated. Where relevant, known local solutions in the vicinity of a singularity or stress concentration may be used as an optional expansion basis to obtain, for example, particular singular corner elements, elements presenting circular holes, etc. Thus a high degree of accuracy may be achieved without a troublesome mesh refinement. Another important advantage of the formulation is the possibility of generating by a single element subroutine a large number of various elements (triangles, quadrilaterals, etc.), presenting an increasing degree of accuracy. The paper summarizes the results of numerical studies and shows the excellent accuracy and efficiency of the new elements. The conclusions present some ideas concerning the adaptive version of the new elements, extension to nonlinear problems and some other developments.
TL;DR: In this article, the authors present numerical results that describe the nature of the focusing singularity in the cubic Schrodinger equation, where singularities occur at points of intense self-focusing.
Abstract: The cubic Schr\"odinger equation has singular solutions in two or more space dimensions. The singularities occur at points of intense self-focusing. In this paper we present numerical results that describe the nature of the focusing singularity.
TL;DR: In this paper, it was shown that there is a quadratic function 4 : H 1 (L), +Q/H such that 4(5*+9)-4(5)-4 (rl)=1(5,~).
TL;DR: In this article, a global analysis of the double forking is made possible by the realization that the reduced phase space consists of bundles of two-dimensional spheres, and extensive numerical integrations illustrate the transitions in the phase flow on the spheres as the system passes through the Hopf bifurcations.
Abstract: Certain it is that the critical inclination in the main problem of artificial satellite theory is an intrinsic singularity. Its significance stems from two geometric events in the reduced phase space on the manifolds of constant polar angular momentum and constant Delaunay action. In the neighborhood of the critical inclination, along the family of circular orbits, there appear two Hopf bifurcations, to each of which there converge two families of orbits with stationary perigees. On the stretch between the bifurcations, the circular orbits in the planes at critical inclinmation are unstable. A global analysis of the double forking is made possible by the realization that the reduced phase space consists of bundles of two-dimensional spheres. Extensive numerical integrations illustrate the transitions in the phase flow on the spheres as the system passes through the bifurcations.
TL;DR: In this article, all solutions of the form u=rkf(φ) to the p-harmonic equation, div(|∇u|p−2∇ u)=0, (p>2) in the plane are determined.
Abstract: Here, all solutions of the form u=rkf(φ) to the p-harmonic equation, div(|∇u|p−2∇u)=0, (p>2) in the plane are determined. One main result is a representation formula for such solutions. Further, solutions with an isolated singularity at the origin are constructed (Theorem 1). Graphical illustrations are given at the end of the paper. Finally, all solutions u=rkf(φ) of the limit equation for p=∞, ux2uxx+2uxuyuxy+uy2uyy=2, are constructed, some of which have a “strong” singularity at the origin (Theorem 2).
TL;DR: It is shown that the symmetry-breaking pattern of this model predicts the correct analytical structure of the two-point Green's functions of the disordered system, which are expected to be singular (finite) for localized states in the limit of vanishing frequency.
Abstract: The critical behavior of disordered single-particle systems without time-reversal invariance is described by a nonlinear \ensuremath{\sigma} model formulated in terms of graded pseudounitary matrices. It is shown that the symmetry-breaking pattern of this model predicts the correct analytical structure of the two-point Green's functions of the disordered system, which are expected to be singular (finite) for localized (extended) states in the limit of vanishing frequency. The main purpose of the paper is to obtain exact solutions for the graded nonlinear \ensuremath{\sigma} model on a Bethe lattice. A combination of analytical and numerical methods is used to determine the critical behavior of all two-point Green's functions. In contrast with other work on the graded nonlinear \ensuremath{\sigma} model, no minimum metallic conductivity is found. Instead, the averaged inverse conductivity has an exponential singularity at the critical point.
TL;DR: In this article, the authors developed the principle of virtual power for finite velocity fields for simple materials (or first-gradient theory) without further constitutive assumptions when the body is swept out by a singular surface which is either a free singular surface or athermodynamical singular surface.
Abstract: The work develops the principle of virtual power for finite velocity fields for so-called simple materials (or first-gradient theory) without further constitutive assumptions when the body is swept out by a singular surface which is either afree singular surface (such as usual strong discontinuities of continuum mechanics) or athermodynamical singular surface (a so-called interface between phases). The formulation given on exemplary cases first shows how to systematically construct the new “internal” contact forces which exist at the discontinuity, as well as the new inertial contributions which arise from mass transfer across the singular surface and the acceleration of particles attached to it. Then it is shown how various virtual velocity fields generate all the dynamical field equations as well as transversality conditions when the description of external forces allows for them. The principle of virtual power here is so formulated that, when combined, forreal velocity fields, with the first principle of thermodynamics in global form, it yields directly the socalled energy theorem both in the bulk and at the singular surface. Then the corresponding rates of entropy production are deduced after introduction of the second principle of thermodynamics. While one does not claim to obtain here essentially new equations, the present formulation of the principle of virtual power paves the way for useful complex extensions which are difficult to deal with through other avenues (e.g., electromagnetic continua with “junctions” such as piezoelectric semiconductors).
TL;DR: All static and cylindrically symmetric vacuum solutions of Einstein's field equation with cosmological constant ..lambda.. are found and are used to represent the exterior metric of a cosmic string.
Abstract: All static and cylindrically symmetric vacuum solutions of Einstein's field equation with cosmological constant ..lambda.. are found. We use these solutions to represent the exterior metric of a cosmic string. If ..lambda.. is negative, the exterior approaches the anti--de Sitter metric away from the string. If ..lambda.. is positive, the exterior metric is either an scrR /sup 3/ x scrS /sup 1/ universe with a curvature singularity, an scrR /sup 2/ x scrS /sup 2/ universe with a conical singularity, or an scrR /sup 2/ x scrS /sup 2/ universe with no singularity.
TL;DR: In this paper, an integrable equation due to Drinfel-d and Sokolov [Sov. Math. Dokl. 23, 457 (1981) and Wilson [Phys. Lett. A 89, 332 (1982)] (DSW) is studied in detail.
Abstract: An integrable equation due to Drinfel’d and Sokolov [Sov. Math. Dokl. 23, 457 (1981)] and Wilson [Phys. Lett. A 89, 332 (1982)] (DSW) is studied in detail. It is shown how this system can be obtained as a six‐reduction of the Kadomtsev–Petviashvili hierarchy. This equation presents a novel type of solutions called static solitons: they are static solutions that interact with moving solitons without deformations. Examples of such solutions are given, together with a general procedure for their construction. Finally the Painleve analysis of the DSW equation is performed directly on the bilinear form, which constitutes a new application of the singularity analysis method.
TL;DR: A critical review of the literature on similarity solutions of nonlinear Schroedinger equations is presented in this article, where it is shown that the self-similar blow-up solutions discovered hitherto are all associated either with a simple stretching invariance, or with a slightly more complicated conformal invariance and generalizations of the latter.
Abstract: A critical review of the literature on similarity solutions of nonlinear Schroedinger equations is presented. We demonstrate that the self-similar blow-up solutions discovered hitherto are all associated either with a simple stretching invariance, or with a slightly more complicated conformal invariance and generalizations of the latter. This generalized "quasi-invariance" reveals the nature of the blow-up singularity and resolves an old controversy. Most of the previous work has been done on the cubic nonlinearity. We generalize the results to an arbitrary power nonlinearity.
TL;DR: In this paper, the generic behavior of vacuum inhomogeneous Kaluza-Klein cosmologies is studied in the vicinity of the cosmological singularity and the collision law for the Kasner exponents is calculated in any number of spatial dimensions d.
TL;DR: In this paper, it was shown that for certain topologically trivial deformations of an isolated hypersurface singularity, the multiplicity does not change for first order deformations.
Abstract: We show that for certain topologically trivial deformations of an isolated hypersurface singularity the multiplicity does not change. This applies to all μ-constant “first order” deformations and to all μ-constant deformations of a quasihomogeneous singularity.
TL;DR: In this article, the free energy and spectrum of the transfer matrix are modified in a specific way for nonunitary conformal invariant 2D systems with the Lee-Yang edge singularity.
Abstract: Finite-size effects for the free energy as well as the spectrum of the transfer matrix are modified in a specific way for nonunitary conformal invariant 2d-systems. We illustrate this behaviour in the case of the Lee-Yang edge singularity.
TL;DR: On formule des equations d'Einstein a symetrie spherique en coordonnees doubles nulles et on etudie l'approximation haute-frequence a un flot radial unidirectionnel de rayonnement non polarise.
Abstract: Einstein's equations with spherical symmetry are formulated in double-null coordinates, and the high-frequency approximation to a unidirectional radial flow of unpolarized radiation (the Vaidya metric) is studied in detail. For this case the Einstein equations reduce to a single first-order nonlinear partial differential equation. Integration of this equation introduces an arbitrary function (of one null variable) which must be chosen so as to regularize the metric across horizons. Although the problem is, in general, not analytically solvable, we are able to extend the class of known analytic solutions from the constant-mass case (Kruskal-Szekeres metric) to linear and exponential mass functions. In the linear case we give the first explicit regular covering of a spacetime with a naked shell-focusing singularity.
TL;DR: In this article, some properties of geodesies and other curves lying in the spatial sectionst = const. of the Curzon solution are derived, which allow one to build up a new coordinate system in which the singularity appears unambiguously as a ring.
Abstract: Some new properties of geodesies and other curves lying in the spatial sectionst = const. of the Curzon solution are derived. These are shown to allow one to build up a new coordinate system in which the singularity appears unambiguously as a ring. A new region of spacelike infinity is also revealed on the “other side” of this ring, which can be approached by spatial geodesies threading through the ring.
01 Aug 1986
TL;DR: The singularity expansion method (SEM) as discussed by the authors is based on the observation that the transient response of complex electromagnetic scatterers appeared to be dominated by a small number of damped sinusoids.
Abstract: The singularity expansion method (SEM) arose from the observation that the transient response of complex electromagnetic scatterers appeared to be dominated by a small number of damped sinusoids. In the complex frequency plane, these damped sinusoids are poles of the Laplace-transformed response. The question is then one of characterizing the object response (time and frequency domains) in terms of all the singularities (poles, branch cuts, entire functions) in the complex frequency plane (hence singularity expansion method). Building on the older concept of natural frequencies, formulae were developed for the pole terms from an integral-equation formulation of the scattering process. The resulting factoring of the pole terms has important application consequences. Later developments include the eigenmode expansion method (EEM) which diagonalizes the integral-equation kernels and which can be used as an intermediate step in ordering the SEM terms. Additional concepts which have appeared include eigenimpedance synthesis and equivalent electrical networks. Of current interest is the use of the theoretical formulae to efficiently analyze and order experimental data, Related to this is the application of SEM results to target identification. This paper does not delve into the mathematical details; it presents an overview of the history and major concepts and results in SEM and EEM and related matters.
TL;DR: In this paper, a linear slip, Basset-type, boundary condition having an experimentally adjustable phenomenological slip coefficient is used to remove the contact-line singularity that would otherwise prevent the movement of a partially penetrating sphere normal to a planar free surface bounding a semi-infinite viscous fluid.
Abstract: A linear slip, Basset‐type, boundary condition having an experimentally adjustable phenomenological slip coefficient is used to remove the contact‐line singularity that would otherwise prevent the movement of a partially penetrating sphere normal to a planar free surface F bounding a semi‐infinite viscous fluid. Stokes flow calculations are presented for the quasistatic hydrodynamic force and torque resistance matrix for a half‐submerged sphere that is instantaneously translating and rotating with vector velocities that are arbitrarily oriented relative to the free‐surface unit normal vector. The singular components of this material matrix (arising either during translational motion normal to F or rotational motion about an axis lying within F) are shown to be finite for finite slip coefficients β, and to become logarithmically infinite in the traditional nonslip limit β→∞. The relative weakness of this logarithmic singularity suggests that a degree of slip as small as, say, 0.01%—which would presumably be kinematically indistinguishable from the no‐slip case—could easily masquerade as a conventional ‘‘wall effect’’ on the Stokes drag. A small degree of slip is thus hypothesized as a mechanism that would permit the observed transport of Brownian corpuscles across interfacial regions.
TL;DR: In this paper, a new set of moment equations describing plasma transport along the field lines of a space and time-dependent magnetic field is derived, which are valid from collisional to weakly collisional limits.
Abstract: A new set of two‐fluid equations that are valid from collisional to weakly collisional limits is derived. Starting from gyrokinetic equations in flux coordinates with no zero‐order drifts, a set of moment equations describing plasma transport along the field lines of a space‐ and time‐dependent magnetic field is derived. No restriction on the anisotropy of the ion distribution function is imposed. In the highly collisional limit, these equations reduce to those of Braginskii, while in the weakly collisional limit they are similar to the double adiabatic or Chew, Goldberger, and Low (CGL) equations [Proc. R. Soc. London, Ser. A 236, 112 (1956)]. The new set of equations also exhibits a physical singularity at the sound speed. This singularity is used to derive and compute the sound speed. Numerical examples comparing these equations with conventional transport equations show that in the limit where the ratio of the mean free path λ to the scale length of the magnetic field gradient LB approaches zero, ther...
TL;DR: In this article, an exact solution for colliding plane impulsive gravitational waves accompanied by shock waves was obtained, which, in contrast to other known solutions, results in the development of a null surface which acts like an event horizon.
Abstract: An exact solution is obtained for colliding plane impulsive gravitational waves accompanied by shock waves, which, in contrast to other known solutions, results in the development of a null surface which acts like an event horizon. The analytic extension of the solution across the null surface reveals the existence of time-like curvature singularities along two hyperbolic arcs in the extended domain, reminiscent of the ring singularity of the Kerr metric. Besides, the space-time, in the region of the interaction of the colliding waves, is of Petrov-type D and locally isometric to the Kerr space-time in a region interior to the ergosphere. Various other aspects of the solution are also discussed.