Showing papers on "Singularity published in 1976"
TL;DR: In this paper, the inverse square root singularity of the plane eight noded isoparametric element has been used to calculate stress intensity factors of elastic fracture mechanics, including constant strain and rigid body motion modes.
Abstract: Quadratic isoparametric elements which embody the inverse square root singularity are used in the calculation of stress intensity factors of elastic fracture mechanics. Examples of the plane eight noded isoparametric element show that it has the same singularity as other special crack tip elements, and still includes the constant strain and rigid body motion modes. Application to three-dimensional analysis is also explored. Stress intensity factors are calculated for mechanical and thermal loads for a number of plane strain and three-dimensional problems.
1,418 citations
01 Jan 1976
874 citations
TL;DR: In this paper, a renormalization group transformation is introduced with the help of which critical properties of infinite systems can be related to finite systems, and applied to the two-dimensional Ising model.
Abstract: A renormalization group transformation is introduced with the help of which critical properties of infinite systems can be related to finite systems. As a numerical example the method is applied to the two-dimensional Ising model. The critical point and critical point exponent are computed in addition to the amplitude of the logarithmic singularity in the specific heat.
370 citations
TL;DR: In this article, the authors considered the case when the Milnor's number of the singularity at the origin does not change in analytic families of n-dimensional hypersurfaces.
Abstract: Introduction. We are interested in analytic families of n-dimensional hypersurfaces having an isolated singularity at the origin. In this paper we only consider the case when the Milnor's number of the singularity at the origin does not change in this family. Under this hypothesis with n = 1, H. Hironaka conjectured that the topological type of the singularity does not change. We give a proof of this conjecture in the more general case of C?? family of n-dimensional hypersurfaces of dimension n=?2. The hypothesis n7^2 comes from the fact we are using n-cobordism theorem. Actually a more general conjecture should be the following:
291 citations
TL;DR: In this article, it was shown that a force singularity parallel to the two planes has a far field behaviour of source and image for the parallel components (a two-dimensional source doublet of height-dependent strength) whereas the normal component, and all fields due to a forces singularity normal to the planes, die out exponentially.
Abstract: Velocity and pressure fields for Stokes flow due to a force singularity of arbitrary orientation and arbitrary distance between two parallel plates are found, using the image technique and a Fourier transform. Two alternative expressions for the solution, one in terms of infinite integrals and the other in terms of infinite series, are given. The infinite series solution is especially suitable for computation purposes being an exponentially decreasing series. From the series the “far field” behaviour is extracted. It is found that a force singularity parallel to the two planes has a far field behaviour of source and image for the parallel components (a two dimensional source doublet of height-dependent strength) whereas the normal component, and all fields due to a force singularity normal to the planes, die out exponentially. Velocity fields are compared with those of the one plane case. An estimate of the influence of the second wall and when its effect can be disregarded is obtained.
266 citations
TL;DR: In this paper, a procedure for generating two-dimensional conforming singularity elements from standard conforming elements is given, based on the use of elements defined by numerically integrated shape functions.
Abstract: A procedure is given for generating two-dimensional conforming singularity elements from standard conforming elements. Three new elements with 0(r-p) derivative singularities are introduced. The technique is based on the use of elements defined by numerically integrated shape functions. Special quadrature rules are suggested for triangular elements. The degeneration of these elements to crack tip elements and the direct evaluation near the singularity of element quantities, such as the stress intensity factors, is discussed.
184 citations
TL;DR: In this paper, the charged C-metric is transformed into another exact solution of the Einstein-Maxwell field equations corresponding to a massive charged particle accelerated by an electric field.
Abstract: The charged C‐metric is transformed into another exact solution of the Einstein–Maxwell field equations corresponding to a massive charged particle accelerated by an electric field. When the appropriate equations of motion are satisfied, the nodal singularity associated with the C‐metric disappears.
117 citations
TL;DR: In this paper, the application of different difference schemes (box, trapezoidal, Euler and backward Euler) to numerical solution of boundary value problems for nonlinear first order systems of ordinary differential equations with a singularity of the first kind is examined.
Abstract: The application of certain difference schemes (box, trapezoidal, Euler and backward Euler) to the numerical solution of boundary value problems for nonlinear first order systems of ordinary differential equations with a singularity of the first kind is examined. The solution of the linear eigenvalue problem is also considered.
112 citations
TL;DR: In this article, a proper prescription for the in-plane self-force on each element of a plane dislocation loop is developed by computing the first variation of the loop self-energy during an arbitrary virtual planar change in the loop configuration.
Abstract: A proper prescription for the in-plane self-force on each element of a plane dislocation loop is developed by computing the first variation of the loop self-energy during an arbitrary virtual planar change in the loop configuration. The appropriate self-energy is defined to be the strain energy exterior to a tube of radius e surrounding the loop. The expression derived for the self-force on a loop element ds depends on the local curvature at ds , on certain elastic data for an infinite straight dislocation tangent to the loop at ds , and only weakly on the ‘cut-off’ radius e. The theory of stress fields of dislocations in anisotropic media is sufficiently advanced to permit easy numerical evaluation of the self-force expression. The analysis further reveals that the singular behavior of the self-stresses in the plane of the loop near an element ds is that of an infinite straight dislocation tangent to ds plus a curvature-dependent logarithmic singularity which is proportional to the line-tension of this tangent dislocation.
107 citations
TL;DR: In this article, the authors discussed the nature of the singularity that arises at a sharp edge in contact problems with friction, and the theoretical treatment is based on the Mellin transform of the elastic fields.
Abstract: The paper discusses the nature of the singularity that arises at a sharp edge in contact problems with friction. The theoretical treatment is based on the Mellin transform of the elastic fields. The results regarding the power singularities confirm the previous work of Gdoutos and Theocaris, but it is shown that logarithmic singularities are always present. Some experimental observations in photoelasticity are also presented.
TL;DR: By distributing the concentrated singularities such as a Kelvin doublet along the axis of symmetry, the authors describes the displacement field, in an elastic medium, for various modes of rotation and translation for a rigid prolate and oblate spheroid.
Abstract: By distributing the concentrated singularities such as a Kelvin doublet along the axis of symmetry we describe the displacement field, in an elastic medium, for various modes of rotation and translation for a rigid prolate and oblate spheroid. The limiting cases of a sphere, a slender body and a thin circular disk are also discussed. All the solutions are presented in a closed form.
TL;DR: In this paper, it was shown that the singularity acts as an impenetrable barrier if the potential is not integrable up to a singularity, but that the behavior is not essentially different from that of a potential which does not become infinite.
Abstract: In quantum mechanics of one dimension it is shown for potentials which become infinite at a point but are continuous elsewhere, that the singularity acts as an impenetrable barrier if the potential is not integrable up to the singularity, but if the potential is integrable the behavior is not essentially different from that of a potential which does not become infinite.
TL;DR: In this paper, a general curved element of arbitrary shape for both thick and thin shells is proposed for the linear fracture analysis of a through crack in a shell or a plate, which is derived from a degenerate 20-noded solid isoparametric element using reduced integration technique.
Abstract: A general curved element of arbitrary shape for both thick and thin shells is proposed for the linear fracture analysis of a through crack in a shell or a plate. The element is derived from a degenerate 20-noded solid isoparametric element using reduced integration technique. The 1/√(r) singularity of the strains is obtained by the same procedure proposed earlier for two- and three-dimensional problems,1,2viz. by placing the mid-side nodes near the crack at the quarter points.
Several illustrated examples ranging from classical solutions to practical problems are given to assess the accuracy of solution attainable.
TL;DR: In this article, the singularity strength of the surface charge density at the vertex of a sectorial conducting plate in an electrostatic field was calculated numerically for different vertex angles.
Abstract: In the present work we calculate the strength of the singularity in surface charge density at the vertex of a sectorial conducting plate in an electrostatic field, including the case in which the media occupying the half-spaces above and below the plate have different dielectric constants. The singularity strength is calculated numerically for different vertex angles $\chi $. In addition, for a “spike” $( {\chi \ll 1} )$, a “nearly straight edge” $( {| {\chi - \pi } | \ll 1} )$ and a “slit” $( {2\pi - \chi \ll 1} )$ , analytic expressions for the singularity strength are found by singular perturbation techniques. The results are relevant to the calculation of circuit board stray capacitances, where one must determine the charge densities on rectilinear sheet conductors, mounted on a dielectric board, for constant conductor potentials. For this case, it is found that the charge singularity, inversely proportional to the square root of the distance from the edge for points far from the vertex, is proportion...
TL;DR: It is well known that solutions u(t, x) of quasi-linear homogeneous hyperbolic equations in one space dimension tend to develop singularities after a finite time.
Abstract: It is well known that solutions u(t, x) of quasi-linear homogeneous hyperbolic equations in one space dimension tend to develop singularities after a finite time (see [1], [2]).
TL;DR: In this article, the electrical conductivity σ of several liquid mixtures was measured near their respective critical points, and it was shown that the singularity could be characterized by a critical exponent ϑ=0.70+0.15−0.10.
Abstract: The electrical conductivity σ of several liquid mixtures was measured near their respective critical points. The systems studied were pure phenol–water, KCl‐doped phenol–water, and isobutyric acid–water. The measurements spanned the reduced temperature range 10−5≲e≲10−2, where e≡ (T‐Tc)/Tc. All systems clearly showed the existence of a singular contribution to the conductivity. A least‐squares fit of the data to the equation, (σ‐σc)/σc∝eϑ+ background terms, showed that the singularity could be characterized by a critical exponent ϑ=0.70+0.15−0.10. A possible explanation of this singularity is offered which uses percolation theory in the mean field limit. This percolation approach yields ϑ=2β, in accord with our own observations. The exponent β (?1/3) characterizes the shape of the coexistence curve.
TL;DR: In this paper, the Galitskii-Feynman T matrix, which sums the infinite ladder series in a many-fermion system for both particle-particle and hole-hole scattering, is studied in detail for a family of realistic He-He interactions.
Abstract: The Galitskii-Feynman T matrix, which sums the infinite ladder series in a many-fermion system for both particle-particle and hole-hole scattering, is studied in detail for a family of realistic He-He interactions. The structure of the S-wave bound-state singularity, reported previously, and its dependence on the bare interaction are documented at length. Special attention is devoted to the T matrix in the scattering region, where the c.m. energy of the interacting pair is positive. In particular, the on-energy-shell T matrix in this region is parametrized in terms of real “effective” phase shifts incorporating many-body effects. The critical behavior discussed previously in the bound-state region manifests itself clearly in the zero-energy limit of these phase shifts for the S wave. Below (above) a certain critical density, which is a function of both temperature and c.m. momentum, this limit approaches the value 0(−π) radians. A generalized Levinson's theorem relates this behavior to the existence of fermion-fermion pairing. An especially striking feature of these many-body phase shifts is the cusp behavior exhibited at the Fermi surface in the lowtemperature limit, which turns out to arise essentially from the structure of the particle and hole occupation probabilities. Throughout this study the temperature dependence of the T matrix is particularly emphasized.
TL;DR: An approach based on extrapolation is described that leads to a method not unlike the Romberg integration, and a set of useful expansions are stated.
TL;DR: In this article, the authors reviewed the known cosmological solutions of the ECSK field equations and explained the prevention of singularities by means of the extension of the Hawking-Penrose singularity theorems to the EDSK theory.
01 Jan 1976
TL;DR: In this article, the effect of laminar boundary layers on the lift of thin wings in subsonic flow at high Reynolds numbers is considered, and the boundary value problem is formulated in the framework of the triple-deck theory of Brown and Stewartson.
Abstract: The problem of determining the effect of laminar boundary layers on the lift of thin wings in subsonic flow at high Reynolds numbers is considered. The boundary value problem is formulated in the framework of the triple-deck theory of Brown and Stewartson. The resulting fourth-order boundary value was solved by an iterative finite-difference technique. An inverse iteration procedure provides proper treatment of the trailing-edge singularity, and asymptotic far-field expansions and coordinate stretchings are used to deal with the problem of the slow algebraic decay of the solution.
TL;DR: The density of boson and fermion pairs, created near the singularity in isotropic homogeneous cosmological models, was calculated in this paper, and the rate of creation of fermions was found to be considerably higher than that of bosons.
TL;DR: Weyl's theory for singular ordinary second order differential equations is presented as a numerical method for analyzing the spectral properties of the Schrodinger operator associated with predissociation by rotation in diatomic molecules as mentioned in this paper.
Abstract: Weyl’s theory for singular ordinary second order differential equations is presented as a numerical method for analyzing the spectral properties of the Schrodinger operator associated with predissociation by rotation in diatomic molecules. It is pointed out that the poles of Weyl–Titchmarsh’s m function characterize both the discrete eigenvalues and the resonances in the continuous spectrum, thus allowing a unified treatment of the entire spectrum. The conventional hard core treatment of the origin is compared to two alternative approaches, both applicable to more general singular behavior of the potential. A detailed discussion of the other singularity, infinity, is given and Kodaira’s theorem is generalized to an expression involving Wronskian’s between the regular and the asymptotic solutions. The calculation of the Weyl–Titchmarsh m function is based on numerical Runge–Kutta integrations of both independent solutions, combined with asymptotic information derived from a Riccati expansion. Im(m) is shown to have its physical significance as flux of predissociating particles. The resonances in the continuous spectrum are obtained by numerical analytical continuation of Im(m) across the real energy axis. The method is applied to Stwalley’s isotopically combined (IC1) potential for HgH. All resonance energies and lifetimes are reported and compared to Stwalley’s phase shift results. Excellent agreement is found for sharp resonances whereas considerable discrepancies occur for the broader ones. This latter fact is attributed to the failure of the Breit–Wigner fit to account for asymmetric non‐Lorentzian shapes.
TL;DR: In this article, a number of exact solutions of Einstein's equations are obtained, which describe the collision and subsequent interaction of two plane parallel waves, and it is shown that two such waves mutually focus each other with the focus usually appearing as a singularity in space-time.
TL;DR: In this paper, the axis of symmetry is at the same time the direction of the magnetic field, and of the aligned spins in axially symmetric cosmological models of the Einstein-Cartan theory.
Abstract: In axially-symmetric cosmological models of the Einstein-Cartan theory (which may be briefly called ‘general relativity plus spin’), the axis of symmetry is at the same time the direction of the magnetic field, and of the aligned spins. The general set of relevant equations is given. Some exact solutions of this set constitute quasi-Euclidean and semiclosed cosmologies with a uniform magnetic field and aligned spinning matter. In contrast to the situation in the framework of general relativity, one may obtain non-singular solutions. Such a behaviour of the solutions of the Einstein-Cartan theory is rendered possible by the specific spin-spin repulsive interaction which is inherent in the theory.
01 Feb 1976
TL;DR: In this article, a number of exact solutions of Einstein's equations are obtained, which describe the collision and subsequent interaction of two plane parallel waves, and it is shown that two such waves mutually focus each other with the focus usually appearing as a singularity in space-time.
Abstract: A number of exact solutions of Einstein's equations are obtained, which describe the collision and subsequent interaction of two plane parallel waves. Gravitational waves, null electromagnetic fields, and neutrino fields are all considered with collisions between any two types. It is shown that two such waves mutually focus each other with the focus usually appearing as a singularity in space-time. Further conclusions are made regarding the qualitative nature of the interactions, and it is argued that these also apply in more realistic physical situations.
TL;DR: In this article, series expansions are investigated for spin-glass Ising models with nearest-neighbour interactions, and it is shown that the free energy has a singularity at about w = µ-1/2 (w = tanh β J) where µ is the self-avoiding walk limit.
Abstract: Series expansions are investigated for a spin-glass Ising model with nearest-neighbour interactions J which can be randomly positive or negative. For the high-temperature phase the following conclusions result: (i) the free energy has a singularity at about w = µ-1/2 (w = tanh β J) (where µ is the self-avoiding walk limit), (ii) the magnetic susceptibility corresponds to uncoupled spins, (iii) the second derivative of susceptibility is simply related to the susceptibility of the standard Ising model and has a singularity at w = wc1/2 (where wc refers to the standard model).
TL;DR: A short survey of results on non-trivial models (that are not integrable analytically) in general relativity and gas dynamics can be found in this article.
Abstract: The paper begins with a short survey of results on non-trivial models (that is, those that are not integrable analytically) in general relativity and gas dynamics. The investigation of these models is carried out by the methods of the qualitative theory of many-dimensional dynamical systems, using geometrical and topological ideas. The first section deals with the results of research on the evolution of homogeneous cosmological models with a hydrodynamic energy tensor - the impulse about a singularity. In the second section similar models are applied to the study of the complex oscillating regimes of a classical ideal compressible fluid. The Appendix contains new, unpublished results due to one of the authors, describing stochastic perturbation of a completely integrable Toda chain.
01 Jul 1976
TL;DR: In this article, a second-order, discontinuity-fitting, finite-difference approach was used to determine the flow field resulting from the interaction of a moving planar shock wave with a compression corner.
Abstract: The unsteady, two-dimensional flowfield resulting from the interaction of a moving planar shock wave with a compression corner is determined using a second-order, discontinuity-fitting, finite-difference approach. The time-dependent Euler equations are transformed to normalize the distance between the body and peripheral shock and to include the existing self-similar property of the flow. The resulting set of partial differential equations in conservation-law form is then solved in a time-dependent fashion using MacCormack's scheme. The vortical singularity, which lies on the body surface, and the single reflected shock are both treated as discontinuities in the numerical procedure. The results of the numerical simulation compare quite favorably with existing experimental interferograms and yield better flowfield resolution than previous first-order, shock-capturing, numerical solutions.