Neighborhood Of Singularity Research Articles

Singular full-field stresses in composite laminates with open holes

A method of the superposition of a hybrid and displacement approximation was developed to provide the accurate stress fields in a multilayered composite laminate, including the singular neighborhood of the ply interface and the hole edge. Asymptotic analysis was used to derive the hybrid stress functions. The displacement approximation is based on the polynomial B-spline functions. The method provides the determination of the coefficient of the singular term along with convergent stress components including the singular regions. Reissner’s variational principle was employed. Simple [45/−45] s and quasi-isotropic IM7/5250 [45/90/−45/0] s laminates were analyzed. Uniaxial loading and residual stress calculation (quasi-isotropic laminate) were considered. A convergence study showed that accurate values of the coefficient of the singular term of the asymptotic stress expansion could be obtained with coarse out-of-plane and in-plane subdivisions. The interaction between the singular terms on the neighboring interfaces was found to be important for the convergence with coarse subdivisions. Converged transverse interlaminar stress components as a function of the distance from the hole edge, were shown for all examples.

Motion Control Strategy for Robot Manipulators in the Neighbourhood of Singularities

This work presents a control strategy applied to robot manipulators operating close to kinematic singularities. Considering the robot to be in contact with the environment, a PD controller is iInplelnented with inverse dynamics compensation to control the robot manipulator. In order to accomlTIodate the forces originated froln the interaction with the environment, the control systelTI includes an impedance loop that modifies the robot's references accordingly. In the vicinity of a singularity, the algorithm generates a fictitious force to which a second impedance function is applied. This causes a small deviation to the original robot trajectory to avoid the singularity region. The algoritlun is based on the minimwn singular value of the Jacobian matrix in order to generate the module and direction of the fictitious force. This strategy is applied to a teleoperation system without time delay

Asymptotically exact stresses in laminates with a rigid fastener

A method of superposition of a hybrid and displacement approximation has been extended to provide accurate stress fields in a multilayered composite laminate including the singular neighborhood of the ply interface and the edge of a hole containing a rigid fastener. Asymptotic analysis was used to derive the hybrid stress functions. Multiple singular terms at each interface were incorporated. The displacement approximation is based on polynomial B-spline functions. The method provides determination of the coefficient of the singular term along with convergent stress components, including the singular regions. Reissner's variational principle was employed. Coefficients of the singular term of the asymptotic expansion were determined for a [45/-45] s laminate under bearing loading introduced through a rigid fastener.

Computation of free surface flows with a Taylor-Galerkin/pressure-correction algorithm

A semi-implicit Taylor–Galerkin/pressure–correction finite element scheme (STGFEM) is developed for problems that manifest free surfaces associated with the incompressible creeping flow of Newtonian fluids. Such problems include stick–slip and die-swell flows, both with and without a superimposed drag flow, and for plane, axisymmetric and annular systems. The numerical solutions are compared with available analytical and numerical solutions, both in the neighbourhood of singularities and elsewhere. Close correspondence in accuracy is extracted from the literature for both stick–slip and die-swell flows. Stick–slip flow is used as a precursor study to the more complex free surface calculations involved for die-swell in extrudate flow. Two different free surface techniques are reported and results are analysed with mesh refinement and varying structure. Copyright © 2000 John Wiley & Sons, Ltd.

Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities

The paper addresses the occurrence of possible restrictions on the flows defined by scalar retarded functional differential equations (FDEs), locally around certain simple singularities, compared with the possible flows of ordinary differential equations (ODEs) with the same singularities. It is found that for the Hopf and the Bogdanov-Takens singularities, there are no restrictions on the local flows defined by scalar FDEs, even when the nonlinearities depend on just one delayed value of the solutions. On the other hand, for the singularity associated with a zero and a conjugated pair of pure imaginary numbers as simple eigenvalues, it is shown that there occur restrictions on the flows defined by scalar FDEs with nonlinearities involving just one delay, as well as two delays satisfying a certain resonance condition. These restrictions are of geometric significance, since they amount to the impossibility of observing the homoclinic orbits that occur in arbitrarily small neighborhoods of the singularity for ODEs. Versal unfolfings for the considered singularities by FDEs and the possible restrictions on the associated flows are also studied.

Computations in the neighbourhood of algebraic singularities

It is known that finite precision versus exact computation is a crucial issue only when the computation takes place in the neighbourhood of a singularity. In such a situation, it is essential to know the distance to singularity. Many attention has been dedicated to the relationship between the distance to singularity δ and the condition number K of the problem under study. The well-known Turing theorem states that, for a linear system Ax = b the distance to singularity, in a normwise measure, is the reciprocal of the normwise condition number ‖A −1‖‖A‖ In this Paper, we examine the possibility of extending this theorem for nonlinear problems in the neighbourhood of algebraic singularities. After reviewing the literature on that topic ([Demmel 1987, 1990], [Shub and Smale 1992]), we propose and check on the computer a conjecture which makes more explicit Demmel's bounds on the distance to singularity.

Review of the damped least-squares inverse kinematics with experiments on an industrial robot manipulator

The goal of this paper is to present experimental results on the implementation of the damped least-squares method for the six-joint ABB IRb2000 industrial robot manipulator. A number of inverse kinematics schemes are reviewed which allow robot control through kinematic singularities. The basic scheme adopts a damped least-squares inverse of the manipulator Jacobian with a varying damping factor acting in the neighborhood of singularities. The effect of a weighted damped least-squares solution is investigated to provide user-defined accuracy capabilities along prescribed end-effector space directions. An online estimation algorithm is employed to measure closeness to singular configurations. A feedback correction error term is introduced to ensure algorithm tracking convergence and its effect on the joint velocity solution is discussed. Computational aspects are discussed in view of real-time implementation of the proposed schemes. Experimental case studies are developed to investigate manipulator performance in the case of critical end-effector trajectories passing through and near the shoulder and wrist singularities of the structure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A differential game of simple approach in manifolds

A game situation is considered in which the players are two points of a manifold with a non-degenerate metric, each with controllable velocity. The payoff in the game is the minimum distance between the players in a semi-infinite interval of the time of motion. The first player minimizes the payoff, the second maximizes it. The phase space of the game is divided into subdomains. In one (the primary domain) the value of the game is the initial distance between the players, in the other (the secondary domain) it is less than the initial distance. It is shown that the boundary of the primary domain consists of singular optimal paths [1], and the regular paths approach it from both sides. Necessary conditions are established for the singular surface to be optimal and the equations of the singular paths are derived. They are of the same form as the analogous relationships in the game of pursuit [2]. A necessary optimality condition, formulated in terms of the geodesic distance between players, is found for the primary domain in the form of an inequality, enabling the boundary of the singular surface to be constructed. The existence of this boundary is a necessary condition for the secondary domain to be non-empty. A generalization of Bellman's equation is obtained; it is shown that the value of the game is constant along secondary optimal paths. On singular paths the distance between the players remains constant. The necessary conditions obtained here provide the basis for an algorithm for constructing optimal paths and the value of the game in the neighbourhood of singularities. The algorithm is then used to work out a complete solution of the problem of approach on a two-dimensional cone, constructing the level curves of the value of the game and an optimal phase portrait. The set of cones for which the secondary domain is empty, i.e. for which the distance between the players is the value throughout the game space, is determined.

Nonlinear vortex trail dynamics. II. Analytic solutions

Spatially periodic large amplitude solutions of the von Karman model are obtained in the neighborhood of singularities. These singularities correspond to vortex clusters in the physical plane. The quasi-periodic and unbounded solutions found analytically confirm earlier numerical work and show qualitative agreement with experimental observations of large-scale phenomena of vortex trails. Separatrices or heteroclinic orbits were explicitly found for an integrable approximate equation, which indicate that the von Karman model itself supports chaotic solutions.

Regularization and stability of the constraints in the dynamics of multibody systems

In the analysis of multibody dynamics, we are often required to deal with singularity problems where the constraint Jacobian matrix may become less than full rank at some instantancous configurations. This creates numerical instability which will affect the performance of the mechanical system. A modification procedure of the constraints when they vanish or become linearly dependent is proposed to regularize the dynamics of the system. A distinction between the asymptotic stability due to the representation of the constraints (at the velocity and acceleration level), and the one due to the singularity is discussed in full in this paper. It is shown that Baumgarte technique could be extended to accommodate the representation of the constraints in the neighborhood of singularity. A two link planar manipulator undergoing large motion and passing through a singular configuration is used to illustrate the proposed stability technique.

Asymptotic behavior of the gravitational field and the nature of singularities in gowdy spacetimes

In this work, we present results which support the Strong Cosmic Censorship Conjecture and some of the ideas of Belinskii, Khalatnikov, and Lifschitz. The results concern the behavior of the gravitational field in the neighborhood of singularities in the polarized Gowdy spacetimes. We rigorously show (using certain “energy” estimates) that as one approaches the singularity, the metric field of any of these vacuum solutions of Einstein's equations asymptotically approaches a solution of a truncated system of equations in which spatial curvature terms have been dropped. Hence, the gravitational fields, in a sense, spatially decouple near the singularity. Based on this asymptotic behavior, we also show that only in a very small subclass of the polarized Gowdy spacetimes is the curvature bounded near the singularity. Hence, the generic polarized Gowdy spacetime cannot be extended across a Cauchy horizon.

Scaling integration method for singular integrals and its application to BEM

A new, accurate, and efficient algorithm is developed for the numerical evaluation of a class of singular integrals which arise in the BEM (boundary element method) analyses. The algorithm is based on an idea of scaling integration that utilizes a proportional relationship inherent in the numerical computation of the singular integrals over flat triangular elements. The present algorithm (scaling integration method) avoids the direct numerical evaluation of the integrand in the neighborhood of singularities. It is shown and demonstrated that the method is simpler, faster, and more accurate than standard Gaussian integration algorithms. An application of the method to the BEM analysis is also described and the results are favorably compared with the exact solution.

The dynamics of multidimensional cosmological models in the presence of quantum effects. I. The case FRW ×T D at high temperatures

The dynamical system approach is applied to the study of dynamics of multidimensional cosmological models with topology FRW ×TD (D-dimensional torus) in the presence of high-temperature quantum effects. The stability methods developed in the paper of Szydlowski (Gen. Rel. Grav.,20, 221, 1988) are used in the analysis of typical states of the metric in the neighborhood of singularities and for large time values. The problems of dynamical dimensional reduction, structure of singularities, isotropization, etc., are discussed in this context.

Motion Control of Manipulators with Closed-Form Inverse Solutions Near Wrist Singularities

Two algorithms, the degenerate axis and iterative methods, are developed for the motion control of manipulators with closed-form solutions in the neighborhood of singularities. These two methods theoretically guarantee a robot’s position accuracy. The degenerate axis method may not work well when a robot’s orientation and location increments become finite. If a robot is moving with slow speed or the interpolation time is in the order of microsecond, the location and orientation increments are small. In this case, the degenerate axis method is favored for it has less computation than that of the iterative method. Two examples are given to illustrate the concepts presented in this paper. Although it cannot be proved that the iterative scheme gives the required position accuracy and minimizes the orientation error, the results seem to show that this scheme converges to an acceptable solution. It is believed that the iterative method is the first of its kind to solve the singular motion control problem by using a robot’s closed-form inverse kinematics. Simple computation for the iterative scheme makes it possible to be implemented in many industrial robots.

Treatment of singularities in the physical theory of diffraction

The diffraction coefficients describing scattering from an edged body in the physical theory of diffraction are reformulated so as to provide improved behavior in the neighborhood of singularities. Numerical results show that as singularities are approached, the reformulated coefficients are better behaved than the original by several orders of magnitude.

General solution of systems of nonlinear functional equations in neighborhoods of singularities

General solution of systems of nonlinear functional equations in neighborhoods of singularities

Padé-approximants in operator theory for the solution of nonlinear differential and integral equations

The Padé approximation problem in operator theory and its solution will be repeated briefly together with some properties of the Padé approximants. Effective methods for the solution of differential and integral equations are of great practical importance and there is a vast literature devoted to this subject. Here a few methods, resulting from the use of Padé approximants in operator theory, are introduced and illustrated by means of some typical examples (initial value problems, boundary value problems, partial differential equations, nonlinear integral equations). Among the new methods is an iterative scheme which we will call Halley's method and which proves to be very useful in the neighbourhood of singularities. Well known methods such as Newton's and Chebyshev's method prove to be special cases of the class of iterative procedures.

Interpolation mit regulären splines

This paper is concerned with interpolation of real functions on compact intervals by nonlinear splines called regular splines introduced by R. Schaback and H. Werner. Regular splines contain the classical polynomial splines and represent a more flexible class; for example in the neighborhood of singularities one can use rational splines. We consider boundary value and periodic interpolation problems. In the case of boundary value problems interpolation yields unique results. With regard to the existence of interpolants we develop a nonlinear system of equations that can be written as the sum of a linear system of equations, known from the theory of polynomial splines, and a perturbation caused by the nonlinearities. The main tool in achieving this is a theorem which states that divided differences may be written as convex combinations of special divided differences that operate only on knot intervals. Convergence properties conclude this paper.

Continuum eigenfunctions in the neighborhood of singularities for a uniform hard-sphere gas

Approximate solutions in the continuum region are obtained in the neighborhood of singularities both for the neutral thermalization problem and the classical hard-sphere gas problem by using a differential equation approach.