Showing papers on "Tangent published in 1999"

Deformable Smooth Surface Design

Abstract: A new paradigm for designing smooth surfaces is described. A finite set of points with weights specifies a closed surface in space referred to as skin . It consists of one or more components, each tangent continuous and free of self-intersections and intersections with other components. The skin varies continuously with the weights and locations of the points, and the variation includes the possibility of a topology change facilitated by the violation of tangent continuity at a single point in space and time. Applications of the skin to molecular modeling and to geometric deformation are discussed.

Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets

Abstract: In this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints. Our analysis is based on various concepts of second order tangent sets and parametric duality. We introduce a condition, called second order regularity, under which there is no gap between the corresponding second order necessary and second order sufficient conditions. We show that the second order regularity condition always holds in the case of semidefinite programming.

Self-consistent polycrystal models: a directional compliance criterion to describe grain interactions

Abstract: Viscoplastic self-consistent polycrystal models have been successful in addressing and explaining features of plastic deformation which cannot be treated with the Taylor condition of isostrain. In particular, these models have been applied to the simulation of plastic deformation and texture development in materials with hexagonal, trigonal, orthorhombic and triclinic symmetry. An important assumption required to solve the equilibrium equation within self-consistent formulations is that the strain-rate varies linearly with the stress in the homogeneous effective medium surrounding the inclusion. The characteristic of such a linear relation has been a matter of debate and two extreme cases can be identified: the tangent and the secant approaches. The secant approach has associated with it a stiffer inclusion-matrix interaction than the tangent approach and is closer to the Taylor approach. In this work we perform a systematic study of the implications of both assumptions on the response of cubic and hexagonal materials (texture development, system activity, stress and strain-rate deviations). In addition, we argue that the strength of the matrix-inclusion interaction should not be constant but should depend on the capability of each orientation to accommodate the particular deformation mode imposed externally. As a consequence, we propose a relative directional compliance (RDC) criterion for defining a variable interaction between grain and matrix depending on their relative compliances and compare the predictions of the RDC approach with the predictions of the secant and the tangent schemes.

Optimal Time Computation of the Tangent of a Discrete Curve: Application to the Curvature

Abstract: With the definition of discrete lines introduced by REveilles [REV91], there has been a wide range of research in discrete geometry and more precisely on the study of discrete lines. By the use of the linear time segment recognition algorithm of Debled and REveilles [DR94], Vialard [VIA96a] has proposed a O(l) algorithm for computing the tangent in one point of a discrete curve where l is the average length of the tangent. By applying her algorithm to n points of a discrete curve, the complexity becomes O(n.l). This paper proposes a new approach for computing the tangent. It is based on a precise study of the tangent evolution along a discrete curve. The resulting algorithm has a O(n) complexity and is thus optimal. Some applications in curvature computation and a tombstones contours study are also presented.

Motion planning and nonlinear simulations for a tank containing a fluid

Abstract: We consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by Saint-Venant's equations. We show how to parameterize the trajectories of the linearized system thanks to the horizontal coordinate of a particular point in the system — the “flat output”, see figure 2- and a periodic function. The motion planning problem of the linearized model is solved in the general case of joining two steady states. Next we provide an algorithm, based on Godunov scheme, with a dedicated way of dealing with boundary conditions, to numerically simulate the evolution of the nonlinear system. Nonlinear simulations provide a way of checking the accuracy of the motion planning based on the tangent linear system.

Symplectic Tangent map for Planetary Motions

Abstract: Equations are presented for the computation of tangent maps for use in nearly Keplerian motion, approximated by use of a symplectic leapfrog map. The resulting algorithms constitute more accurate and efficient methods to obtain the Liapunov exponents and the state transition matrix, and can be used to study chaos in planetary motions, as well as in orbit determination procedures from observations. Applications include planetary systems, satellite motions and hierarchical, nearly Keplerian systems in general.

Lateral dynamics of a railway vehicle in tangent track and curve: tests and simulation

Abstract: A mathematical model for the simulation in time domain of the dynamic behaviour of a railway vehicle running in tangent track and curve is presented in the paper. The main features of the model, with particular emphasis on the subject of wheel-rail contact, are described, and some comparisons with experimental data coming from line tests are presented. The topics of wheel rail contact modelling and of defining a suitable, frequency dependent model for the anti-yaw dampers appear from these comparisons to be critical in order to obtain satisfactory results. For the covering abstract see ITRD E117109.

Tools for computing tangent curves for linearly varying vector fields over tetrahedral domains

Abstract: We present some very efficient and accurate methods for computing tangent curves for three-dimensional flows. Our methods work directly in physical coordinates, eliminating the usual need to switch back and forth with computational coordinates. Unlike conventional methods, such as Runge-Kutta, for computing tangent curves which give only approximations, our methods produce exact values based upon piecewise linear variation over a tetrahedrization of the domain of interest. We use balycentric coordinates in order to efficiently track cell-to-cell movement of the tangent curves.

Safety and the Choice of Degree of Curve

Abstract: Consider a horizontal curve set between two tangents forming a given deflection angle. Choosing a larger degree of curve (shorter radius) will make the tangents longer and the curve shorter and sharper. The question examined here is how this simultaneous change of several geometric elements affects the overall accident frequency. After a review of the relevant literature it is shown that increasing the degree of curve always increases the expected accident frequency. Also a simple way to compute the approximate amount of accident increase is described. It can be shown that the choice of degree of curve has large safety consequences when the deflection angle is large. This is not currently reflected in geometric design standards. It can also be shown that, contrary to commonly held opinion, adding x meters to the radius has the same effect on accident frequency regardless of whether the radius is 100 m or 1000 m. Finally, the effect of tangent length on curve accidents is added to the mix and a computational scheme to aid the selection of the degree of curve in specified circumstances of tangent lengths and deflection angle is provided.

High-Order Approximations and Generalized Necessary Conditions for Optimality

Abstract: In this paper we derive generalized necessary conditions for optimality for an optimization problem with equality and inequality constraints in a Banach space. The equality constraints are given in operator form as $Q=\{x\in X:F(x)=0\}$ where $F:X\rightarrow Y$ is an operator between Banach spaces; the inequality constraints are given by smooth functionals or by closed convex sets. Models of this type are common in the optimal control problem. The paper addresses the case when the Frechet-derivative F '(x*)is not onto and hence the classical Lyusternik theorem does not apply to describe the tangent space to Q. In this case the classical Euler--Lagrange type necessary conditions are trivially satisfied, generating abnormal cases. A high-order generalization of the Lyusternik theorem derived earlier [U. Ledzewicz and H. Schattler, Nonlinear Anal., 34 (1998), pp. 793--815] is used to calculate high-order tangent cones to the equality constraint at points $x_{*}\in Q$ where F '(x*) is not onto. Combining these with high-order approximating cones related to the other constraints of the problem (feasible cones respectively cones of decrease) a high-order generalization of the Dubovitskii--Milyutin theorem is given and then applied to derive generalized necessary conditions for optimality. These conditions reduce to classical conditions for normal cases, but they give new and nontrivial conditions for abnormal cases.

Automated calculation of sight distance from horizontal geometry

Abstract: This paper describes a method for automatic calculation of sight distance, subject only to constraints of horizontal geometry, along an arbitrary alignment of tangent sections, circular curves, and spiral transition curves. A parametric representation is used to describe the alignment and its tangent and normal vectors at all points. From this information, other parametric forms are derived that describe the curve along the center of the traveled lane and the lateral obstruction curve. Finally, an automatic method to determine sight distance is described, which is analogous to the manual “graphical method” often employed in practice. An example alignment is considered for demonstration purposes. The method is accurate, efficient, and consistent with manual calculations. It might be used for situations where detailed information about possible sight distance obstructions is not available, such as planning, preliminary design, or to support passing behavior modules of microscopic simulation models.

On the intrinsic reconstruction of shape from its symmetries

Abstract: We address the issue of the use of symmetry-based representations, such as the medial axis and an augmented form of it, the shock structure, to regenerate shapes. First, we address pointwise reconstruction of the boundary from points of the medial axis. As classified into three generic types (A/sup 2//1 mid-branch, A/sub 3/ end point of a branch, and A/sub 1//sup 3/ junction). Second, we examine the intrinsic reconstruction of shape when differential properties of the axis are also available. We show the surprising result that the tangent and curvature of the medial axis, coupled with the speed and acceleration of the shock flowing along the's axis, i.e., first and second order properties, are sufficient to determine the boundary tangents and curvatures at corresponding points of the boundary. This implies that for a rather coarse sampling of the symmetry axis, the location together with its tangent, curvature: speed, and acceleration is sufficient to accurately regenerate a local neighborhood of shape at this point. Together with reconstruction properties at junction (A/sup 3//sub 1/) and end points (A/sub 3/), these results lead to the full intrinsic regeneration of a shape from a representation of it as a directed planar graph (where the links represent curvature and acceleration functions, and where the nodes contain tangent and speed information): a representation ideally suited for the design and manipulation of free-form shape.

VU -Decomposition Derivatives for Convex Max-Functions

Abstract: For minimizing a convex max-function f we consider, at a minimizer, a space decomposition. That is, we distinguish a subspace V, where f’s nonsmoothness is concentrated, from its orthogonal complement, U. We characterize smooth trajectories, tangent to U, along which f has a second order expansion. We give conditions (weaker than typical strong second order sufficient conditions for optimality) guaranteeing the existence of a Hessian of a related U-Lagrangian. We also prove, under weak assumptions and for a general convex function, superlinear convergence of a conceptual algorithm for minimizing f using VU-decomposition derivatives.

Cohen-Macaulayness of tangent cones

Abstract: We give a criterion for checking the Cohen-Macaulayness of the tangent cone of a monomial curve by using the Gr6bner basis For a family of monomial curves, we give the full description of the defining ideal of the curve and its tangent cone at the origin By using this family of curves and their extended versions to higher dimensions, we prove that the minimal number of generators of a Cohen-Macaulay tangent cone of a monomial curve in an affine i-space can be arbitrarily large for l > 4 contrary to the l = 3 case shown by Robbiano and Valla We also determine the Hilbert series of the associated graded ring of this family of curves and their extended versions

Symmetry of Tangent Stiffness Matrices of 3D Elastic Frame

Abstract: In the literature, the symmetry of the element tangent stiffness matrix of a spatial elastic beam has been a subject of debate. The symmetry of the tangent stiffness matrices derived by some researchers are tenuously attributed to the use of Lagrangian formulations, while the asymmetry of corotational tangent stiffness matrices is commonly attributed to the noncommutativity of spatial rotations. In this paper, the inconsistency regarding the symmetry of element tangent stiffness matrices formulated in the Lagrangian and the corotational frameworks is resolved. It is shown that, irrespective of the formulation framework, the element tangent stiffness matrix is invariably asymmetric. A “correction matrix” that enforces the proper rotational behavior of nodal moments into the conventional geometric stiffness matrix of an Updated Lagrangian spatial beam element is presented. It is demonstrated through a numerical example that adoption of this correction matrix is necessary for the detection of the lowest buck...

Tangency, Flow Invariance for Differential Equations, and Optimization Problems

Abstract: Tangent vectors to closed sets flow-invariant sets second order differential equations and flow-invariance flow-invariant sets with respect to semilinear differential equations a transversability approach to flow-invariance optimization and optimal control via tangential cones critical point theory on flow-invariant sets elements of nonlinear analysis the approximate difference scheme and nonlinear semigroups Banach manifolds and vector fields generalized gradients.

Atmospheric ray path modeling for radiative transfer algorithms.

Abstract: A new method for the determination of ray paths as well as resulting path segments and partial gas columns within a layered atmosphere is presented. Any singularity at the tangent point is avoided. No use is made of the gross spherical symmetry of the Earth's atmosphere. Using this approach we examine the impact of the Earth's oblate shape and horizontal atmospheric inhomogeneities on infrared limb spectra.

Combinatorics of geometrically distributed random variables: New q-tangent and q-secant numbers

Abstract: Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all coincide with the classical version. In this way, we get some new q-tangent and q-secant functions. Some of them also have nice continued fraction expansions; in one particular case, we could not find a proof for it. Divisibility results a la Andrews/Foata/Gessel are also discussed.

Extensions of the self-consistent tangent model

Abstract: The overall response of heterogeneous viscoplastic materials is obtained using self-consistent schemes with the local response governed by convex flow potentials. The tangent model, originally proposed by Molinari et al is reviewed and extended and a discussion of the nonlinear inclusion problem is presented. A new general formulation of the self-consistent scheme is proposed and algorithms of calculations are suggested.

On Singularities of a Boundary of the Stability Domain

Abstract: This paper deals with the study of generic singularities of a boundary of the stability domain in a parameter space for systems governed by autonomous linear differential equations $\dot y=Ay$ or x(m) + a1x(m-1) + . . . + amx=0. It is assumed that elements of the matrix A and coefficients of the differential equation of mth order smoothly depend on one, two, or three real parameters. A constructive approach allowing the geometry of singularities (orientation in space, magnitudes of angles, etc.) to be determined with the use of tangent cones to the stability domain is suggested. The approach allows the geometry of singularities to be described using only first derivatives of the coefficients ai of the differential equation or first derivatives of the elements of the matrix A with respect to problem parameters with its eigenvectors and associated vectors calculated at the singular points of the boundary. Two methods of study of singularities are suggested. It is shown that they are constructive and can be applied to investigate more complicated singularities for multiparameter families of matrices or polynomials. Two physical examples are presented and discussed in detail.

Shock spectra and damage boundary curves for non-linear package cushioning systems†

Abstract: This paper suggests a general approach for obtaining shock spectra and damage boundary curves for cushioning packaging systems. This approach can treat both linear and non-linear cushioning systems such as cubic, tangent and hyperbolic tangent systems. Corresponding software has been developed for analysing different cushioning systems, and the shock spectra and the damage boundary curves are given for a tangent non-linear cushioning system under the action of a rectangular, half-sine, terminal-peak saw-tooth and initial-peak saw-tooth pulse, respectively. It is worth noting that the shock spectrum is affected not only by the damping parameter but also by the dimensionless pulse peak, and that both the damping parameter and the dimensionless fragility also influence the damage boundary curve. These are important features of non-linear cushioning system. Copyright © 1999 John Wiley & Sons, Ltd.

Don’t panic … just do it in parallel!

Abstract: Parallel coordinates is a methodology for visualizing N-dimensional geometry and multivariate problems. In this self-contained up-to-date overview the aim is to clarify salient points causing difficulties, and point out more sophisticated applications and uses in statistics which are marked by **. Starting from the definition of the parallel-axes multidimensional coordinate system, where a point in Euclidean N-space RN is represented by a polygonal line, it is found that a point ↔ line duality is induced in the Euclidean plane R2. This leads to the development in the projective, P2, rather than the Euclidean plane. Pointers on how to minimize the technical complications and avoid errors are provided. The representation (i.e. visualization) of 1-dimensional objects is obtained from the envelope of the polygonal lines representing the points on their points. On the plane R2 there is a inflection-point ↔ cusp, conies ↔ conies and other potentially useful dualities. A line l ⊂ RN is represented by N − 1 points with a pair of indices in [1, 2, …, N]. This representation also enables the visualization and computation of proximity properties like the minimum distance between pairs of lines [18]. The representation of objects of dimension ≥ 2 is obtained recursively. Specifically, the representation of a p-flat, a plane of dimension 2 ≤ p ≤ N − 1 in RN is obtained from the (p−1)-flats it contains, and which are obtained from the (p−2)-flats and so on all the way down from the points (0-dimensional); hence the recursion. A p-flat is represented by p-points each with (p+1) indices. This is the key message: ** high-dimensional objects may be visualized recursively, in terms of their higher dimensional components, rather than directly from their points. Further, this process is robust so that “near” p-flats are also detected in the same way and very useful tight error bounds are available. The representation of a smooth hypersurface in RN is obtained as the envelope of the tangent hyperplanes. The set of points obtained in this way visually reveal properties like convexity, whether the surface is developable, or ruled. A simpler but ambiguous representation for hypersurfaces is also given together with modeling applications of an algorithm for computing and displaying interior, exterior or surface points.