On the geometry of polar varieties

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular, we show the non–emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment.     

On the geometry of entangled states

The basic question that is addressed in this paper is finding the closest separable state for a given entangled state, measured with the Hilbert-Schmidt distance. While this problem is in general very difficult, we show that the following strongly related problem can be solved: find the Hilbert-Schmidt distance of an entangled state to the set of all partially transposed states. We prove that this latter distance can be expressed as a function of the negative eigenvalues of the partial transpose of the entangled state, and show how it is related to the distance of a state to the set of positive, partially transposed (PPT) states. We illustrate this by calculating the closest biseparable state to the W state and give a simple and very general proof for the fact that the set of W-type states is not of measure zero. Next we show that all surfaces with states whose partial transposes have constant minimal negative eigenvalue are similar to the boundary of PPT states. We illustrate this with some examples on bipartite qubit states, where contours of constant negativity are plotted on two-dimensional intersections of the complete state space.     

On the line geometry of rigid-body inertia

In this work, several classical ideas concerning the geometry of the inertia of a rigid body are revisited. This is done using a modern approach to screw theory. A screw, or more precisely a twist, is viewed as an element of the Lie algebra to the group of proper rigid-body displacements. Various moments of inertia, about lines, planes and points are considered as geometrical objects resulting from least-squares problems. This allows relations between the various inertias to be found quite simply. A brief review of classical line geometry is given; this includes an outline of the theory of the linear line complex and a brief introduction to quadratic line complexes. These are related to the geometry of the inertia of an arbitrary rigid body. Several classical problems concerning the mechanics of rigid bodies subject to impulsive wrenches are reviewed. We are able to correct a small error in Ball’s seminal treatise. The notion of spatial percussion axes is introduced, and these are used to solve a problem concerning the diagonalisation of the mass matrix of a two-joint robot.     

On the geometry and dynamics of floating four-bar linkages

In this paper, we investigate the kinematics and dynamics of floating, planar four-bar linkages. The geometry of configuration space is analyzed in part through the classical theory of mechanisms due to Grashof. The techniques of symplectic and Poisson reduction are used to understand the dynamics of the system. Bifurcations of relative equilibria for linkages admitting symmetric shapes are studied using the techniques of singularity theory. The problem of reconstruction of the full dynamics and its relation to geometric phases is discussed through some examples. This research reveals that a coupled mechanical system with kinematic loops possesses richer and more complicated dynamical aspects in comparison with systems which have the same number of degrees of freedom, but no kinematic loops     

On the optimal nose geometry for a rigid penetrator

A variational formulation for the net force on the nose of a rigid projectile normally penetrating a compliant target is given. Frictional effects are negligible in this formulation. The variational problem is solved and the result compared to several popular nose geometries. For blunt tipped projectiles, the optimal geometry can significantly enhance penetration by reducing the net force of resistance. For long penetrator noses, the effect has much less value. The most interesting conclusion is that all the optimal geometries have blunt tips.     

On the geometry of radiative and thermally conducting gas flows

After formulation of the various dynamical and kinematical relations connecting the flow quantities with the geometrical parameters of the streamline trajectories, the expressions for the tangent, principal normal and binormal vectors and the curvature and torsion of the streamlines have been obtained in terms of the velocity components, pressure, density, thermal conductivity and radiation variables. This is followed by the determination of the pressure gradient along the streamlines, their principal normals and binormals. It has been shown that the radiating character of the gas decreases the pressure gradient whereas thermal conductivity increases it along the streamlines but the pressure remains constant along the binormals and if the streamlines are straight lines, the trajectories of the principal normals lie on the surface of the constant pressure. Finally, the expressions for the vorticity components in terms of the curvature of the streamline and the velocity gradients along the streamline, their principal normals and binormals have been obtained.     

On the space-charge longitudinal wake field in perfectly conducting pipe of planar geometry

Longitudinal space-charge wake field is considered for a beam of rectangular cross-section propagating between two perfectly conducting planes. It is shown that the forces driving negative mass instability are weaker compared to circular configuration. A similar effect takes place for a flattened (ribbon) beam.     

On the relationship between the geometry of deformed crack tip and crack parameters

An experimental investigation into the relationship between fracture mechanics parameters (J integral and crack tip opening displacement) determined by mechanics tests and geometrical parameters of deformed crack tip (stretched zone width,SZW and stretched zone depth,SZD) derived from microfractography is described. On the basis of these correlations and results of stereoscopic analysis ofSZ, it is proved thatSZD instead ofSZW is equivalent toCTOD and theSZW = 1/2CTOD is not true.SZ can be divided into two part: the horizontal part (HSZ) nearly parallel to the fatigue precracked surface and the tilt part (TSZ) which intersects the fatigue precracked surface at a blunting angle . The occurence ofHSZ is related to the deformation at crack flank behind crack tip and leads to the inconsistence betweenSZW andCTOD. The connection between the stage 11 fatigue crack growth by the ductile striation andSZ formation process is discussed. The rate ofFCG should be directly related to the width ofSZ formed during fatigue loading, and not be dominated byCTOD.

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Résumé On décrit une étude expérimentale de la relation entre les paramètres de mécanique de rupture (IntégraleJ et déplacement d'ouverture de l'extrémité d'une fissure CTOD) déterminés par des essais mécaniques et les paramètres géométriques de l'extrémité déformée de la fissure (largeur de zone déforméeLZD et profondeur de zone déforméePZD) déterminés par microfractographie. Sur base de ces relations et des résultats d'une analyse spectroscopique de la zone déformée, on établit que c'est laPZD plutôt que laLZD qui est équivalente auCTOD, et que la relationLZD = 1/2CTOD n'est pas exacte. La zone déformée peut être divisée en deux parties: la portion horizontale, sensiblement parallèle au plan de la préfissure de fatigue, et la portion transversale, qui intersecte le plan de la préfissure sous un angle solide .L'apparition de déformation dans la portion horizontale est liée à la déformation sur les bords de la fissure en arrière de l'extrémité de celle-ci, et conduit à une non relation entreLZD etCTOD.On discute la relation entre le stade II de la croissance de la fissure par fatigue par l'apparition de striures ductiles et la formation de la zone déformée. La vitesse de croissance de la fissure de fatigue devrait être en relation directe avec la largeur de la zone déformée survenant au cours de la sollicitation de fatigue, et non être déterminée par leCTOD.

     

任意结构基阵波束优化的稳健性分析

分析了任意结构基阵波束优化的稳健性。首先从理论上推导了阵列误差、权向量误差和权向量范数等因素与波束旁瓣畸变之间的关系,并利用优化波束的白噪声增益和权向量范数之间的关系,提出了任意结构基阵波束优化时权向量范数约束上限的选取方法。针对圆阵进行了大量的计算机仿真,分析了不同权向量范数约束条件下获得的优化波束的稳健性,证明了理论推导的正确性。     

On the sharpness of straight edge blades in cutting soft solids: Part II - Analysis of blade geometry

In Part I of this paper a new metric, titled the “blade sharpness index” or “BSI”, for quantifying the sharpness of a straight edge blade when cutting soft solids was derived from first principles and verified experimentally by carrying out indentation type cutting tests with different blade types cutting different target or substrate materials. In this Part II companion paper, a finite element model is constructed to examine the effect of different blade variables including tip radius, wedge angle and blade profile on the BSI developed in Part I. The finite element model is constructed using ABAQUS implicit and experiments are performed to characterise the non-linear material behaviour observed in the elastomeric substrate. The model is validated against the experiments performed in Part I and a suitable failure criterion is determined by carrying out experiments on blades with different tip radii. The paper finds that a simple maximum stress criterion is a good indicator for predicting the onset of cutting. The validated model is then used to examine blade geometry. It is shown that finite element analysis is an important tool in helping to understand the mechanics of indentation. Furthermore, the study finds that all the blade geometric variables have an influence on the sharpness of a blade, with the BSI being most sensitive to tip radius. Increasing the tip radius and wedge angle decreases the sharpness of the blade.     

On the classical shell model underlying bilinear degenerated shell finite elements: general shell geometry

We study the shell models arising in the numerical modelling of shells by geometrically incompatible finite elements. We build a connection from the so‐called bilinear degenerated 3D FEM to the classical 2D shell theory of Reissner–Naghdi type showing how nearly equivalent finite element formulations can be constructed within the classical framework. The connection found here facilitates the mathematical error analysis of the bilinear elements based on the degenerated 3D approach. In particular, the connection reveals the ‘secrets’ that relate to the treatment of locking effects within this formulation. Copyright © 2002 John Wiley & Sons, Ltd.     

Optimisation of the protrusion geometry in Comeld™ joints

Design solutions using composite laminates frequently require joining the laminates to metal. A novel approach to such bonding using a sculpted metal surface has been proposed which can be described as a combination of mechanical and adhesive bonding. Optimisation of the protrusion geometry on the sculpted metal surface has been studied numerically with the finite element method using a simplified model for the end protrusion where initial failure is observed. The angle and height of the protrusion are found to significantly affect the stress concentrations around the protrusion which initiate failure. It is predicted that the optimum angle is opposing the shear and that higher protrusions are more likely to prevent joint failure.     

Local environmental geometry of atoms in the Lennard-Jones systems

The distributions of topological and metrical characteristics of the Voronoi polyhedra have been obtained for the Lennard-Jones systems of atoms modelling a crystal, a liquid and a fluid. The distributions of the number of faces, the number of edges per face, the number of pentagonal faces, the histograms of topological types, the radial distributions of the Voronoi polyhedron neighbours, the distribution of edge lengths, face areas and the sphericity factor have been investigated for the Voronoi polyhedra. The local atomic structure is discussed for various phase states and various temperatures and densities. The importance is noted of a simultaneous analysis of topological and metrical characteristics of the Voronoi polyhedra.     

Integral transformation of the Navier-Stokes equations in cylindrical geometry

The Generalized Integral Transform Technique (G.I.T.T.) is extended to handle the incompressible Navier-Stokes equations for two-dimensional steady laminar flow in cylindrical geometries. Hybrid numerical-analytical solutions with controlled accuracy are obtained, as a result of an appropriate choice of the associated eigenfunction expansion basis, extracted from the diffusion operator of the stream function-only formulation for this class of problems. The approach is illustrated for developing laminar flow within an annular channel and numerical results are obtained to demonstrate the excellent convergence characteristics of this hybrid method. Critical comparisons against the boundary layer formulation are provided, and a set of benchmark results is produced, for different values of Reynolds number and aspect ratio.     

Dislocation dynamics in confined geometry

A simulation of dislocation dynamics has been used to calculate the critical stress for a threading dislocation moving in a confined geometry. The optimum conditions for conducting simulations in systems of various sizes, down to the nanometer range, are defined. The results are critically compared with the available theoretical and numerical estimates for the problem of dislocation motion in capped layers. This revised version was published online in July 2006 with corrections to the Cover Date.     

Differential geometry of the coincidence site lattice

The tensor properties of simple internal surfaces, such as two-phase interfaces and grain boundaries, have been studied in detail. In particular, these tensor quantities have been defined with respect to the original crystal lattice as well as to a common coincidence site lattice that is a characteristic of the boundary. Such lattices allow a given type of distortion to be represented in either a Riemannian or a non-Riemannian (dislocated) space. This particular generality provides a powerful method of anlysing problems in continuum mechanics.The present research was supported by the National Science Foundation under Grant no. DMR-7202944.     

A theoretical treatment of void electromigration in the strip geometry

The void electromigration process in the strip geometry is investigated analytically and numerically. The void is assumed to travel either along the axis of symmetry of the metal strip or at the boundary. In each case, the shape, the velocity of the void and the characteristic electrical current are predicted.