Curvature Discontinuities Research Articles

Softening of edges of solids by surface tension

Surface tension tends to minimize the area of interfaces between pieces of matter in different thermodynamic phases, be they in the solid or the liquid state. This can be relevant for the macroscopic shape of very soft solids and lead to a roughening of initially sharp edges. We calculate this effect for a Neo-Hookean elastic solid, with assumptions corresponding to actual experiments, namely the case where an initially sharp edge is rounded by the effect of surface tension felt when the fluid surrounding the soft solid (and so surface tension) is changed at the solid/liquid boundary. We consider two opposite limits where the analysis can be carried to the end, the one of a shallow angle and the one of a very sharp angle. Both cases yield a discontinuity of curvature in the state with surface tension although the initial state had a discontinuous slope.

The lemon illusion: seeing curvature where there is none.

Curvature is a highly informative visual cue for shape perception and object recognition. We introduce a novel illusion—the Lemon Illusion—in which subtle illusory curvature is perceived along contour regions that are devoid of physical curvature. We offer several perceptual demonstrations and observations that lead us to conclude that the Lemon Illusion is an instance of a more general illusory curvature phenomenon, one in which the presence of contour curvature discontinuities lead to the erroneous extension of perceived curvature. We propose that this erroneous extension of perceived curvature results from the interaction of neural mechanisms that operate on spatially local contour curvature signals with higher-tier mechanisms that serve to establish more global representations of object shape. Our observations suggest that the Lemon Illusion stems from discontinuous curvature transitions between rectilinear and curved contour segments. However, the presence of curvature discontinuities is not sufficient to produce the Lemon Illusion, and the minimal conditions necessary to elicit this subtle and insidious illusion are difficult to pin down.

Shock formation in Lovelock theories

We argue that Lovelock theories of gravity suffer from shock formation, unlike General Relativity. We consider the propagation of (i) a discontinuity in curvature, and (ii) weak, high frequency, gravitational waves. Such disturbances propagate along characteristic hypersurfaces of a "background" spacetime and their amplitude is governed by a transport equation. In GR the transport equation is linear. In Lovelock theories, it is nonlinear and its solutions can blow up, corresponding to the formation of a shock. We show that this effect is absent in some simple cases e.g. a flat background spacetime, and demonstrate its presence for a plane wave background. We comment on weak cosmic censorship, the evolution of shocks, and the nonlinear stability of Minkowski spacetime, in Lovelock theories.

Numerical and Theoretical Investigations Concerning the Continuous-Surface-Curvature Effect in Compressor Blades

Though the importance of curvature continuity on compressor blade performances has been realized, there are two major questions that need to be solved, i.e., the respective effects of curvature continuity at the leading-edge blend point and the main surface, and the contradiction between the traditional theory and experimental observations in the effect of those novel leading-edge shapes with smaller curvature discontinuity and sharper nose. In this paper, an optimization method to design continuous-curvature blade profiles which deviate little from datum blades is proposed, and numerical and theoretical analysis is carried out to investigate the continuous-curvature effect on blade performances. The results show that the curvature continuity at the leading-edge blend point helps to eliminate the separation bubble, thus improving the blade performance. The main-surface curvature continuity is also beneficial, although its effects are much smaller than those of the blend-point curvature continuity. Furthermore, it is observed that there exist two factors controlling the leading-edge spike, i.e., the curvature discontinuity at the blend point which dominates at small incidences, and the nose curvature which dominates at large incidences. To the authors’ knowledge, such mechanisms have not been reported before, and they can help to solve the sharp-leading-edge paradox.

Corner rounding of linear five-axis tool path by dual PH curves blending

The widespread linear five-axis tool path (G01 blocks) is usually described by two trajectories. One trajectory describes the position of the tool tip point, and the other one describes the position of the second point on the tool axis. The inherent disadvantages of linear tool path are tangential and curvature discontinuities at the corners in five-axis tool path, which will result in feedrate fluctuation and decrease due to the kinematic constraints of the machine tools. In this paper, by using a pair of quintic PH curves, a smoothing method is proposed to round the corners. There are two steps involved in our method. Firstly, according to the accuracy requirements of the tool tip contour and tool orientation tolerances, the corner is rounded with a pair of PH curves directly. Then, the control polygon lengths of PH curves are adjusted simply to guarantee the continuous variation of the tool orientation at the junctions between the transition curves and the remainder linear segments. Because the PH curves for corner rounding can be constructed without any iteration, and those two rounded trajectories are synchronized linearly in interpolation, which makes this smoothing method can be applied in a high efficiency way. Its high computational efficiency allows it to be implemented in real-time applications. This method has been integrated into a CNC system with an open architecture to implement on-line linear five-axis tool path smoothing. Simulations and experiments validate its practicability and reliability.

Development of Real-Time Look-Ahead Methodology Based on Quintic PH Curve with G<sup>2</sup> Continuity for High-Speed Machining

Curving tool paths composed of straight lines, which are often represented as G01 blocks, are still the most widespread format form in the machining process chain of CAD/CAM/CNC. At the junctions between consecutive segments, the tangency and curvature discontinuities may lead to feedrate fluctuation and acceleration oscillation, which would deteriorate the machining efficiency and quality. In this paper, a real-time look-ahead interpolation methodology is proposed, which adopts a curvature-continuous PH curve as a transition to blend corner at the junction of adjacent lines in the tool path. The blending algorithm can guarantee the approximation error exactly, and the control points of the curve can be calculated analytically. On the other hand, the arc length and the curvature of the transition curve, which are important items in speed planning, also have analytical expressions. All the advantages are the guarantee of calculation efficiency during the interpolation. Except for a curvature-continuous tool path, our look-ahead algorithm adopts a speed planning window strategy to achieve a balance between the calculation capabilities and the real-time interpolation requirements. In this window, the corner transition algorithm and speed planning are implemented simultaneously and dynamically during the interpolation. By defining the width of this window, which is actually the number of linear segments contained in this window, can adjust the time consuming of speed planning. Simulation and experiments on our own developed CNC platform are conducted. The results demonstrate the feasibility and efficiency of the proposed algorithms.

Non-smooth bending and buckling of a strain gradient elastic beam with non-convex stored energy function

Non-smooth strain gradient fields are studied in bending beams, in the context of strain gradient elasticity. It is found that strain fields with continuous curvature, but discontinuous curvature derivatives (evolutes) are possible. The pure bending and buckling problems of a simply supported beam are investigated.

Variational formulation of curved beams in global coordinates

In this paper we derive a variational formulation for a linear curved beam which is natively expressed in global Cartesian coordinates. During derivation the beam midline is assumed to be implicitly described by a vector distance function which eliminates the need for local coordinates. The only geometrical information appearing in the final expressions for the governing equations is the tangential direction, and thus there is no need to introduce normal directions along the curve. As a consequence zero or discontinuous curvature, for example at inflection points, pose no difficulty in this formulation. Kinematic assumptions encompassing both Timoshenko and Euler--Bernoulli beam theories are considered. With the exception of truly three dimensional formulations, models for curved beams found in literature are typically derived in the Frenet frame defined by the geometry of the beam midline. While it is intuitive to formulate curved beam models in these local coordinates, the Frenet frame suffers from ambiguity and sudden changes of orientation in straight sections of the beam. Based on the variational formulation we implement finite element models using global Cartesian degrees of freedom and discuss curvature coupling effects and locking. Numerical comparisons with classical solutions for both straight and curved cantilever beams under a tip load are given, as well as numerical examples illustrating curvature coupling effects.

Searching for bumps and ellipses on the ground and in the sky: No advantage for the ground plane

A staple of modern theories of vision is that the visual system has evolved to perceive cues containing the most predictive information about the layout of the environment. This entails the prediction that - other things being equal - visual performance in a familiar setting should be superior to performance in an unfamiliar one. Visual performance should therefore be better on the familiar ground plane compared to an implied sky or wall plane. We tested this comparing visual search for stimuli presented in an implied ground plane with search on a 180° rotated search display so that the stimuli appeared in an implied "sky" plane, and with search in a random layout implying no depth. This was tested for stimuli with, or without, curvature discontinuities, that have previously been shown to be strong cues for shape analysis. Surprisingly, no advantage of the ground plane over the sky plane was observed, while a strong effect of layout regularity was seen. Similarly, in experiment 2 there was little effect of placing the stimuli on an implied wall plane compared to the ground or the sky. The results are not explained by assuming that curvature discontinuities are such strong cues that they overshadow any effect of depth-plane, since there was a strong effect of regular versus random layout, which should also have disappeared under this account. The results argue instead for a very strong effect of layout regularity, unrelated to environmental regularities in evolutionary history, since there was no ground-plane benefit.

Damage detection and full surface characterization of a wind turbine blade using three-dimensional digital image correlation

The increasing demand for wind power has led to a significant increase in the number and size of wind turbine blades manufactured globally. As the number and physical size of turbines deployed grow, the probability of manufacturing defects being present in composite turbine blades also increases. As capital blade costs and operational and maintenance expenses increase in ever larger turbine systems, the need for inspection of the structural health of large-scale turbine blades during operation critically increases. One method for locating and quantifying manufacturing defects, while also allowing for the in situ measurement of the structural health of blades, is monitoring the full-field deformation and strain of a blade. In a demonstration of this methodology, static tests were performed on a Sandia National Laboratories CX-100 9-m composite turbine blade to extract full-field displacement and strain measurements. Three-dimensional digital image correlation was used. Measurements were taken at previously identified damaged areas near the blade root, along the high- and low-pressure surfaces. The results indicate that the measurement approach can clearly identify failure locations and discontinuities in the blade curvature under load. Postprocessing of the data, using a stitching technique of digital image correlation snapshots taken along the length of the blade, allows observation of the shape and curvature of the entire blade. The experiment demonstrates the feasibility of the approach and reveals that the technique can be readily scaled to accommodate utility-scale blades. As long as a trackable pattern is applied to the surface of the blade, measurements can be made in situ when a blade is on a manufacturing floor, installed in a test fixture, or installed on a rotating turbine. The results demonstrate the potential of the optical measurement technique for use in the wind industry.

ANALYTICAL AND EXPERIMENTAL STUDY OF FEED RATE IN HIGH-SPEED MILLING

In the context of high-speed milling (HSM), during the machining process dynamic machine response has to be identified. To achieve this, we have to calculate the feed rate evolution in linear and circular interpolation according to dynamic performance of machine. In addition to that, actual trajectory for transition passages between two interpolations must be estimated with take into account of specific machining tolerances. This article proposes a model of machine tool behavior for a tool path with linear and circular interpolations and machining cycle time prediction. The method involves subdividing the trajectories into elementary geometries according to the type of interpolation (circular or linear). At points where different trajectories meet, there is often a discontinuity in curvature or in tangency, which decreases the feed rate. At the points of discontinuity in tangency, a fillet radius is inserted. In this article, the influence of the geometry for elementary trajectories was determined. Then, the value of the fillet radius between linear and circular contours in different combinations was modeled. An industrial application was carried out in order to validate models and to determine the influence of feed rate evolution on the machining cycle time.

Une formulation hybride du modèle de coque de Naghdi

We present a new version of the Naghdi model for shells with curvature discontinuities. The unknowns – the displacement and the rotation of the normal to the shell midsurface – are described respectively in Cartesian and local covariant or contravariant basis. Our purpose here is to consider a constraint-free formulation instead of the one introduced by Blouza et al. (Two finite element approximation of Naghdiʼs shell model in Cartesian coordinates, SIAM J. Numer. Anal. 44 (2) (2006) 636–654), where the tangency character of the rotation is enforced by penalization or by duality. This new version enables us, in particular, to approximate by conforming finite elements the solution with less degrees of freedom compared to the method of Blouza et al.

Model based real-time collision-free motion planning for nonholonomic mobile robots in unknown dynamic environments

In this paper, by explicitly considering a dynamic model of the robots, the coefficients of trajectories are determined by boundary conditions, optimal performance index and collision avoidance conditions. The planned trajectory is feasible and has a closed loop expression, which is efficient for real-time updating. There are two main improvements compared with existing parametric approaches. Firstly, most of existing methods use the kinematic models of the robots, which could cause curvature discontinuities when trajectories are updated in real-time. Secondly, in some existing parametric methods, the initial position and ending waypoints cannot be aligned vertically due to singularities. The approach proposed in this paper overcomes this limitation. Computer simulations verified the effectiveness of the proposed method.

Microfluidics of imbibition in random porous media

A free-energy lattice Boltzmann approach has been used to perform simulations of liquid penetration into random porous media. We focus our study on the effects of microstructures, particularly microtopography, on liquid penetration driven by capillary force and external pressure. For this purpose we set up a model structure that consists of a network of interconnected capillaries with varying pore geometries. The results showed that the discontinuities in the solid-surface curvature, as are present as corners on the capillary surfaces, have strong influences on liquid penetration through their pinning effects and interactions with local geometry.

Continuous Curvature Path Planning using Voronoi diagrams and Fermat's spirals

This paper presents a two-dimensional curvature-continuous path planning algorithm based on Voronoi diagrams and Fermat's spiral segments. The map and the obstacles position are assumed to be known a-priori and static. Despite the disposition of the obstacles, the Voronoi diagram always presents at least one collision-free path, maximally distant from all the obstacles. If more than one flyable path is available, the shortest one is selected. The result is further refined to obtain a more practical path that is piecewise linear with discontinuous curvature and velocity. Fermat's spirals are used to smooth the path and provide curvature-continuity. A maximum threshold for the curvature is set so that the result of the algorithm respects kinematics and dynamics constraints of the vehicle. Moreover a minimum clearance from the obstacles can be chosen to respect additional safety constraints. The final result of the algorithm is a simple and intuitive path composed only by straight lines and spiral segments.

Defects and boundary layers in non-Euclidean plates

We investigate the behaviour of non-Euclidean plates with constant negative Gaussian curvature using the Föppl–von Kármán reduced theory of elasticity. Motivated by recent experimental results, we focus on annuli with a periodic profile. We prove rigorous upper and lower bounds for the elastic energy that scales like the thickness squared. In particular we show that are only two types of global minimizers—deformations that remain flat and saddle shaped deformations with isolated regions of stretching near the edge of the annulus. We also show that there exist local minimizers with a periodic profile that have additional boundary layers near their lines of inflection. These additional boundary layers are a new phenomenon in thin elastic sheets and are necessary to regularize jump discontinuities in the azimuthal curvature across lines of inflection. We rigorously derive scaling laws for the width of these boundary layers as a function of the thickness of the sheet.

A real-time look-ahead interpolation methodology with curvature-continuous B-spline transition scheme for CNC machining of short line segments

Straight lines, or G01 blocks, are the most widespread representation form for the tool path in CNC machining. At the junctions between consecutive segments, the tangency and curvature discontinuities may lead to feedrate fluctuation and acceleration oscillation, which would deteriorate the machining efficiency and quality. To solve this problem, a real-time path-smoothing method is proposed, which adopts a curvature-continuous B-spline with five control points to blend the adjacent straight lines. The advantage of the transition scheme is that, G2 continuity, analytical calculation of the curvature extrema, approximation error control and real-time performance are considered simultaneously. Then, a bidirectional scanning algorithm for jerk limited S-shape feedrate profile is proposed to evaluate the feedrate constraints. On this basis, a real-time look-ahead scheme, which comprises of path-smoothing, bidirectional scanning and feedrate scheduling, is developed to acquire a feedrate profile with smooth acceleration. Also, an arc-length based interpolation algorithm for mixed linear and parametric segments is proposed to overcome the difficulty of crossing different segments. With these schemes, the smoothness of both tool path and feedrate is guaranteed. Simulation and experiments on an X–Y–Z platform are conducted. The results demonstrate the feasibility and efficiency of the present algorithms.

Feed rate modeling in circular–circular interpolation discontinuity for high-speed milling

In this paper, a modeling approach is presented in order to evaluate feed rate during a circular interpolation in high-speed milling. The developed model depends on the type of discontinuity and the kinematic performance of the machine tool. To begin with, a feed rate modeling for circular interpolation with continuity in tangency is developed. After, the discontinuity in tangency between two circular interpolations is replaced by discontinuity in curvature by adding a fillet which is in relation to the functional tolerance e imposed in the part design. An experimental study has been carried out to validate the models.

Emergence of a thin shell structure during collapse in isotropic coordinates

Numerical studies of gravitational collapse in isotropic coordinates have recently shown an interesting connection between the gravitational Lagrangian and black hole thermodynamics. A study of the actual spacetime was not the main focus of this work and, in particular, the rich and interesting structure of the interior has not been investigated in much detail and remains largely unknown. We elucidate its features by performing a numerical study of the spacetime in isotropic coordinates during gravitational collapse of a massless scalar field. The most salient feature to emerge is the formation of a thin shell of matter just inside the apparent horizon. The energy density and Ricci scalar peak at the shell and there is a jump discontinuity in the extrinsic curvature across the apparent horizon, the hallmark that a thin shell is present in its vicinity. At late stages of the collapse, the spacetime consists of two vacuum regions separated by the thin shell. The interior is described by an interesting collapsing isotropic universe. It tends towards a vacuum (never reaches a perfect vacuum) and there is a slight inhomogeneity in the interior that plays a crucial role in the collapse process as the areal radius tends to zero. The spacetime evolves towards a curvature (physical) singularity in the interior, both a Weyl and Ricci singularity. In the exterior, our numerical results match closely the analytical form of the Schwarzschild metric in isotropic coordinates, providing a strong test of our numerical code.

An analytical curvature-continuous Bézier transition algorithm for high-speed machining of a linear tool path

An analytical curvature-continuous path-smoothing algorithm is developed for the high speed machining of a linear tool path. The algorithm can be used in a post-processing stage or NC unit. Every segment junction of the linear tool path, which is the point of tangent discontinuity, is blended by inserting two cubic Bézier spiral curves. A tool path, which is composed of cubic Bézier curves and lines, is then obtained to replace the linear tool path. The new tool path is everywhere curvature-continuous, and both the tangent and curvature discontinuities are avoided. The feed motion will be more stable since the discontinuities are the most important sources of feed fluctuation. In the blending algorithm, the approximation error at the segment junction is accurately guaranteed and the control points of the two cubic Bézier transition curves are all analytically computed. The maximal curvature in every Bézier transition pair, which is critical for velocity planning, is also analytically computed. The analytical expressions provide a way to optimize the curvature radii of the transition curves to pursue the high feed speed. The path-smoothing methods for the post-processing stage and NC unit are both developed. The computational examples confirm the validity of the algorithm. The transition algorithm has been integrated into an open NC system. Cutting experiments show that the curvature-continuous tool path generates smoother feed and consumes shorter machining time than the original linear tool path.