Direction Of Curvature Research Articles

Impact of pulse dynamic loading direction and surface curvature on the stress-strain state of a three-layered spherical shell

On two size types of the semispherical three-layered structures, impact of the surface curvature and impact of the pulse dynamic loading on the stress-strain state (SSS) of these structures has been investigated. The layered hemispheres have been analyzed with the clamped footing and diameters (D1=0.30 m, D2=0.60 m), which had similar bearing layers’ thickness (h1=h3=0.010 m), polymeric filler with h2=0.020 m thickness, reinforced with the 5 discrete stiffening rings rigidly bound to the bearing layers.
 Values of normal stresses and vertical displacements of the structures’ bearing layers and distribution of these indicators along the spatial coordinate have been determined. Distribution of displacements’ and stresses’ magnitudes along the spatial coordinate α was determined by the software complex Nastran through the direct transient dynamic process algorithm within the time interval 0 ≤ t ≤ 10T. The time interval step duration was 0.25*10-6 s and the total number of steps was 200. The detailed and accurate calculation results have determined the choice of the solid finite element type.
 Value of the Gaussian curvature of the layered shell structures impacts their stress-strain state. Increase of the surface curvature of spherical structures under the impact of the dynamic pulse load increases the displacement and stress of their bearing layers.
 In addition to the vertical displacements and normal stresses of the bearing layers of the analyzed semispherical three-layered structures, their first natural frequency (f1) was also calculated.

Convergence to a second-order critical point by a primal-dual interior point trust-region method for nonlinear semidefinite programming

In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems, in which the iterates converge to a point that satisfies the first-order and second-order optimality conditions. The method consists of the outer iteration (SDPIP-revised) that finds a Karush-Kuhn-Tucker (KKT) point which satisfies the second-order optimality condition, and the inner iteration (SDPTR-revised) that calculates an approximate barrier KKT point. Algorithm SDPTR-revised uses a commutative class of Newton-like directions within the framework of the trust-region method in the primal-dual space. In addition, we also use a direction of negative curvature when it exists. The proposed algorithm employs a new method that generates negative-curvature directions in the existence of -type penalty term for equality constraints. It is proved that there exists a limit point of the generated sequence which satisfies the second-order optimality condition along with the barrier KKT conditions.

Computational Design of Self‐Actuated Surfaces by Printing Plastic Ribbons on Stretched Fabric

AbstractWe introduce a new mechanism for self‐actuating deployable structures, based on printing a dense pattern of closely‐spaced plastic ribbons on sheets of pre‐stretched elastic fabric. We leverage two shape‐changing effects that occur when such an assembly is printed and allowed to relax: first, the incompressible plastic ribbons frustrate the contraction of the fabric back to its rest state, forcing residual strain in the fabric and creating intrinsic curvature. Second, the differential compression at the interface between the plastic and fabric layers yields a bilayer effect in the direction of the ribbons, making each ribbon buckle into an arc at equilibrium state and creating extrinsic curvature. We describe an inverse design tool to fabricate low‐cost, lightweight prototypes of freeform surfaces using the controllable directional distortion and curvature offered by this mechanism. The core of our method is a parameterization algorithm that bounds surface distortions along and across principal curvature directions, along with a pattern synthesis algorithm that covers a surface with ribbons to match the target distortions and curvature given by the aforementioned parameterization. We demonstrate the flexibility and accuracy of our method by fabricating and measuring a variety of surfaces, including nearly‐developable surfaces as well as surfaces with positive and negative mean curvature, which we achieve thanks to a simple hardware setup that allows printing on both sides of the fabric.

Solving phase retrieval with random initial guess is nearly as good as by spectral initialization

The problem of recovering a signal x∈Rn from a set of magnitude measurements yi=|〈ai,x〉|,i=1,…,m is referred as phase retrieval, which has many applications in fields of physical sciences and engineering. In this paper we show that the smoothed amplitude flow based model for phase retrieval has benign geometric structure under the optimal sampling complexity. In particular, we show that when the measurements ai∈Rn are Gaussian random vectors and the number of measurements m≥Cn, our smoothed amplitude flow based model has no spurious local minimizers with high probability, i.e., the target solution x is the unique global minimizer (up to a global phase) and the loss function has a negative directional curvature around each saddle point. Due to this benign geometric landscape, the phase retrieval problem can be solved by the gradient descent algorithms without spectral initialization. Numerical experiments show that the gradient descent algorithm with random initialization performs well even comparing with state-of-the-art algorithms with spectral initialization in empirical success rate and convergence speed.

Uniformity Prediction of Bending Stiffness of Composite Space Mirrors With Ply Angle Misalignments by Deflection Angle

Based on the classical lamination theory, the principal moment direction of the laminate is derived. It is 45° apart from the generalized ply angle, which is the orientation of the ply where the ply angle misalignment occurs nominally. After adding a fixed one-ply angle or random multiple ply angle misalignments to six quasi-isotropic and symmetric stacking sequences, out-of-plane deformations of mirrors were simulated and compared. The result shows an angle difference between the principal moment and curvature directions. It is referred to as the deflection angle, approximately appearing as a trigonometric curve, which varies with laminates and is dependent on the generalized ply angle within each laminate. The deflection angle directly reflects the consistency of bending stiffness or laminate bending anisotropy; the smaller the deflection angle, the better will be the uniformity of bending stiffness. Consequently, it is an efficient approach to obtain a laminate with more uniformity of bending stiffness by comparing the deflection angle. These results enrich the research of out-of-plane deformation and could objectively promote the study of quasi-isotropic laminates for bending stiffness.

The Effect of Bulk Residual Stress on Milling-Induced Residual Stress and Distortion

BackgroundDistortion arises during machining of metallic parts from two main mechanisms: 1) release of bulk residual stress (BRS) in the pre-form, and 2) permanent deformation induced by cut tools. Interaction between these mechanisms is unexplored.ObjectiveAssess this interaction using aluminum samples that have a flat surface with variations of BRS, where that surface is subsequently milled, and we observe milling-induced residual stress (MIRS) and distortion.MethodsPlate samples are cut from two kinds of large blocks, one kind stress-relieved by stretching and a second kind solution heat treated, quenched and aged. The BRS field in the plates is known from a recent series of measurements, being small in the stress relieved plates (within ±20 MPa) and large (±100 MPa) in the quenched plates, varying from tension to compression over the surface that is milled. MIRS is measured following milling using hole-drilling. Distortions of thin wafers cut at the milled surfaces are used to elucidate BRS/MIRS interactions. A finite element (FE) model and a strength of materials model are each used to assess consistency between wafer distortion and measured MIRS.ResultsMilling in samples with high BRS magnitude changes the directions of MIRS and distortion relative to the milling direction, with the direction of maximum curvature rotating toward or away from the milling direction depending on the sign and direction of BRS. High magnitude BRS was also found to increase the wafer peak arc height, nearly doubling the amount found in low BRS samples.ConclusionMeasured residual stress and observed wafer distortion both show interactions between MIRS and BRS. Stress analysis models show that the differences in measured MIRS are consistent with the differences in observed distortion.

Gradient-driven turbulence in Texas Helimak

We investigate the turbulence level dependence on plasma profiles in experiments in Texas Helimak, a toroidal basic plasma device, with long stable electron cyclotron resonant heating (ECRH) discharges and great flexibility to alter the equilibrium magnetic field. A large set of Langmuir probes is used to obtain the turbulence level and also the plasma radial profiles for several magnetic field intensities with the same safety factor and field line pitch profiles. As a consequence of the ECRH heating, changing the toroidal magnetic field, the equilibrium density profiles are radially displaced. For all the analyzed discharges, with constant magnetic field curvature and shear profiles, we verify that the plasma turbulence has a critical dependence on the equilibrium density profile. Namely, radial regions with negative density radial gradient, i.e., in the opposite direction of the magnetic curvature, present high turbulence level. By properly comparing the turbulence radial profiles with the density peak position, we show that the negative density gradient is the main cause of high amplitude turbulence, in agreement with predictions for ideal interchange modes. Furthermore, intermittence analysis shows that the extreme events (bursts) contribution for the probability density functions (PDFs) is also related to the relative position with respect to the density peak, and that the turbulence level enhancement is likely due to the increase in burst occurrence.

Outcomes of Collagenase Clostridium Histolyticum in Men with Congenital Penile Curvature

ABSTRACT Introduction Congenital penile curvature is a relatively uncommon condition characterized by penile angulation. Similar to Peyronie's Disease (PD), this condition has historically been managed with various surgical techniques. Given the efficacy of Collagenase Clostridium Histolyticum (CCH) in men with PD, our team has elected to treat men with CPD with CCH injections. Objective To describe outcomes of CPD men treated with CCH injections and to compare against a traditional PD cohort. Methods A prospective database has been maintained of all men undergoing CCH injections by our team since Jan 2014. The specific CCH administration protocol has been modified over time and currently includes combined use of RestoreX traction therapy, sildenafil, 0.9 mg of CCH diluted into 0.7 ml, injections performed at the point of maximal curvature, and aggressive manual modeling. Curvatures were measured in two planes (dorsal and lateral), with the combined measure reported as a composite. Outcomes were reported based on the most recent curvature assessment obtained minus baseline. Men were included if they received at least one series of CCH injections and had at least one baseline and follow-up curvature assessment available. Results A total of 272 men comprised the current cohort, with 23 exhibiting CPD and 249 PD. Mean age was higher in PD men (57.4 vs 38.4, p<0.0001), and mean PD duration was 18 months in PD men. Median baseline curvature was non-statistically higher in PD (65 vs 55 degrees, p=0.06). Baseline curvature directions for PD and CPD, respectively were: dorsal (82% vs 35%, p<0.0001), lateral (58% vs 78%, p<0.05), ventral (11% vs 35%, p<0.01). Following treatment with CCH, median curvature improvement was similar between cohorts, with 20 degrees (IQR 5, 35) for PD and 20 degrees (IQR 5, 30) for CPD, p=0.62. Percentage improvement from baseline was non-statistically higher in PD men (33% vs 29%, p=0.99). Conclusions Treatment of CPD men with CCH is a viable therapeutic option and provides similar benefits to those observed in men with PD. Disclosure No

Corrected trapezoidal rules for singular implicit boundary integrals

We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in Kublik et al. (2013) [7]. In this formulation, a boundary integral over a smooth, closed hypersurface is transformed into an equivalent volume integral defined in a sufficiently thin tubular neighborhood of the surface. The volumetric formulation makes it possible to use the simple trapezoidal rule on uniform Cartesian grids and relieves the need to use parameterization for developing quadrature. Consequently, typical point singularities in a layer potential extend along the surface's normal lines. We propose new higher-order corrections to the trapezoidal rule on the grid nodes around the singularities. This correction is based on local decompositions of the singularity and is dependent on the angle of approach to the singularity relative to the surface's principal curvature directions. The proposed decomposition, combined with the volumetric formulation, leads to a special quadrature error cancellation.

Free vibration analysis of a circular cylindrical shell using a new four-variable refined shell theory

A theory is proposed to study the vibrations of homogenous isotropic circular cylindrical shells based on a developed plate vibration theory. By proposing a novel distribution of the shear stress and strain across the thickness of the shell, the arguments are adjusted such that the transverse shear strains and stresses equal zero at the shell surfaces. Due to the existing curvature in the circumferential direction, the defined shear stress and strain distributions consider a change in the position of the neutral plane toward the center of curvature. In the presented Two-Separated Lateral Displacement (TSLD) theory, the lateral displacement is defined by two bending and shear parts to provide an analytical model, referred to as the four-variable refined plate theory. The similarity to classical theory and involving only four unknowns are significant features of the introduced theory. The TSLD theory is verified against the available data in the literature. It is shown that the provided stress and stress distributions lead to more accurate results, especially in moderately thick cylinders. The proposed theory is remarkably more precise than classical shell theory and First-order Shear Deformation Theory (FSDT). Results of the TSLD theory are as accurate as the Higher-Order Shear Deformation Theory (HOST12) for the higher frequency parameters and more thick shells, while the TSLD theory implementation is much simpler than the HOST12.

Some notes on directional curvature of a convex body in ℝ n

Take a point ξ on the boundary of a convex body F in , near which the boundary is given by an implicit equation. We present some notes on the formula, proposed in Pereira [A directional curvature formula for convex bodies in . J Math Anal Appl. 2022;506(1):125656.], for calculating the curvature of F at ξ in the direction of its any tangent vector. Namely, we see that our formula is equivalent to the existing one for the curvature of a certain curve given by the intersection of n−1 implicit equations, but it is easier to apply. Furthermore, we show that when the directional curvature of F is positive, there is the directional derivative of the Minkowski functional of the polar set , and we propose a formula to calculate it.

Coarse-grained curvature tensor on polygonal surfaces

Using concepts from integral geometry, we propose a definition for a local coarse-grained curvature tensor that is well-defined on polygonal surfaces. This coarse-grained curvature tensor shows fast convergence to the curvature tensor of smooth surfaces, capturing with accuracy not only the principal curvatures but also the principal directions of curvature. Thanks to the additivity of the integrated curvature tensor, coarse-graining procedures can be implemented to compute it over arbitrary patches of polygons. When computed for a closed surface, the integrated curvature tensor is identical to a rank-2 Minkowski tensor. We also provide an algorithm to extend an existing C++ package, that can be used to compute efficiently local curvature tensors on triangulated surfaces.

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input that connect the PQ strips. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling-based representations of developable surfaces while still obtaining developable, curvature-aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its isolines are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight isolines is non-linear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle-based interactive freeform modeling of developable shapes.

Directional wavelet modal curvature method for damage detection in plates

Abstract Vibration-based damage detection using the 2D modal curvature has become a research focus. However, the traditional methods based on modal curvature lack the sensitivity to direction of the damage. To address this problem, a directional 2D Morlet wavelet modal curvature is proposed in this paper. The directional wavelet modal curvatures show the following merits: robust to noise interference and sensitive to direction of damage. By the numerical simulation, the efficiency of the proposed method is demonstrated in the plate-like structure.

An analytical framework of the kinematic geometry for general point-contact gears from contact path

Based on the idea that the relative curvature relationships along total contact path will better reflect the contact characteristics of gears, this study attempts to develop an analytical framework for the kinematic geometry of point-contact gears from the contact paths. First, a general method for obtaining the contact path of a gear from a pinion is proposed. Second, a normal curvature and geodesic torsion calculation method in the contact path direction is developed. Then, based on the linearly dependent condition of the tangent vectors of the two contact paths, two relative normal curvature analysis methods are proposed by applying the fundamental form theory. Finally, a sample case is presented to show the application and complete the verification of the kinematic geometry methods. The sample case agrees with the analytical model.

Optical scanning holography for a three-dimensional equal scale display.

We propose an optical scanning holography (OSH) for optical reconstruction with equal scale magnification or demagnification ratio along the transverse and longitudinal directions when the magnification or demagnification ratio of the hologram is priori known. First, we review the principles of OSH. Second, we propose an equal scale OSH that encodes a 3D distribution of a 3D scene with a scanning beam pattern that is formed by the superposition of two spherical waves with the same curvature direction. The hologram recorded by the equal scale OSH reconstructs the 3D scene with equal scale magnification or demagnification along the transverse and longitudinal directions. Finally, we provide simulation and experiment results to illustrate and clarify the proposed idea.

Energy consumption modelling in milling of variable curved geometry

The accurate estimation of energy consumption is beneficial to manufacturing enterprises economically as well as to overcome global energy crisis. The present work concentrates on developing an energy consumption model in milling of variable curved geometries where magnitudes and directions of workpiece curvature vary along tool contact path of a component. The current work deals with estimation and analysis of energy consumption in peripheral milling of variable curved surfaces where cutting forces differ along tool contact path in the presence of workpiece curvature. The proposed hybrid model developed in MATLAB involves process mechanics, cutting forces and energy consumption and has modules for idle, auxiliary and cutting power. The proposed model is validated by the experimental work. The model is generic and versatile in nature and is useful for milling of straight, circular and curved surfaces. In addition to it, the influence of workpiece curvature on power consumption has been investigated to realize the variation of power consumption along the tool contact path. The developed model offers a basic platform to understand and characterize the energy consumption for general peripheral milling considering workpiece geometry. The comparison of predicted and measured results indicates that the model is capable to estimate the power consumption accurately. The proposed model will be used by the practitioners to find the optimum cutting conditions to reduce power consumption during the machining of curved geometries – a pragmatic condition but not much researched condition in machining.

The role of color in the perception of three-dimensional shape

SummaryThe human visual system can derive information about three-dimensional (3D) shape from the structure of light reflected by surfaces. Most research on single static images has focused on the 3D shape information contained in variations of brightness caused by interactions between the illumination and local surface orientation (“shading”).1, 2, 3, 4, 5, 6 Although color can enhance the recovery of surface shading when color and brightness vary independently,7, 8, 9 there is no evidence that color alone provides any information about 3D shape. Here, we show that the wavelength-dependent reflectance of chromatic materials provides information about the 3D shape of translucent materials. We show that different wavelengths of light undergo varying degrees of subsurface light transport, which generates multiple forms of spatial structure: wavelengths that are weakly reflected generate shading-like image structure, linked to 3D surface orientation, whereas wavelengths that penetrate more deeply into the material are primarily constrained by the direction of surface curvature (convexities and concavities).10 Psychophysical experiments demonstrate that the enhanced perception of 3D shape in chromatic translucent surfaces arises from the shading structure generated by weakly reflected wavelengths, which, in turn, generates correlated spatial variations in saturation. These results demonstrate a new functional role for color in the perception of the 3D shape of translucent materials.

Spontaneous local membrane curvature induced by transmembrane proteins

The (local) curvature of cellular membranes acts as a driving force for the targeting of membrane-associated proteins to specific membrane domains, as well as a sorting mechanism for transmembrane proteins, e.g., by accumulation in regions of matching spontaneous curvature. The latter measure was previously experimentally employed to study the curvature induced by the potassium channel KvAP and by aquaporin AQP0. However, the direction of the reported spontaneous curvature levels as well as the molecular driving forces governing the membrane curvature induced by these integral transmembrane proteins could not be addressed experimentally.Here, using both coarse-grained and atomistic molecular dynamics (MD) simulations, we report induced spontaneous curvature values for the homologous potassium channel Kv 1.2/2.1 Chimera (KvChim) and AQP0 embedded in unrestrained lipid bicelles that are in very good agreement with experiment. Importantly, the direction of curvature could be directly assessed from our simulations: KvChim induces a strong positive membrane curvature (≈0.036 nm−1) whereas AQP0 causes a comparably small negative curvature (≈−0.019 nm−1).Analyses of protein-lipid interactions within the bicelle revealed that the potassium channel shapes the surrounding membrane via structural determinants. Differences in shape of the protein-lipid interface of the voltage-gating domains between the extracellular and cytosolic membrane leaflets induce membrane stress and thereby promote a protein-proximal membrane curvature. In contrast, the water pore AQP0 displayed a high structural stability and an only faint effect on the surrounding membrane environment that is connected to its wedge-like shape.

Characteristic parameterizations of surfaces with a constant ratio of principal curvatures

Motivated by applications in architectural geometry, we study and compute surfaces with a constant ratio of principal curvatures (CRPC surfaces) based on their characteristic parameterizations. For negative Gaussian curvature K, these parameterizations are asymptotic. For positive K they are conjugate and symmetric with respect to the principal curvature directions. CRPC surfaces are described by characteristic parameterizations whose parameter lines form a constant angle. We use them to derive characteristic parameterizations of rotational CRPC surfaces in a simple geometric way. Pairs of such surfaces with principal curvature ratio κ1/κ2=±a can be seen as equilibrium shapes and reciprocal force diagrams of each other. We then introduce discrete CRPC surfaces, expressed via discrete isogonal characteristic nets, and show how to efficiently compute them through numerical optimization. In particular, we derive discrete helical and spiral CRPC surfaces. We provide various ways how these and other special types of CRPC surfaces can serve as a basis for computational design of more general CRPC surfaces. Our computational tools may also serve as an experimental basis for mathematical studies of the largely unexplored class of CRPC surfaces.