Impact of backwater on water surface profile in curved channels

Owing to extensive construction of dams, the impact of backwater on flow may lead to navigation or flood control issues in curved channels. To date, the impact of backwater on the water surface profile in curved channels remains unknown and requires investigation. In this study, experiments were conducted in a glass-walled recirculating flume with a length of 19.4 m, a width of 0.6 m, and a depth of 0.8 m, and the impact of backwater on the water surface profile in a 90° channel bend was investigated. The experimental results showed that the backwater degree had a significant impact on the transverse and longitudinal flow depth distributions in the bend. The transverse slope of the flow (Jr) increased linearly with an increase in the Froude number of the approach flow upstream of the bend. Jr increased with the longitudinal location parameter ξ when −0.2 < ξ < 0.5, and decreased with ξ when 0.5 < ξ < 1.2. Furthermore, the results showed that Jr asymptotically decreased to zero with an increase in the degree of backwater. An equation was formulated to estimate the transverse slope of the flow in a 90° bend in backwater zones.

A New Dynamics Analysis Model for Five-Axis Machining of Curved Surface Based on Dimension Reduction and Mapping

The equipment used in various fields contains an increasing number of parts with curved surfaces of increasing size.Five-axis computer numerical control(CNC)milling is the main parts machining method,while dynamics analysis has always been a research hotspot.The cutting conditions determined by the cutter axis,tool path,and workpiece geometry are complex and changeable,which has made dynamics research a major challenge.For this reason,this paper introduces the innovative idea of applying dimension reduction and mapping to the five-axis machining of curved surfaces,and proposes an efficient dynamics analysis model.To simplify the research object,the cutter position points along the tool path were discretized into inclined plane five-axis machining.The cutter dip angle and feed deflection angle were used to define the spatial position relationship in five-axis machining.These were then taken as the new base variables to construct an abstract two-dimensional space and establish the mapping relationship between the cutter position point and space point sets to further simplify the dimensions of the research object.Based on the in-cut cutting edge solved by the space limitation method,the dynamics of the inclined plane five-axis machining unit were studied,and the results were uniformly stored in the abstract space to produce a database.Finally,the prediction of the milling force and vibration state along the tool path became a data extraction process that significantly improved efficiency.Two experiments were also conducted which proved the accuracy and efficiency of the proposed dynamics analysis model.This study has great potential for the online synchronization of intelligent machining of large surfaces.

Unsteady Flow and Heat Transfer of a Casson Micropolar Nanofluid over a Curved Stretching/Shrinking Surface

We present the results of an investigation into the behavior of the unsteady flow of a Casson Micropolar nanofluid over a shrinking/stretching curved surface,together with a heat transfer analysis of the same problem.The body force acting perpendicular to the surface wall is in charge of regulating the fluid flow rate.Curvilinear coordinates are used to account for the considered curved geometry and a set of balance equations for mass,momentum,energy and concentration is obtained accordingly.These are turned into ordinary differential equations using a similarity transformation.We show that these equations have dual solutions for a number of different combinations of various parameters.The stability of such solutions is investigated by applying perturbations on the steady states.It is found that high values of the Micropolar and Casson parameters cause the flow to move more slowly.However,when compared to a shrunken surface,a stretched surface produces a greater Micro-rotation flux.

High-Order Method with Moving Frames to Compute the Covariant Derivatives of Vectors on General 2D Curved Surfaces

The covariant derivative is a generalization of differentiating vectors.The Euclidean derivative is a special case of the covariant derivative in Euclidean space.The covariant derivative gathers broad attention,particularly when computing vector derivatives on curved surfaces and volumes in various applications.Covariant derivatives have been computed using the metric tensor from the analytically known curved axes.However,deriving the global axis for the domain has been mathematically and computationally challenging for an arbitrary two-dimensional(2D)surface.Consequently,computing the covariant derivative has been difficult or even impossible.A novel high-order numerical scheme is proposed for computing the covariant derivative on any 2D curved surface.A set of orthonormal vectors,known as moving frames,expand vectors to compute accurately covariant derivatives on 2D curved surfaces.The proposed scheme does not require the construction of curved axes for the metric tensor or the Christoffel symbols.The connectivity given by the Christoffel symbols is equivalently provided by the attitude matrix of orthonormal moving frames.Consequently,the proposed scheme can be extended to the general 2D curved surface.As an application,the Helmholtz‐Hodge decomposition is considered for a realistic atrium and a bunny.

Effective dynamics for a spin-1/2 particle constrained to a curved layer with inhomogeneous thickness

We derive an effective Hamiltonian for a spin-1/2 particle confined within a curved thin layer with non-uniform thickness using the confining potential approach.Our analysis reveals the presence of a pseudo-magnetic field and effective spin–orbit interaction(SOI)arising from the curvature,as well as an effective scalar potential resulting from variations in thickness.Importantly,we demonstrate that the physical effect of additional SOI from thickness fluctuations vanishes in low-dimensional systems,thus guaranteeing the robustness of spin interference measurements to thickness imperfection.Furthermore,we establish the applicability of the effective Hamiltonian in both symmetric and asymmetric confinement scenarios,which is crucial for its utilization in one-side etching systems.

A novel approach of jet polishing for interior surface of small-grooved components using three developed setups

It is a challenge to polish the interior surface of an additively manufactured component with complex structures and groove sizes less than 1 mm.Traditional polishing methods are disabled to polish the component,meanwhile keeping the structure intact.To overcome this challenge,small-grooved components made of aluminum alloy with sizes less than 1 mm were fabricated by a custom-made printer.A novel approach to multi-phase jet(MPJ)polishing is proposed,utilizing a self-developed polisher that incorporates solid,liquid,and gas phases.In contrast,abrasive air jet(AAJ)polishing is recommended,employing a customized polisher that combines solid and gas phases.After jet polishing,surface roughness(Sa)on the interior surface of grooves decreases from pristine 8.596μm to 0.701μm and 0.336μm via AAJ polishing and MPJ polishing,respectively,and Sa reduces 92%and 96%,correspondingly.Furthermore,a formula defining the relationship between linear energy density and unit defect volume has been developed.The optimized parameters in additive manufacturing are that linear energy density varies from 0.135 J mm^(-1)to 0.22 J mm^(-1).The unit area defect volume achieved via the optimized parameters decreases to 1/12 of that achieved via non-optimized ones.Computational fluid dynamics simulation results reveal that material is removed by shear stress,and the alumina abrasives experience multiple collisions with the defects on the heat pipe groove,resulting in uniform material removal.This is in good agreement with the experimental results.The novel proposed setups,approach,and findings provide new insights into manufacturing complex-structured components,polishing the small-grooved structure,and keeping it unbroken.

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

Study on curved surface fitting model using GPS and leveling in local area

The height anomaly surface is fitt and the quasi-geoid can be obtained when the height anomaly is determined with the geometric analytic method. Therefore, some mathematical models to fit height anomaly surface using GPS, leveling and terrain data in a local area, including the polynomial fitting model, the multi-surface function fitting model, the motion surface fitting model and the fitting model of little flexibility deformation of thin board, are given. Then the digital characteristics are analyzed with the curved surface theory. The General curvature and the mean curvature of surface are concluded. The advantage, disadvantage and application of the above models are discussed. The effect of terrain undulation on height anomaly is considered in the surface fitting models. The practical case indicates that these models are of validity and practicability. It is concluded that the above models can give the good fitting results at the centimeter level. But the polynomial fitting model is worse than the other models.

Programmable robotized‘transfer-and-jet’printing for large,3D curved electronics on complex surfaces

Large,3D curved electronics are a trend of the microelectronic industry due to their unique ability to conformally coexist with complex surfaces while retaining the electronic functions of 2D planar integrated circuit technologies.However,these curved electronics present great challenges to the fabrication processes.Here,we propose a reconfigurable,mask-free,conformal fabrication strategy with a robot-like system,called robotized‘transfer-and-jet’printing,to assemble diverse electronic devices on complex surfaces.This novel method is a ground-breaking advance with the unique capability to integrate rigid chips,flexible electronics,and conformal circuits on complex surfaces.Critically,each process,including transfer printing,inkjet printing,and plasma treating,are mask-free,digitalized,and programmable.The robotization techniques,including measurement,surface reconstruction and localization,and path programming,break through the fundamental constraints of 2D planar microfabrication in the context of geometric shape and size.The transfer printing begins with the laser lift-off of rigid chips or flexible electronics from donor substrates,which are then transferred onto a curved surface via a dexterous robotic palm.Then the robotic electrohydrodynamic printing directly writes submicrometer structures on the curved surface.Their permutation and combination allow versatile conformal microfabrication.Finally,robotized hybrid printing is utilized to successfully fabricate a conformal heater and antenna on a spherical surface and a flexible smart sensing skin on a winged model,where the curved circuit,flexible capacitive and piezoelectric sensor arrays,and rigid digital–analog conversion chips are assembled.Robotized hybrid printing is an innovative printing technology,enabling additive,noncontact and digital microfabrication for 3D curved electronics.

Curved surface effect and emission on silicon nanostructures

The curved surface （CS） effect on nanosilicon plays a main role in the activation for emission and photonic manipulation. The CS effect breaks the symmetrical shape of nanosilicon on which some bonds can produce localized electron states in the band gap. The investigation in calculation and experiment demonstrates that the different curvatures can form the characteristic electron states for some special bonding on the nanosilicon surface, which are related to a series of peaks in photoluminecience （PL）, such as LN, LNO, Lo1, and Lo2 lines in PL spectra due to Si-N, Si-NO, Si=O, and Si-O-Si bonds on curved surface, respectively. Si-Yb bond on curved surface of Si nanostructures can provide the localized states in the band gap deeply and manipulate the emission wavelength into the window of optical communication by the CS effect, which is marked as the Lyb line of electroluminescence （EL） emission.

Residual strain evaluation of curved surface by grating-transferring technique and GPA

This paper investigates an advanced grating-transferring technique combined with geometric phase analysis (GPA) for residual strain evaluation of curved surface.A standard holographic grating is first transferred to a pre-produced epoxy resin film and then consolidated to a test region of curved surface.With a rubber mold and silicone rubber the deformed grating is replicated to a sheet metal after hole-drilling for release of residual stress.After that the grating is transferred from the sheet metal to the glass plate,which would be served as an analyzer grating (specimen grating).By GPA the local strain distributions related to the phase difference between the reference grating and analyzer grating for the released stress can be evaluated.A validation test has been conducted on the weld joint of a stainless steel tube and the obtained results demonstrate the ability of the method in measuring the residual strain of curved surface.

Conformal manufacturing of soft deformable sensors on the curved surface

Health monitoring of structures and people requires the integration of sensors and devices on various 3D curvilinear,hierarchically structured,and even dynamically changing surfaces.Therefore,it is highly desirable to explore conformal manufacturing techniques to fabricate and integrate soft deformable devices on complex 3D curvilinear surfaces.Although planar fabrication methods are not directly suitable to manufacture conformal devices on 3D curvilinear surfaces,they can be combined with stretchable structures and the use of transfer printing or assembly methods to enable the device integration on 3D surfaces.Combined with functional nanomaterials,various direct printing and writing methods have also been developed to fabricate conformal electronics on curved surfaces with intimate contact even over a large area.After a brief summary of the recent advancement of the recent conformal manufacturing techniques,we also discuss the challenges and potential opportunities for future development in this burgeoning field of conformal electronics on complex 3D surfaces.

Reconstruction of B-spline surface by interpolating boundary curves and approximating inner points

In this paper, we present an algorithm for reconstruction of B-spline surface such that it interpolates the four given bound- ary curves and simultaneously approximates some given inner points. The main idea of our method is： first, we construct an initial surface which interpolates the four given boundary curves; then, while keeping the boundary control points of the initial surface un- changed, we reposition the inner control points of the surface with energy optimization method. Examples show that our algorithm is practicable and effective.

Magnetorheological finishing of low-gradient curved surfaces based on four-axis linkage technique

Based on the distribution characteristic of magnetic field along the polish wheel,the four-axis linkage technique is advanced to replace a standard five-axis one to figure low-gradient optical surfaces with a raster tool-path in magnetorheological finishing(MRF).After introducing the fundaments of such simplification,the figuring reachability of a four-axis system for the low-gradient optics was theoretically analyzed.Further validation including magnetic field intensity and influence function characteristic was performed to establish its application.To demonstrate the correctness,feasibility and applicability of such technique,a K4 spherical part was figured by two iterations of MRF with surface form error improved to 0.219λPV and 0.027λRMS.Meanwhile,the surface roughness was also improved a lot in MRF process.These theoretical analyses and experimental results both indicate that high form accuracy and excellent surface quality can be obtained by using the four-axis linkage technique in the process of figuring low-gradient optical elements,and the four-axis linkage system undoubtedly is much more easy to control and much more economical.

Numerical analysis of 3D spot continual induction hardening on curved surface of AISI 1045 steel

In order to investigate and predict the material properties of curved surface AISI 1045 steel component during spot continual induction hardening(SCIH),a 3D model for curved surface workpieces which coupled electromagnetic,temperature and phase transformation fields was built by finite element software ANSYS.A small size inductor and magnetizer were used in this model,which can move along the top surface of workpiece flexibly.The effect of inductor moving velocity and workpiece radius on temperature field was analyzed and the heating delay phenomenon was found through comparing the simulated results.The temperature field results indicate that the heating delay phenomenon is more obvious under high inductor moving velocity condition.This trend becomes more obvious if the workpiece radius becomes larger.The predictions of microstructure and micro-hardness distribution were also carried out via this model.The predicted results show that the inductor moving velocity is the dominated factor for the distribution of 100% martensite region and phase transformation region.The influencing factor of workpiece radius on 100% martensite region and phase transformation region distribution is obvious under relatively high inductor moving velocity but inconspicuous under relatively low inductor moving velocity.

ON CURVED SURFACE APPROXIMATION AND HIDDEN SURFACE ELIMINATION IN 3-D COMPUTER AIDED GARMENT DESIGN

An approximation method for curved surface mannequin and hidden surface eliminationin 3-D computer aided garment design system is described. The mannequin is the basis of the3-D modeling for clothes. In terms of the requirements of computer aided garment design,the authors put forward a method for curved surface approximation in the meaning of leastsquares. Using. this method the computation of geometric modeling is simple andefficient. It is also convenient for curved surface modification and shading.

DIRECTIVITY OPTIMIZATION OF CURVED SURFACE DIPOLE ANTENNAS

Based on the calculation of the characteristic parameters by moment method, the curved surface dipoles are optimized by an optimization method, the maximum directivities of some V-curved and Gauss-curved surface dipoles are given.

Research on Planar Development of Curved Surfaces

Development of curved surface is a useful tool in CAD(computer aided design) and CAGD(computer aided geometric design).This paper presents the algorithms for developing (flattening) a smooth continuous curved surface embedded in three dimensional space into a planar shape. First the definition of planar development of a curved surface is presented, and the distortions (at length and area) of development are discussed in this paper. Then several planar flattening methods of curved surface, and their advantages and disadvantages are analyzed in detail. For NURBS(non uniform rational basic spline) surfaces, which are broadly used in CAGD, we put forward a new planar developing algorithm, i.e. hybrid developing, and present the steps of the algorithm. At last, some examples are used to show the effectiveness of the algorithm.

RECONSTRUCTION OF SYMMETRIC B-SPLINE CURVES AND SURFACES

A method to reconstruct symmetric B-spline curves and surfaces is presented. The symmetry property is realized by using symmetric knot vector and symmetric control points. Firstly, data points are divided into two parts based on the symmetry axis or symmetry plane extracted from data points. Then the divided data points are parameterized and a symmetric knot vector is selected in order to get symmetric B-spline basis functions. Constraint equations regarding the control points are deduced to keep the control points of the B-spline curve or surface to be symmetric with respect to the extracted symmetry axis or symmetry plane. Lastly, the constrained least squares fitting problem is solved with the Lagrange multiplier method. Two examples from industry are given to show that the proposed method is efficient, robust and able to meet the general engineering requirements.

Shape gradient and classical gradient of curvatures:driving forces on micro/nano curved surfaces

Recent experiments and molecule dynamics simulations have shown that adhesion droplets on conical surfaces may move spontaneously and directionally. Besides, this spontaneous and directional motion is independent of the hydrophilicity and hydrophobicity of the conical surfaces. Aimed at this important phenomenon, a gen- eral theoretical explanation is provided from the viewpoint of the geometrization of micro/nano mechanics on curved surfaces. In the extrinsic mechanics on micro/nano soft curved surfaces, we disclose that the curvatures and their extrinsic gradients form the driving forces on the curved spaces. This paper focuses on the intrinsic mechanics on micro/nano hard curved surfaces and the experiment on the spontaneous and directional motion. Based on the pair potentials of particles, the interactions between an isolated particle and a micro/nano hard curved surface are studied, and the geometric foundation for the interactions between the particle and the hard curved surface is analyzed. The following results are derived： （a） Whatever the exponents in the pair potentials may be, the potential of the particle/hard curved surface is always of the unified curvature form, i.e., the potential is always a unified function of the mean curvature and the Gaussian curvature of the curved surface. （b） On the basis of the curvature-based potential, the geometrization of the micro/nano mechanics on hard curved surfaces may be realized. （c） Similar to the extrinsic mechanics on micro/nano soft curved surfaces, in the intrinsic mechanics on micro/nano hard curved surfaces, the curvatures and their intrinsic gradi- ents form the driving forces on the curved spaces. In other words, either on soft curved surfaces or hard curved surfaces and either in the extrinsic mechanics or the intrinsic mechanics, the curvatures and their gradients are all essential factors for the driving forces on the curved spaces. （d） The direction of the driving force induced by the hard curved surface is independent of the hydrophilieity and hydrophobicity of the curved surface, explaining the experimental phenomenon of the spontaneous and directional motion.