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.

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.

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.

Wrinkling modes of graphene oxide assembled on curved surfaces

Assembling two-dimensional(2D)sheets into macroscopic three-dimensional(3D)forms has created a promising material family with rich functionalities.Multiscale wrinkles are intrinsic features of 2D sheets in their 3D assembles.Therefore,the precise wrinkling modulation optimizes the transition of outstanding properties of 2D sheets to expected performances of assembled materials and dominates their fabrication process.The wrinkling evolution of 2D sheets assembling onto flat surfaces has been extensively understood,however,the wrinkling behaviors on the more generally curved surface still remain unclear.Here,we investigate the wrinkling behaviors of graphene oxide sheets assembled onto curved surfaces and reveal the selection rule of wrinkling modes that determined by the curvature mismatch between 2D sheets and target surfaces.We uncover that three wrinkling modes including isotropic cracked land,labyrinth,and anisotropic curtain phases,respectively emerge on flat,spherical,and cylindrical surfaces.A favorable description paradigm is offered to quantitatively measure the complex wrinkling patterns and assess the curvature mismatch constraint underlying the wrinkling mode selection.This research provides a general and quantitative description framework of wrinkling modulation of 2D materials such as high performance graphene fibers,and guides the precise fabrication of particles and functional coatings.

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.

TFM imaging of aeroengine casing ring forgings with curved surfaces using acoustic field threshold segmentation and vector coherence factor

The aeroengine casing ring forgings have complex cross-section shapes,when the conventional ultrasonic or phased array is applied to detect such curved surfaces,the inspection images always have low resolution and even artifacts due to the distortion of the wave beam.In this article,taking a type of aeroengine casing ring forging as an example,the Total Focusing Method(TFM)algorithms for curved surfaces are investigated.First,the Acoustic Field Threshold Segmentation(AFTS)algorithm is proposed to reduce background noise and data calculation.Furthermore,the Vector Coherence Factor(VCF)is adopted to improve the lateral resolution of the TFM imaging.Finally,a series of 0.8 mm diameter Side-Drilled Holes(SDHs)are machined below convex and concave surfaces of the specimen.The quantitative comparison of the detection images using the conventional TFM,AFTS-TFM,VCF-TFM,and AFTS-VCF-TFM is implemented in terms of data volume,imaging Signal-to-Noise Ratio(SNR),and defect echo width.The results show that compared with conventional TFM,the data volume of AFTS-VCF-TFM algorithm for convex and concave is decreased by 32.39%and 73.40%,respectively.Moreover,the average SNR of the AFTS-VCF-TFM is gained up to 40.0 dB,while the average 6 dB-drop echo width of defects is reduced to 0.74 mm.

Flexible Scanner Works on Curved Surfaces

你印象中传统的扫描仪(scanner)是何种模样?也许在你的脑海中会跳出这些形容词:方方的,较大的,有一定重量。然而,日本科学家最新研发了一种扫描仪,它比一张普通的信用卡还要小,能够装在手机里。更值得一提的是,它可以清晰地扫描出褶皱物品的表面。本文详细介绍了这款扫描仪的原理及功能特性。

Development and Implementation of a Rotating Nanoimprint Lithography Tool for Orthogonal Imprinting on Edges of Curved Surfaces

Uniform molding and demolding of structures on highly curved surfaces through conformal contact is a crucial yet often-overlooked aspect of nanoimprint lithography(NIL).This study describes the development of a NIL tool and its integration into a nanopositioning and nanomeasuring machine to achieve high-precision orthogonal molding and demolding for soft ultraviolet-assisted NIL(soft UV-NIL).The process was implemented primarily on the edges of highly curved plano-convex substrates to demonstrate structure uniformity on the edges.High-resolution nanostructures of sub-200-nm lateral dimension and microstructures in the range of tens of microns were imprinted.However,the nanostructures on the edges of the large,curved substrates were difficult to characterize precisely.Therefore,microstructures were used to measure the structure fidelity and were characterized using profilometry,white light interferometry,and confocal laser scanning microscopy.Regardless of the restricted imaging capabilities at high inclinations for high-resolution nanostructures,the scanning electron microscope(SEM)imaging of the structures on top of the lens substrate and at an inclination of 45°was performed.The micro and nanostructures were successfully imprinted on the edges of the plano-convex lens at angles of 45°,60°,and 90°from the center of rotation of the rotating NIL tool.The method enables precise imprinting at high inclinations,thereby presenting a different approach to soft UV-NIL on curved surfaces.

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.

Extension of covariant derivative(Ⅱ): From flat space to curved space

This paper extends the classical covariant deriva tive to the generalized covariant derivative on curved sur faces. The basement for the extension is similar to the pre vious paper, i.e., the axiom of the covariant form invariabil ity. Based on the generalized covariant derivative, a covari ant differential transformation group with orthogonal duality is set up. Through such orthogonal duality, tensor analy sis on curved surfaces is simplified intensively. Under the covariant differential transformation group, the differential invariabilities and integral invariabilities are constructed on curved surfaces.

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.

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.

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.

Judging or setting weight steady-state of rational Bézier curves and surfaces

Many works have investigated the problem of reparameterizing rational B^zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.

ANOTHER TYPE OF GENERALIZED BALL CURVES AND SURFACES

In 2000, Wu presented two new types of generalized Ball curves, one of which is called an NB1 curve located between the Wang-Ball curve and the Said-Ball curve. In this article, the authors aim to discuss properties of NB1 curves and surfaces, including the recursive algorithms, conversion algorithms between NB1 and Bezier curves and surfaces, etc. In addition the authors compare the computation efficiency of recursive algorithms for the NB1 and above mentioned two generalized Ball curves and surfaces.

Schrdinger Equation of a Particle on a Rotating Curved Surface

We derive the Schr6dinger equation of a particle constrained to move on a rotating curved surface S. Using the thin-layer quantization scheme to confine the particle on S, and with a proper choice of gauge transformation for the wave function, we obtain the well-known geometric potentiM Vg and an additive Coriolis-induced geometric potential in the co-rotationM curvilinear coordinates. This novel effective potential, which is included in the surface Schr6dinger equation and is coupled with the mean curvature of S, contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian. We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.

Flow of Eyring-Powell liquid due to oscillatory stretchable curved sheet with modified Fourier and Fick’s model

This study deals with the features of the mass and heat transport mechanism by adopting a modified version of Fourier and Fick’s model known as the CattaneoChristov double diffusive theory.The time-dependent magnetohydrodynamic(MHD)flow of the Eyring-Powell liquid across an oscillatory stretchable curved sheet in the presence of Fourier and Fick’s model is investigated.The acquired set of flow equations is transformed into the form of nonlinear partial differential equations(PDEs)by applying appropriate similarity variables.A convergent series solution to the developed nonlinear equations is accomplished with the help of an analytical approach,i.e.,the homotopy analysis method(HAM).The consequences of diverse parameters,including the dimensionless EyringPowell liquid parameter,the radius of curvature,the Schmidt/Prandtl numbers,the ratio of the oscillatory frequency of the sheet to its stretchable rate constant,the mass and thermal relaxation variables involved in the flow,and the heat and mass properties,are displayed through graphs and tables.It is noted from this study that the amplitude of the pressure distribution rises for the high parametric values of the Eyring-Powell parameter.