Principal Curvatures Research Articles

Gradient estimates for the insulated conductivity problem: The non-umbilical case

We study the insulated conductivity problem with inclusions embedded in a bounded domain in Rn, for n≥3. The gradient of solutions may blow up as ε, the distance between the inclusions, approaches to 0. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on Sn−2.

Oscillating water column wave energy converter arrays coupled with a parabolic-wall energy concentrator in regular and irregular wave conditions

Based upon previous research on a single, high-efficiency Oscillating Water Column (OWC) wave energy capture system (Mayon et al., 2021), this work further extends the numerical investigation to the layout design and performance of the wave energy convertor (WEC) array coupled with a parabolic, energy concentrating wall in both regular and irregular incident wave conditions. A heuristic method to identify the optimal siting of the component, cylindrical OWC WECs in the array, installed in the concave opening of the wall is presented. The most advantageous location of the chambers is found to lie on a parabolic curvature line which is inset from the wall. Two separate arrays composed of three and five component OWCs are investigated in a range of regular wave conditions. The hydrodynamic power and efficiency of each chamber in each of the arrays is determined, and subsequently the aggregated array performance is established. It is found that the primary OWC chamber in the array configuration can attain approximately the same hydrodynamic power output as a single, isolated OWC chamber located the parabolic wall focus, albeit with a narrower energy capture bandwidth. The secondary and tertiary component chambers in the arrays contribute a lesser, yet still considerable quantity of hydrodynamic power to the consolidated system. The cumulative hydrodynamic efficiency of the collective arrays is less than the hydrodynamic efficiency of a single OWC chamber at the wall focus, but more efficient than an isolated OWC chamber positioned in open-sea conditions. Moreover, the hydrodynamic efficiency of the arrays exhibits better stability across the range of incident wave periods investigated, denoting that the individual component chambers in the array are efficacious at different incident wave conditions. The five chamber array is comprehensively analysed in irregular incident wave conditions. The array system is demonstrated to maintain a high power output and efficient behaviour in irregular incident wave conditions-an effect attributable to the reflecting wall influence. In summary, the five-chamber array configuration yields a higher power output and improved stability in terms of efficiency performance when compared with the three-chamber array configuration in regular waves. The array maintains this exceptional performance in irregular incident wave conditions.

Statistical Killing vector fields on the Hopf hypersurfaces in the complex space forms

Statistical Killing vector field is introduced and Hopf hypersurfaces of complex space forms with the condition of being structural statistical Killing vector field are studied. It is shown that these hypersurfaces have at most three distinct constant principal curvatures.

Field Line Curvature (FLC) Scattering in the Dayside Off‐Equatorial Minima Regions

AbstractMagnetic field line curvature (FLC) scattering is an effective mechanism for collisionless particle scattering. In the terrestrial magnetosphere, the FLC scattering plays an essential role in shaping the outer boundary of protons radiation belt, the rapid decay of ring current, and the formation of proton isotropic boundary (IB). However, previous studies have yet to adequately investigate the influence of FLC scattering on charged particles in the Earth's dayside magnetosphere, particularly in the off‐equatorial magnetic minima regions. This study employs T89 magnetic field model to investigate the impacts of FLC scattering on ring current protons in the dayside magnetosphere, with a specific focus on the off‐equatorial minimum regions. We analyze the spatial distributions of single and dual magnetic minima regions, adiabatic parameter, and pitch angle diffusion coefficients due to FLC scattering as functions of Kp. The results show that the effects of FLC scattering are significant not only on the dusk and dawn sides but also in the off‐equatorial minima regions on the noon. Additionally, we investigate the role of dipole tilt angle in the hemispheric asymmetry of FLC scattering effects. The dipole tilt angle controls the overall displacement of the dayside magnetosphere, resulting in different FLC scattering effects in the two hemispheres. Our study holds significance for understanding the FLC scattering effects in the off‐equatorial region of Earth's dayside magnetosphere and for constructing a more accurate dynamic model of particles.

Design and analysis of bending-active configurations through an initial flat-cut pattern based on the curvature lines

Bending active structures involve the deformation of originally flat structures during the erection process. These structures offer a promising approach to enhance design sustainability. However, despite advancements in structural computational methods, their implementation is still limited. This paper examines the constraints associated with this design approach from a designer’s perspective and proposes potential solutions. From this analysis, a design method is developed that utilizes additive manufacturing to explore different solutions during the concept stage. Prior to the form-finding process, a flat, variable-density pattern is created within a parametric environment. This pattern, referred to as the “Soft Pattern based on Control Points” (SPSCP), incorporates multiple design variables to accommodate diverse design requirements. In this preliminary study, we compare the impact of varying the design parameters on a small-scale model, as observed by the naked eye, and through finite element analysis.

Geometric modeling of phase ordering for the isotropic–smectic A phase transition

BackgroundLiquid crystal (LC) mesophases have an orientational and positional order that can be found in both synthetic and biological materials. These orders are maintained until some parameter, mainly the temperature or concentration, is changed, inducing a phase transition. Among these transitions, a special sequence of mesophases has been observed, in which priority is given to the direct smectic liquid crystal transition. The description of these transitions is carried out using the Landau–de Gennes (LdG) model, which correlates the free energy of the system with the orientational and positional order.MethodologyThis work explored the direct isotropic-to-smectic A transition studying the free energy landscape constructed with the LdG model and its relation to three curve families: (I) level-set curves, steepest descent, and critical points; (II) lines of curvature (LOC) and geodesics, which are directly connected to the principal curvatures; and (III) the Casorati curvature and shape coefficient that describe the local surface geometries resemblance (sphere, cylinder, and saddle).ResultsThe experimental data on 12-cyanobiphenyl were used to study the three curve families. The presence of unstable nematic and metastable plastic crystal information was found to add information to the already developed smectic A phase diagram. The lines of curvature and geodesics were calculated and laid out on the energy landscape, which highlighted the energetic pathways connecting critical points. The Casorati curvature and shape coefficient were computed, and in addition to the previous family, they framed a geometric region that describes the phase transition zone.Conclusion and significanceA direct link between the energy landscape’s topological geometry, phase transitions, and relevant critical points was established. The shape coefficient delineates a stability zone in which the phase transition develops. The methodology significantly reduces the impact of unknown parametric data. Symmetry breaking with two order parameters (OPs) may lead to novel phase transformation kinetics and droplets with partially ordered surface structures.

RANSAC-Based Planar Point Cloud Segmentation Enhanced by Normal Vector and Maximum Principal Curvature Clustering

Abstract. Planar feature segmentation is an essential task for 3D point cloud processing, finding many applications in various fields such as robotics and computer vision. The Random Sample Consensus (RANSAC) is one of the most common algorithms for the segmentation, but its performance, as given by the original form, is usually limited due to the use of a single threshold and interruption of similar planar features presented close to each other. To address these issues, we present a novel point cloud processing workflow which aims at developing an initial segmentation stage before the basic RANSAC is performed. Initially, normal vectors and maximum principal curvatures for each point of a given point cloud are analyzed and integrated. Subsequently, a subset of normal vectors and curvature is utilized to cluster planes with similar geometry based on the region growing algorithm, serving as a coarse but fast segmentation process. The segmentation is therefore refined with the RANSAC algorithm which can be now performed with higher accuracy and speed due to the reduced interference. After the RANSAC process is applied, resultant planar point clouds are built from the sparse ones via a point aggregation process based on geometric constraints. Four datasets (three real and one simulated) were used to verify the method. Compared to the classic segmentation method, our method achieves higher accuracy, with an RMSE from fitting equal to 0.0521 m, along with a higher recall of 93.31% and a higher F1-score of 95.38%.

Structural Analysis of Glazed Tubular Tiles of Oriental Architectures Based on 3D Point Clouds for Cultural Heritage

Abstract. Laser scanning, along with its resultant 3D point clouds, constitutes a prevalent method for the documentation of cultural heritage. This paper introduces a novel workflow for the structural analysis of glazed tubular tiles that adorn the roofs of historical buildings in the Orient, utilizing 3D point clouds. The workflow integrates a robust segmentation algorithm utilizing the maximum principal curvature and normal vectors. Moreover, clustering algorithms, including DBSCAN, are incorporated to refine the clusters and thus increase segmentation accuracy. Structural analysis is enabled by cylindrical model fitting, which allows for the estimation of parameters and residuals. While the results exhibit commendable performance in individual tile segmentation, it is imperative to address the impact of substantial variations in scanning range and incident angles before engaging in the structural analysis fitting process. The results of experiment demonstrate that under conditions of significantly large scanning angles, the root mean square error (RMSE) for inadequately fitted tiles can extend to 0.066 m, surpassing twice the RMSE observed for well-fitted tiles. The proposed workflow proves to be applicable and exhibits significant potential to advance practices in cultural heritage documentation.

A Molecular Expression for "Line Tension".

"Line tension", a concept that features in an additional term to the Young's equation, was introduced to describe the size dependence of contact angles of nanodroplets on surfaces. Although this concept describes the observations in a succinct, elegant manner, theorists have long had misgivings about the physical interpretation of the phenomenon. Papers have been published that attempt to nail down its value, which is reportedly very small (∼10 pN) and evidently even the sign has been uncertain. Attempts to interpret it in a mechanical manner analogous to interfacial tension, i.e., due to the curvature of the three-phase contact line, have run into conceptual problems that require invocations of ever more complex models. In this work, we have used molecular simulations to systematically relate "line tension" to the additional free energy per unit length of the three-phase line and found no direct relation. However, when we rederived the Young's equation without ignoring the interfacial molecules, we found a physically satisfying explanation for the size dependence of the contact angle of nanodroplets without invoking the curvature of the three-phase contact line. The new model does not have the elegant form of the modified Young's equation, but each parameter in it has an unambiguous physical interpretation. An approximate form of this model, linearized in the inverse droplet radius, yields a quantity that is mathematically analogous to what is conventionally called "line tension", but unpacked at the molecular level, showing that it is unrelated to a restoring force associated with the curvature of the macroscopic three-phase contact line.

Hypersurfaces in spaces of constant curvature satisfying a particular Roter type equation

Let M be a hypersurface isometrically immersed in an (n+1)-dimensional semi-Riemannian space of constant curvature, n>3, such that its shape operator A satisfies A3=ϕA2+ψA+ρId, where ϕ, ψ and ρ are some functions on M and Id is the identity operator. The main result of this paper states that on the set U of all points of M at which the square S2 of the Ricci operator S of M is not a linear combination of S and Id, the Riemann-Christoffel curvature tensor R of M is a linear combination of some Kulkarni-Nomizu products formed by the metric tensor g, the Ricci tensor S and the tensor S2 of M, i.e., the tensor R satisfies on U some Roter type equation. Moreover, the (0,4)-tensor R⋅S is on U a linear combination of some Tachibana tensors formed by the tensors g, S and S2. In particular, if M is a hypersurface isometrically immersed in the (n+1)-dimensional Riemannian space of constant curvature, n>3, with three distinct principal curvatures and the Ricci operator S with three distinct eigenvalues then the Riemann-Christoffel curvature tensor R of M also satisfies a Roter type equation of this kind.

The architectural application of shells whose boundaries subtend a constant solid angle

Surface geometry plays a central role in the design of bridges, vaults and shells, using various techniques for generating a geometry which aims to balance structural, spatial, aesthetic and construction requirements.In this paper we propose the use of surfaces defined such that given closed curves subtend a constant solid angle at all points on the surface and form its boundary. Constant solid angle surfaces enable one to control the boundary slope and hence achieve an approximately constant span-to-height ratio as the span varies, making them structurally viable for shell structures. In addition, when the entire surface boundary is in the same plane, the slope of the surface around the boundary is constant and thus follows a principal curvature direction. Such surfaces are suitable for surface grids where planar quadrilaterals meet the surface boundaries. They can also be used as the Airy stress function in the form finding of shells having forces concentrated at the corners.Our technique employs the Gauss-Bonnet theorem to calculate the solid angle of a point in space and Newton's method to move the point onto the constant solid angle surface. We use the Biot-Savart law to find the gradient of the solid angle. The technique can be applied in parallel to each surface point without an initial mesh, opening up for future studies and other applications when boundary curves are known but the initial topology is unknown.We show the geometrical properties, possibilities and limitations of surfaces of constant solid angle using examples in three dimensions.

Utjecaj ekscentrične porivne sile na odabir statvenog kliznog ležaja

Stern tube slide bearing of the shaft line of the ship's propulsion system causes losses during operation. Friction and material wear of the tribosystem occur. One of the reasons is the different navigating regimes of the ship, which have an influence on the formation of the velocity field on the ship's propeller. The variable velocity field defines the magnitude and the position of the thrust force. The methodology presented in this paper is based on the fact of the existence of the eccentricity of the thrust force, i.e. loads due to the action of the moment of the eccentric thrust force. Due to the changing position of the thrust force, the curvature of the elastic line of the sleeve inside the stern tube bearing, which is mostly spatial, changes. Spatial elastic line of the sleeve has an influence on the working area of the slide bearing. Therefore, the importance of a sufficient thickness of the lubricant layer for lubrication in relation to these changes was highlighted. In this paper, this is the initial criterion for the selection of materials and lubricants of the stern tube bearing. An equally important criterion would be to define the possible harmful impact of lubricant leakage with particles of bearing bush material on the ecosystem. In order to facilitate the whole process, the simplified analytical model is presented, which can be used in the initial stage of stern tube bearing selection. Such a model allows a simpler analysis of the influence of the elastic line on the condition of the stern tube bearing due to the loading by the weight of the ship's propeller and by the moment of the eccentric thrust force. The procedure for the initial selection of the stern tube bearing created by this methodological approach could have its practical application. For this reason, the paper analyzed different navigating regimes and their influence on the working area of the stern tube slide bearing. The obtained results can be useful in the selection of materials and lubricants for the stern tube bearing tribosystem.

Meshing principle and numerical study of classic conic worm drive

To explore the meshing mechanism of classic conic worm drive and provide a theoretical basis for creating other types of worm drive, a complete meshing theory on the proposed worm pair and its meshing performance are investigated. A new mathematical model containing a full set of geometric parameters on this drive is established using differential geometry. The formulae for calculating the reference point coordinates in different coordinate systems are attained, and then the blank size of the conical worm wheel is given. To build tooth surface boundaries, the meshing theory based on reference point is applied, which ensures the location of this point on the tooth surface. By solving multivariate nonlinear equation system in MATLAB environment, these conjugate regions and instantaneous contact lines are plotted within the axial section of the conic worm pair, at the same time, these values of sliding angle and induced principal curvature at meshing points are calculated. Finally, a numerical study is performed and the results verify that the conic worm pair has a relatively high tooth surface utilization ratio and favorable conditions for contact stress and lubrication, which is consistent with the practical application.

Biconservative Riemannian Submanifolds in Minkowski 5-Space

In this study, we focus biconservative Riemannian submanifolds with parallel normalized mean curvature vector field (PNMCV) in $\mathbb{E}^5_1$. We obtain explicit classifications for these submanifolds with exactly two distinct principal curvatures of the shape operator along the mean curvature vector field. In particular, we investigate these submanifolds which have time-like and space-like mean curvature vector field in $\mathbb{E}^5_1$.

On discrete constant principal curvature surfaces

It has been recently discovered that a certain class of nanocarbon materials has geometrical properties related to the geometry of discrete surfaces with a pre-constant discrete curvature, based on a discrete surface theory for trivalent graphs proposed in 2017 by Kotani et al. In this paper, with the aim of an application to the nanocarbon materials, we will study discrete constant principal curvature (CPC) surfaces. Firstly, we develop the discrete surface theory on a full 3-ary oriented tree so that we define a discrete analogue of principal directions on them and investigate it. We also construct some interesting examples of discrete constant principal curvature surfaces, including discrete CPC tori.

Experimental study and numerical simulation of morphing characteristics of bistable laminates embedded with 3D printed shape memory polymers

Bistable laminates have potential applications in areas such as buffer and soft robotics due to their two stable states and ability to deform through both snap-through and snap-back processes. However, conventional orthotropic bistable laminates with two stable state configurations and snap loads are similar in magnitude. There is a problem that the deformation cannot be easily driven when the laminate stiffness is high. To solve this problem, a clever design is needed to reduce the snap-back load and increase the snap-through load so that the driven deformation can be realized more easily. In this paper, we propose a design method to suppress and control the deformation of bistable laminates by utilizing the shape memory of 3D-printed shape memory polymers (SMP), which enables the laminates to increase the stiffness while reducing the driving force. First, a numerical model of viscoelasticity of the laminate is established to study its deformation characteristics, and the numerical results are compared with experimental results with good agreement. The effect of SMPs on the snap load and principal curvature of bistable laminates is also investigated. Finally, the interlayer interface bonding of the bistable laminates is examined in microscopic perspective. The results demonstrate that 3D printed SMPs can effectively enhance the snap-through load and reduce the snap-back load of bistable laminates, achieving deformation suppression and control while maintaining good interlaminar bonding with carbon fiber composites. This study provides new insights and practical significance for the deformation suppression and active control of bistable structures.

Simplified Formulation for Bush, Gibson, and Thomas Model

Abstract The random process model, also known as the multi-asperity contact model or statistical model, is one of the dominant methodologies for analyzing the contact between nominally flat rough surfaces. In 1975, Bush, Gibson, and Thomas developed a complete random process model (known as the BGT model) assuming that solid–solid contacts occur on the summits of the asperities, which are semi-ellipsoids uniquely described by three random variables (the asperity height and two principal peak curvatures). The Hertzian elliptic contact theory states that the contact area and normal load are implicit functions of the random variables, which results in an original BGT model with a complex formulation. In this study, we used an adapted Hertzian elliptic contact theory to simplify the formulation of the BGT model. The relative contact area to normal load relation predicted by the simplified BGT model perfectly agrees with that of the original formulation. It is anticipated that rough surface contact models with complex asperity interactions can be effectively built under the framework of this simplified BGT model.

On the large solutions to a class of k-Hessian problems

Abstract In this paper, we consider the k-Hessian problem S k (D 2 u) = b(x)f(u) in Ω, u = +∞ on ∂Ω, where Ω is a C ∞-smooth bounded strictly (k − 1)-convex domain in R N ${\mathbb{R}}^{N}$ with N ≥ 2, b ∈ C∞(Ω) is positive in Ω and may be singular or vanish on ∂Ω, f ∈ C[0, ∞) ∩ C 1(0, ∞) (or f ∈ C 1 ( R ) $f\in {C}^{1}\left(\mathbb{R}\right)$ ) is a positive and increasing function. We establish the first expansions (equalities) of k-convex solutions to the above problem when f is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of f and principal curvatures of ∂Ω on the first expansion of solutions. For the latter, we find the principal curvatures of ∂Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including k = N). Moreover, we know the existence of k-convex solutions to the above problem (including k = N) is still an open problem when b possesses high singularity on ∂Ω and f satisfies Keller–Osserman type condition. For the radially symmetric case in the ball, we give a positive answer to this open problem, and then we further show the global estimates for all radial large solutions.

On the principal curvatures of a class of Finsler metrics

On the principal curvatures of a class of Finsler metrics

Modeling Field Line Curvature Scattering Loss of 1–10 MeV Protons During Geomagnetic Storms

AbstractThe proton radiation belt contains high fluxes of adiabatically trapped protons varying in energy from ∼one to hundreds of megaelectron volts (MeV). At large radial distances, magnetospheric field lines become stretched on the nightside of Earth and exhibit a small radius of curvature RC near the equator. This leads protons to undergo field line curvature (FLC) scattering, whereby changes to the first adiabatic invariant accumulate as field strength becomes nonuniform across a gyroorbit. The outer boundary of the proton belt at a given energy corresponds to the range of magnetic L shell over which this transition to nonadiabatic motion takes place, and is sensitive to the occurrence of geomagnetic storms. In this work, we first find expressions for nightside equatorial RC and field strength Be as functions of Dst and L* to fit the TS04 field model. We then apply the Tu et al. (2014, https://doi.org/10.1002/2014ja019864) condition for nonadiabatic onset to solve the outer boundary L*, and refine our expression for RC to achieve agreement with Van Allen Probes observations of 1–50 MeV proton flux over the 2014–2018 era. Finally, we implement this nonadiabatic onset condition into the British Antarctic Survey proton belt model (BAS‐PRO) to solve the temporal evolution of proton fluxes at L ≤ 4. Compared with observations, BAS‐PRO reproduces storm losses due to FLC scattering, but there is a discrepancy in mid‐2017 that suggests a ∼5 MeV proton source not accounted for. Our work sheds light on outer zone proton belt variability at 1–10 MeV and demonstrates a useful tool for real‐time forecasting.