Smooth Curvature Research Articles

A TMPS-designed personalized mandibular scaffolds with optimized SLA parameters and mechanical properties

With the rapid development of 3D printing technology, porous titanium scaffolds have provided a new restoration method to repair bone defects. Compared with the traditional body-centered cubic (bcc) dot matrix structure with a simple arrangement and repetitive structure, the topology-driven properties of triply periodic minimal surfaces (TPMS) can offer a continuous surface and smooth curvature, an excellent platform for cell proliferation. In this study, we used reverse engineering techniques to model the mandible. Sheet and solid networks of gyroid structure, the most common type of TPMS, were selected for porous design and then molded using metal 3D printing technology. At the same time, the surface treatment parameters of sandblasted, large-grit, and acid-etched (SLA) were optimized by orthogonal experimental design. Then, the optimized SLA parameter was used to treat the gyroid with 70% porosity. The result showed that reverse engineering reconstructed the TPMS-based mandibular model had good formability. Furthermore, the best surface morphology, wettability, and roughness were obtained for 3D printed Ti6Al4V under the treatment of 80 mesh Al2O3, blasting distances of 4 cm, and a 1:1:2 acid ratio. Moreover, the mechanical properties of Sheet-Gyroid and Solid-Gyroid were significantly different at 70% porosity. The porosity of the scaffolds was close to the design porosity after SLA treatment. However, no significant changes were found in its mechanical properties, all matching the mandible’s mechanical properties to meet the implantation conditions.

Characterisation of valuations and curvature measures in euclidean spaces

Valuations constitute a class of functionals on convex bodies which include the Euler-characteristic, the surface area, the Lebesgue-measure, and many more classical functionals. Curvature measures may be regarded as “localised” versions of valuations which yield local information about the geometry of a body’s boundary. A complete classification of continuous translation-invariant SO(n)-invariant valuations and curvature measures with values in ℝ was obtained by Hadwiger and Schneider, respectively. More recently, characterisation results have been achieved for curvature measures with values in Symp ℝn and Sym2Λqℝn for p, q ≥ 1 with varying assumptions as for their invariance properties. In the present work, we classify all smooth translation-invariant SO(n)-equivariant curvature measures with values in any SO(n)-representation in terms of certain differential forms on the sphere bundle Sℝn and describe their behaviour under the globalisation map. The latter result also yields a similar classification of all continuous SO(n)-equivariant valuations with values in any SO(n)-representation. Furthermore, a decomposition of the space of smooth translation-invariant ℝ-valued curvature measures as an SO(n)-representation is obtained. As a corollary, we construct an explicit basis of continuous translation-invariant ℝ-valued valuations.

On the Bövik-Benveniste methodology and related approaches for modelling thin layers.

This paper reviews several leading approaches for asymptotic modelling of thin layers in elastostatics and wave propagation phenomena. The issues related to applications of the so-called 'equivalent' or 'effective' boundary conditions and their interpretations are highlighted. Comparative analysis of asymptotic models is performed for a two-dimensional elastostatic case using a novel complex variables-based modelling tool. Its implementation allows for straightforward derivations of higher order boundary conditions for problems with layers of arbitrary sufficiently smooth curvatures. Explicit expressions for the conditions up to the third order are provided. All models are tested using available benchmark solutions and the solutions for the limiting cases of the layer parameters. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

Optimization of Lateral Collision Risk of Aircraft Based on the Skid-Slip Event Model

With the development of aircraft flow under the existing airspace capacity now, the shortage of airspace resources and flight delays have become significantly severe. Therefore, building a safe and efficient mathematical model from quantitative analysis and improving the scientificity of route planning and management are essential propositions for future research on new navigation systems. Based on the traditional event collision risk assessment model, the collision module is upgraded to an ellipsoid according to the performance of aircraft, integrating the boundary curvature optimization characteristics of the TSRRT (task-space rapidly-exploring random trees) algorithm. An aircraft event lateral conflict resolution model based on the TSRRT algorithm is proposed. Taking the A320 aircraft and 737-800 aircraft as experimental subjects, the corresponding collision coefficients are imported into software tools for the simulation and the Kalman filtering is combined to verify the smoothness of the front and rear boundary curvatures. The result proves that the event side collision risk of aircraft based on TSRRT is 15% of the traditional event model and the smooth curvature error is reduced by 80% due to the improvement. Therefore, the improved event model is practical and valuable, providing a theoretical basis for future track-based operations.

A fractional Michael–Simon Sobolev inequality on convex hypersurfaces

The classical Michael-Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, but displays on the right-hand side an additional $L^p$ term weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo semi-norm of the function, as well as its $L^p$ norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.

On the construction of Riemannian three-spaces with smooth inverse mean curvature foliation

Consider a one-parameter family of smooth Riemannian metrics on a two-sphere, {mathscr {S}}. By choosing a one-parameter family of smooth lapse and shift, these Riemannian two-spheres can always be assembled into smooth Riemannian three-space, with metric h_{ij} on a three-manifold Sigma foliated by a one-parameter family of two-spheres {mathscr {S}}_rho . It is shown first that we can always choose the shift such that the {mathscr {S}}_rho surfaces form a smooth inverse mean curvature foliation of Sigma . An integrodifferential expression, referring only to the area of the level sets and the lapse function, is also derived that can be used to quantify the Geroch mass. If the constructed Riemannian three-space happens to be asymptotically flat and the rho -integral of the integrodifferential expression is non-negative, then not only the positive mass theorem but, if one of the {mathscr {S}}_{rho } level sets is a minimal surface, the Penrose inequality also holds. Notably, neither of the above results requires the scalar curvature of the constructed three-metric to be non-negative.

Associations between self-reported sleep duration and abnormal serum lipids in eastern China: a population-based cross-sectional survey

ObjectiveTo evaluate the associations between sleep duration and total cholesterol (TC), triglycerides (TG), high-density lipoprotein cholesterol (HDL-C), and low-density lipoprotein cholesterol (LDL-C). MethodsThe present study included 60,283 adults aged ≥18 years from the Chronic Disease and Risk Factor Surveillance in Nanjing. Generalized additive model (GAM) with forward stepwise selection method was used to analyze the nonlinear relationships between sleep duration and abnormal serum lipids. The reported effective degree of freedom (EDF) values in GAM indicates the degree of smooth curvature. EDF = 1 is a sign of linear correlation between predictors and outcome. EDF >1 is the sign of a more complex relationship between sleep duration and abnormal serum lipids. ResultsThe evaluation of interaction of sex and sleep duration by using multivariable GAM revealed a U-shaped correlation between sleep duration and dyslipidemia (EDF = 4.60, P < 0.001), high TC (EDF = 3.38, P < 0.001), and high LDL-C (EDF = 3.67, P < 0.001) in male, and a U-shaped correlation between sleep duration and dyslipidemia (EDF = 4.69, P < 0.001), high TC (EDF = 3.33, P < 0.001), and high LDL-C (EDF = 3.21, P < 0.001) in female. There was a U-shaped correlation between sleep duration and high TG in male (EDF = 3.84, P < 0.001) and semi-linear correlation in female (EDF = 1.82, P = 0.028). Moreover, there was a linear correlation between sleep duration and low HDL-C in men (EDF = 1.04, P = 0.002), but no significant correlation in women (EDF = 3.18, P = 0.080). ConclusionsBoth shorter and longer sleep durations were associated with abnormal serum lipid profiles in men and women.

Driving Lane Generation Method using Deflection Curve of a Beam in High-Precision Digital Road Map

In this study, we propose a method to automatically generate driving lanes in the section where no lane lines exist, such as intersections, with the intension of using it in the process of producing high-precision digital road maps for automated driving vehicles. The lanes are generated by setting the entrances and exits of intersections based on the location information of landmarks such as curbstones and white lines recorded in the high-precision digital road map, and connecting the entrances and exits using deflection curves of beams to generate travel lanes with smooth curvature changes. The travel lanes generated by the proposed method were compared with the routes in previous studies, and it was confirmed that the curvature change of the lanes was smooth.

Driving Lane Design Method using Deflection Curve of a Distributed Load Beam in High-Precision Digital Road Map

In this study,we propose a method to automatically design driving lanes in the section where no lane lines exist, such as intersections, with the intension of using it in the process of producing high-precision digital road maps for automated driving vehicles. The lanes are generated by setting the entrances and exits of intersections based on the location information of landmarks such as curbstones and white lines recorded in the high-precision digital road map, and connecting the entrances and exits using deflection curves of beams to which distributed loads are added to generate travel lanes with smooth curvature changes. The travel lanes generated by the proposed method were compared with the routes in previous studies, and it was confirmed that the curvature change of the continuous curvature lane at the start and end points is smooth.

A Liquid Hydrogel to Restore Long Term Corneal Integrity After Perforating and Non-Perforating Trauma in Feline Eyes.

Purpose: To evaluate long-term in vivo functionality of corneas regenerated using a cell-free, liquid hydrogel filler (LiQD Cornea) after deep corneal trauma in the feline model. Methods: Two healthy cats underwent 4 mm diameter stepwise 250/450 µm deep surgical corneal ablation with and without needle perforation. The filler comprising 10% (w/w) collagen-like peptide conjugated to polyethylene glycol (CLP-PEG) and 1% fibrinogen and crosslinked with 2% (w/w) 4-(4,6-dimethoxy-1,3,5-triazin-2-yl)-4-methylmorpholinium chloride (DMTMM), was applied to the wound bed previously coated with thrombin (250 U/ml). In situ gelation occurred within 5 min, and a temporary tarsorrhaphy was performed. Eyes were examined weekly for 1 month, then monthly over 12 months. Outcome parameters included slit-lamp, Scheimpflug tomography, optical coherence tomography, confocal and specular microscopy, and immunohistochemistry studies. Results: The gelled filler was seamlessly incorporated, supporting smooth corneal re-epithelialization. Progressive in-growth of keratocytes and nerves into the filler corresponding to the mild haze observed faded with time. The regenerated neo-cornea remained stably integrated throughout the 12 months, without swelling, inflammation, infection, neovascularization, or rejection. The surrounding host stroma and endothelium remained normal at all times. Tomography confirmed restoration of a smooth surface curvature. Conclusion: Biointegration of this hydrogel filler allowed stable restoration of corneal shape and transparency in the feline model, with less inflammation and no neovascularization compared to previous reports in the minipig and rabbit models. It offers a promising alternative to cyanoacrylate glue and corneal transplantation for ulcerated and traumatized corneas in human patients.

Multiple synergistic effect and mechanical enhancement of lotus petiole

The curved petiole of lotus makes it more flexible, absorbs much more energy, resists shear forces and torsion performance. Multiple synergies effectively ensure the integrity of the overall lotus, no matter how the lotus leaf deforms, the petiole is hard to break. In addition, the surface of the petiole is covered by millimetre cone (MC) arrays. The root of the MC has smooth curvature and forms a stable transition in the interface of cone and petiole, the load transfer from stalk to MC could inhibit the deformation of petiole and increase the fracture limit. Thus, the stability of lotus leaves and petiole is guaranteed by the synergy effect between the curve section and MC on the surface. In this study, a polymethyl methacrylate -based bionic fabricated curved bar with MC arrays is fabricated by 3D printing technology. Compared with the straight bar, this designed bar shows good performance in anti-bending, anti-twisting and anti-stretching. Further, this bar is mounted on a stool to achieve a stable mechanical function. In the application, this design not only improves the comfort of the experience, but also improves the safety factor. This research has potential applications in the biomimetic field, ski pole, and construction.

DNA Mutations via Chern\u2013Simons Currents

We test the validity of a possible schematization of DNA structure and dynamics based on the Chern–Simons theory, that is a topological field theory mostly considered in the context of effective gravity theories. By means of the expectation value of the Wilson Loop, derived from this analogue gravity approach, we find the point-like curvature of genomic strings in KRAS human gene and COVID-19 sequences, correlating this curvature with the genetic mutations. The point-like curvature profile, obtained by means of the Chern–Simons currents, can be used to infer the position of the given mutations within the genetic string. Generally, mutations take place in the highest Chern–Simons current gradient locations and subsequent mutated sequences appear to have a smoother curvature than the initial ones, in agreement with a free energy minimization argument.

Investigation on Deformation of the Flex-Gear and Optimization Design of Compound Curve Cam Wave Generator with Spiral

The deformation shape and mechanical property of flex-gear greatly affect the meshing accuracy and service life of harmonic drive, and the stress wave generator plays a decisive role in the deformation shape and mechanical properties of the flexible gear. The investigation on deformation of the flex-gear around the major axis is developed with comparative analysis with three kinds of traditional and commonly used wave generators. Based on the influence of the deformation around the major axis on meshing accuracy and service life, the eccentric arc cam wave generator improved by transition curve for controlling the deformation shape and reducing the deformation stress is developed. Considering the achievable flexibility of spiral with smooth curvature change rate, the contour of the compound curve cam wave generator with arc and spiral is proposed. The deformation of the flex-gear under the influence of this compound curve cam and the standard elliptic cam is further comparative analyzed. The conclusion shows that this compound curve cam can indeed reduce the deformation stress in the meshing area by increasing the meshing width, and correct deformation shape can also be gained by shape control on the whole circumference of the flex-gear’s neutral layer.

Molecular Insights on the Wetting Behavior of a Surface Corrugated with Nanoscale Domed Pillars.

Using all-atom molecular dynamics simulation, we investigated the wettability of a surface texturized with nanoscale pillars of domed, rectangular, or cylindrical shapes. The dewetted and wetted states of the gaps between the pillars were related to the Cassie-Baxter (CB) and Wenzel (WZ) states of a macroscopic water droplet resting on top of the pillars. We uncovered the structures and free energies of the intermediate states existing between the CB and WZ states. The contact line of the liquid-vapor-solid interface could not be depinned for the domed pillars due to their smooth curvatures unlike for the rectangular or cylindrical pillars. The liquid symmetrically penetrated down into the gap between the domed pillars by a liquid-vapor interface shape like a paraboloid, while the penetration for the rectangular or cylindrical pillars was often asymmetrical, giving a half-tubular liquid-vapor interface.

On Ilmanen’s multiplicity-one conjecture for mean curvature flow with type-$I$ mean curvature

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in ${\\mathbb R}^3$ is of type-$I$, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumption that the mean curvature is of type-$I$. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in ${\\mathbb R}^3$ is at least of type-$I$.

Discrete curvatures of triangle meshes: From approximation of smooth surfaces to digital terrain data

The emergence of high resolution LiDAR measurements raises the challenge of extracting accurate geometric information from 3D point cloud data. The standard approach is to generate digital elevation models in the form of regular grids and to compute numerical descriptors such as slope and curvatures. An alternative is the use of triangular representations, which preserve many details that might be lost when passing to regular grids. It is therefore a natural question whether triangulated terrains benefit of appropriate counterparts for the Gaussian and mean curvatures. A possible answer might be provided by an interdisciplinary perspective: triangle meshes are popular in computer graphics and, in this case, several approaches were taken into account for defining suitable concepts of curvature. The aim of this study is to compare methods for computing curvatures of triangle meshes, with an emphasis on the case of terrain data. We first consider several particular surfaces and a synthetic terrain generated by a smooth function and we assess whether the methods provide values that are correlated to the true smooth curvatures. Then we address two study cases relying on point clouds acquired through in situ measurements and we compare the methods with the regular grid approach. The outcomes of the numerical experiments are consistent to each other and lead to the same conclusions. Thus, for the Gaussian curvature the best approximation is provided by two methods derived from the Gauss–Bonnet theorem and by the jet fitting method. In the case of mean curvatures, one of the Gauss–Bonnet schemes, the method based on Euler’s theorem and the tensor approach yield the best matches with the reference methods. We also illustrate how curvatures for triangle meshes can highlight specific terrain features.

The role of gender and academic degree on preference for smooth curvature of abstract shapes.

BackgroundPreference for smooth contours occurs for a variety of visual stimuli. However, there are individual differences. Openness to experience, a trait associated with aesthetic appreciation, emotional sensitivity and abstract thinking, correlates with this preference. The evaluation of meaningless stimuli entails automatic associations influenced by knowledge, intellectual interests and individual experiences which are diverse. However, it is difficult to capture this variability in studies restricted to Undergraduate students in Psychology with a prevalence of female participants.MethodsHere we examined preference for curvature with 160 undergraduate students in Psychology, Mathematics, Engineering and Computer Science, balanced for gender. Participants viewed abstract shapes varying for contour (angular vs. curved). The shapes presented variations in Vertices (10, 20, 30) and Concavity (30%, 40%, 50%) to increase complexity. Participants rated how much they liked each shape on a 0 (dislike) to 100 (like) scale. Furthermore, because students in pure Science disciplines present autistic-like traits as measured with the Autism Quotient (AQ), and there is evidence that individuals with autism respond positively to edgy abstract shapes, participants also completed the AQ.ResultsOverall participants preferred curved shapes to angular shapes. We confirmed past research showing that complexity played a key role, with simple shapes with less vertices (10 vertices) being preferred over shapes with larger number of vertices (20 and 30 vertices). Furthermore, simple shapes (10 vertices) were preferred more with more concavities (50%). Importantly, an interaction between academic degree and gender revealed that preference for smooth curvature was stronger for Psychology female participants. Science students scored higher than Psychology students on the AQ. Interestingly, multilevel analyses showed that the variability of AQ traits in the sample did not contribute to this interaction. The results are discussed in relation to theories of preference formation and individual differences.

Discrete curvature and torsion from cross-ratios

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature.

Shape-programmable magneto-active elastomer composites for curve and biomimetic behavior imitation.

A programming methodology, which can be applied to soft-magnetic-material-based magneto-active elastomers (MAEs), to catch the predefined specific objective curves is proposed in this study. The objective curves have been equally separated into a couple of segments, which will be filled by the designed MAE elements. Furthermore, the designed MAE segments with different chain angles, in which the deformation orientation of each element under applied homogeneous magnetic fields has been investigated based on the designed experimental setup, are arrayed based on the proposed programming methodology to constitute the MAE composite to catch the orientation of the objective curve. The experimental results show that based on the proposed programming methodology, the MAE composites can describe different curves, which include harmonic, tangential and arc tangential functions under applied homogeneous magnetic fields with good agreement. Furthermore, on the basis of the proposed programming methodology, the MAE composites are utilized to mimic the typical biomimetic behavior (the peeking-up behavior of snakes and the flapping behavior of birds) with smooth curvature properties, in which the dynamic procedures present continuous curves.

Multiscalar Geomorphometric Generalization for Soil-Landscape Modeling by Random Forest: A Case Study in the Eastern Amazon

Multiscalar topography influence on soil distribution has a complex pattern that is related to overlay of pedological processes which occurred at different times, and these driving forces are correlated with many geomorphologic scales. In this sense, the present study tested the hypothesis whether multiscale geomorphometric generalized covariables can improve pedometric modeling. To achieve this goal, this case study applied the Random Forest algorithm to a multiscale geomorphometric database to predict soil surface attributes. The study area is in phanerozoic sedimentary basins, in the Alter do Chão geological formation, Eastern Amazon, Brazil. The multiscale geomorphometric generalization was applied at general and specific geomorphometric covariables, producing groups for each scale combination. The modeling was run using Random Forest for A-horizon thickness, pH, silt and sand content. For model evaluation, visual analysis of digital maps, metrics of forest structures and effect of variables on prediction were used. For evaluation of soil textural classifications, the confusion matrix with a Kappa index, and the user’s and producer’s accuracies were employed. The geomorphometry generalization tends to smooth curvatures and produces identifiable geomorphic representations at sub-watershed and watershed levels. The forest structures and effect of variables on prediction are in agreement with pedological knowledge. The multiscale geomorphometric generalized covariables improved accuracy metrics of soil surface texture classification, with the Kappa Index going from 43% to 62%. Therefore, it can be argued that topography influences soil distribution at combined coarser spatial scales and is able to predict soil particle size contents in the studied watershed. Future development of the multiscale geomorphometric generalization framework could include generalization methods concerning preservation of features, landform classification adaptable at multiple scales.