Two-dimensional Curvature Research Articles

In-Fiber Hybrid Structure Sensor Based on the Vernier Effect for Vector Curvature and Temperature Measurement

A vector curvature and temperature sensor based on an in-fiber hybrid microstructure is proposed and experimentally demonstrated. The proposed scheme enables the dimensions of the Fabry–Perot and Mach–Zehnder hybrid interferometer to be adjusted for the formation of the Vernier effect by simply changing the length of a single optical fiber. The sensor is fabricated using a fiber Bragg grating (FBG), multimode fiber (MMF), and a single-hole dual-core fiber (SHDCF). The sensor exhibits different curvature sensitivities in four vertical directions, enabling two-dimensional curvature sensing. The temperature and curvature sensitivities of the sensor were enhanced to 100 pm/°C and −25.55 nm/m−1, respectively, and the temperature crosstalk was minimal at −3.9 × 10−3 m−1/°C. This hybrid microstructure sensor technology can be applied to high-sensitivity two-dimensional vector curvature and temperature detection for structural health monitoring of buildings, bridge engineering, and other related fields.

Long-Period Grating with Asymmetrical Modulation for Curvature Sensing

We propose and demonstrate a curvature sensor based on long-period fiber grating (LPFG) with asymmetric index modulation. The LPFG is fabricated in single-mode fiber with femtosecond laser micromachining. The grating structure is not introduced in the central fiber core, but is located off-axis with a distance of a few micrometers. Experimental results indicate that the offset distance has direct influence on the grating spectra. By utilizing such an asymmetric structure, two-dimensional vector curvature sensing can be realized. For an LPFG with an offset distance of 6 μm, the curvature sensitivity is around 29 nm/m−1 in the 0° and 180° direction and about 20 nm/m−1 in the 90° and 270° direction. The difference in curvature sensitivity in different bending directions makes the sensor capable of distinguishing the curvature orientation. The temperature response of the sensor is also experimentally investigated, and results indicate that the sensor has a very low temperature cross-sensitivity of 0.003 m−1/°C. The characteristics of high curvature sensitivity, two-dimensional bending direction identification, and compact structure make the device an ideal candidate to be applied in the field of power grid health monitoring and intelligent robotics.

Mechanism of Silylation of Vinyl Arenes by Hydrodisiloxanes Driven by Stoichiometric Amounts of Sodium Triethylborohydride-A Combined DFT and Experimental Study.

The reactions of vinyl arenes with hydrodisiloxanes in the presence of sodium triethylborohydride were studied using experimental and computational methods. The expected hydrosilylation products were not detected because triethylborohydrides did not exhibit the catalytic activity observed in previous studies; instead, the product of formal silylation with dimethylsilane was identified, and triethylborohydride was consumed in stoichiometric amounts. In this article, the mechanism of the reaction is described in detail, with due consideration given to the conformational freedom of important intermediates and the two-dimensional curvature of the potential energy hypersurface cross sections. A simple way to reestablish the catalytic character of the transformation was identified and explained with reference to its mechanism. The reaction presented here is an example of the application of a simple transition-metal-free catalyst in the synthesis of silylation products, with flammable gaseous reagents replaced by a more convenient silane surrogate.

Damage detection in slab structures based on two-dimensional curvature mode shape method and Faster R-CNN

This paper proposes a novel method based on the two-dimensional (2D) curvature mode shape method, Convolutional Neural Networks (CNN), and Faster Region-based Convolutional Neural Networks (faster R-CNN) for detecting damage in slab structures. The 2D curvature mode shapes could be measured directly or calculated from the measured mode shapes using the central difference method. The damage indicator is defined as the absolute differences between the 2D curvature mode shapes of the damaged and intact slabs. The contour plot is chosen to convert the damage indicators into images. Four hundred damage scenarios are created using a Finite Element (FE) model of the slab. Images created from those damage scenarios are then used to train CNN and faster R-CNN. Four damage types are considered in this research, namely a single small hole, a single big hole, two small holes, and two large holes. After training, CNN can predict the damage types and faster R-CNN can predict the bounding boxes around the damaged areas. A test sample set is created to test the performance of the proposed method. The effect of noise in the mode shape data is considered. Results show that the classification accuracy for damage type is high. The overlap ratios between the predicted bounding boxes and the real damaged areas are more than 40% for 80% of tested scenarios. Furthermore, the low influence of noise on the predicted results is investigated. The proposed method is robust and has great potential for application to real structures.

Relations Between Statistics of Three-Dimensional Flame Curvature and its Two-Dimensional Counterpart in Turbulent Premixed Flames

The relations between the actual flame curvature probability density function (PDF) evaluated in three-dimensions and its two-dimensional counterpart based on planar measurements have been analytically derived subject to the assumptions of isotropy and statistical independence of various angles and two-dimensional curvature. These relations have been assessed based on Direct Numerical Simulation (DNS) databases of turbulent premixed (a) statistically planar and (b) statistically axisymmetric Bunsen flames. It has been found that the analytically derived relation interlinking the PDFs of actual three-dimensional curvature and its two-dimensional counterpart holds reasonably well for a range of curvatures around the mean value defined by the inverse of the thermal flame thickness for different turbulence intensities across different combustion regimes. The flame surface is shown to exhibit predominantly two-dimensional cylindrical curvature but there is a significant probability of finding saddle type flame topologies and this probability increases with increasing turbulence intensity. The presence of saddle type flame topologies affects the ratios of second and third moments of two-dimensional and three-dimensional curvatures. It has been demonstrated that the ratios of second and third moments of two-dimensional and three-dimensional curvatures cannot be accurately predicted based on two-dimensional measurements. The ratio of the third moments of two-dimensional and three-dimensional curvatures remains positive and thus the qualitative nature of curvature skewness can still be obtained based on two-dimensional curvature measurements. As the curvature skewness is often taken to be a marker of the Darrius-Landau instability, the conclusion regarding the presence of this instability can potentially be taken from the two-dimensional curvature measurements.

Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-set Method

We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-C\'ardenas and Gibou, J. Comput. Phys. (May 2022), 10.1016/j.jcp.2022.111291] that relies on numerical schemes to enable machine-learning operations on demand. In particular, our routine features double predicting to harness curvature symmetry invariance in favor of precision and stability. The core of this solver is a multilayer perceptron trained on circular- and sinusoidal-interface samples. Its role is to quantify the error in numerical curvature approximations and emit corrected estimates for select grid vertices along the free boundary. These corrections arise in response to preprocessed context level-set, curvature, and gradient data. To promote neural capacity, we have adopted sample negative-curvature normalization, reorientation, and reflection-based augmentation. In the same manner, our system incorporates dimensionality reduction, well-balancedness, and regularization to minimize outlying effects. Our training approach is likewise scalable across mesh sizes. For this purpose, we have introduced dimensionless parametrization and probabilistic subsampling during data production. Together, all these elements have improved the accuracy and efficiency of curvature calculations around under-resolved regions. In most experiments, our strategy has outperformed the numerical baseline at twice the number of redistancing steps while requiring only a fraction of the cost.

A note on the two-dimensional Lagrangian mean curvature equation

In this note, we use Warren-Yuan's super isoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike the previous approach, the simplified approach in this proof does not require the Michael-Simon mean value and Sobolev inequalities on generalized submanifolds of $\mathbb{R}^n$.

Baseline-free structural damage identification for plate-like structures based on two-dimensional curvature propagating flexural waves

Waveforms of propagating flexural waves can reveal plentiful information about local anomalies caused by damage, and the local anomalies can be used for damage identification. However, the local anomalies can be masked by the interference of measurement noise and global trends of the waveforms, and they may be able to indicate only fractions of the size and extent of the damage. In this paper, an effective noise-robust damage identification method for plate-like structures is proposed based on the fact that damage can introduce imminent local anomalies to two-dimensional curvature propagating flexural waves (2D-CPFW). The 2D-CPFW can be alternated using two-dimensional continuous wavelet transform (2D-CWT) to lower adverse effects of measurement noise and errors. To suppress global trends and intensify the local anomalies in the 2D-CPFW, 2D-CWT with a higher-order Laplacian of Gaussian function and the Teager energy operator are applied. The fundamental mechanism of how 2D-CWT with different orders of Laplacians of Gaussian function can suppress global trends of 2D-CPFW is investigated for the first time. It is found that the rth-order Laplacian of Gaussian function can well suppress the global trend of a 2D-CPFW within a finite interval if the trend can be well approximated by a 2r−1th-order bi-variant polynomial. The modal assurance criterion and a statistical criterion are used to assist the selection of the proper order of Laplacian of Gaussian function. An accumulative damage index is proposed, and the locations and extent of the damage can be identified within neighborhoods with high damage index values. If an intact plate-like structure is geometrically smooth and made of materials that have no mass and stiffness discontinuities, the proposed method does not require any waveforms of the intact structure serving as a baseline, and therefore, the method can be considered baseline-free. Effectiveness and noise-robustness of the proposed method are investigated in two numerical examples. Two experimental investigations were conducted on two aluminum plates under different damage scenarios. Both the numerical and experimental examples verify that the proposed method is effective and noise-robust in identifying the location and extent of the damage.

Damage Detection in Steel-Concrete Composite Structures by Impact Hammer Modal Testing and Experimental Validation.

Steel–concrete composite systems are an efficient alternative to mid- and high-rise building structures because of their high strength-to-weight ratio when compared to traditional concrete or steel constructive systems. Nevertheless, composite structural systems are susceptible to damage due to, for example, deficient construction processes, errors in design and detailing, steel corrosion, and the drying shrinkage of concrete. As a consequence, the overall strength of the structure may be significantly decreased. In view of the relevance of this subject, the present paper addresses the damage detection problem in a steel–concrete composite structure with an impact-hammer-based modal testing procedure. The mathematical formulation adopted in this work allows for the identification of regions where stiffness varies with respect to an initial virgin state without the need for theoretical models of the undamaged structure (such as finite element models). Since mode shape curvatures change due to the loss of stiffness at the presence of cracks, a change in curvature was adopted as a criterion to quantify stiffness reduction. A stiffness variability index based on two-dimensional mode shape curvatures is generated for several points on the structure, resulting in a damage distribution pattern. Our numerical predictions were compared with experimentally measured data in a full-scale steel–concrete composite beam subjected to bending and were successfully validated. The present damage detection strategy provides further insight into the failure mechanisms of steel–concrete composite structures, and promotes the future development of safer and more reliable infrastructures.

Thermodynamic geometry of spin-one lattice models. I. Spin and quadrupolar orders and critical scaling functions in one dimension.

State space Riemannian geometry is obtained for the one-dimensional Blume-Emery-Griffiths model and its Blume-Capel and Griffiths model limits, and its (pseudo)critical as well as noncritical parameter regimes are extensively investigated. Two codimension one geometries are obtained by taking suitable hypersurfaces in the three-dimensional state space manifold, and the induced thermal metrics are accordingly interpreted in terms of constrained fluctuations. The three-dimensional scalar curvature and the two two-dimensional curvatures are shown to be consistent with Ruppeiner's conjecture relating the inverse of the singular free energy to the thermodynamic scalar curvature. Moreover, they are found to be in an excellent agreement over a greater part of the noncritical region with the corresponding correlation lengths for the spin and the quadrupolar order parameters. The scaling function for the free energy near the pseudocritical and tricritical points is obtained thermodynamically by using Ruppeiner's conjecture. A connection is made between the sign change in the curvatures and the change in fluctuation patterns of the order parameters. In the accompanying paper we shall analyze the geometry of the spin-one model in its mean field approximation.

Vector curvature sensor based on asymmetrically polished long-period fiber grating

In this paper, we experimentally propose and theoretically investigate a two-dimensional vector curvature sensor based on asymmetrically polished long period fiber grating (AP-LPFG). This sensor is fabricated by a high-frequency CO2 laser in order to polish periodic asymmetrical V-shaped grooves on a single mode fiber (SMF). Such grooves efficiently enhances the bending sensitivity of sensor and shorten the length of the structure to 7.39 mm. Experimental results show that the sensitivity of AP-LPFG reaches 30.85 nm/m−1 in the direction of the X-axis and −0.97 nm/m−1 in the direction of the Y-axis. Meanwhile, the temperature response of AP-LPFG is carried out, which is 76 pm/℃ in range of 30-160℃. Moreover, the AP-LPFG has the features of compact length and simple structure, which shows the potential in curvature sensing.

Two-dimensional vector bending sensor based on Fabry-Pérot cavities in a multicore fiber.

In this work we demonstrate the fabrication and characterization of a temperature insensitive, two-dimensional curvature sensor using a resin based Fabry-Pérot interferometer, constructed using a multicore fiber (MCF). The fabrication simplicity makes this fiber device very attractive compared to the already reported technologies. Furthermore, the sensitivity reached (>400 pm/m-1), 7 times higher than the one reported for fiber Bragg gratings written on a similar MCF. The reconstruction of the amplitude and curvature has been performed for, showing errors lower than 4%. A numerical study has also been developed, allowing us to understand the sensor response at different fiber sensor geometries.

Anomalous planar Hall effect in two-dimensional trigonal crystals

The planar Hall effect (PHE) is the appearance of an in-plane transverse voltage in the presence of coplanar electric and magnetic fields. Its hallmark is a characteristic $\pi$-periodic, i.e. even under a magnetic field reversal, angular dependence with the transverse voltage that exactly vanishes when the electric and magnetic fields are aligned. Here we demonstrate that in two-dimensional trigonal crystals Zeeman-induced non-trivial Berry curvature effects yield a previously unknown anomalous PHE that is odd in the magnetic field and independent of the relative angle with the driving electric field. We further show that when an additional mirror symmetry forces the transverse voltage to vanish in the linear response regime, the anomalous PHE can occur as a second-order response at both zero and twice the frequency of the applied electric field. We demonstrate that this non-linear PHE possesses an antisymmetric quantum contribution that originates from a Zeeman-induced Berry curvature dipole.

Extension of Two-Dimensional Mean Curvature Flow with Free Boundary

Given a mean curvature flow of compact, embedded $C^2$ surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean curvature and perimeter stay uniformly bounded along the flow.

Effects of surface diffusion length on steady-state persistence probabilities

The dependence of the steady-state persistence probabilities (Ps) and exponents (θs) on surface diffusion length (l) for four discrete growth models is investigated. The persistence exponents which describe the decay of the persistence probabilities, the probabilities of the average of all initial height (h0), are increased as l is increased for all models. The results of one-dimensional Family ((1+1)-Family) and Das Sarma-Tamborenea ((1+1)-DT) models with kinetically rough film surface show the decrease of the growth exponent (β) with l. The l>1 results preserve the relation . In contrast, β is observed to increase with l for the two-dimensional larger curvature ((2+1)-LC) and Wolf-Villain ((2+1)-WV) models with mounded morphology. Our results show that the relation is not valid in l>1 cases in models with mounded surfaces. The persistence probabilities of a specific value of initial height (Ps(h0)) for l>1 are found to behave differently between mounded and kinetically rough models.

Novel automated vessel pattern characterization of larynx contact endoscopic video images

Purpose Contact endoscopy (CE) is a minimally invasive procedure providing real-time information about the cellular and vascular structure of the superficial layer of laryngeal mucosa. This method can be combined with optical enhancement methods such as narrow band imaging (NBI). However, these techniques have some problems like subjective interpretation of vascular patterns and difficulty in differentiation between benign and malignant lesions. We propose a novel automated approach for vessel pattern characterization of larynx CE + NBI images in order to solve these problems.Methods In this approach, five indicators were computed to characterize the level of vessel’s disorder based on evaluation of consistency of gradient and two-dimensional curvature analysis and then 24 features were extracted from these indicators. The method evaluated the ability of the extracted features to classify CE + NBI images based on the vascular pattern and based on the laryngeal lesions. Four datasets were generated from 32 patients involving 1485 images. The classification scenarios were implemented using four supervised classifiers.Results For classification of CE + NBI images based on the vascular pattern, polykernel support vector machine (SVM), SVM with radial basis function (RBF), k-nearest neighbor (kNN), and random forest (RF) show an accuracy of 97%, 96%, 96%, and 96%, respectively. For the classification based on the histopathology, Polykernel SVM showed an accuracy of 84%, 86% and 84%, RBF SVM showed an accuracy of 81%, 87% and 83%, kNN showed an accuracy of 89%, 87%, 91%, RF showed an accuracy of 90%, 88% and 91% for classification between benign histopathologies, between malignant histopathologies and between benign and malignant lesions, respectively.Conclusion These promising results show that the proposed method could solve the problem of subjectivity in interpretation of vascular patterns and also support the clinicians in the early detection of benign, pre-malignant and malignant lesions.

Off-Design Analysis of Transonic Bypass Fan Systems Using Streamline Curvature Through-Flow Method

Abstract The two-dimensional streamline curvature through-flow modeling of turbomachinery is still a key element for turbomachinery preliminary analysis. Basically, axisymmetric swirling flow field is solved numerically. The effects of blades are imposed as sources of swirl, work input/output and entropy generation. Although the topic is studied vastly in the literature for compressors and turbines, combined modeling of the transonic fan and the downstream splitter of turbofan engine configuration, to the authors’ best knowledge, is limited. In a prior study, the authors presented a new method for bypass fan modeling for inverse design calculations. Moreover, new set of practical empirical correlations are calibrated and validated. This paper is an extension of this study to rapid off-design analysis of transonic by-pass fan systems. The methodology is validated by two test cases: NASA 2-stage fan and GE-NASA bypass fan case. The proposed methodology is a simple extension for streamline curvature method and can be applied to existing compressor methodologies with minimum numerical effort.

Asymptotic and scattering behaviour for degenerate multi-solitons in the Hirota equation

We construct all higher order conserved charges from a general two-dimensional zero curvature condition using a Gardner transformation. Employing two of those charges in the definition of a Hamiltonian allows to view the Hirota equations as an integrable $\mathcal{PT}$-symmetric extension of the nonlinear Schr\"{o}dinger equation. We construct new degenerate multi-soliton solutions from Hirota's direct method as well as Darboux-Crum transformations based on Jordan states. We study the properties of these solutions, computing their asymptotic time-dependent displacements and also show that their scattering process has a distinct characteristic behaviour different from the nondegenerate counterparts allowing only for interactions of absorb-emit type.

Trace formula for the magnetic Laplacian

The Guillemin–Uribe trace formula is a semiclassical version of the Selberg trace formula and the more general Duistermaat–Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian) associated with the magnetic field. This paper gives a survey of basic notions and results related to the Guillemin–Uribe trace formula and provides concrete examples of its computation for two-dimensional constant curvature surfaces with constant magnetic fields and for the Katok example. Bibliography: 53 titles.

Topological dyonic dilaton black holes in AdS spaces

We construct a new class of dyonic dilaton black hole solutions in the background of Anti-de Sitter (AdS) spacetime. In order to find an analytical solution which satisfy all the field equations, we should consider the string case where the dilaton coupling is $\alpha=1$. The asymptotic behaviour of the solution ($r\rightarrow\infty$) is exactly AdS, where the dilaton field becomes zero and the metric function reduces to $f(r)\rightarrow -\Lambda r^2/3$. In this spacetime, black hole horizons and cosmological horizons, can be a two-dimensional positive, zero or negative constant curvature surface. We study the physical properties of the solution and show that depending on the metric parameters, these solutions can describe black holes with one or two horizons or a naked singularity. We investigate thermodynamics of the solutions by calculating the charge, mass, temperature, entropy and the electric potential of these solutions and disclose that these conserved and thermodynamic quantities satisfy the first law of black hole thermodynamics. We also analyze thermal stability of the solutions and find the conditions for which we have a stable dyonic dilaton black hole.