Corner Function Research Articles

Mindlin cracked plates modelling and implementation in train-track coupled dynamics

Nowadays, the widely constructed ballastless track in high-speed railways will inevitably suffer from localized damage cracks induced by complicate environmental effects, which would change its designed dynamic performance under train dynamic loads and further affect its service life and durability. To this end, a semi-analytical approach of the in-plane and the out of plane forced vibration for Mindlin plates with a side crack are derived, and is further implemented into the train-track coupled dynamic (TTCD) system which comprehensively considering nonlinear wheel-rail contact and 3D spatial flexibility of the cracked track slab. The dynamic partial differential equation of the cracked slab is derived by Hamilton principle and discretized by Galerkin method. Then the eigenvalues and forced vibrations of the cracked slabs for both in-plane and flexural vibrations are tackled utilizing a set of powerful trigonometric modified by special corner function, which can effectively explain the singularity of stress at the crack tip and the discontinuity of deformation on both sides of the crack. The eigenvalue analysis and mode shapes of the cracked plate mode in free boundary condition are compared with the finite element method to verify its accuracy. Meanwhile, the reliability of the proposed dynamics model is verified by comparing with the responses predicted by the co-simulation technology. Finally, the discrepancies of dynamic responses of the track slab with or without a side crack are analyzed, and the effects of various operation speeds and support stiffness on the dynamic behaviour of both slabs are revealed. Results indicates that the track slab with a side crack has significant distinctions on the slab dynamic signals in both time and frequency domains, and the amplitude amplification phenomenon occurs in some specific areas compared with the intact slab; The type-Ⅱ stress intensity factor (SIF) is more sensitive to train speeds, while the support stiffness is highly correlated to all types of SIFs. This work may contribute to the scientific maintenance and health monitoring of high-speed railway tracks.

O3o3: A Variant of Spectral Elements with a Regular Collocation Grid

AbstractIn this study, an alternative local Galerkin method (LGM), the o3o3 scheme, is proposed. o3o3 is a variant or generalization of the third-order spectral element method (SEM3). It uses third-order piecewise polynomials for the representation of a field and piecewise third-degree polynomials for fluxes. For the discretization, SEM3 uses the irregular Legendre–Gauss–Lobatto grid while o3o3 uses a regular collocation grid. o3o3 can be regarded as an inhomogeneous finite-difference scheme on a uniform grid, which means that the finite-difference equations are different for each group with three points. A particular version of o3o3 is set as an example of many possibilities to construct LGM schemes on piecewise polynomial spaces in which the basis functions used are continuous at corner points and function spaces having continuous derivatives are shortly discussed. We propose a standard o3o3 scheme and a spectral o3o3 scheme as alternatives to the standard method of using the quadrature approximation. These two particular schemes selected were chosen for ease of implementation rather than optimal performance. In one dimension, compared to standard SEM3, o3o3 has a larger CFL condition benefiting from the use of a regular collocation grid. While SEM3 uses the irregular Legendre–Gauss–Lobatto collocation grid, o3o3 uses a regular grid. This is considered an advantage for physical parameterizations. The shortest resolved wave is marginally smaller than that with SEM3. In two dimensions, o3o3 is implemented on a sparse grid where only a part of the points on the underlying regular grid are used for forecasting.

Holographic entanglement entropy in AdS4/BCFT3 and the Willmore functional

We study the holographic entanglement entropy of spatial regions having arbitrary shapes in the AdS4/BCFT3 correspondence with static gravitational backgrounds, focusing on the subleading term with respect to the area law term in its expansion as the UV cutoff vanishes. An analytic expression depending on the unit vector normal to the minimal area surface anchored to the entangling curve is obtained. When the bulk spacetime is a part of AdS4, this formula becomes the Willmore functional with a proper boundary term evaluated on the minimal surface viewed as a submanifold of a three dimensional flat Euclidean space with boundary. For some smooth domains, the analytic expressions of the finite term are reproduced, including the case of a disk disjoint from a boundary which is either flat or circular. When the spatial region contains corners adjacent to the boundary, the subleading term is a logarithmic divergence whose coefficient is determined by a corner function that is known analytically, and this result is also recovered. A numerical approach is employed to construct extremal surfaces anchored to entangling curves with arbitrary shapes. This analysis is used both to check some analytic results and to find numerically the finite term of the holographic entanglement entropy for some ellipses at finite distance from a flat boundary.

Corner contributions to holographic entanglement entropy in AdS4/BCFT3

We study the holographic entanglement entropy of spatial regions with corners in the AdS4/BCFT3 correspondence by considering three dimensional boundary conformal field theories whose boundary is a timelike plane. We compute analytically the corner function corresponding to an infinite wedge having one edge on the boundary. A relation between this corner function and the holographic one point function of the stress tensor is observed. An analytic expression for the corner function of an infinite wedge having only its tip on the boundary is also provided. This formula requires to find the global minimum among two extrema of the area functional. The corresponding critical configurations of corners are studied. The results have been checked against a numerical analysis performed by computing the area of the minimal surfaces anchored to some finite domains containing corners.

Entanglement negativity in a two dimensional harmonic lattice: area law and corner contributions

We study the logarithmic negativity and the moments of the partial transpose in the ground state of a two dimensional massless harmonic square lattice with nearest neighbour interactions for various configurations of adjacent domains. At leading order for large domains, the logarithmic negativity and the logarithm of the ratio between the generic moment of the partial transpose and the moment of the reduced density matrix at the same order satisfy an area law in terms of the length of the curve shared by the adjacent regions. We give numerical evidence that the coefficient of the area law term in these quantities is related to the coefficient of the area law term in the Rényi entropies. Whenever the curve shared by the adjacent domains contains vertices, a subleading logarithmic term occurs in these quantities and the numerical values of the corner function for some pairs of angles are obtained. In the special case of vertices corresponding to explementary angles, we provide numerical evidence that the corner function of the logarithmic negativity is given by the corner function of the Rényi entropy of order 1/2.

Spatiotemporal feature extraction for facial expression recognition

A key issue regarding feature extraction is the capability of a technique to extract distinctive features to represent facial expressions while requiring a low computational complexity. In this study, the authors propose a novel approach for appearance-based facial feature extraction to perform the task of facial expression recognition on video sequences. The proposed spatiotemporal texture map (STTM) is capable of capturing subtle spatial and temporal variations of facial expressions with low computational complexity. First, face is detected using Viola–Jones face detector and frames are cropped to remove unnecessary background. Facial features are then modelled with the proposed STTM, which uses the spatiotemporal information extracted from three-dimensional Harris corner function. A block-based method is adopted to extract the dynamic features and represent the features in the form of histograms. The features are then classified into classes of emotion by the support vector machine classifier. The experimental results demonstrate that the proposed approach shows superior performance compared with the state-of-the-art approaches with an average recognition rate of 95.37, 98.56, and 84.52% on datasets containing posed expressions, spontaneous micro-expressions, and close-to-real-world expressions, respectively. They also show that the proposed algorithm requires low computational cost.

Exact results for corner contributions to the entanglement entropy and Rényi entropies of free bosons and fermions in 3d

In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Rényi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient σ for the EE and σn for the Rényi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for σ and σn for all n=2,3,… . The results for σ agree with a recent universality conjecture of Bueno, Myers, and Witczak-Krempa that σ/CT=π2/24 in all 3d CFTs, where CT is the central charge. For the Rényi entropies, the ratios σn/CT do not indicate similar universality. However, in the limit n→∞, the asymptotic values satisfy a simple relationship and equal 1/(4π2) times the asymptotic values of the free energy of free scalars/fermions on the n-covered 3-sphere.

Fourier series method for plane elastic problems of polygonal domain

This study employs the Fourier series method based on the edge function approach to solve the plane elastic problem of polygonal domain described by Navier equations. The analytical solutions serve as a set of fundamental solutions for each edge. Superposing the solution function and matching the prescribed boundary conditions in each edge allows the solving of the unknown variables and the analysis of the problem. An extra corner function and regularization technique is utilized to enhance convergence rate. Only one element is required to analyze which polygon domain is convex, however, by dividing a non-convex shape into several convex shapes, the proposed method can be extended to irregular geometrical shape. Numerical examples demonstrate the merits of the developed scheme, as well as its efficiency and accuracy.

Some analytical solutions to the general approximation problem for feedforward neural networks

The general approximation problem of interest to the area of feedforward neural networks is stated. Solutions for some special cases are given, which include an upper bound on the number of nodes in hidden layer(s) and the weights for that configuration. Analytical solutions to the general feedforward neural network problem in one-dimensional cases requiring an infinite number of nodes are presented. The practical solutions (not requiring an infinite number of nodes) in one-dimensional cases are derived under piecewise constant approximations with constant width partitions, under piecewise constant approximations with variable width partitions, and under piecewise linear approximations using ramps instead of sigmoids. A four layer solution to the general feedforward neural network problem in the n-dimensional case is presented. A three layer solution to the general feedforward neural network problem in the n-dimensional case with piecewise constant approximation requires the use of the corner function as the activation function. The corner function, a special case of n dimensional sigmoid function, is found to have desirable characteristics, and can be used to approximate functions with much weaker requirements (only boundedness and piecewise continuity. ) Concave regions can be formed with a single layer of nodes with the corner function.

Microstrip transmission line analysis by the use of the least-squares collocation method

A simple method of analysis of microstrip transmission lines by the corner function method (eigenfunction approach) is presented. The inhomogeneous dielectric system of the microstrip is treated as a mixed boundary-value problem and is solved by the use of the least-squares collocation method. The results obtained by this method agree well with previous theoretical results and the method of analysis is much less laborious, simpler and faster.

Application of the ‘corner function approach’ to strip-line problems

A new method of ‘Corner Function Approach’ is being proposed and applied to strip line problems. Numerical results obtained by this method are compared with previous theoretical and experimental results.