In this paper, the transverse vibration of a fractional viscoelastic beam is studied based on the fractional calculus, and the corresponding scheme of a viscoelastic beam is established by using the mixed finite volume element method. The stability and convergence of the algorithm are analyzed. Numerical examples demonstrate the effectiveness of the algorithm. Finally, the values of different parameter sets are tested, and the test results show that both the damping coefficient and the fractional derivative have significant effects on the model. The results of this paper can be used for the damping modeling of viscoelastic structures.

This paper discusses the Crank–Nicolson compact difference method for the time-fractional damped plate vibration problems. For the time-fractional damped plate vibration equations, we introduce the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems. We discretize the space derivative via compact difference and approximate the time-integer-order derivative and fraction-order derivative via central difference and L1 interpolation, respectively, to obtain the compact difference formats with fourth-order space precision and $3-\alpha$($1<\alpha <2$)-order time precision. We apply the energy method to analyze the stability and convergence of this difference format. We provide numerical cases, which not only validate the convergence order and feasibility of the given difference format, but also simulate the influence of the damping coefficient on the amplitude of plate vibration.

In this paper, we propose and analyze the compact finite difference scheme of the two-dimensional Cattaneo model. The stability and convergence of the scheme are proved by the energy method, the convergence orders are 2 in time and 4 in space. We also use the variables separation method to find the true solution of the problem. On this basis, the validity and accuracy of the scheme are verified by numerical experiments.

Optical flow estimation is a basic task in self-driving and robotics systems, which enables to temporally interpret traffic scenes. Autonomous vehicles clearly benefit from the ultra-wide Field of View (FoV) offered by 360{\deg} panoramic sensors. However, due to the unique imaging process of panoramic cameras, models designed for pinhole images do not directly generalize satisfactorily to 360{\deg} panoramic images. In this paper, we put forward a novel network framework--PanoFlow, to learn optical flow for panoramic images. To overcome the distortions introduced by equirectangular projection in panoramic transformation, we design a Flow Distortion Augmentation (FDA) method, which contains radial flow distortion (FDA-R) or equirectangular flow distortion (FDA-E). We further look into the definition and properties of cyclic optical flow for panoramic videos, and hereby propose a Cyclic Flow Estimation (CFE) method by leveraging the cyclicity of spherical images to infer 360{\deg} optical flow and converting large displacement to relatively small displacement. PanoFlow is applicable to any existing flow estimation method and benefits from the progress of narrow-FoV flow estimation. In addition, we create and release a synthetic panoramic dataset FlowScape based on CARLA to facilitate training and quantitative analysis. PanoFlow achieves state-of-the-art performance on the public OmniFlowNet and the established FlowScape benchmarks. Our proposed approach reduces the End-Point-Error (EPE) on FlowScape by 27.3%. On OmniFlowNet, PanoFlow achieves a 55.5% error reduction from the best published result. We also qualitatively validate our method via a collection vehicle and a public real-world OmniPhotos dataset, indicating strong potential and robustness for real-world navigation applications. Code and dataset are publicly available at https://github.com/MasterHow/PanoFlow.