Using Python to Solve the Navier-Stokes Equations - Applications in the Preconditioned Iterative Methods

This article describes a new numerical solver for the Navier-Stokes equations. The proposed solver is written in Python which is a newly developed language. The Python packages are built to solve the Navier-Stokes equations with existing libraries. We have created discretized coefficient matrices from systems of the Navier-Stokes equations by the finite difference method. In addition we focus on the preconditioned Krylov subspace iterative methods in the linearized systems. Numerical results of performances for the Preconditioned iterative methods are demonstrated. The comparison between Python and Matlab is discussed at the end of the paper

Transition to amplitude death in scale-free networks

Transition to amplitude death in scale-free networks of nonlinear oscillators is investigated. As the coupling strength increases, the network will undergo three stages in approaching to the state of complete amplitude death. The first stage is featured by a \emph{"stair-like"} distribution of the node amplitude, and the transition is accomplished by a \emph{hierarchical death} of the amplitude stairs. The second and third stages are characterized by, respectively, a continuing elimination of the synchronous clusters and a fast death of the non-synchronized nodes.Comment: 4 pages, 4 figure

Comparative Analysis on Elastic-plastic Analytical Methods for Tunnel Surrounding Rocks

Elastic-plastic analysis is an important way to obtain the mechanical characteristics of surrounding rocks. However, selecting a reasonable analytical method is difficult. To explore the differences between the surrounding rocks constitutive relation and yield criterion analytical methods, the stress distribution and deformation characteristics of tunnel surrounding rocks at the −817 m firefighting material chamber of the Wangfenggang Well, Xieyi Mine, Huainan Coal Industry Group (China) were analyzed by using the bilinear constitutive relation and unified strength criterion analytical methods. Calculated results of the two analytical methods were compared. The radius and displacement evolution laws of the plastic region on tunnel surrounding rocks with in situ rock stress and supporting resistance were discussed. Results demonstrate that compared with the unified strength criterion analytical method, the bilinear constitutive relation analytical method avoids the influences of intermediate principal stress coefficient on results and the tangential stress distribution curve is smooth. The calculated radius of the plastic region and periphery displacement of the tunnel are 4,365 m and 87,373 mm, which are higher than those of the unified strength criterion analytical method. Stress difference is a major factor that influences the mechanical characteristics of tunnel surrounding rocks. As the stress difference decreases from 20,4 MPa to 16,4 MPa, the radius of the plastic region and periphery displacement of the tunnel decrease by 0,697 m and 26,73 mm, respectively. This study provides theoretical references for the practical selection of elastic-plastic analytical method for tunnel surrounding rocks

Voxurf: Voxel-based Efficient and Accurate Neural Surface Reconstruction

Neural surface reconstruction aims to reconstruct accurate 3D surfaces based on multi-view images. Previous methods based on neural volume rendering mostly train a fully implicit model with MLPs, which typically require hours of training for a single scene. Recent efforts explore the explicit volumetric representation to accelerate the optimization via memorizing significant information with learnable voxel grids. However, existing voxel-based methods often struggle in reconstructing fine-grained geometry, even when combined with an SDF-based volume rendering scheme. We reveal that this is because 1) the voxel grids tend to break the color-geometry dependency that facilitates fine-geometry learning, and 2) the under-constrained voxel grids lack spatial coherence and are vulnerable to local minima. In this work, we present Voxurf, a voxel-based surface reconstruction approach that is both efficient and accurate. Voxurf addresses the aforementioned issues via several key designs, including 1) a two-stage training procedure that attains a coherent coarse shape and recovers fine details successively, 2) a dual color network that maintains color-geometry dependency, and 3) a hierarchical geometry feature to encourage information propagation across voxels. Extensive experiments show that Voxurf achieves high efficiency and high quality at the same time. On the DTU benchmark, Voxurf achieves higher reconstruction quality with a 20x training speedup compared to previous fully implicit methods

Global solutions to a nonlinear Fokker-Planck equation

In this paper, we construct global solutions to the Cauchy problem on a nonlinear Fokker-Planck equation near Maxwellian with small-amplitude initial data in Sobolev space $H^2_{x}L^2_v$ by a refined nonlinear energy method. Compared with the results of Liao et al. (Global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations, J. Stat. Phys., 173 (2018), 222–241.), the regularity assumption on the initial data is much weaker