On the Regularity for 3D Navier-Stokes Equation

This paper deals with the influence of boundary values on the regularity of 3D incompressible Navier-Stokes equation, which is separated into two parts: First part, we consider the potential theory of unsteady 3D Stokes flow. In order to consider Dirichlet and Neumann boundary values, we establish the Green's formula, and then give the double layer and single layer potentials. Second part, based on problems separated and the potential theory we reach the integral representations of the solutions with respect to Dirichlet and Neumann boundary conditions. Consequently we change the integral representations of solutions from velocity form into vorticity form so that we can construct schemes in favor of numerical computation in this way. Finally we analyze the regularity of the initial-boundary value problems for Navier-Stokes equation, and obtain regular solutions by assuming the regularity on the boundary and data.Comment: modificatio

A Multi-level Correction Scheme for Eigenvalue Problems

In this paper, a new type of multi-level correction scheme is proposed for solving eigenvalue problems by finite element method. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving eigenvalue problems by finite element method.Comment: 16 pages, 5 figure

A Parallel Method for Population Balance Equations Based on the Method of Characteristics

In this paper, we present a parallel scheme to solve the population balance equations based on the method of characteristics and the finite element discretization. The application of the method of characteristics transform the higher dimensional population balance equation into a series of lower dimensional convection-diffusion-reaction equations which can be solved in a parallel way.Some numerical results are presented to show the accuracy and efficiency.Comment: 10 pages, 0 figur

The well-posedness and solutions of Boussinesq-type equations

We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive sufficient conditions for solution blow-up in finite time.Secondly, a generalized Jacobi/exponential expansion method for finding exact solutions of non-linear partial differential equations is discussed. We use the proposed expansion method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified Korteweg-de Vries equations. We also apply it to the shallow water long wave approximate equations. New solutions are deduced for this system of partial differential equations.Finally, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem, which can be solved using many accurate numerical methods. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior

Towards an optimization theory for deforming dense granular materials: Minimum cost maximum flow solutions

We use concepts and techniques of network optimization theory to gain a better understanding of force transmission in dense granular materials. Specifically, we represent a deforming granular material over the different stages of a quasi-static biaxial compression test as a series of representative flow networks, and analyze force transmission through these networks. The forces in such a material are transmitted through the contacts between the constituent grains. As the sample deforms during the various stages of the biaxial test, these grains rearrange: while many contacts are preserved in this rearrangement process, some new contacts form and some old contacts break. We consider the maximum flow problem and the minimum cost maximum flow (MCMF) problem for the flow networks constructed from this evolving network of grain contacts. We identify the flow network bottleneck and establish the sufficient and necessary conditions for a minimum cut of the maximum flow problem to be unique. We also develop an algorithm to determine the MCMF pathway, i.e. a set of edges that always transmit non-zero flow in every solution of the MCMF problem. The bottlenecks of the flow networks develop in the locality of the persistent shear band, an intensively-studied phenomenon that has long been regarded as the signature failure microstructure for dense granular materials. The cooperative evolution of the most important structural building blocks for force transmission, i.e. the force chains and 3-cycles, is examined with respect to the MCMF pathways. We find that the majority of the particles in the major load-bearing columnar force chains and 3-cycles consistently participate in the MCMF pathways