Yufeng Nie

Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains

In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme. A finite element method is developed to solve 2D-space fractional diffusion equations with nonlinear source term.We develop a finite element method for FPDEs on irregular domain.We obtain explicit expressions for fractional derivatives of shape functions.We give the details on how to compute fractional stiffness matrix.Fractional FitzHugh-Nagumo model is solved on circular domain.

A Node Placement Method with high quality for mesh generation

The quality of nodes distribution has great influence on the accuracy of numerical solution of partial differential equations system (PDEs), whether a mesh-based method or a meshless method is used. So here, we focus on how to place nodes properly in the domain on which the PDEs is defined. In this paper, the previous 2D Node Placement method by Bubble Simulation (NPBS) has been further researched. It is proved that the average speed of bubbles tends to be zero during dynamic simulation, i.e. the bubble system could converge to a stable configuration. Some examples are given to demonstrate the quality of resulting meshes in 2D and 3D.

An edge fusion scheme for image denoising based on anisotropic diffusion models

Image information are classified as smooth regions, edges, corners and isolated noises.A reasonable diffusion tensor is defined to conduct the adaptive diffusion.An improved anisotropic diffusion denoising model for image enhancement is proposed.An edge fusion scheme is posed to preserve edges after denoising.Every mode of edge fusion scheme can preserve more edges than single denoising method. In this paper, we propose an enhanced anisotropic diffusion model. The improved model can classify finely image information as smooth regions, edges, corners and isolated noises by characteristic parameters and gradient variance parameter. And for different image information the eigenvalues of diffusion tensor are designed to conduct adaptive diffusion. Moreover, an edge fusion scheme is posed to preserve edges after denoising by combing different denoising and edge detection methods. Firstly, different denoising methods are applied for noisy image to obtain denoised images, and the best method among them is selected as main method. Then edge images of denoised images are obtained by edge detection methods. Finally, by fusing edge images together more integrated edges can be achieved to replace edges of denoised image obtained by main method. The experimental results show the proposed model can denoise meanwhile preserve edges and corners, and the edge fusion scheme is accurate and effective.

Parallel node placement method by bubble simulation

Abstract An efficient Parallel Node Placement method by Bubble Simulation (PNPBS), employing METIS-based domain decomposition (DD) for an arbitrary number of processors is introduced. In accordance with the desired nodal density and Newton’s Second Law of Motion, automatic generation of node sets by bubble simulation has been demonstrated in previous work. Since the interaction force between nodes is short-range, for two distant nodes, their positions and velocities can be updated simultaneously and independently during dynamic simulation, which indicates the inherent property of parallelism, it is quite suitable for parallel computing. In this PNPBS method, the METIS-based DD scheme has been investigated for uniform and non-uniform node sets, and dynamic load balancing is obtained by evenly distributing work among the processors. For the nodes near the common interface of two neighboring subdomains, there is no need for special treatment after dynamic simulation. These nodes have good geometrical properties and a smooth density distribution which is desirable in the numerical solution of partial differential equations (PDEs). The results of numerical examples show that quasi linear speedup in the number of processors and high efficiency are achieved.

The Thermo-Mechanics Coupled Model of Polycrystalline Aggregates Based on Plastic Slip System in Crystals and their Interfaces

This paper presents the thermal-mechanical coupled model of polycrystalline aggregates based on plastic slip theory inside crystals and on the interfaces of crystals. It involves the mechanics and heat conduction behaviors caused by both force loads and temperature changing in the polycrystalline aggregates. At first, the constitutive relationship inside single crystal, and the moment equations and energy equations are derived by means of rate-dependent plastic deformation theory and the formulas of elastic-plastic tangent modulus depended on temperature. And those on crystal interfaces are also given. Based on the ABAQUS software [1], the subroutines to calculate the tension, torsion and bending strength of polycrystalline copper are coded. The numerical simulation results show that breakages occurred more easily on the interfaces than other areas of the polycrystalline aggregates, especially for bending loading and torsion loading, and that’s consistent with results by molecular dynamics but their computing cost are less and less than MD simulation.

Second-order two-scale analysis and numerical algorithm for the damped wave equations of composite materials with quasi-periodic structures

In this paper, we perform a second-order two-scale analysis and introduce a numerical algorithm for the damped wave equations of composite materials with a quasi-periodic structure. Firstly, second-order two-scale asymptotic expansion solutions for these problems are constructed by a multiscale asymptotic analysis. In addition, we explain the importance of the second-order two-scale solutions by the error analysis in the pointwise sense. Moreover, explicit convergence rates of these second-order two-scale solutions are obtained in the integral sense. Then a second-order two-scale numerical method based on a Newmark scheme is presented to solve these multiscale problems. Finally, some numerical examples show the effectiveness and efficiency of the multiscale numerical method we proposed.

A third-order entropy stable scheme for hyperbolic conservation laws

A third-order entropy stable scheme for nonlinear hyperbolic conservation laws is proposed here. This scheme contains two main ingredients: a fourth-order entropy conservative flux and a third-order numerical diffusion operator. A piecewise-quadratic reconstruction from pointwise values is developed in order to approximate the third-order dissipative term. To guarantee a non-oscillating property, a nonlinear limiter is employed and, furthermore, the scheme is proven to be entropy stable. Finally, numerical experiments are presented and demonstrate the accuracy, high-resolution, and robustness of our method.

A new boundary condition for homogenization of high-contrast random heterogeneous materials

Approximate effective coefficients of random heterogeneous materials could be obtained by solving auxiliary problems with certain boundary condition. When the coefficients of the auxiliary problems have great jump caused by the random heterogeneous materials with high-contrast properties, Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) provide broad upper and lower bounds of effective coefficients, respectively. Two possible factors that result in inaccurate approximations of effective coefficients are discussed in this paper. Effect of large condition number of stiffness matrix caused by the high contrast on the numerical accuracy of approximate effective coefficients is analysed. Since DBC and NBC are not effective for the high-contrast materials, an alternative Robin boundary condition (RBC) is presented to provide much better approximations of effective coefficients. Convergence of the approximate effective coefficients under RBC is proved. Numerical examples indicate that proper adjusting factor introduced in RBC makes it more flexible than other boundary conditions. RBC is more suitable for the high-contrast materials and has potential to be an optimal boundary condition.

NUMERICAL SIMULATION OF 2D LIQUID SLOSHING

In this paper, we present an extended ghost fluid method (GFM) for computations of liquid sloshing in incompressible multifluids consisting of inviscid and viscous regions. That is, the sloshing interface between inviscid and viscous fluids is tracked by the zero contour of a level set function and the appropriate sloshing interface conditions are captured by defining ghost fluids that have the velocities and pressure of the real fluid at each point while fixing the density and the kinematic viscosity of the other fluid. Meanwhile, a second order single-fluid solver, the central-weighted-essentially-nonoscillatory(CWENO)-type central-upwind scheme, is developed from our previous works. The high resolution and the nonoscillatory quality of the scheme can be verified by solving several numerical experiments. Nonlinear sloshing inside a pitching partially filled rectangular tank with/without baffles has been investigated.

Parallel adaptive mesh refinement method based on bubble-type local mesh generation

Abstract An efficient parallel adaptive mesh refinement method based on bubble-type local mesh generation (BLMG), employing ParMETIS-based dynamic domain decomposition method, is developed. The BLMG method is applied to generate the mesh with high quality, and the local mesh for each processor can be generated simultaneously without communication. The node-based distributed mesh structure is designed to reduce the communication amount spent in mesh generation and finite element calculation. To gain the load balance, a new load balancing algorithm for the new mesh structure is devised to make sure the whole algorithm is efficient. Several numerical examples are carried out to verify the high efficiency of the algorithm.