Moving Mesh Discontinuous Galerkin Method for Hyperbolic Conservation Laws

In this paper, a moving mesh discontinuous Galerkin (DG) method is developed to solve the nonlinear conservation laws. In the mesh adaptation part, two issues have received much attention. One is about the construction of the monitor function which is used to guide the mesh redistribution. In this study, a heuristic posteriori error estimator is used in constructing the monitor function. The second issue is concerned with the solution interpolation which is used to interpolates the numerical solution from the old mesh to the updated mesh. This is done by using a scheme that mimics the DG method for linear conservation laws. Appropriate limiters are used on seriously distorted meshes generated by the moving mesh approach to suppress the numerical oscillations. Numerical results are provided to show the efficiency of the proposed moving mesh DG method.     

A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains

Numerical techniques for moving meshes are many and varied. In this paper we present a novel application of a moving grid finite element method applied to biological problems related to pattern formation where the mesh movement is prescribed through a specific definition to mimic the growth that is observed in nature. Through the use of a moving grid finite element technique, we present numerical computational results illustrating how period doubling behaviour occurs as the domain doubles in size.     

Variable time-step method with coordinate transformation

The variable time-step methods for solving moving boundary problems are presented by transforming the variable space domain. This results in dissociating the mode of advancement of the boundary from the size of the space mesh. That is, a small movement of the moving boundary may be chosen for computing the time interval while the space domain is subdivided into larger space meshes. As a consequence, an enormous amount of saving in computer time may be achieved by using the proposed method. Two sample problems are selected for the illustration of the method.     

Level Set Calculations for Incompressible Two-Phase Flows on a Dynamically Adaptive Grid

We present a coupled moving mesh and level set method for computing incompressible two-phase flow with surface tension. This work extends a recent work of Di et al. [(2005). SIAM J. Sci. Comput. 26, 1036–1056] where a moving mesh strategy was proposed to solve the incompressible Navier–Stokes equations. With the involvement of the level set function and the curvature of the interface, some subtle issues in the moving mesh scheme, in particular the solution interpolation from the old mesh to the new mesh and the choice of monitor functions, require careful considerations. In this work, a simple monitor function is proposed that involves both the level set function and its curvature. The purpose for designing the coupled moving mesh and level set method is to achieve higher resolution for the free surface by using a minimum amount of additional expense. Numerical experiments for air bubbles and water drops are presented to demonstrate the effectiveness of the proposed scheme.     

平面及空间区域渐变无结构网格的自动生成

１．前 言 网格生成是许多数值计算首先要解决的问题．规则区域上的均匀网格,比较容易生成．但许多工程实际问题,求解区域边界形状极不规则,且由于物理参数的剧烈变化及解的性态复杂,对求解区域的网格疏密变化有某些特殊的要求．如何在非规则区域上自动生成符合使用者特殊要求的网格,是科学计算过程中人们所关心的问题． 本文基于 Ｄｅｌａｕｎａｙ三角划分原理,在二维（三维）区域给出一种自动生成渐变无结构三角形（四面体）网格的方法．对事先给定的指定结点集合和对应的间隔值集合,算法将首先自动生成全部边界结点和内部结点,然…     

On marching cubes

A characterization and classification of the isosurfaces of trilinear functions is presented. Based upon these results, a new algorithm for computing a triangular mesh approximation to isosurfaces for data given on a 3D rectilinear grid is presented. The original marching cubes algorithm is based upon linear interpolation along edges of the voxels. The asymptotic decider method is based upon bilinear interpolation on faces of the voxels. The algorithm of this paper carries this theme forward to using trilinear interpolation on the interior of voxels. The algorithm described here will produce a triangular mesh surface approximation to an isosurface which preserves the same connectivity/separation of vertices as given by the isosurface of trilinear interpolation.     

Simultaneous Refinement and Coarsening for Adaptive Meshing

In the numerical simulation of the combustion process and microstructural evolution, we need to consider the adaptive meshing problem for a domain that has a moving boundary. During the simulation, the region ahead of the moving boundary needs to be refined (to satisfy stronger numerical conditions), and the submesh in the region behind the moving boundary should be coarsened (to reduce the mesh size). We present a unified scheme for simultaneously refining and coarsening a mesh. Our method uses sphere packings and guarantees that the resulting mesh is well-shaped and is within a constant factor of the optimal possible in the number of mesh elements. We also present several practical variations of our provably good algorithm.     

Application of mortar elements to diffuse-interface methods

An adaptive 2D mesh refinement technique based on mortar spectral elements applied to diffuse-interface methods is presented. The refinement algorithm tracks the movement of the 2D diffuse-interface and subsequently refines the mesh locally at that interface, while coarsening the mesh in the rest of the computational domain, based on error estimators. Convergence of the method is validated using a Gaussian distribution problem and results are presented for a Cahn–Hilliard diffuse-interface model applied to capture the transient dynamics of polymer blends.     

移动最小二乘代理模型支持域半径的优化方法

移动最小二乘代理模型描述局部波动的能力优于一般的代理模型,但其精度受支持域半径的影响。在经验公式的基础上提出了一种针对移动最小二乘代理模型支持域半径的优化方法。对支持域内抽样点数寻优获取最佳半径值,提高近似精度进而达到减少抽样点的目的。数值实验结果表明,对于不同基函数阶次和权函数的情况,提出的方法大大提高了移动最小二乘代理模型的近似精度,与基于经验公式的移动最小二乘代理模型相比,其仅需较少的抽样点即可达到相同的近似精度。     

A posteriori error analysis for a fractional differential equation

Numerical treatment for a fractional differential equation (FDE) is proposed and analysed. The solution of the FDE may be singular near certain domain boundaries, which leads to numerical difficulty. We apply the upwind finite difference method to the FDE. The stability properties and a posteriori error analysis for the discrete scheme are given. Then, a posteriori adapted mesh based on a posteriori error analysis is established by equidistributing arc-length monitor function. Numerical experiments illustrate that the upwind finite difference method on a posteriori adapted mesh is more accurate than the method on uniform mesh.     

Adaptive moving mesh methods for two-dimensional resistive magneto-hydrodynamic PDE models

An adaptive moving mesh technique is applied to magneto-hydrodynamics (MHD) model problem. The moving mesh strategy is based on the approach proposed in Li et al. [Li R, Tang T, Zhang P. Moving mesh methods in multiple dimensions based on harmonic maps. J Comput Phys 2001;170:562-88] to separate the mesh-moving and PDE evolution at each time step. The Magneto-hydrodynamic equations are discretized by a finite-volume method in space, and the mesh-moving part is realized by solving the Euler-Lagrange equations to minimize a certain variation with the directional splitting monitor function. A conservative interpolation is used to redistribute the numerical solutions on the new meshes. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm.     

A Parallel Implementation of ALE Moving Mesh Technique for FSI Problems using OpenMP

This paper investigates a high performance implementation of an Arbitrary Lagrangian Eulerian moving mesh technique on shared memory systems using OpenMP environment. Moving mesh techniques are considered an integral part of a wider class of fluid mechanics problems that involve moving and deforming spatial domains, namely, free-surface flows and Fluid Structure Interaction (FSI). The moving mesh technique adopted in this work is based on the notion of nodes relocation, subjected to a certain evolution as well as constraint conditions. A conjugate gradient method augmented with preconditioning is employed for solution of the resulting system of equations. The proposed algorithm, initially, reorders the mesh using an efficient divide and conquer approach and then parallelizes the ALE moving mesh scheme. Numerical simulations are conducted on the multicore AMD Opteron and Intel Xeon processors, and unstructured triangular and tetrahedral meshes are used for the 2D and 3D problems. The quality of generated meshes is checked by comparing the element Jacobians in the reference and current meshes, and by keeping track of the change in the interior angles in triangles and tetrahedrons. Overall, 51 and 72% efficiencies in terms of speedup are achieved for both the parallel mesh reordering and ALE moving mesh algorithms, respectively.     

基于视频的快速运动检测方法及其应用研究

介绍了一种以视频图像为检测源的快速运动检测方法并给出了该方法在闯红灯监控系统中的应用实例。与传统的运动检测方法相比,该方法采用视频处理技术;通过先进的运动检测算法快速识别运动物体,并能判断出运动物体的运动方向。该方法在闯红灯监控系统中的应用结果表明该方法的快速性和可行性。     

Isotropic Surface Remeshing Using Constrained Centroidal Delaunay Mesh

We develop a novel isotropic remeshing method based on constrained centroidal Delaunay mesh (CCDM), a generalization of centroidal patch triangulation from 2D to mesh surface. Our method starts with resampling an input mesh with a vertex distribution according to a user‐defined density function. The initial remeshing result is then progressively optimized by alternatively recovering the Delaunay mesh and moving each vertex to the centroid of its 1‐ring neighborhood. The key to making such simple iterations work is an efficient optimization framework that combines both local and global optimization methods. Our method is parameterization‐free, thus avoiding the metric distortion introduced by parameterization, and generating more well‐shaped triangles. Our method guarantees that the topology of surface is preserved without requiring geodesic information. We conduct various experiments to demonstrate the simplicity, efficacy, and robustness of the presented method.     

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a \(C^0\) finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.     

Free-surface flow simulations in the presence of inclined walls

Difficulties associated with free-surface finite element flow simulations are related to (a) nonlinear and advective nature of most hydrodynamic flows, (b) requirements for compatibility between velocity and pressure interpolation, (c) maintaining a valid computational mesh in the presence of moving boundaries, and (d) enforcement of the kinematic conditions at the free surface. Focusing on the last issue, we present an extension of the free-surface elevation equation to cases where the prescribed direction of the surface node motion is not uniformly vertical. The resulting hyperbolic generalized elevation equation is discretized using a Galerkin/least-squares formulation applied on the surface mesh. The elevation field so obtained is then used to impose displacement boundary conditions on the elastic mesh update scheme that governs the movement of interior mesh nodes. The proposed method is used to solve a two-dimensional problem of sloshing in a trapezoidal tank, and a three-dimensional application involving flow in a trapezoidal channel with bridge supports.     

一维热传导问题时变边界上热通量重构问题

具有Neumann边界条件的抛物方程的初边值问题是偏微分方程研究领域的一类经典问题.正问题是由已知的边界条件和初始条件来求区域温度场的问题.如果边界条件不足,但给出了区域内部的一些额外信息,这样便构成了一类热通量重构的反问题.本文讨论了一维热传导问题时动边界上的热通量重构问题,借助于位势理论方法,引入密度函数,将反问题本质上转化为一类关于密度函数的具有弱奇性核的第一类Volterra积分方程,采用了Tikhonov正则化,在正则化参数的选取上采用了后验的模型函数方法,数值结果验证了反演方法的有效性.     

The simulation of 3D unsteady incompressible flows with moving boundaries on unstructured meshes

The development of a computational model for the simulation of three-dimensional unsteady incompressible viscous fluid flows with moving boundaries is presented. The numerical model is based upon the solution of the Navier–Stokes equations on unstructured meshes using the artificial compressibility approach. An ALE formulation is adopted and the equations are discretized using a cell vertex finite volume method. The formulation ensures the satisfaction of the geometric conservation law when the mesh is allowed to move. An implicit time discretization is adopted and a dual time approach is employed. Explicit relaxation is used for the sub-iterations, with multigrid acceleration. For moving geometries, the mesh is deformed by adopting a spring analogy, combined with a wall distance function approach. The numerical procedure is validated on a standard problem and is then used for the simulation of flow over a flexible fish-like body.     

An arbitrary Lagrangian Eulerian formulation for residual distribution schemes on moving grids

The arbitrary Lagrangian Eulerian formulation is derived for the residual distribution method on moving meshes. The system of Euler equations is discretized on moving meshes and in case of deforming meshes a geometrical source term has to be taken into account. A conservative linearization guarantees the conservation property of the discretized equations.From the geometric conservation law we obtain the appropriate integration points in time for the cell fluctuation and a guideline for how to distribute the geometrical source term.Testcases include the flow around a transonic oscillating airfoil and a convected vortex. In the first case a rigidly moving mesh is employed, while in the other testcase a deforming mesh is used to investigate the influence of the geometrical source term on the solution.     

Improvement of the precision of the gradient method and object tracking using optical flow

In this study, the spatial local optimization method was improved to obtain high precision of optical flow for cases in which the object movement changes substantially and a method to trace the loci of moving objects was considered. In the spatial local optimization method, the precision of the optical flow when the object movement changes substantially becomes a problem. Therefore, to make the object movement relatively small, we obtained flow vectors from the image sequence to drop the resolution of the original input image sequence to half the initial resolution. flow vectors were then obtained from the original input image sequence that were smaller than the threshold value. We show that the precision of the optical flow when the object movement changes substantially is improved by this method. Method used to trace the loci of moving objects was demonstrated. We obtained clusters from histograms of flow vectors and pursued each cluster. We show that it is possible to trace moving objects by this method. This work was presented, in part, at the 7th International Symposium on Artificial Life and Robotics, Oita, Japan, January 16–18, 2002