Numerical Solution of Convection–Diffusion Equations Using a Nonlinear Method of Upwind Type

This paper is devoted to the numerical solution of convection–diffusion equations using the Mizukami–Hughes method which is a nonlinear method of upwind type using conforming piecewise linear triangular finite elements. We extend this method to the whole range of the diffusion parameter whereas the original method was introduced for the convection-dominated regime only. We prove that the extended method satisfies the discrete maximum principle and illustrate its properties by means of numerical results.     

一种用于分布参数系统移动边界的自适应剖分数值方法

本文提出通过对具有移动边界分布参数系统中的移动边界的一步预报,自适应生成剖分网格,然后通过系统的焓方程应用有限元方法求解,得到具有移动边界的分布参数系统的数值解.结果表明,这种方法较好地解决了用有限元方法求解该类系统的数值解时遇到的移动边界附近数值解精度与网格剖分过细所导致的计算量过大的矛盾.为具有移动边界的分布参数系统的建模和仿真提供了一种有效的数值计算方法,同时也为研究系统的控制、估计、辨识等问题的数值方法打下了基础.     

一种基于非线性扩散方程的PET重建方法

正电子发射断层重建是一类不适定问题,因此正则化方法常用来抑制噪声提高重建的质量.依据Mumford-Shah泛函模型提出了该类重建问题的统一变分模型.基于这一模型,提出了一种新的基于扩散方程的PET重建算法,同时给出了线性和非线性两种方法.实验结果表明,两种方法均有良好的重建性能,并且非线性方法通过合理的设置参数,能体现出较为优越的抗噪声性能和边缘保持特性.     

A High-Order Moving Mesh Kinetic Scheme Based on WENO Reconstruction for Compressible Flows on Unstructured Meshes

In this paper, we present a high-order moving mesh (HMM) kinetic scheme for compressible flow computations on unstructured meshes. To construct the scheme, we employ the frame of the remapping-free ALE-type kinetic method (Ni et al. in J Comput Phys 228:3154–3171, 2009) to get the discretization of compressible system. For the space accuracy, we use the weighted essential non-oscillatory reconstruction on the adaptive moving mesh from Tang and Tang (SIAM J Numer Anal 41:487–515 2003) to achieve time accuracy,we make use of the kinetic flux which includes time accurate integral, and thus obtain a HMM scheme. A number of numerical examples are given, especially an isentropic vortex problem to show the convergence order of the scheme. Numerical results demonstrate the accuracy and robustness of the scheme.     

一个适用于大型三角网格的鲁棒水印算法

针对大型三角网格的特点,提出一个基于外存的鲁棒水印算法。首先对大规模三角网格顶点的重新排序和分组,然后对由分组中顶点到原点的距离组成的序列进行DCT变换,最后将由二值图像调制的水印序列嵌入到经过量化的DCT高频系数中。实验表明,该算法能够抵抗网格噪声、简化、剪切以及仿射变换等攻击,具有良好的不可见性和鲁棒性。     

An Unconditionally Stable Quadratic Finite Volume Scheme over Triangular Meshes for Elliptic Equations

In this note, we present and analyze a special quadratic finite volume scheme over triangular meshes for elliptic equations. The scheme is designed with the second degree Gauss points on the edges and the barycenters of the triangle elements. With a novel from-the-trial-to-test-space mapping, the inf–sup condition of the scheme is shown to hold independently of the minimal angle of the underlying mesh. As a direct consequence, the \(H^1\) norm error of the finite volume solution is shown to converge with the optimal order.     

Solution Of Bounded Nonlinear Systems Of Equations Using Homotopies With Inexact Restoration

Nonlinear systems of equations often represent mathematical models of chemical production processes and other engineering problems. Homotopic techniques (in particular, the bounded homotopies introduced by Paloschi) are used for enhancing convergence to solutions, especially when a good initial estimate is not available. In this paper, the homotopy curve is considered as the feasible set of a mathematical programming problem, where the objective is to find the optimal value of the homotopic parameter. Inexact restoration techniques can then be used to generate approximations in a neighborhood of the homotopy, the size of which is theoretically justified. Numerical examples are given.     

The Superconvergence Phenomenon and Proof of the MAC Scheme for the Stokes Equations on Non-uniform Rectangular Meshes

For decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the \(L^2\) norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.     

径向基函数及移动网格在Euler方程数值计算中的应用

基于二维Euler方程,在利用弹簧技术的移动非结构三角形网格上给出了一种基于紧支径向基函数重构的ENO型有限体积格式,方法的主要思想是先对每一个三角形单元构造插值径向基函数,而在计算交界面的流通量采用两点高斯积分公式以保证格式的整体精度,时间离散采用三阶TVD Runge-Kutta方法。最后用该格式对一些典型算例进行了数值模拟,结果表明该方法计算速度快,对间断有很好的分辨能力。     

Adaptive Nonlinear Finite Elements for Deformable Body Simulation Using Dynamic Progressive Meshes

Realistic behavior of deformable objects is essential for many applications such as simulation for surgical training. Existing techniques of deformable modeling for real time simulation have either used approximate methods that are not physically accurate or linear methods that do not produce reasonable global behavior. Nonlinear finite element methods (FEM) are globally accurate, but conventional FEM is not real time. In this paper, we apply nonlinear FEM using mass lumping to produce a diagonal mass matrix that allows real time computation. Adaptive meshing is necessary to provide sufficient detail where required while minimizing unnecessary computation. We propose a scheme for mesh adaptation based on an extension of the progressive mesh concept, which we call dynamic progressive meshes.     

A Colour Interpolation Scheme for Topologically Unrestricted Gradient Meshes

Gradient meshes are a 2D vector graphics primitive where colour is interpolated between mesh vertices. The current implementations of gradient meshes are restricted to rectangular mesh topology. Our new interpolation method relaxes this restriction by supporting arbitrary manifold topology of the input gradient mesh. Our method is based on the Catmull‐Clark subdivision scheme, which is well‐known to support arbitrary mesh topology in 3D. We adapt this scheme to support gradient mesh colour interpolation, adding extensions to handle interpolation of colours of the control points, interpolation only inside the given colour space and emulation of gradient constraints seen in related closed‐form solutions. These extensions make subdivision a viable option for interpolating arbitrary‐topology gradient meshes for 2D vector graphics.     

Extrapolation of Nystrom Solution for Two-Dimensional Nonlinear Fredholm Integral Equations

In this paper, we analyze the existence of asymptotic error expansion of Nystrom solution for two-dimensional nonlinear Fredholm integral of the second kind. We show that the Nystrom solution admits an error expansion in powers of the step-size h and the step-size k. For a special choice of the numerical quadrature, the leading terms in the error expansion for the Nystrom solution contain only even powers of h and k, beginning with terms h ^2p and k ^2q . These expansions are useful for the application of Richardson extrapolation and for obtaining sharper error bounds. Numerical examples show that how Richardson extrapolation gives a remarkable increase of precision, in addition to faster convergence.     

A RT Mixed FEM/DG Scheme for Optimal Control Governed by Convection Diffusion Equations

In this paper, we provide a numerical scheme—RT mixed FEM/DG scheme for the constrained optimal control problem governed by convection dominated diffusion equations. A priori and a posteriori error estimates are obtained for both the state, the co-state and the control. The adaptive mesh refinement can be applied indicated by a posteriori error estimator provided in this paper. Numerical examples are presented to illustrate the theoretical analysis.     

A Very High-Order Accurate Staggered Finite Volume Scheme for the Stationary Incompressible Navier–Stokes and Euler Equations on Unstructured Meshes

We propose a sixth-order staggered finite volume scheme based on polynomial reconstructions to achieve high accurate numerical solutions for the incompressible Navier–Stokes and Euler equations. The scheme is equipped with a fixed-point algorithm with solution relaxation to speed-up the convergence and reduce the computation time. Numerical tests are provided to assess the effectiveness of the method to achieve up to sixth-order convergence rates. Simulations for the benchmark lid-driven cavity problem are also provided to highlight the benefit of the proposed high-order scheme.     

Multilevel Diffusion Schemes for Repartitioning of Adaptive Meshes

For a large class of irregular mesh applications, the structure of the mesh changes from one phase of the computation to the next. Eventually, as the mesh evolves, the adapted mesh has to be repartitioned to ensure good load balance. If this new graph is partitioned from scratch, it may lead to an excessive migration of data among processors. In this paper, we present schemes for computing repartitionings of adaptively refined meshes that perform diffusion of vertices in a multilevel framework. These schemes try to minimize vertex movement without significantly compromising the edge-cut. We present heuristics to control the tradeoff between edge-cut and vertex migration costs. We also show that multilevel diffusion produces results with improved edge-cuts over single-level diffusion, and is better able to make use of heuristics to control the tradeoff between edge-cut and vertex migration costs than single-level diffusion.     

基于几何特征的三维网格数字水印算法

首先把网格上的点到网格中心的距离看成一维的空域数字信号,并用一维DCT变化到频域;然后在频域信号上加入水印,再通过逆向DCT转回空域信号;最后根据新的空域信号的值修改网格上点的坐标得到加入数字水印的网格．实验结果表明：该算法实现简单,在简化攻击、噪音攻击、二次水印攻击和剪切攻击情况下具有良好的鲁棒性．