Parallel 3D Mortar Element Method for Adaptive Nonconforming Meshes

We present a new two-step “intermediate mortar” approach that efficiently extends the Mortar Element Method (MEM) to 3D meshes with nonconforming discretizations. The technique utilizes projection matrices derived in 2D, thereby avoiding the need to explicitly form large matrices for the 3D cases. The method also allows more flexibility for both h- and p-type adaptation. Parallel implementations with OpenMP and multithreading compiler directives are used to solve a moving heat source problem on SGI Origin and Cray MTA machines. Numerical results demonstrate the advantages of adaptive nonconforming meshes with MEM over uniformly fine meshes.     

A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations

We consider Lagrangian reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.     

Adaptive Spectral Element Simulations of Thin Premixed Flame Sheet Deformations

The high order Spectral Element Method is used to solve the reaction-diffusion-advection problem as described by the premixed flame case. An h–p adaptive refinement-coarsening algorithm is developed based on a posteriori error estimators. Simulations of the wrinkling of a premixed flame front are used to illustrate how the mesh is adaptively refined. A similar, idealized heat transfer problem is used to show the adaptive refinement and coarsening of the mesh. Adaptivity efficiently provides high resolution in areas of the domain where large or rapidly varying physical changes exist, while saving unnecessary computation where the solution is smooth or physical phenomena have passed by.     

Higher-Order Level-Set Method and Its Application in Biomolecular Surfaces Construction

We present a general framework for a higher-order spline level-set(HLS) method and apply this to biomolecule surfaces construction. Starting from a first order energy functional,we obtain a general level set formulation of geometric partial differential equation,and provide an efficient approach to solving this partial differential equation using a C2 spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Z-transform,which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. One example of our HLS method is demonstrated,which is the construction of biomolecule surfaces(an implicit solvation interface) with their individual atomic coordinates and solvated radii as prerequisites.     

切比契夫谱元素局部混合基函数构造

针对切比契夫谱方法,该文首次构造了两类局部混合基函数,据此发展了一种新的谱元素方法:在元素端点采用局部拉格朗日插值基,元素内部采用经调整后的切比契夫多项式。这里的两类混合基函数在计算精度上可与传统的拉格朗日基相媲美,而且元素矩阵具有稀疏特征和数据重用性。该文给出的局部混合基函数对传统的谱元素方法进行了扩充。     

有限元数值解法在MATLAB中的实现及可视化

偏微分方程的数值解法在数值分析中占有很重要的地位,很多科学技术问题的数值计算包括了偏微分方程的数值解问题。在学习初等函数时,总是先画出它们的图形,因为图形能帮助我们了解函数的性质。而对于偏微分方程,画出它们的图形并不容易,尤其是没有解析解的偏微分方程,画图就显得更加不容易了。为了从偏微分方程的数学表达式中看出其所表达的图形、函数值与自变量之间的关系,通过MATLAB编程,用有限元数值解法求解了偏微分方程,并将其结果可视化。     

边界满足C0或C1连续条件的光顺过渡面的构造

针对联结两个曲面片的过渡曲面构造问题,使用偏微分方程中的热传导方程,详细研究了如何构造边界满足C⁰或C¹连续条件的光顺过渡面的偏微分方程模型,讨论了初始曲面及方程参数对光顺过渡面的影响。     

光顺过渡面的热传导方程-能量优化构造方法

首先运用热传导方程、初值条件和带有固定边界的边值条件,建立过渡面的偏微分方程模型.再对方程像空间中的曲面簇,采用关于时间t的曲面能量表达式作为曲面光顺准则,建立起满足热传导方程的光顺过渡面的优化构造模型,即带有固定边界的光顺过渡面的热传导方程-能量优化构造模型,此模型的解即为满足连续性和光顺性要求的过渡面.数值实验表明这种光顺过渡面的构造方法是可行与有效的.     

具有变动边界的过渡面的热传导方程构造法

提出过渡面的热传导方程构造方法。使用热传导方程、初值条件和变动的边界条件,得到过渡面的偏微分方程模型,再利用分离变量法或者数值方法得到模型的解,则方程的像空间就是满足连续性要求的过渡面。数值实验也表明这种方法是可行的、有效的。     

基于水平集接力的图像自动分割方法

为了实现图像的完全分割,基于无须重新初始化的水平集方法提出了一种接力水平集方法.该方法在待分割图像中自动交替地创建嵌套子区域和相应的初始水平集函数,使水平集函数在其中演化并收敛,然后重复这个过程直到子区域面积为0.与原始算法及经典的基于区域的水平集方法相比,该方法具有如下优点:1) 自动完成,无须交互式的初始化;2) 多次分割图像,能够比原始算法检测到更多的边缘;3) 对于非匀质的图像,能够取得比经典的基于区域的水平集方法更好的分割效果;4) 提供一个开放的分割算法框架,其他单水平集方法稍作修改后也可替换这里所使用的单水平集方法.实验结果表明,此算法对人造图像和医学影像实现了无须交互的完全分割,对非匀质图像分割表现出更好的鲁棒性.     

Distributed feedback design for systems governed by the wave equation

This article deals with the geometric control of a one-dimensional non-autonomous linear wave equation. The idea consists in reducing the wave equation to a set of first-order linear hyperbolic equations. Then, based on geometric control concepts, a distributed control law that enforces the exponential stability and output tracking in the closed-loop system is designed. The presented control approach is applied to obtain a distributed control law that brings a stretched uniform string, modelled by a wave equation with Dirichlet boundary conditions, to rest in infinite time by considering the displacement of the middle point of the string as the controlled output. The controller performances have been evaluated in simulation by considering both tracking and disturbance rejection problems. The robustness of the controller has also been studied when the string tension is subjected to sudden variations.     

Steady Gap Flows by the Spectral and Mortar Element Method

This work aims at observing the effect of the mortar element method applied to a geometry requiring refinement in the vicinity of singularities induced by the presence of sharp corners. We solve the two-dimensional incompressible Navier–Stokes equations with a spectral element method. Mortar elements allow for local polynomial refinement, since they allow for functional nonconformity. The problem solved is the flow in a channel partially obstructed by an obstacle representing a rectangular blade.     

A Robust Spectral Element Method for Simulations of Time-Dependent Viscoelastic Flows, Derived from the Brownian Configuration Field Method

This paper is an extension of previous work [4], where a robust numerical method derived from the Brownian configuration field method [8] was introduced in order to simulate the flows of dilute polymeric solutions. In [4], we limited our study to solutions of dumbbells having infinite extensibility (Oldroyd-B model), whereas in this paper, we tackle the more difficult problem of dumbbells having finite extensibility (FENE-P model).     

A Spectral Mortar Element Discretization of the Poisson Equation with Mixed Boundary Conditions

In this paper, we study a spectral mortar element discretization of the Poisson equation on a square subject to mixed boundary conditions of Dirichlet and Neumann type. We carry out the numerical analysis of the method and derive error estimates. An efficient algorithm for the solution of the problem is proposed and numerical tests confirming the theoretical results are presented.     

一种自适应的图像平滑技术

图像平滑是大多数图像分析和计算机视觉问题中必需的环节。文中探讨了噪声图像的噪声抑制方法，提出了基于各向异性分布方程的平滑技术。该方法的优点在于可以在消除噪声的同时有效地保持空间分辨率。最后采用了真实数据验证了方法的有效性。     

Analysis of a Discontinuous Least Squares Spectral Element Method

This paper addresses the development of a Discontinuous Spectral Least-Squares method. Based on pre-multiplication with a mesh-dependent function a discontinuous functional can be set up. Coercivity of this functional will be established. An example of the approximation to a continuous solution and a solution in which a jump is prescribed will be presented. The discontinuous least-squares method preserves symmetry and positive definiteness of the discrete system.     

一种基于偏微分方程的图像超分辨方法

在图像超分辨方法中,经典插值方法存在块状效应和图像模糊的情况,而以经典插值方法为基础,用偏微分方程进行后处理的方法则较为复杂。运用各向同性方程与各向异性方程进行耦合得到了一个新的偏微分方程模型,用该模型对图像进行超分辨处理,既降低了计算复杂度,又能保证得到较好的图像超分辨效果。实验结果表明了本文方法的优越性。视觉感知具有良好的一致性。     

基于梯度场的拼接缝消除方法

提出了一种基于图像梯度场的拼接缝消除方法--GFBSE.在一般的整体变分模型中引入源图像的梯度场,建立一个基于梯度场的能量函数,通过求解非线性偏微分方程以优化该能量函数来实现拼接缝消除.该方法能够在全局消除图像拼接中的颜色不一致,同时能够较好地处理由于图像配准造成的几何结构的错位.最后与当前常用的几种方法进行了理论和实验上的讨论和比较,表明了GFBSE方法的有效性.     

分布系统的频域极点配置方法

本文就一类分布参数系统的有限极点配置问题进行了研究.文中给出了两种闭环控制设计方法,由此得到的控制算法简单、结构紧凑可实现,并适用于自身不稳定的对象.     

多小波在一维有限元方法中的应用

有限元方法（FEM）是建立在变分原理基础上的一种频域数值计算方法。其基函数的选取相当重要,既影响到计算结果的精度也影响到计算效率。通常情况下,都是利用拉格朗日线性插值函数作为基函数。文中利用了多尺度函数。由于多尺度函数及它的偏导数的差值特性,可以快速逼近某个函数。同时这个新的基函数的一阶偏导数在相邻节点上是连续的。最后得到的数值结果显示：在保证一定计算效率的基础上,使得精度大幅度提高。因此采用多尺度函数作为基函数具有很多优势。