一类新的分层基以及非线性Galerkin方法

本文首先讨论了一类新的分层基函数，该基函数具有小波和有限元两方面的优点，特别适用于用来构造耗散型发展方程的非线性Ｇａｌｅｒｋｉｎ算法．本文利用该基函数对一类抽象发展方程给出了其非线性Ｇａｌｅｒｋｉｎ算法。并分析了该算法的收敛性，给出了误差估计。     

半离散非线性Galerkin算法的收敛性

本文给出了求解非线性发展方程的半离散非线性Ｇａｌｅｒｋｉｎ算法，即用非线性Ｇａｌｅｒｋｉｎ谱元法对空间变量进行离散。     

功率失配误差限的几种表达方式

不确定度评定中常需对功率失配的不确定度进行评定,这就需要我们了解其误差限。实际情况中,有多种功率失配误差限的表达方式,它们的本质相同,形式不同,故在此做一下分析和总结。     

加罚纳维尔——斯托克斯方程的全离散非线性伽辽金方法

讨论了非线性伽辽金方法全离散格式用于加罚纳维尔－斯托克斯方程，这里用到了近似惯性流形方法和算子分裂方法，并证明了方法的收敛性和误差估计。     

冻结与解浆过程热分析的Galerkin方法

本文建立了二维轴对称冻结问题的Ｇａｌｅｒｋｉｎｇ该有限元数值分析法,利用该方法可分析一维问题及具有轴对称特性的任何复杂形状和复杂换热条件４的二维冻结及解冻问题。证明该方法具有精度高,收敛速度快和稳定性好的特点,是处理非线性问题的有效方法。     

非线性Galerkin方法在梁的非线性动力学性态分析中的应用

本文将非线性Ｇａｌｅｒｋｉｎ方法引入到梁的非线性动力学性状分析之中，首先，将描述梁振动的非线性偏微分方程转化为三个不同的离散化模型，其中模型Ⅱ是基于非线性Ｇａｌｅｒｋｉｎ方法得到的，它由一组常微分方程和一组代数方程组成，其次，对两组不同的轴向力，分别计算分析了上述三个模型的长期性态（时间历程，频谱，相平面图）最后，对所得结果进行了讨论。     

Helmholtz方程基于变限积分法的数值求解

Helmholtz方程是一类描述电磁波的椭圆型偏微分方程,在力学、声学和电磁学等领域应用广泛。为了消除因高波数引起的污染效应,数值求解Helmholtz方程的传统方法是对网格进行加密,网格加密不仅增加了时间复杂度,且离散后的矩阵通常是病态的。因此,寻求对任意波数都有效的方法是必要的。在有限体积法的基础上,引入变限因子,将微分方程完全转换成积分方程,利用一元三点和二元九点Lagrange插值公式,构造含三对角矩阵的离散格式,分别对一维和二维Helmholtz方程进行变限积分法的数值求解。该方法适用于任意波数,求解过程物理意义明确,数值格式简单。对于一维Helmholtz方程研究了变限因子对误差的影响,利用Taylor展式及Lagrange插值余项公式进行误差估计,证明离散格式的截断误差达到二阶。数值实例表明该离散格式的变限因子和步长相等时,误差阶较低。对二维Helmholtz方程,探究不同波数对数值解的影响,证明离散格式的截断误差达到三阶。数值实例表明,对于不同的波数,数值格式都有较好的精度,高波数没有引起污染效应。     

激光测振仪误差限的测量不确定度

介绍了评价激光测振仪误差限指标时,测量结果的不确定度分析和评价过程;讨论了影响评价结果不确定度的几个主要误差来源,包括信号源误差、测量仪器的测量误差、测量重复性的影响等等.在一个实际评价例子上,给出了误差限指标不确定度分析和评价结果;该过程及结论可应用在对于计量标准进行误差限指标的不确定度分析上,也可用于估计误差限指标本身的测量不确定度.     

对JJG577—94中基本误差限和检定流量点规定的探讨

新规程ＪＪＧ５７７－９４中的基本误差限规定与标准不符，检定Ｑｍｉｎ流量点的规定与实际情况不符。建议修改。     

基于多重有限体积法解决含有耗散项的双曲扩散问题

双曲扩散方程在数学上是一类重要的偏微分方程,在众多工程领域中有着广泛的应用.其常用于描述声波和电势场中电流的传播问题,也用于模拟计算流体力学中的对流扩散以及热传导模型.本文研究了一类具有耗散项的双曲扩散方程,并通过数值方法对其初边值问题进行求解.针对已有的有限体积法在求解该问题时精度无法提高的问题,本文基于变限积分的方法提出了新的高精度有限体积格式,并利用Fourier分析法得到其离散格式无条件稳定的结论,最后利用数值实验证实了理论分析结果.     

四阶Cahn-Hilliard方程的间断有限元方法

Cahn-Hilliard方程是一类非常重要的四阶扩散方程,具有深刻的物理背景和丰富的理论内涵,对其设计高精度的数值格式具有重要的工程实践价值和科学意义．在本文中我们对四阶Cahn-Hilliard方程设计一种高精度的间断有限元,该方法不同于传统的局部间断有限元方法,不需要引进另外的辅助变量或将方程转化为一阶方程组,能够显著降低计算量和存储量．通过选取合适的数值流通量,我们证明了方法的稳定性和收敛性．数值实验结果表明该方法求解Cahn-Hilliard方程是收敛的和有效的．     

Discontinuous Galerkin method for the forward modelling in optical diffusion tomography

We propose a discontinuous Galerkin discretization scheme for the forward modelling in optical diffusion tomography in highly scattering media to facilitate dynamic mesh adaptation for complex domains. In addition, the numerical method is also shown to effectively deal with inhomogeneities in optical properties and refractive index mismatch at material interfaces. The accuracy of the method is demonstrated over a model concentric spherical layers problem where the discontinuous Galerkin method is compared against an analytical solution. Copyright © 2010 John Wiley & Sons, Ltd.     

Discontinuous Galerkin method based on peridynamic theory for linear elasticity

Modeling of discontinuities (shock waves, crack surfaces, etc.) in solid mechanics is one of the major research areas in modeling the mechanical behavior of materials. Among the numerical methods, the discontinuous Galerkin method (DGM) poses some advantages in solving these problems. In this study, a novel formulation for DGM is derived for elastostatics based on the peridynamic theory. Derivation of the proposed formulation is presented. Numerical analyses are performed for different problems, and the numerical results are compared to that of the known exact solutions of the problems. The proposed weak formulation is stable and coercive. Peridynamic discontinuous Galerkin formulation is found to be robust and successful in modeling elastostatic problems. Copyright © 2011 John Wiley & Sons, Ltd.     

粘弹流动的间断有限元模拟

针对传统有限元法求解Oldroyd-B本构方程时需加入稳定化方案的缺点,本文基于非结构网格给出了统一间断有限元求解框架.该框架包含采用IPDG(interior penalty discontinuous Galerkin)求解质量方程和动量方程,与采用RKDG(RungeKutta discontinuous Galerkin)求解本构方程这两个核心.数值结果表明:该方法在求解Oldroyd-B本构方程时无需加入稳定化方案,实施比有限元法简便,且具有较高的计算精度,可有效地模拟包含应力奇异点的复杂粘弹流动问题,进而揭示非牛顿粘弹流动的基本特征.     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.     

不连续平面的平面度误差评定方法研究

提出了一种基于包容直线评定不连续平面的方法.首先,基于主成分分析对点云数据进行全局噪声剔除;然后,采用最小二乘平面拟合算法、多阈值点云提取算法及平行直线拟合算法获取符合条件的包容直线,并以空间中2条平行的包容直线代替包容平面建立最小区域;最后实现不连续平面的平面度误差评定及指导平面调整.实验结果表明:该方法与传统的特征...     

自然对流问题两重网格算法的残量型后验误差估计(英文)

本文得到了自然对流问题基于牛顿迭代两重网格算法的残量型后验误差估计.相对于标准有限元一层方法的后验误差估计,牛顿迭代两重网格算法的后验误差估计多了一些额外项.通过研究这些额外项的渐近行为,本文得到了这些额外项在误差估计中所起的作用.对于牛顿迭代两重网格方法的最优粗细网格匹配尺寸,这些额外项的收敛阶不高于离散解的收敛阶.数值算例验证了理论分析结论.     

Stability properties of the Discontinuous Galerkin Material Point Method for hyperbolic problems in one and two space dimensions

In this paper, stability conditions are derived for the Discontinuous Galerkin Material Point Method (DGMPM) on the scalar linear advection equation for the sake of simplicity and without loss of generality for linear problems. The discrete systems resulting from the application of the DGMPM discretization in one and two space dimensions are first written. For these problems, a second-order Runge-Kutta and the forward Euler time discretizations are respectively considered. Moreover, the numerical fluxes are computed at cell faces by means of either the Donor-Cell Upwind or the Corner Transport Upwind methods for multidimensional problems. Second, the discrete scheme equations are derived assuming that all cells of a background grid contain at least one particle. Although a Cartesian grid is considered in two space dimensions, the results can be extended to regular grids. The von Neumann linear stability analysis then allows the computation of the critical Courant number for a given space discretization. Although the DGMPM is equivalent to the first-order finite volume method if one particle lies in each element, so that the Courant number can be set to unity, other distributions of particles may restrict the stability region of the scheme. The study of several configurations is then proposed.     

可靠性工程中参数的一种估计方法

提出了可靠性工程中参数的一种估计方法——新Bayes估计法，给出了失效概率、失效率的新Bayes估计的定义及其新Bayes估计。最后，结合实际问题的数据，进行了具体计算和分析，结果表明所提出的新Bayes估计法有效、可行，便于工程技术人员在工程中应用。     

对流扩散方程的特征线EFG算法

为了消除对流扩散方程因对流占优引起的数值震荡,本文首先将其转化为特征形式,并利用移动最小二乘基函数,构建了特征线无单元Galerkin方法．再对新建方法进行收敛性分析,分别给出关于支持域半径和时间步长的两种误差估计．最后,分别针对一维和二维算例进行了数值计算,并与有限元法进行了比较．数值结果表明,本文算法收敛性好,可以消除数值震荡,且通过选取合适的罚因子和支持域的无量纲尺寸,计算精度比有限元法更高,是求解对流占优扩散方程的一种有效程数值计算方法．