离散系统运动方程的Galerkin有限元EEP法自适应求解

对于结构动力分析中的离散系统运动方程,现有算法的计算精度和效率均依赖于时间步长的选取,这是时间域问题求解的难点.基于EEP(element energy projection)超收敛计算的自适应有限元法,以EEP超收敛解代替未知真解,估计常规有限元解的误差,并自动细分网格,目前已对诸类以空间坐标为自变量的边值问题取得成功.对离散系统运动方程建立弱型Galerkin有限元解,引入基于EEP法的自适应求解策略,在时间域上自动划分网格,最终得到所求时域内任一时刻均满足给定误差限的动位移解,进而建立了一种时间域上的新型自适应求解算法.     

基于EEP法的一维有限元自适应求解

基于新近提出的一维有限元后处理超收敛算法——单元能量投影（EEP）法,将有限元自适应求解问题转化为对超收敛解答的自适应分段多项式插值问题;对于大多数问题,一步便可获得满意的有限元网格划分,在该网格上再次进行有限元计算,一般即可获得满足用户给定的误差限的有限元解答．即便未能完全满足精度要求,一般只需局部细分加密网格一至二步即可．该法简单实用、高效可靠,是一个颇具优势和潜力的自适应方法．以二阶椭圆型常微分方程模型问题为例,对该法的基本思想、实施策略及具体算法做一介绍,并给出有代表性的数值算例用以展示该法的优良性能和效果．     

杆件轴向受迫振动的Galerkin有限元EEP法自适应求解

基于单元能量投影(element energy projection,EEP)法自适应分析在杆件静力问题以及离散系统运动方程组中所取得的成果,以直杆轴向受迫振动为例,研究并建立了一种在时间域和一维空间域同时实现自适应分析的方法.该方法在时间和空间两个维度都采用连续的Galerkin有限元法(finite element method,FEM)进行求解,根据半离散的思想,由空间有限元离散将模型问题的偏微分控制方程转化为离散系统运动方程组,对该方程组进行时域有限元自适应求解;然后再基于空间域超收敛计算的EEP解对空间域进行自适应,直至最终的时空网格下动位移解答的精度逐点均满足给定误差限要求.文中对其基本思想、关键技术和实施策略进行了阐述,并给出了包括地震波输入下的典型算例以展示该法有效可靠.     

Timoshenko梁单元超收敛结点应力的EEP法计算

将新近提出的单元能量投影(Element Energy Projection,简称EEP)法应用于Timoshenko梁单元的超收敛结点应力计算．根据单元投影定理具体推导了一般单元的计算公式,并对两个有代表性的单元给出了数值算例．分析和算例表明,EEP法对于解答是向量函数(即常微分方程组)的问题具有同样优良的表现,不仅能给出与结点位移精度同阶、同量级的超收敛结点应力,而且在位移出现了剪切闭锁的情况下仍能有效地克服应力的剪切闭锁．该研究为EEP法广泛应用于一般的一维常微分方程组问题的有限元解答的超收敛计算打下了良好的基础．     

基于改进位移模式的二维有限元线法超收敛算法

提出了基于改进位移模式的二维有限元线法超收敛算法．利用单元内部需满足平衡方程的条件,推导了超收敛计算的解析公式的显式,即将高阶有限元线法解的位移模式用常规有限元线法解的位移模式表示．用常规有限元线法解的位移模式与高阶有限元线法解的位移模式之和构造新的位移模式,基于线性形函数,采用变分形式推导了有限元线法求解的修正的常微分方程组．该算法在前和后处理同时使用超收敛计算公式,在原有试函数的基础上,增加了高阶试函数．使得单元内平衡方程的残差减少,从而达到提高精度的目标．对于二维Poisson方程问题,给出了有代表性的算例,结点和单元内的位移、导数的收敛精度得到了极大的提高．     

一维有限元后处理的EEP法的数学分析

利用一维投影型插值与有限元超收敛基本估计,对一类两点边值问题,严格证明了袁驷等人由单元能量投影(EEP)法获得的节点恢复导数,当有限元空间的次数不超过4时,具有最佳阶超收敛．理论分析圆满地解释了已有的数值结果．     

加罚Navier—Stokes方程的最佳非线性Galerkin算法

该文提出了求解二维加罚Navier-Stokes方程的最佳非线性Galerkin算法．这个算法在于在粗网格有限元空间上求解一非线性子问题，在细网格增量有限元空间Wh上求解一线性子问题．如果线性有限元被使用及，则该算法具有和有限元Galerkin算法同阶的收敛速度．然而该文提出的算法可以节省可观的计算时间．     

一阶双曲问题的间断流线扩散法

1．引言众所周知，求解一阶双曲问题的Galerkin有限元法，仅具有次最优LZ一收敛阶估计，难于建立H＼误差估计【‘1，且Gderkill有限元解常呈现伪数值振荡．为改善计算精度与稳定性，诸多非标准有限元解法相继提出，其中，间断（Discontinuous）Galerkin有限元法（以下简称DG方法）与流线扩散（StreamlineDiffusion）有限元法（以下简称SD法）是两种具有鲜明特点，较为成功的算法．具体地，DG方法是一种迎风型显式算法，它从入流边界开始，沿流场方向，自上游往下游，逐个单元进行解算，计算十分简便且可局部并行化．SD方法则是一种P…     

Cahn-Hilliard方程的有限元分析

建立了求解非线性发展型Cahn-Hilliard方程的有限元方法,借助于一个双调和问题的有限元投影逼近,给出了最优阶L_2模误差估计。特别对于3次Hermite型有限元,导出了L_∞模和W_∞~1模的最优阶误差估计和导数逼近的超收敛结果。     

拟等级网格下非线性延迟微分方程间断有限元法

本文利用间断有限元法求解非线性延迟微分方程,在拟等级网格下.给出非线性延迟微分方程间断有限元解的整体收敛阶和局部超收敛阶,数值实验验证了理论结果的正确性.     

Implementation and efficiency analysis of an adaptive <Emphasis Type="Italic">hp</Emphasis>-finite element method for solving boundary value problems for the stationary reaction–diffusion equation

An iterative process implementing an adaptive hp-version of the finite element method (FEM) previously proposed by the authors for the approximate solution of boundary value problems for the stationary reaction–diffusion equation is described. The method relies on piecewise polynomial basis functions and makes use of an adaptive strategy for constructing a sequence of finite-dimensional subspaces based on the computation of correction indicators. Singularly perturbed boundary value test problems with smooth and not very smooth solutions are used to analyze the efficiency of the method in the situation when an approximate solution has to be found with high accuracy. The convergence of the approximate solution to the exact one is investigated depending on the value of the small parameter multiplying the highest derivative, on the family of basis functions and the quadrature formulas used, and on the internal parameters of the method. The method is compared with an adaptive h-version of FEM that also relies on correction indicators and with its nonadaptive variant based on the bisection of grid intervals.     

Error Analysis of SAV Finite Element Method to Phase Field Crystal Model

In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the $L^2$ and $H^{-1}$ gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e., $L^2$ norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.     

求解流固耦合问题的一种四步分裂有限元算法

基于arbitrary Lagrangian Eulerian (ALE) 有限元方法,发展了一种求解流固耦合问题的弱耦合算法．将半隐式四步分裂有限元格式推广至求解ALE描述下的Navier-Stokes（N-S）方程,并在动量方程中引入迎风流线（streamline upwind/Petrov-Galerkin, SUPG）稳定项以消除对流引发的速度场数值振荡;采用Newmark-β法对结构方程进行时间离散;运用经典的Galerkin有限元法求解修正的Laplace方程以实现网格更新,每个计算步施加网格总变形量防止结构长时间、大位移运动时的网格质量恶化．运用上述算法对弹性支撑刚性圆柱体的流致振动问题进行了数值模拟,计算结果与已有结果相吻合,初步验证了该算法的正确性和有效性．     

Alternating direction implicit OSC scheme for the two-dimensional fractional evolution equation with a weakly singular kernel

In this paper, a new kind of alternating direction implicit (ADI) Crank-Nicolson-type orthogonal spline collocation (OSC) method is formulated for the two-dimensional fractional evolution equation with a weakly singular kernel arising in the theory of linear viscoelasticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for the temporal component. The stability of proposed scheme is rigourously established, and nearly optimal order error estimate is also derived. Numerical experiments are conducted to support the predicted convergence rates and also exhibit expected super-convergence phenomena.     

Static and dynamic analysis of two-layer Timoshenko composite beams by weak-form quadrature element method

To improve the efficiency in predicting the dynamic mode and static response of the two-layer partial interaction composite beams, this paper utilizes the differential quadrature technique to approximate derivatives of the primary unknowns with adaptive order of precision, rather than the low and constant order of interpolation used in the conventional finite element method (FEM). A degree-of-freedom-adaptive weak-form quadrature element (WQE) for dynamic analysis is formulated and implemented based on the principle of virtual work. For the purpose of comparison, a parabolic displacement-based finite element is also provided, thus (1) the predicted deflections and natural frequencies of the composite beams are verified; (2) the smoothness of the internal forces and stresses generated by WQE method and FEM are compared, and (3) the convergent rates of higher order free vibration modes are also examined. Numerical results show that the efficiency of the proposed WQE method has, on the one hand, significantly triumphed over that of FEM on analyses including static response, natural frequencies and higher order free vibration modes, on the other hand, the smoothness of results, including internal forces and stresses, is greatly refined.