一维C~1有限元EEP超收敛位移计算简约格式的误差估计

该文对一维C1有限元后处理超收敛计算的EEP(单元能量投影)法简约格式中的位移解给出误差估计的数学证明,即对足够光滑问题的m(3)次单元的有限元解答,采用EEP法简约格式得到的单元内任一点位移超收敛解均可以达到hm+2的收敛阶,比常规有限元位移解的收敛阶至少高一阶。     

基于EEP法的三维有限元超收敛计算初探

二维有限元法（FEM）的超收敛计算,借助有限元线法（FEMOL）作为桥梁,分两步采用单元能量投影（EEP）法导出超收敛公式,初步形成“逐维离散、逐维恢复”的方案。然而这一思路直接应用于三维问题却遇到了困扰:一维问题的EEP解（位移和导数）均可达到相同的超收敛阶,而二维问题却难以做到。研究发现,为了得到三维问题的EEP超收敛位移,只需提供二维问题最低阶的超收敛位移即可。该文按此思路推导了非规则网格下三维六面体单元的EEP超收敛位移公式,给出了一个实施方案,并通过数值算例验证了此方案的有效性。     

二维变分不等式问题的自适应有限元分析

有限元后处理超收敛计算的EEP(单元能量投影)法以及基于该法的自适应有限元分析已在一维变分不等式问题的求解中取得显著成功。以此为基础,该文对二维变分不等式问题成功地实现了自适应有限元分析。该文提出二维区域二分法和二维C 检验技术,有效地提升了松弛迭代的收敛速率,进而应用EEP 超收敛公式计算超收敛解答,用其检验误差并指导网格细分。该文给出的典型数值算例表明该文算法高效、稳定、精确,解答可逐点以最大模度量满足精度要求,堪称为数值精确解。     

二维自由振动问题的自适应有限元分析初探

自由振动反映结构动力特性,是抗震分析和结构设计的重要基础。近年来,基于单元能量投影（EEP）法的自适应有限元分析已在一系列线弹性及非线性问题中取得成功,而有限元线法（FEMOL）自适应分析在二维自由振动问题中的应用也被证实是有效的。在此基础上,该文进一步提出二维自由振动问题的自适应有限元分析方法。通过将特征值问题线性化,合理引入二维线性问题的EEP超收敛计算和自适应求解技术,该法可得到满足精度要求的自振频率和按最大模度量满足用户给定误差限的振型。该文以弹性薄膜为例,介绍了这一进展,并给出数值算例以表明该方法的有效性和可靠性。     

一维EEP自适应技术新进展:从线性到非线性

有限元后处理中超收敛计算的EEP(单元能量投影)法以及基于该法的自适应分析方法对线性ODE(常微分方程)问题的求解已经获得了全面成功,也推动了非线性ODE问题自适应求解的研究。经过研究,已经实现了一维有限元自适应分析技术从线性到非线性的跨越,该文意在对这方面的进展作一简要综述与报道。该文提出一种基于EEP法的一维非线性有限元自适应求解方法,其基本思想是通过线性化,将现有的线性问题自适应求解方法直接引入非线性问题求解,而无需单独建立非线性问题的超收敛计算公式和自适应算法,从而构成一个统一的、通用的非线性问题自适应求解算法。该文给出的数值算例表明所提出的算法高效、稳定、通用、可靠,解答可逐点按最大模度量满足用户给定的误差限,可作为先进高效的非线性ODE求解器的核心理论和算法。     

具有最佳超收敛阶的EEP法计算格式:Ⅲ数学证明

对一维C0问题的高次有限元后处理中超收敛计算的EEP(单元能量投影)法提出改进的最佳超收敛计算格式,即用m次单元对足够光滑问题的有限元解答,采用该格式计算的任一点的位移和应力都可以达到h2m阶的最佳超收敛结果。整个工作分为3个部分,分别给出算法公式、数值算例和数学证明。该文是系列工作的第三部分,对所提出的最佳的EEP超收敛格式给出数学证明。     

固端法:二维有限元先验定量误差估计与控制

该文基于有限元超收敛计算的单元能量投影(Element Energy Projection,简称EEP)法,尝试将一维有限元中新近提出的先验定量误差估计的“固端法”拓展到二维有限元分析,以Poisson方程为例,用EEP公式预先估算出各单元的误差,可以不经有限元求解计算而直接给出满足精度要求的网格划分。该文给出的初步数值算例验证了该法的有效性。     

一维Ritz有限元超收敛计算的EEP法简约格式的误差估计

该文对一维问题Ritz有限元后处理超收敛计算的EEP(单元能量投影)法简约格式给出误差估计的数学证明,即对足够光滑问题的(>1)次单元的有限元解答,采用EEP法简约格式计算得到的单元内任一点位移和应力(导数)超收敛解均可以达到的收敛阶,即位移比常规有限元解的收敛阶至少高一阶,而应力则至少高二阶。     

膜结构极小曲面找形的一种自适应有限元分析

找形分析是膜结构设计中的关键环节,但在数学上,膜结构的极小曲面找形分析是一个高度非线性问题,一般无法求得其解析解,因此数值方法成为重要工具。近年来,基于单元能量投影法（EEP法）的一维非线性有限元的自适应分析已经取得成功,基于EEP法的二维线性有限元自适应分析也被证实是有效、可靠的。在此基础上,该文提出一种基于EEP法的二维非线性有限元自适应方法,并成功将之应用于膜结构的找形分析。其主要思想是,通过将非线性问题用Newton法线性化,引入现有的二维线性问题的自适应求解技术,进而实现二维有限元自适应分析技术从线性到非线性的跨越,将非线性有限元的自适应分析求解从一维问题拓展到二维问题。该方法兼顾求解的精度和效率,对网格自适应地进行调整,最终得到优化的网格,其解答可按最大模度量逐点满足用户设定的误差限。该文综述介绍了这一进展,并给出数值算例用以表明该方法的可行性和可靠性。     

具有最佳超收敛阶的Galerkin有限元EEP法计算格式

对二阶非自伴问题的一维Galerkin有限元法提出其后处理超收敛计算的EEP(单元能量投影)法改进的最佳超收敛计算格式,即用m次单元对足够光滑问题的Galerkin有限元解答,采用该格式计算的任一点的位移和应力都可以达到h2m阶的最佳超收敛结果。该文首先针对高次单元提出了凝聚试探形函数和凝聚检验形函数的概念,证明了相关的逼近定理和等价定理,然后给出了具体的算法公式。最后给出了一系列典型的数值算例用以验证这种最新的EEP法改进格式确实能够使位移和导数逐点达到最佳收敛阶。     

H(curl)空间中椭圆最优控制问题的自适应有限元算法

针对H(curl)空间椭圆型最优控制问题提出了一种自适应有限元方法。首先将H(curl)空间Maxwell方程的最优控制模型转化为偏微分方程组,给出了偏微分方程组的解得正则性。其次利用自适应有限元方法求解此偏微分方程组,同时讨论了方法的后验误差及收敛性。最后通过数值算例给出了该方法的数值结果,验证了有限元方法的有效性和可靠性。这一方法可以应用于更复杂的最优控制问题的求解。     

弹性力学问题自适应有限元及其局部多重网格法

针对平面弹性力学问题,利用最新顶点二分法,设计了一种不需要标记振荡项和加密单元不需要满足“内节点”性质的自适应有限元法;利用自适应加密过程中每层网格上只有局部单元需要加密这一特性,设计了一种基于局部松弛的多重网格法.数值实验结果表明:该文设计的自适应有限元法具有一致收敛性和拟最优计算复杂度,基于局部松弛的多重网格法对求解弹性力学问题自适应网格下的有限元方程具有很好的计算效率和鲁棒性.     

关于单元能量投影法的两点注记

最近,袁驷等基于力学原理提出了一种一维有限元超收敛后处理计算格式,称为单元能量投影(EEP)法。大量数值例子显示:若真解充分光滑,对m次有限元解,EEP法后处理节点恢复导数具有h2m阶精度。首先利用限元超收敛理论中的一个基本估计式证明了线性元(m=1)节点恢复导数具有h2阶精度。另外,对EEP法高次元的内点计算公式提出了一点简化。     

基于EEP技术的一维有限元结点位移误差计算

Using super-convergent solutions calculated by the Element Energy Projection (EEP) method, equivalent nodal load vectors from the residual load term were derived in this paper without changing the finite element (FE) meshes and the global stiffness matrices. The subsequent back-substitutions can generate highly accurate estimates for the errors of nodal displacements and hence greatly improve the nodal accuracy. Taking a general second-order ordinary differential equation as the model problem, the algorithm of the proposed method and associated numerical examples were given to show that the proposed method is simple and effective, and that using elements of degree m≥1, the improved nodal displacements can gain the super-super-convergence orders h2m+2 and h3m+mod(m, 2) for simplified and condensed EEP forms, respectively. A variety of significant further extensions and applications were also discussed.     

一维C~1有限元超收敛解答计算的EEP法

将新近提出的C0有限元后处理中超收敛解答计算的单元能量投影(Element Energy Projection,简称EEP)法推广到一维C1类有限元。根据单元投影定理具体推导了一般梁单元的计算公式,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法在一维C1类有限元中再次获得令人满意的效果,即对任一单元中的任一点,从位移一直到三阶导数(如梁的挠度、转角、弯矩、剪力),匀可获得与结点位移精度相当的超收敛结果,而且可精确满足自然边界条件。     

Smart computational fracture of materials and structures

The use of the finite element method for complex engineering problems is now common. To ease the burden on the engineer the development of smart or adaptive computational methods is now required to model complex problems. In this paper we investigate the development of an adaptive finite element method for fracture-related problems. The adaptive method involves various stages which include the finite element analysis, error estimation/indication, mesh refinement and fracture/failure analysis in a loop. Some simple error estimators, based on stress projection, are used to investigate the adaptive finite element process. Element refinement is based on three schemes; the first and second are a simple and hierarchical refinement scheme with transitioning which avoids the need for constraint equations between element boundaries. Another scheme based on constraint equations between elements is also examined. The energy norm is used to estimate the element error. The software has the ability to introduce a discrete fracture in the structure according to standard fracture analysis practice. Crack tip parameters are calculated using a least-squares fit of the displacements into the asymptotic crack tip displacement field. Some simple examples are used to investigate the adaptive process, its behavior and some of the practical problems encountered. The convergence and equilibrium of the adaptive process, in terms of global error in the energy norm, are investigated. In the example the same problem is analyzed using both a fine computational grid and a coarse one. The coarse mesh is then adapted using the three different procedures available. The estimated error in the solution and the stress intensity are shown against the number of elements and number of iterations. Some further areas of research in adaptive finite element analysis are discussed.     

Automatic adaptive refinement for plate bending problems using Reissner–Mindlin plate bending elements

The influence of the presence of singular points and boundary layers associated with the edge effects in a Reissner–Mindlin (RM) plate in the design of an optimal mesh for a finite element solution is studied, and methods for controlling the discretization error of the solution are suggested. An effective adaptive refinement strategy for the solution of plate bending problems based on the RM plate bending model is developed. This two-stage adaptive strategy is designed to control both the total and the shear error norms of a plate in which both singular points and boundary layers are present. A series of three different order assumed strain RM plate bending elements has been used in the adaptive refinement procedure. The locations of optimal sampling points and the effect of element shape distortions on the theoretical convergence rate of these elements are given and discussed. Numerical experiments show that the suggested refinement procedure is effective and that optimally refined meshes can be generated. It is also found that all the plate bending elements used can attain their full convergence rates regardless of the presence of singular points and boundary layers inside the problem domain. Boundary layer effects are well captured in all the examples tested and the use of a second stage of refinement to control the shear error is justified. In addition, tests on the Zienkiewicz–Zhu error estimator show that their performances are satisfactory. Finally, tests of the relative effectiveness of the plate bending elements used have also been made and it is found that while the higher order cubic element is the most accurate element tested, the quadratic element tested is the most efficient one in terms of CPU time used. © 1998 John Wiley & Sons, Ltd.     

Efficient adaptive integration of functions with sharp gradients and cusps in n‐dimensional parallelepipeds

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a C⁰ function (derivative discontinuity at a point), a rate of convergence of n + 1 is obtained in . Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function—a strongly localized function that is used for modeling sharp gradients. Then the adaptive quadratures are employed in the enriched finite element solution of the all‐electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in n‐dimensional parallelepiped elements. Copyright © 2012 John Wiley & Sons, Ltd.