二维泊松方程问题Lagrange型有限元p型超收敛算法

该文针对二维泊松方程问题的Lagrange型有限元法提出了一种p型超收敛算法。该法受有限元线法对二维问题降维思想的启发,基于网格结点位移的天然超收敛性,通过从网格中取出一行对边相邻的单元作一子域,将子域内各单元另一对边解答取为原有限元解答,在子域上建立真解近似满足的局部偏微分方程边值问题,对该局部边值问题,沿对边方向单向提高单元阶次进行有限元求解获得单元对边上的超收敛解。单元另一对边上的超收敛解可通过另一方向的单元行类似获得。在单元边超收敛解的基础上,依次取出各个单元,以单元边位移超收敛解为Dirichlet边界条件,双向提高单元阶次对原泊松方程问题进行有限元求解即可获得全域超收敛解。数值算例表明,通过简单的后处理计算本法可显著提高解答的精度和收敛阶。     

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。     

基于EEP超收敛解的自适应有限元法特性分析

基于EEP （单元能量投影）超收敛计算的自适应有限元法,已对一系列问题取得成功,但其自适应特性尚缺乏相关研究。该文以二阶常微分方程为模型问题,同时考察基于EEP和SPR （超收敛分片恢复）超收敛解的自适应分析方法,与有限元最优网格进行了比较分析,进而提出反映自适应有限元收敛特性的估计式,并给出了自适应收敛率β的定义。该文给出的数值试验表明:采用m次单元,对于解答光滑的问题,SPR法与EEP法均可有效用于自适应求解,其位移可按最大模获得m+1的自适应收敛率;对于奇异因子为α（<1）的奇异问题,SPR法失效,而基于EEP法的自适应求解,其位移按最大模可获得m+α的自适应收敛率,远高于α的常规有限元收敛率。     

基于p型超收敛计算的一维有限元自适应分析

基于提高单元阶次的p型超收敛算法,可以在有限元解答基础上求得超收敛解。用该超收敛解代替精确解可以对有限元解答进行可靠的误差估计。对Zienkiewicz网格划分策略进行一定的改进,得到一种更有效的网格划分策略。基于可靠的误差估计和高效的网格划分,可以进行有限元自适应求解。数值试验表明,该文的自适应求解方案能够得到较优的网格和满足误差限的解答。     

变截面变曲率梁振型的有限元超收敛拼片恢复解和网格自适应分析

该文提出变截面变曲率梁振型的有限元后处理超收敛拼片恢复方法,建立各阶振型的超收敛解,并基于振型超收敛解进行变截面曲梁面内和面外自由振动的自适应分析。在位移型有限元后处理阶段,引入超收敛拼片恢复方法和高阶形函数插值技术,得到振型(位移)的超收敛解。利用振型超收敛解估计当前网格下振型有限元解的能量模形式下的误差,并指导网格进行自适应细分加密分析,获得优化的网格和满足预设误差限的高精度解答。数值算例表明该算法适于求解不同曲线形态、多类边界条件、变截面、变曲率形式的曲梁面内和面外自由振动连续阶频率和振型,解答精确、分析过程高效可靠。     

自适应有限元-离散元算法、ELFEN软件及页岩体积压裂应用

该文介绍流体-固体-断裂耦合分析的自适应有限元（FE）-离散元（DE）算法,引进一款新近基于该方法研发的数值计算软件ELFEN,并将其应用于页岩分段体积压裂的三维数值计算和机理分析。该方法引入有限元应力恢复的超收敛拼片恢复（SPR）法,获得应力的超收敛SPR解,利用SPR解估计常规有限元解的误差,通过裂纹尖端局部区域的自适应网格重划分获得高精度应力解答并得以有效描述裂纹动态扩展,形成分析策略和求解方案。数值算例表明该算法和软件分析流体-固体-断裂耦合作用下单一、多水平井分段体积压裂的可靠性、有效性和实用性。     

结构自由振动问题有限元新型超收敛算法研究

该文以杆件轴向自由振动问题为例提出一个结构自由振动问题的新型超收敛计算方法。该法基于有限元解答中频率和振型结点位移的超收敛特性,建立了单元上振型近似满足的线性常微分方程边值问题,对该线性边值问题采用更高次数的多项式进行有限元求解获得各单元上振型的超收敛解,将振型的超收敛解代入Rayleigh商,获得结构频率的超收敛解。该法简单、直接,通过很少量的计算即能显著提高频率和振型的精度和收敛阶。数值算例显示,该法高效、可靠,是一个颇具潜力的新方法。     

一维变分不等式问题的自适应有限元分析新探

结构工程中的弹性薄膜接触和杆件弹塑性扭转等问题是典型的变分不等式问题,对其高效精确求解,特别是满足给定精度要求下的自适应求解,是挑战性课题。该文作者新近成功实现了一维变分不等式问题的自适应有限元分析,该文对此进展作一报道。对于变分不等式的有限元求解,该文提出区域二分法和C检验技术,极大提升了松弛迭代的收敛速度,一般4次~5次线性解即可得到收敛的有限元解答,进而采用作者提出的EEP(单元能量投影)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,逐步得到堪称为数值精确解的解答,亦即得到按照最大模度量逐点满足精度要求的解答。该文给出的数值算例表明所提出的算法具有高效、可靠、精确的优良特性。     

二维有限元线法超收敛解答计算的EEP法

有限元线法(FEMOL)是一种优良的半解析、半离散方法,但其解答存在解析方向和离散方向的精度不相称的弱点。本文提出将二维有限元线法比拟为广义一维问题的概念,遂可将新近提出的一维有限元超收敛计算的单元能量投影(EEP)法推广到二维有限元线法分析中。经有限元线法后处理中EEP超收敛计算而获得的解答,继承和保留了一维有限元中的出色表现,不但使任意一点的位移和应力的解答在两个方向具有相当的精度,而且都具有超收敛性质。文中以二维Poisson方程问题为例,具体给出了有限元线法EEP超收敛的公式,并给出了数值算例,用以表明本法的可行性和有效性。     

平面曲梁面内自由振动有限元分析的p型超收敛算法

该文提出一种求解平面曲梁面内自由振动问题的p型超收敛算法。该法基于有限元解答中频率和振型结点位移的固有超收敛特性,在单个单元上建立了振型近似满足的线性常微分方程边值问题,对该局部线性边值问题采用单个高次元进行有限元求解获得该单元上振型的超收敛解,逐单元计算完毕后,将振型的超收敛解代入Rayleigh商,获得频率的超收敛解。该法为后处理法,且后处理计算仅在单个单元上进行,通过少量计算即能显著提高频率和振型的精度和收敛阶。数值算例表明,该法可靠、高效,值得进一步研究和推广。     

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。     

自适应有限元线法在二维无穷域问题中的应用

无穷域问题广泛存在于实际工程中,半解析、半离散的数值计算方法—有限元线法（Finite ElementMethod of Lines,简称FEMOL）对其具有较好的适应性。在已有的映射型FEMOL无穷单元理论的基础上,基于单元能量投影（Element Energy Projection,简称EEP）法的自适应FEMOL被应用于二维无穷域问题的求解。用户只需输入稀疏的初始网格和误差限,算法即自动生成优化的FEMOL网格,该网格上常规单元和无穷单元的FEMOL解均按最大模度量满足给定误差限。文中首先介绍二维FEMOL的原理策略、无穷单元的构建,然后概述基于EEP法的自适应FEMOL算法,并讨论其对无穷域问题的适用性,之后对圆柱绕流的Poisson方程问题、带孔无穷大板单向拉伸的弹性力学平面问题、受圆形均布荷载半空间体的三维轴对称问题进行了自适应分析,最终不仅给出了满足误差限的函数（位移）解,也给出了具有优良性态的导数（应力）解,从而为无穷域问题的求解提供了一种高效可靠的新途径。     

A locking‐free selective smoothed finite element method using tetrahedral and triangular elements with adaptive mesh rezoning for large deformation problems

A novel finite element (FE) formulation with adaptive mesh rezoning for large deformation problems is proposed. The proposed method takes the advantage of the selective smoothed FE method (S‐FEM), which has been recently developed as a locking‐free FE formulation with strain smoothing technique. We adopt the selective face‐based smoothed/node‐based smoothed FEM (FS/NS‐FEM‐T4) and edge‐based smoothed/node‐based smoothed FEM (ES/NS‐FEM‐T3) basically but modify them partly so that our method can handle any kind of material constitutive models other than elastic models. We also present an adaptive mesh rezoning method specialized for our S‐FEM formulation with material constitutive models in total form. Because of the modification of the selective S‐FEMs and specialization of adaptive mesh rezoning, our method is locking‐free for severely large deformation problems even with the use of tetrahedral and triangular meshes. The formulation details for static implicit analysis and several examples of analysis of the proposed method are presented in this paper to demonstrate its efficiency. Copyright © 2014 John Wiley & Sons, Ltd.     

An edge-based smoothed tetrahedron finite element method (ES-T-FEM) for 3D static and dynamic problems

Strain smoothing operation has been recently adopted to soften the stiffness of the model created using tetrahedron mesh, such as the Face-based Smoothed Finite Element Method (FS-FEM), with the aim to improve solution accuracy and the applicability of low order tetrahedral elements. In this paper, a new method with strain smoothing operation based on the edge of four-node tetrahedron mesh is proposed, and the edge-based smoothing domain of tetrahedron mesh is serving as the assembly unit for computing the 3D stiffness matrix. Numerical results demonstrate that the proposed method possesses a close-to-exact stiffness of the continuous system and gives better results than both the FEM and FS-FEM using tetrahedron mesh or even the FEM using hexahedral mesh in the static and dynamic analysis. In addition, a novel domain-based selective scheme is proposed leading to a combined ES-T-/NS-FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed method is an innovative and unique numerical method with its distinct features, which possesses strong potentials in the successful applications for static and dynamics problems.     

SPH and FEM Investigation of Hydrodynamic Impact Problems

Simulation of hydrodynamic impact problems and its effect on surrounding structures, can be considered as a fluid structure coupling problem. The application is mainly used in automotive and aerospace engineering and also in civil engineering. Classical FEM and Finite Volume methods were the main formulations used by engineers to solve these problems. For the last decades, new formulations have been developed for fluid structure coupling applications using mesh free methods as SPH method, (Smooth Particle Hydrodynamic) and DEM (Discrete Element Method). Up to these days very little has been done to compare different methods and assess which one would be more suitable. In this paper the mathematical and numerical implementation of the FEM and SPH formulations for hydrodynamic problem are described. From different simulations, it has been observed that for the SPH method to provide similar results as FEM Lagrangian formulations, the SPH meshing, or SPH particle spacing needs to be finer than FEM mesh. To validate the statement, we perform a simulation of a hydrodynamic impact on an elasto-plastic plate structure. For this simple, the particle spacing of SPH method needs to be at least two times finer than FEM mesh. A contact algorithm is performed at the fluid structure interface for both SPH and FEM formulations. In the paper the efficiency and usefulness of two methods, often used in numerical simulations, are compared.     

Specific features of FE-meshes for the analysis of metal forming processes

Due to steadily increasing demands on the die forging industry, process simulation is of continuous importance. The Finite Element Method (FEM) is frequently used to simulate forming processes for the purpose of optimising the quality of the final product and predicting the loading on the tool. Most forgoing processes reduce the original height of the workpiece by 60–70%. Therefore, considerable distortions of the finite element mesh occur and some elements can even degenerate. Consequently, the results obtained are unreliable and sometimes it is impossible to pursue the calculation any further. In order to continue the analysis, it becomes necessary to define a new mesh during the FE-simulation (remeshing). The loss of volume of the new mesh is a very critical point. In this paper an algorithm to mesh complex 3D geometries with tetrahedron elements (FORM TET) is presented, which is taking into account the specific aspects of metal forming processes and increases the loss of volume in FE-analysis. This algorithm creates a mesh that meets the requirements of the object. As illustrated in the presented examples, the elaborated strategy ensures a robust and efficient meshing scheme. To verify the efficiency of this remeshing concept some fundamental metal forming processes have been simulated.     

A finite element method with mesh‐separation‐based approximation technique and its application in modeling crack propagation with adaptive mesh refinement

This paper presents a FEM with mesh‐separation‐based approximation technique that separates a standard element into three geometrically independent elements. A dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation. The novel technique makes it very easy for mesh generation of problems with complex or solution‐dependent, varying geometry. It offers a flexible way to construct displacement approximations and provides a unified framework for the FEM to enjoy some of the key advantages of the Hansbo and Hansbo method, the meshfree methods, the semi‐analytical FEMs, and the smoothed FEM. For problems with evolving discontinuities, the method enables the devising of an efficient crack‐tip adaptive mesh refinement strategy to improve the accuracy of crack‐tip fields. Both the discontinuities due to intra‐element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level. The effectiveness and robustness of the present method are benchmarked with several numerical examples. The numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test, and it is essential to apply local adaptive mesh refinement for low fracture energy problems. Copyright © 2014 John Wiley & Sons, Ltd.     

二维自由振动的有限元线法自适应分析新进展

有限元线法(FEMOL)是一种优良的半解析、半离散方法,可将其比拟为广义一维问题,进而将一维有限元中单元能量投影(EEP)法及相应的自适应求解技术引入,使FEMOL由半解析方法变为完全解析、数值精确的方法。在对二维线性问题成功地实现了自适应FEMOL分析的基础上,该文进一步报道FEMOL自适应方法在二维自由振动问题中的成功应用和最新进展。该文简要介绍了FEMOL自适应分析二维振动问题的求解策略和技术,整套方法思路清晰、算法严谨、高效可靠,可以得到满足精度要求的自振频率和按最大模度量满足用户事先给定误差限的振型,均为数值精确解。该文给出的数值算例表明所提出的算法具有高效、稳定、通用、可靠的优良特性。     

二维有限元线法自适应分析的若干新进展

有限元线法(FEMOL)是一种优良的半解析、半离散方法,将其比拟为广义一维问题,遂可将一维有限元中十分成功的单元能量投影(EEP)超收敛算法以及基于该法的自适应求解方法推广到二维有限元线法分析中,至今已在二维Poisson方程和弹性力学平面问题中取得了令人满意的进展.该文旨在报道这些进展和成果.该文简要介绍了线法的EE...     

The Discontinuity‐Enriched Finite Element Method

We introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization. At variance with the eXtended/Generalized Finite Element Method (X/GFEM), the new method, named the Discontinuity‐Enriched Finite Element Method (DE‐FEM), adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh. Although general, the method is demonstrated in the context of fracture mechanics, and its versatility is illustrated with a set of traction‐free and cohesive crack examples. We show that DE‐FEM recovers the same rate of convergence as the standard FEM with matching meshes, and we also compare the new approach to X/GFEM.