基于改进位移模式的一维C1有限元超收敛算法

提出了基于改进位移模式的一维C1有限元超收敛算法。利用单元内部需满足平衡方程的条件,推导了超收敛计算解析公式的显式,即将高阶有限元解的位移模式用常规有限元解的位移模式表示。用常规有限元解的位移模式与高阶有限元解的位移模式之和构造新的位移模式。采用积分形式推导了单元刚度矩阵。该算法在前处理阶段使用了超收敛计算公式,在常规试函数的基础上,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Hermite单元,本文的结点和单元的位移、导数都达到了h4阶的超收敛精度。     

二阶非自伴两点边值问题Garlerkin有限元的超收敛算法

提出了基于改进位移模式的二阶非自伴两点边值问题Garlerkin有限元的超收敛算法. 用常规有限元解的位移模式与高阶有限元解的位移模式之和构造新的位移模式,基于Garlerkin 方法,采用积分形式推导了单元平衡方程. 对于线性单元,本文给出了有代表性的算例,结点和单元的位移、导数都达到了h⁴阶的超收敛精度.     

A NEW SEMI-ANALYTIC ALGORITHM OF NEARLY SINGULAR INTEGRALS IN HIGH ORDER BOUNDARY ELEMENT ANALYSIS OF 2D ELASTICITY

针对边界元法中高阶单元中几乎奇异积分计算难题,解剖了二维边界元法高阶单元的几何特征,定义源点相对高阶单元的接近度。将高阶单元上奇异积分核函数用近似奇异函数逼近,从而分离出积分核中主导的奇异函数部分,其奇异积分核分解为规则核函 数和奇异核函数两项积分之和。规则核函数用常规高斯数值积分,再对奇异核函数积分导出解析公式,从而建立了一种新的半解析法,用于高阶边界单元上几乎强奇异和超奇异积分计算。给出3个算例,采用边界元法高阶单元的半解析法计算了弹性力学薄体结构和近边界点位移/应力,并与线性边界元正则化算法结果作了比较,结果表明提出的二次元的半解析算法更加有效。特别是分析薄体结构,采用正则化算法的线性边界元分析比有限元有显著优势,而用提出的二次边界元半解析算法分析比其线性元的有效接近度又减小了4个量级。     

A p-TYPE SUPERCONVERGENT RECOVERY METHOD FOR FE ELASTIC STABILITY ANALYSIS OF EULER BEAMS1)

本文将有限元p型超收敛算法应用于欧拉梁弹性稳定分析。该法基于有限元解答中失稳载荷和失稳模态结点位移的超收敛特性,建立了单元上失稳模态近似满足的线性常微分方程边值问题,在每个单元上,对该边值问题采用一个高次元进行求解,获得失稳模态的超收敛解,再将失稳模态的超收敛解代入瑞利商的解析表达式,最终获得失稳载荷的超收敛解。该法思路简明,通过少量计算即可显著提高失稳载荷和失稳模态的精度与收敛阶。数值算例表明,该法高效、可靠,值得进一步研究和推广到各类杆系结构。     

欧拉梁弹性稳定分析的有限元p型超收敛算法

本文将有限元p型超收敛算法应用于欧拉梁弹性稳定分析。该法基于有限元解答中失稳载荷和失稳模态结点位移的超收敛特性,建立了单元上失稳模态近似满足的线性常微分方程边值问题,在每个单元上,对该边值问题采用一个高次元进行求解,获得失稳模态的超收敛解,再将失稳模态的超收敛解代入瑞利商的解析表达式,最终获得失稳载荷的超收敛解。该法思路简明,通过少量计算即可显著提高失稳载荷和失稳模态的精度与收敛阶。数值算例表明,该法高效、可靠,值得进一步研究和推广到各类杆系结构。     

二维位势边界元法高阶单元几乎奇异积分半解析算法

准确计算几乎奇异积分是边界元法难题之一。目前,对于一般的高阶单元的几乎奇异积分尚缺乏通用高效的计算方法。本文在单元局部坐标系中表征了二维高阶单元的几何特征,提出了源点相对高阶单元的接近度概念。针对二维位势边界元法的3节点二次等参单元,构造出与单元积分核具有相同几乎奇异性的近似奇异核函数。从二维位势几乎奇异积分单元积分核中扣除近似奇异核函数,把几乎奇异积分项转换为规则积分和奇异积分两部分之和,规则积分部分用常规Gauss数值积分计算,奇异积分部分由导出的解析公式计算,从而建立了二维位势问题高阶单元几乎强奇异和超奇异积分的半解析算法。算例结果表明了本文半解析算法的有效性和计算精度。     

一维Ritz有限元EEP超收敛位移计算简约格式的直接推导与证明

一维Ritz有限元后处理超收敛计算的EEP(单元能量投影)法简约格式中,若问题和解答足够光滑,其m(1)次单元的超收敛位移解在单元内任一点均可以达到至少hm+2的超收敛阶。对此,本文提出一套全新的推证方法,通过对单元能量投影的等效变形,直接推导出EEP简约格式位移解的计算公式及其误差项,进而采用更为简单通用的数学证明方法,证明了这一超收敛性。     

高阶数值流形方法的初应力公式

高阶数值流形方法和高阶DDA方法可以显著提高结构变形的计算精度,但目前涉及几何非线性问题的研究成果大都计算精度差甚至不收敛,这是由高阶初应力公式的不准确或不正确引起的。本文介绍数值流形方法的大变形计算格式,基于平面三角形数学网格和多项式覆盖函数,提出高阶流形法的两种初应力处理方法,首次导出了高阶初应力的准确公式。该公式在分步计算的初应力累加中考虑了大变形结构的构形变化,并将初应力表示成多项式函数形式以满足单纯形积分的要求。文中给出的悬臂梁大变形数值算例与理论解的对比结果证明了方法的正确性。本文的方法和公式也适用于三维四面体数学网格,稍加修改后将可应用于高阶DDA方法和常规的有限元方法。     

基于解析试函数三角形带旋转自由度膜元的显式格式研究

三角形单元是有限元分析中常用的单元.在平面单元内引入结点转动自由度,可以提高单元位移场的阶次,在不增加单元结点的前提下提高单元性能.论文利用问题基本解析解作为试函数来构造带旋转自由度的三角形单元ATF-R3H,采用了杂交应力函数单元模式,确保了单元优良的抗畸变性能和较高应力计算精度.论文利用直角坐标与三角形面积坐标的线性关系,以及面积坐标幂函数在三角形域内和边界上的积分公式,直接给出单元刚度矩阵的显式表达式,从而避免了大量数值积分,提高了计算效率.数值算例表明,显式格式的ATF-R3H单元具有良好的性能.     

基于Bézier提取的三维等几何分析

等几何分析(IGA)将非均匀有理B样条(NURBS)函数作为有限元形函数,具有几何精确、高阶连续和精度高等优点。与常规有限元法C0连续的形函数不同,高阶IGA基函数不是定义在一个单元上,而是跨越由几个单元组成的参数空间,因而编程复杂且无法嵌入现有的有限元法计算框架及相应算法。本文建立了基于Bézier提取的三维IGA,将NURBS函数分解成伯恩斯坦多项式的线性组合,从而实现把NURBS单元分解为C0连续Bézier单元,这些单元与Lagrange单元相似,使IGA的实现和常规有限元一样,以便将IGA分析嵌入现有的有限元软件中。两个三维算例结果表明,基于Bézier提取的IGA和传统IGA的收敛性和精度相同。     

Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions

A new type of Galerkin finite element for first-order initial-value problems (IVPs) is proposed. Both the trial and test functions employ the same m-degreed polynomials. The adjoint equation is used to eliminate one degree of freedom (DOF) from the test function, and then the so-called condensed test function and its consequent condensed Galerkin element are constructed. It is mathematically proved and numerically verified that the condensed element produces the super-convergent nodal solutions of O(h^2m+2), which is equivalent to the order of accuracy by the conventional element of degree m + 1. Some related properties are addressed, and typical numerical examples of both linear and nonlinear IVPs of both a single equation and a system of equations are presented to show the validity and effectiveness of the proposed element.

     

二阶非自伴两点边值问题Galerkin有限元后处理超收敛解答计算的EEP法

将一维Ritz有限元法超收敛计算的EEP(单元能量投影)法推广到二阶非自伴常微分方程两点边值问题Galerkin有限元法的超收敛计算。在对精确单元的研究中,发现与Ritz有限元法不同,只要检验函数采用伴随算子方程的解,无论试函数取何形式,在结点处都可得到精确的解函数值。对近似单元的研究表明,EEP法同样适用于Galerkin有限元法,不仅保留了简便易行、行之有效、效果显著的特点,同时也保留了EEP法的特有优点,如:任一点的导数和解函数的误差与结点值的误差具有相同的收敛阶。     

Three-dimensional finite element model for metal displacement and heat transfer in squeeze casting processes

A three-dimensional finite element model for the numerical simulation of metal displacement and heat transfer in the squeeze casting process has been developed. In the model, a numerical approach, termed as ‘Quasi-static Eulerian’, is proposed, in which the dynamic metal displacement process is divided into a certain number of sub-cycles. In each of the sub-cycles, the dieset configuration is assumed to be static and a fixed finite element mesh is created, thus making the Eulerian approach applicable to the solution of metal flow and heat transfer. Mesh-to-mesh data mapping is carried out for any two adjacent sub-cycles to ensure that the physical continuity of the real metal displacement process is represented. A numerical example is presented, which shows the application of the present model to geometrically complex three-dimensional squeeze casting problems. To cite this article: R.W. Lewis et al., C. R. Mecanique 335 (2007).     

平面广义四节点等参元GQ4及其性能探讨

广义节点有限元是将传统有限元方法中的节点广义化,在不增加节点个数的前提下,仅通过提高广义节点的插值函数的阶次,从而达到提高有限元解精度的目的.与现有的p型和hp型有限元不同,在这种新的有限元中,节点自由度全部定义在节点处,在理论与程序实现上与传统有限元方法具有很好的相容性,传统有限元方法是这种新方法的广义节点退化为0阶时的特殊情形.文中主要讨论了这一新方法的四节点等参元（记为GQ4）的形式.对GQ4进行的各种数值试验表明,所发展的广义四节点等参单元具有精度高且无剪切自锁与体积自锁等的特点.     

A hybrid-stress element based on Hamilton principle

A novel hybrid-stress finite element method is proposed for constructing simple 4-node quadrilateral plane elements, and the new element is denoted as HH4-3fl here. Firstly, the theoretical basis of the traditional hybrid-stress elements, i.e., the Hellinger-Reissner variational principle, is replaced by the Hamilton variational principle, in which the number of the stress variables is reduced from 3 to 2. Secondly, three stress parameters and corresponding trial functions are introduced into the system equations. Thirdly, the displacement fields of the conventional bilinear isoparametric element are employed in the new models. Finally, from the stationary condition, the stress parameters can be expressed in terms of the displacement parameters, and thus the new element stiffness matrices can be obtained. Since the required number of stress variables in the Hamilton variational principle is less than that in the Hellinger-Reissner variational principle, and no additional incompatible displacement modes are considered, the new hybrid-stress element is simpler than the traditional ones. Furthermore, in order to improve the accuracy of the stress solutions, two enhanced post-processing schemes are also proposed for element HH4-3β. Numerical examples show that the proposed model exhibits great improvements in both displacement and stress solutions, implying that the proposed technique is an effective way for developing simple finite element models with high performance.     

Exact integration of the von Mises elastoplasticity model with combined linear isotropic-kinematic hardening

The main purpose of this work is to present two semi-analytical solutions for the von Mises elastoplasticity model governed by combined linear isotropic-kinematic hardening. The first solution (SOLε) corresponds to strain-driven problems with constant strain rate assumption, whereas the second one (SOLσ) is proposed for stress-driven problems using constant stress rate assumption. The formulas are derived within the small strain theory Besides the new analytical solutions, a new discretized integration scheme (AMε) based on the time-continuous SOLε is also presented and the corresponding algorithmically consistent tangent tensor is provided. A main advantage of the discretized stress updating algorithm is its accuracy; it renders the exact solution if constant strain rate is assumed during the strain increment, which is a commonly adopted assumption in the standard finite element calculations. The improved accuracy of the new method (AMε) compared with the well-known radial return method (RRM) is demonstrated by evaluating two simple examples characterized by generic nonlinear strain paths.