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。     

基于改进位移模式的一维有限元超收敛算法

将多尺度方法的思想与超收敛计算的解析公式结合起来,提出了改进有限元位移模式的算法。利用超收敛计算的解析公式,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。该算法在前处理和后处理两个阶段都使用超收敛计算公式,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。对于线性单元,本文结点和单元的位移、导数都达到了h4阶的超收敛精度。     

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.     

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。     

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。     

位移贡献法及其级数收敛解

文中的位移贡献法是以结构中两节点间的杆件为基本计算单元,通过这些基本计算单元两端的位移对结构中节点位移的贡献及单元之间位移的传递,可求出结构中节点的位移。当采用适当的方法进行位移传递时,可用一组等比收敛级数对结构中节点的位移进行收敛计算,从而得到精确解.     

基于静动态混合重构的DG/FV混合格式

通过比较紧致格式和间断Galerkin(DG)格式, 提出了``静态重构'和``动态重构'的概念,对有限体积方法和DG有限元方法进行统一的表述. 借鉴有限体积的思想, 发展了基于``混合重构'技术的一类新的DG格式, 称之为间断Galerkin有限元/有限体积混合格式(DG/FV格式). 该类混合格式通过适当地扩展模板(拓展至紧邻单元)重构单元内的高阶多项式分布, 在提高精度的同时, 减少了传统DG格式的计算量和存储量. 通过典型一维和二维标量方程的计算发现新的混合格式在有些情况下具有超收敛(superconvergence)性质.      

纺织物下垂变形数值计算

利用共旋技术提出了一个高柔性梁（布条）大位移大转动计算模型,通过把纺织物沿经向和纬向离散成许多布条,建立了纺织物下垂变形模型,模型仅使用了点平移自由度而没有使用转动自由度,并采用了一种单元应变能积分格式,模型考虑了纺织物面内拉伸、面内剪切以及弯曲;提出了一种控制单步全Newton-Raphson迭代格式步长的方法来确保得到控制方程组的收敛解。数值算例表明模型能够较精确地预测纺织物的最终下垂形状。     

基于多层模拟方法的双层梁断裂有限元分析

本文提出一种用于含分层的双层梁线弹性断裂分析的有限元方法．将上下子梁均模拟为多个子层,采用只有平动位移自由度的新型梁单元,假设单元内的位移沿纵向和横向均线性变化,推导了该单元的单元刚度矩阵．将开裂部分和未开裂部分的子梁进行单元刚度矩阵组装,施加相应的等效结点力,得到整体平衡方程,并结合边界条件进行求解．为验证该方法的有效性和精度,开展非对称双悬臂梁(Asymmetric Double Cantilever Beam, ADCB)和单臂弯曲梁(Single Leg Bending, SLB)试样的断裂分析,利用虚拟裂纹闭合技术(Virtual Crack Closure Technique, VCCT)得到了试样的能量释放率及其分量,并将求得的结果与解析解和二维有限元解进行对比．计算结果表明,相对于传统双层模拟方法,该多层模拟方法能够精确、高效地计算各类梁试样的能量释放率及其分量,并且无需引入界面连续条件．     

Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order

Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method （FEM） is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach.     

Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method

The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.     

有限元计算中疏密网格过渡方法研究

工程计算中出于节省计算量的目的,往往需要在一个有限元模型中布置粗细不同的网格。为保证计算结果的准确性,必须保证网格突变情况下的位移协调问题。本文工作之一是在强天驰界面过渡单元的基础上,引入虚拟节点和子单元,在子单元中应用节理元思想,提出了基于最小势能原理的弹簧节理单元法。简化了积分运算,避免了精度要求极高的坐标转换,从而提高了方法的精度和实用性;二是提出了基于位移约束的主从自由度法,简便实用,只需简单的矩阵运算即可实现。两种方法均实现了不同尺寸网格间位移的协调性和刚度的匹配,从而使之满足有限元收敛准则,且生成的刚度阵具有对称性及带状性。算例证明两种方法精度良好,并可方便地应用于求解大规模工程问题。     

Computation of super-convergent nodal stresses of timoshenko beam elements by EEP method

The newly proposed element energy projection (EEP) method has been applied to the computation of super-convergent nodal stresses of Timoshenko beam elements. Generalformul as based on element projection theorem were derived and illustrative numerical examples using two typical elements were given. Both the analysis and examples show that EEP method also works very well for the problems with vector function solutions. The EEP method gives super-convergent nodal stresses, which are well comparable to the nodal displacements in terms of both convergence rate and error magnitude. And in addition, it can overcome the “ shear locking“ difficulty for stresses even when the displacements are badly affected. This research paves the way for application of the EEP method to general onedimensional systems of ordinary differential equations.     

三维连续与非连续变形分析

将石根华博士所提出的二维非连续变形分析——Discontinuous Deformation Analysis（DDA）方法扩展到三维情况,并对三维不连续块体进行有限元网格剖分,即块体之间的接触采用DDA描述,块体内部的位移场和应力场则采用有限单元法描述,从而将三维DDA与有限元方法结合起来,增强了DDA方法与有限元方法解决实际工程问题的能力,实现了三维连续与非连续变形分析.给出了基本公式的推导过程和各子矩阵的形式.典型接触、碰撞算例证明了所提出方法的有效性和正确性.     

Adaptive boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

This paper deals with the adaptive control problem of the unforced generalized Korteweg?Cde Vries?CBurgers (GKdVB) equation when the spatial domain is [0,1]. Three adaptive control laws are designed for the GKdVB equation when either the kinematic viscosity ?? or the dynamic viscosity ?? is unknown, or when both viscosities ?? and ?? are unknowns. Using the Lyapunov theory, the L ²-global exponential stability of the solutions of this equation is shown for each of the proposed control laws. Also, numerical simulations based on the Finite Element method (FEM) are given to illustrate the analytical results.     

膜-基复合材料界面剪应力分析的影响系数法

提出了一种计算膜 基复合材料界面剪应力的新方法。将薄膜、基体分别作为二维和三维问题进行有限元计算 ,利用影响系数 ,以剪应力为未知量 ,求出同一点处薄膜、基体的位移。通过薄膜、基体在界面上位移相等的条件建立线性方程组 ,联立求解 ,计算出界面剪应力。结果表明 ,本文提出的方法思路清晰 ,易于实现 ,精度令人满意 ,为膜 基复合材料结构强度分析提供了一种新的途径     

Cracking analysis of fracture mechanics by the finite element method of lines （FEMOL）

The Finite Element Method of Lines (FEMOL) is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to a set of examples such as edge-crack plate, the central-crack plate, the plate with cracks emanating from a hole under tensile or under combination loads of tensile and bending. Their dimensionless stress distribution, the stress intensify factor (SIF) and crack opening displacement (COD) are obtained, and comparison with known solutions by other methods are reported. It is found that a good accuracy is achieved by FEMOL. The method is successfully modified to remarkably increase the accuracy and reduce convergence difficulties. So it is a very useful and new tool in studying fracture mechanics problems. The English text was polished by Yunming Chen.