基于单元的子区间摄动有限元方法研究

针对区间相关性导致区间扩张的问题,探讨了区间数之间的相关性并给出了降低区间扩张的子区间摄动方法.文中给出了基于单元的子区间摄动有限元计算公式和子区间划分数目的近似计算公式,同时文中讨论了区间有限元计算精度问题,给出了可提高计算效率的一些措施.对桁架结构和平面应力问题梁结构算例分析结果表明文中方法可以达到一定的计算精度,并且是合理可行的.     

基于区间参数摄动法的三维弹塑性区间有限元探讨

首次探讨了弹塑性问题的区间分析,推导了三维弹塑性问题的区间参数摄动法有限元计算公式,研制了相应的弹塑性区间有限元程序,通过算例分析得到结论:(1)采用增量法对弹塑性问题进行区间分析时,增量位移离差随着增量位移均值的增大而增大.(2)单元高斯点应力没有达到屈服极限之前,高斯点的应力离差很小;当单元高斯点应力达到屈服极限之后,高斯点应力离差逐渐增大.(3)是否考虑塑性矩阵的应力分量为弹塑性参数的隐函数,对应力高差计算的影响较小.     

基于单元分解的结构断裂扩展有限元法

研究并提出了一种基于单元分解的扩展有限元方法,并将其用于求解结构断裂问题。首先,将不含加强节点的四边形单元剖分为四个三角形子域,通过加权平均获得单元中心处局部应变值;其次,基于三角形子域局部应变进一步构造单元刚度矩阵稳定项。最后,将该单元分解法应用到扩展有限元法的分析中。与传统扩展有限元方法相比,该方法可有效减少积分点数量、避免复杂等参变换且能保证裂纹尖端应力强度因子的求解精度。

     

不确定非线性结构动力响应的区间分析方法

研究多自由度非线性不确定参数系统的动力响应问题. 以区间数学为基础,将不确定性参数用区间进行定量化,借助一阶Taylor级数,给出了近似估计非线性振动系统动力响应范围的区间分析方法. 从数学证明和数值算例两方面,将其与概率摄动有限元法进行了比较,结果显示区间分析方法对不确定参数先验信息具有要求较少、精度较高的优点.     

区间参数结构的动力响应优化

讨论区间参数结构的动态响应问题的区间优化方法利用摄动理论和函数区间扩张,将区间优化问题转化为近似的确定性优化问题由于区间设计变量的中值和不确定性半径均可取作优化参数,所以可得到比确定性优化更多的优化信息将该方法应用于桁架结构,算例表明该方法是有效的.     

改进型扩展比例边界有限元法

结合了扩展有限元法(extended finite element methods,XFEM)和比例边界有限元法(scaled boundary finite element methods,SBFEM)的主要优点,提出了一种改进型扩展比例边界有限元法(improved extended scaled boundary finite element methods,i XSBFEM),为断裂问题模拟提供了一条新的途径.类似XFEM,采用两个正交的水平集函数表征材料内部裂纹面,并基于水平集函数判断单元切割类型;将被裂纹切割的单元作为SBFE的子域处理,采用SBFEM求解单元刚度矩阵,从而避免了XFEM中求解不连续单元刚度矩阵需要进一步进行单元子划分的缺陷;同时,借助XFEM的主要思想,将裂纹与单元边界交点的真实位移作为单元结点的附加自由度考虑,赋予了单元结点附加自由度明确的物理意义,可以直接根据位移求解结果得出裂纹与单元边界交点的位移;对于含有裂尖的单元,选取围绕裂尖单元一圈的若干层单元作为超级单元,并将此超级单元作为SBFE的一个子域求解刚度矩阵,超级单元内部的结点位移可通过SBFE的位移模式求解得到,应力强度因子可基于裂尖处的奇异位移(应力)直接获得,无需借助其他的数值方法.最后,通过若干数值算例验证了建议的i XSBFEM的有效性,相比于常规XFEM,i XSBFEM的基于位移范数的相对误差收敛性较好;采用i XSBFEM通过应力法和位移法直接计算得到的裂尖应力强度因子均与解析解吻合较好.     

不确定参数闭环控制系统特征值的区间分析

运用区间理论,讨论了区间参数的结构振动控制问题,给出了求解闭环系统区间特征值的一种方法.基于区间参数导出了区间刚度矩阵和质量矩阵,然后利用矩阵摄动理论和区间扩张理论,推导了复区间特征值上下界估计的算法.这些结果是从二阶系统的左右特征向量出发得到的.将该文方法应用到悬臂梁的控制问题,数值结果表明它是有效的.     

具有区间参数的PCB组件随机振动响应

为降低具有区间参数印刷电路板的动力响应,建立了PCB组件的动力学区间分析模型,采用Block-Lanczos法求解固有频率,采用模态叠加法求解加速度功率谱激励下的区间动力响应.通过区间敏感度分析比较了影响目标变量的设计参数,对关键参数的取值区间进行了优化.算例表明,该方法可以获得此类区间问题的优化近似解.     

一维区间B样条小波单元的构造研究

基于区间B样条小波及小波有限元理论,提出了一种区间B样条小波有限元方法。传统有限元多项式插值被一维区间B样条小波尺度函数取代,进而构造形状函数和单元。与小波Galer-kin方法不同,本文构造的区间B样条小波单元通过转换矩阵将无明确物理意义的小波插值系数转换到物理空间。转换矩阵在小波单元构造过程中起到关键作用,为了保证求解的稳定性,转换矩阵必须非奇异。构造了以区间B样条尺度函数为插值函数的一系列一维区间B样条小波单元。数值算例表明,本文构造的区间B样条小波单元与传统有限元方法相比,在求解变截面,变载荷等问题时具有收敛快和精度高等优势;有效地丰富了小波有限元法单元库。     

多自由度非线性随机参数振动系统响应分析的概率摄动有限元法

提出了一般概率摄动有限元法,并用以解决了具有向量值和矩阵值函数的多自由度非线性随机结构系统承受随机激励的响应分析问题,应用Kronecker代数,矩阵微分理论,向量值和矩阵值函数的二阶矩技术,矩阵摄动理论和概率统计方法系统地扩展了国际上通用的随机有限元法,随机变量和系统导数很方便地排列到二维矩阵中,得到了优美的数学表达式。     

线性区间有限元静力控制方程的组合解法

区间有限元的静力控制方程常被归结为区间方程组来求解。但实际上两者并不等价。本文根据不确定结构有限元分析的力学背景,直接从问题的基本参量的不确定性出发,将基本区间参量的边界组合与求解区间方程组的有关解法相结合,提出了线性区间有限元静力控制方程的两种组合解法－参量边界全组合法和组合迭代法。可以以较小的计算量获得或逼近位移和应力区间的准确界限。且不受基本参量变化范围的限制。算例分析表明文中方法是实用和可行的。     

Interval finite element method and its application on anti-slide stability analysis

The problem of interval correlation results in interval extension is discussed by the relationship of interval-valued functions and real-valued functions. The methods of reducing interval extension are given. Based on the ideas of the paper, the formulas of sub-interval perturbed finite element method based on the elements are given. The sub-interval amount is discussed and the approximate computation formula is given. At the same time, the computational precision is discussed and some measures of improving computational efficiency are given. Finally, based on sub-interval perturbed finite element method and anti-slide stability analysis method, the formula for computing the bounds of stability factor is given. It provides a basis for estimating and evaluating reasonably anti-slide stability of structures.     

FUZZY ARITHMETIC AND SOLVING OF THE STATIC GOVERNING EQUATIONS OF FUZZY FINITE ELEMENT METHOD

The key component of finite element analysis of structures with fuzzy parameters, which is associated with handling of some fuzzy information and arithmetic relation of fuzzy variables , was the solving of the governing equations of fuzzy finite element method. Based on a given interval representation of fuzzy numbers, some arithmetic rules of fuzzy numbers and fuzzy variables were developed in terms of the properties of interval arithmetic. According to the rules and by the theory of interval finite element method, procedures for solving the static governing equations of fuzzy finite element method of structures were presented. By the proposed procedure, the possibility distributions of responses of fuzzy structures can be generated in terms of the membership functions of the input fuzzy numbers. It is shown by a numerical example that the computational burden of the presented procedures is low and easy to implement. The effectiveness and usefulness of the presented procedures are also illustrated.     

基于有限元分析的特征值反问题求解的逆摄动方法

本文研究特征值反问题的求解方法,根据广义特征值反问题理论和有限元法的特点,以转子系统平面梁单元有限元模型结构分析的特征值反问题求解为例,给出一种新的逆摄动方法,给出了本逆摄动法较完整的理论基础,给出了其逆摄动参数的显式计算公式及相应的取值方法,本逆摄动法也可推广到其他单元类型的有限元模型特征值反问题的求解。     

The finite element method based on interpolating with wavelet basis function

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｐｒｏｂｌｅｍｓｗｉｔｈｌａｒｇｅｇｒａｄｉｅｎｔａｒｅｃｏｍｍｏｎｉｎｐｒａｃｔｉｃａｌｅｎｇｉｎｅｅｒｉｎｇｆｉｅｌｄｓ,ｅ．ｇ．ｉｎｍａｔｅｒｉａｌｌｏｃａｌｉｚａｔｉｏｎ,ｗｉｔｈｉｎｔｈｅｌｏｃａｌｉｚａｔｉｏ...     

The random variational principle and finite element method

In this paper, we introduced the random materials, geometrical shapes, force and displacement boundary condition directly into the functional variational formulations and developed a unified random variational principle and finite element method with the small parameter perturbation method. Numerical examples showed that the methods have the advantages of the simple and convenient program implementation, and are effective for the random mechanics problems.     

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.     

Perturbational solutions for fuzzy-stochastic finite element equilibrium equations (FSFEEE)

I.IntroductionInthefieldofengineeringstructureanalysis,suchparametersasmodulusofmaterial-Poissonratio,geometricalsizesofstructure,boundaryconditionsandexternalloadingsprobablyhavefuzzinessor/andrandomness.Thestochasticfiniteelementmethod(SFEM)1'1disposeso…     

一个改进的平面梁单元

根据有限单元法基本原理 ,提出了一个变截面平面梁单元 ,推导了其单元钢度矩阵。这一改进的梁单元用于分析梁高呈线性变化及二次抛物线变化的矩形截面梁 ,将得到准确解。文中给出了一个变截面悬臂梁算例 ,计算表明 ,这一改进的梁单元使变截面梁的分析大大简化     

Stochastic finite element method based on local averages of random fields

The stochastic finite element method (SFEM) based on the local, averages of random fields, which was proposed in [5], is now generalized to analyze the structures with several correlated random parameters. The covariance matrix of the local averages of a random vector field is derived. The SFEM based on the local averages of random vector fields is formulated. The numerical examples show that the generalized SFEM preserves the advantages of the original one, i. e., rapid convergence, good accuracy and insensitivity to the correlation structures of random parameters.Project supported by National Natural Science Foundation of China.