结构动力分析的随机变分原理及随机有限元法

将结构动力系统的参数及激励的随机性直接引入结构的动力泛函变分表达式中，基于瞬时最小势能原理，应用小参数摄动法，建立了随机结构动力分析的随机变分列式及相应的确机有限元法。算例表明，应用此法分析随机结构动力响应，具有程序实施简便，计算效率高的优点。     

The random variational principle in finite deformation of elasticity and finite element method

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PARAMETRIC VARIATIONAL PRINCIPLE BASED ELASTIC-PLASTIC ANALYSIS OF HETEROGENEOUS MATERIALS WITH VORONOI FINITE ELEMENT METHOD

The Voronoi cell finite element method (VCFEM) is adopted to overcome the limitations of the classic displacement based finite element method in the numerical simulation of heterogeneous materials. The parametric variational principle and quadratic programming method are developed for elastic-plastic Voronoi finite element analysis of two-dimensional problems. Finite element formulations are derived and a standard quadratic programming model is deduced from the elastic-plastic equations. Influence of microscopic heterogeneities on the overall mechanical response of heterogeneous materials is studied in detail. The overall properties of heterogeneous materials depend mostly on the size, shape and distribution of the material phases of the microstructure. Numerical examples are presented to demonstrate the validity and effectiveness of the method developed.     

The optimal control variational principle and finite elements analysis for viscoplastic dynamics

This paper presents the optimal control variational principle for Perzyna model which is one of the main constitutive relation of viscoplasticity in dynamics. And it could also be transformed to solve the parametric quadratic programming problem. The FEM form of this problem and its implementation have also been discussed in the paper.     

粘弹性随机分析的虚功原理及随机有限元

利用增量法处理粘弹性本构关系中的遗传积分，将粘弹性材料的随机性、结构几何形状的随机性、外载荷的随机性引入虚功方程，应用摄动方法，研究了粘弹性随机分析的虚功原理和粘弹性随机有限元。研究发现，尽管粘弹性本构关系具有时间相依性，其随机摄动格式并不存在“长期项”的影响，算例表明，应用该方法进行粘弹性结构的随机模拟，计算效率较高、精度较高。     

复固有频率问题的模糊变分原理

模糊变分原理是模糊有限元重要的理论基础之一,模糊有限元的研究已经比较成熟了,然而关于模糊变分原理的研究却非常少.为研究复固有频率问题的模糊变分原理,首先介绍了一些模糊数学的概念,之后推导了非保守系统的拟Hamilton变分原理.接着通过将模糊参数引入到拟Hamilton变分原理,推导了复固有频率问题的模糊变分原理.作为模糊变分原理的应用,又推导了模糊有限元法.该方法可以直接得到问题的模糊解.与传统的模糊有限元方法相比,它避免了先将模糊参数转化为区间形式求解,之后再由区间解构造模糊解的过程.因此,该方法可以很大程度上减少计算量.最后通过数值算例表明了所提方法的可行性.     

索网天线找形分析的参变量变分有限元法

针对柔性空间索网天线的非线性特点,建立了基于参变量变分描述索网拉压非线性和共旋列式描述几何非线性的有限元控制方程,应用Lernke与改进牛顿法进行求解.通过对索网预张力平衡计算,证明改进牛顿法比Newton-Raphson法具有更强的收敛能力.进一步将力密度法迭代原理与有限元法结合应用于索网天线的非线性找形分析中,获得了理想的索网构型.本文的索网找形方法可广泛应用于空间索网天线结构的设计.     

Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.     

波导本征问题分析的比例边界有限元方法

引入了一种求解波导本征值问题的高效而精确算法-比例边界有限元方法SBFEM (Scaled Boundary Finite Element Method).该方法的一个特点是只需在边界上进行离散,问题降低一维,使计算工作量大大减少;另一特点是所建立的控制方程为二阶常微分方程,可以解析地求解,使计算精度得到了保证.论文利用变分原理并通过比例边界坐标变换,推导了TE波和TM波波导的比例边界有限元频域方程以及波导动剐度方程,同时给出了波导动刚度矩阵的连分式解形式,通过引入辅助变量进一步得出波导特征值方程并求出波导本征值.以矩形、L形波导和叶型加载矩形波导的本征问题分析为例,通过与解析解及其他数值方法比较,结果表明,此方法具有精度高、计算工作量小的优点,而且随着连分式阶数增加收敛速度快.进一步分析了一类角切四脊正方形波导的传输特性.     

Daubechies条件小波有限元法研究

为拓展小波理论在结构工程中的应用,提高结构计算精度,提出了以Daubechies条件小波Ritz法为基础的Daubechies条件小波有限元法。该法结合广义变分原理和拉格朗日乘子法构造修正泛函,根据修正泛函的驻值条件得到全域法求解方程矩阵。根据构件的边界条件,按左右边界对求解矩阵进行相应拆分,构建条件小波单元刚度矩阵,并依据公共节点位移相等原则形成总体刚度矩阵,由此解得各单元的小波基待定系数,即可进一步求解位移场函数、内力分布函数及荷载集度函数。以工程中常见的弹性拉压杆及平面弯曲梁为例,详细阐述了该方法的构造过程。并通过典型算例将Daubechies条件小波有限元法计算值与理论解进行了对比,结果表明:在弹性拉压杆算例中,位移、应力、载荷集度的相对误差均在1.22×10-3%以内;在平面弯曲梁算例中,挠度、弯矩、载荷集度的相对误差均在8.91×10-2%以内。     

矩形空腔内Stokes流的状态空间有限元法

基于Hellinger-Reissner二类变分原理,从平面Stokes流问题的平衡方程、连续性要求和边界条件出发,得到相应的Hamilton函数,建立Hamilton正则方程后,采用分离变量法对场变量进行离散求解:在x方向采用有限元插值,在y方向采用状态空间法给出控制坐标方向的解析解。计算过程中的指数矩阵均采用精细积分法求解,使得本文算法具有高效率、高精度、对步长不敏感的优点。通过对侧边自由液面边界条件的单板驱动矩形空腔Stokes流问题的求解,得到与文献相同的结果,从而验证了本文方法的有效性。本文旨在将弹性力学状态空间有限元法的思想引入到低雷诺数流体力学中,为Hamilton体系下研究复杂边界Stokes流问题提供新的途径。     

基于对偶变量变分原理和两端位移独立变量的保辛方法

将广义位移和动量同时用拉格朗日多项式近似,并选择积分区间两端位移为独立变量,然后基于对偶变量变分原理导出了哈密顿系统的离散正则变换和对应的数值积分保辛算法。当位移和动量的拉格朗日多项式近似阶数满足一定条件时,可以自然导出保辛算法的不动点格式。通过数值算例分析了位移和动量采用不同阶次插值所需最少Gauss积分点个数,并讨论了位移插值阶数、动量插值阶数以及Gauss积分点个数对保辛算法精度的影响,说明了上述不动点格式恰好是一种最优格式。     

Fuzzy variational principle and its applications

Linear and non-linear peaky fuzzy numbers and their arithmetic operations are constructed for the analysis of engineering structures with fuzzy characteristic quantities. Fuzziness of the corresponding quantities is consistently incorporated into the functional of the total potential energy. A set of deterministic recursive equations is obtained as the alternative expressions of the fuzzy variational principle by means of the second-order perturbation technique. The fuzzy Ritz method and the fuzzy finite element method are presented as the applications of the fuzzy variational principle. Accordingly, the roundabout procedures frequently used in the formulations of the fuzzy finite element method are avoided. A benchmark problem of a bending beam with fuzzy Young's modulus under fuzzy external loading is solved by the developed fuzzy numerical methods. Numerical examples show that results determined by these two fuzzy methods are both little conservative, and are in good agreement with those obtained by the analytical method. Moreover, the fuzzy Ritz method or the fuzzy finite element method can provide more valuable information than the conventional deterministic methods.     

Variational principles of perforated thin plates and finite element method for buckling and post-buckling analysis

On the basis of the general theory of perforated thin plates under large deflections^{[1, 2]}, variational principles with deflectionw and stress functionF as variables are stated in detail. Based on these principles, finite element method is established for analysing the buckling and post-buckling of perforated thin plates. It is found that the property of element is very complicated, owing to the multiple connexity of the region. Project supported by National Natural Science Foundation of China.     

Nonlinear quasi-conforming finite element method

The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasiconforming finite element. First, the incremental principle of stationary potential energy is discussed. Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.     

P-version hybrid analytical finite element method for plate bending problems

Two sets of trial functions with different variables are constructed for the admissible space of the finite element analysis. The trial functions satisfy the equilibrium differential equation inside elements, while the deflections and rotations on the edges of the elements are approximated by the Peano hierarchical interpolation functions. Then, a generalized variational principle is applied to set up the p-version hybrid analytical finite element method for plate bending problems. The accuracy of finite element computation can be improved by increasing the order of the interpolation polynomials with fixed mesh. In the finite element formulation, to obtain the stiffness matrices and the load vectors, it is only necessary to perform quadrature over the edges of the elements. These matrices and vectors possess an embedding structure. The conformability between the elements can be controlled automatically.This work is supported by the Natural Science Foundation of China and the Aeronautical Science Foundation of China.     

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.     

A variational principle and finite element formulation for multi-physics PLZT ceramics

This research presents an extended variational principle and a finite element formulation for multi-physics analysis of PLZT (lanthanum zirconate titanate) ceramics by including the photovoltaic and optothermic effects. The photo-induced electrical and thermal, and mechanical fields, as well as the heat source effect due to light illumination are all considered in the formulation. A generalized variation approach for advanced multi-physics PLZT problems is developed and then the relevant finite element formulation is established.     

The parametric variational principle for Perzyna model in viscoplasticity

This paper presents the parametric variational principle for Perzyna model which isone of the main constitutive relations of viscoplasticity.The principle,by which the potentialenergy function is minimized under a constrained condition transformed by the constitutiverelations of viscoplasticity,is free from the bound of Drucker’s postulate of plastic flow andconsequently suitable for solving the nonassociated plastic flow problems.Furthermore,the paper has proven the presented principle and discussed the creep problem.