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.     

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。     

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.     

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

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基于参变量变分原理的非均质多边形微结构材料弹塑性分析

为了更好地模拟复合材料及含夹杂非均质材料等的宏观弹塑性力学性能,简化有限元建模时间和减少有限元模拟计算量。本文基于参变量变分原理,提出了一种采用任意多边形弹塑性单元进行结构非线性分析的参数二次规划算法,给出了参变量最小势能原理以及最终的二次规划模型,并在有限元分析与优化设计软件系统JIFEX上进行了程序实现。数值算例证明了本文方法的正确与可行性。     

一类铁电模型的参变量变分原理及其应用

将参变量变分原理引入铁电问题。对一类借用了经典弹塑性理论中的概念和方法的多轴铁电模型建立基于Helmholtz自由能的参变量变分原理,可以有效处理传统变分原理中由非关联流动法则或屈服面不考虑材料系数变化所引起的切线模量非对称困难。相应于参变量变分原理,引入参数二次规划算法,可获得具有可靠数值稳定性的一套铁电算法。将该算法应用于一个具体的铁电模型,数值计算结果表明本文方法的有效性。     

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.     

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.     

Extended multiscale finite element method for mechanical analysis of heterogeneous materials

An extended multiscale finite element method （EMsFEM） is developed for solving the mechanical problems of heterogeneous materials in elasticity.The underlying idea of the method is to construct numerically the multiscale base functions to capture the small-scale features of the coarse elements in the multiscale finite element analysis.On the basis of our existing work for periodic truss materials, the construction methods of the base functions for continuum heterogeneous materials are systematically introduced. Numerical experiments show that the choice of boundary conditions for the construction of the base functions has a big influence on the accuracy of the multiscale solutions, thus,different kinds of boundary conditions are proposed. The efficiency and accuracy of the developed method are validated and the results with different boundary conditions are verified through extensive numerical examples with both periodic and random heterogeneous micro-structures.Also, a consistency test of the method is performed numerically. The results show that the EMsFEM can effectively obtain the macro response of the heterogeneous structures as well as the response in micro-scale,especially under the periodic boundary conditions.     

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.     

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.     

Rigid finite element and limit analysis

According to the lower bound theorem of limit analysis the Rigid Finite Element Method (RFEM) is applied to structural limit analysis and the linear programmings for limit analysis are deduced in this paper. Moreover, the Thermo-Parameter Method (TPM) and Parametric Variational principles (PVP) are used to reduce the computational effort while maintaining the accuracy of solutions. A better solution is also obtained in this paper. The project supported by National Natural Science Foundation of China     

On parametric minimum complementary energy variational principle in soil mechanics

The parametric minimum complementary energy variational principle is given in this paper. To the problems of nonassociative flow rule in the theory of plasticity, the Drucker postulation can no longer be applied and the classical variational principles fail, the parametric variation principles can play the role instead. The parametric variational principle can be used for the finite element solution with sequential quadratic programming method to soil mechanics problems.     

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。     

A natural neighbour method based on Fraeijs de Veubeke variational principle for materially non-linear problems

Abstract The natural neighbour method can be considered as one of many variants of the meshless methods. In the present paper, a new approach based on the Fraeijs de Veubeke （FdV） functional, which is initially developed for linear elasticity, is extended to the case of geometrically linear but materially non-linear solids. The new approach provides an original treatment to two classical problems： the numerical evaluation of the integrals over the domain A and the enforcement of boundary conditions of the type ui = hi on Su. In the absence of body forces （Fi = 0）, it will be shown that the calculation of integrals of the type fA .dA can be avoided and that boundary conditions of the type ui = hi on Su can be imposed in the average sense in general and exactly if hi is linear between two contour nodes, which is obviously the case for tTi = O.     

基于参变量变分原理的直升机系留载荷高性能计算方法

为了保证直升机在舰船上的安全性,必须使用系留设备将直升机系留在舰船上。直升机的系留问题可简化为由机身刚体、索具和起落架组成的杆件系统,索具只承受拉力而不承受压力,起落架只承受压力而不承受拉力。因此,直升机系留问题为典型的强非线性问题,需要发展有效的求解算法。在考虑大变形的情况下,基于参变量变分原理建立了求解直升机系留载荷的高性能计算方法。该方法利用参变量变分原理能够准确判断索具和起落架的拉压状态,并将材料非线性静力问题转换为线性静力互补问题求解,极大地提高了结果的收敛性。数值算例中,通过与有限元通用软件NASTRAN和ABAQUS计算结果比较,证实了该方法的精确性、收敛性及高效性。     

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.     

基于等几何分析的比例边界有限元方法

提出了一种具有比例边界有限元的半解析特性和等几何分析的几何特性的新方法。该新方法是在比例边界有限元框架中用NURBS曲线或曲面精确描述域边界几何形状,同时域边界位移场采用描述几何形状的NURBS形函数等参构造。这种新方法具有比例边界有限元固有的径向解析特性和NURBS的高阶连续性的优点。数值算例显示,与传统的比例边界有限元相比,基于等几何分析的比例边界有限元方法提高了域边界单元和域内应力场的连续性,减少了计算自由度。应用此方法可以用较少的计算自由度获得更高连续阶和更高精度的位移、应力和应变场。     

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。     

参变量变分原理的提出、发展与应用

参变量变分原理及其参数二次规划算法是由钟万勰院士1985年针对弹性接触边界非线性问题首次提出来的,经过将近40年的不断发展,目前参变量变分原理已经成功应用于各个领域,其中包括弹塑性分析、接触问题、润滑力学、岩土力学、变刚度杆系结构、先进材料性能分析、材料的蠕变与损伤、柔性结构力学和LQ最优控制等各个工程领域。本文首先回顾了参变量变分原理的起源,介绍了参变量变分原理的基本概念,然后以弹塑性分析问题为例,阐明建立参变量变分原理的理论模型以及实现数值参数二次规划求解原理,最后详细回顾了参变量变分原理的基本理论与相应数值算法在各个领域的发展及其工程应用,展示了参变量变分原理在求解各类非线性问题的特色与优势。