Orthogonal basic deformation mode method for zero-energy mode suppression of hybrid stress elements

A set of basic deformation modes for hybrid stress finite elements are directly derived from the element displacement field. Subsequently, by employing the so-called united orthogonal conditions, a new orthogonalization method is proposed. The resulting orthogonal basic deformation modes exhibit simple and clear physical meanings. In addition, they do not involve any material parameters, and thus can be efficiently used to examine the element performance and serve as a unified tool to assess different hybrid elements. Thereafter, a convenient approach for the identification of spurious zero-energy modes is presented using the positive definiteness property of a flexibility matrix. Moreover, based on the orthogonality relationship between the given initial stress modes and the orthogonal basic deformation modes, an alternative method of assumed stress modes to formulate a hybrid element free of spurious modes is discussed. It is found that the orthogonality of the basic deformation modes is the sufficient and necessary condition for the suppression of spurious zero-energy modes. Numerical examples of 2D 4-node quadrilateral elements and 3D 8-node hexahedral elements are illustrated in detail to demonstrate the efficiency of the proposed orthogonal basic deformation mode method.     

一种抑制杂交元零能模式的假设应力场方法

基于杂交元位移场直接导出可以表示单元任意变形的简单变形模式,同时指出与所有假设应力模式正交的非零变形为零能机动模式,从而可以用简单变形模式方便地识别和抑制单元零能模式。在此基础上利用初始应力模式与简单变形模式的正交性提出一种假设杂交元应力场的有效方法,结合等函数法应力模式组成初始应力模式,不仅可以根据实际问题需要灵活地假设不同分布规律的应力场,而且所形成杂交元完全避免零能机动模式。在数值算例中采用本文方法分别形成了2D-4节点杂交元和3D-8节点杂交元的多种假设应力场,表明本文所提出方法是有效可行的。     

杂交元零能模式抑制的改进基本变形模式方法

基本变形模式方法是构造杂交元假设应力场的一种有效方法,但只给出了识别零能模式的必要条件.论文指出基本变形模式方法由于没有考虑耦合影响而存在应用上的局限,在此基础上利用柔度矩阵的正定性给出了判别零能机动模式的充要条件,并且通过内积向量无关性进一步完善了假设杂交元应力场的基本变形模式方法,可以适用于基本变形模式之间耦合的任...     

杂交元本征应力模式和应力子空间的性质研究

详细讨论了有限元本征应力模式和应力子空间的性质，并着重讨论和进一步完善了与杂交应力有限元应力子空间有关的一些定理，为提出新方法提供了理论基础，主要包括：（1）证明了杂交元特征值不大于对应位移元的特征值；（2）证明了矩阵H非奇异的充分必要条件是假设应力模式线性无关；（3）证明了杂交元所对应位移元的本征应力模式形成的杂交元与该位移元相同；（4）证明了等价假设应力模式形成相同的杂交元；（5）证明了确定杂交元本征应力模式的充分必要条件是其范数平方等于所形成杂交元的变形模态特征值；（6）证明了杂交元假设应力模式与变形模态的能量一一对应的充分必要条件是假设应力模式彼此正交且与所对应位移元的本征应力模式除了一一对应者之外都正交。     

一种近场动力学非普通状态理论零能模式控制方法

近场动力学非普通状态理论在采用节点积分时将引起零能模式,造成位移场、应力应变场的数值不稳定性,影响计算精度甚至会导致完全错误的结果,因此必须对其进行控制.目前国际上还没有十分有效的零能模式控制方法.本文针对零能模式问题,提出了一种通用的、高效的控制方法.根据近场动力学线性键理论,确定非均匀变形对应弹性张量的具体形式,考虑了微模量随不同作用键的变化.通过最小位能原理推导出非均匀变形引起的力状态,结合近场动力学力状态,得到稳定力状态表达式.从而建立起基于线性键理论的稳定关联材料模型,并应用于含圆孔平板、三点弯试件线弹性变形和损伤破坏过程模拟.数值结果表明,本文模型能有效抑制近场动力学非普通状态理论中的零能模式现象.与已有零能模式控制方法相比,其物理意义明确,不包含控制参数,避免了复杂的零能模式参数调节过程,提高了计算效率.      

轴对称实体迴转杂交应力元

用杂交应力法导出一系列新的八节点、四边形轴对称实体迴转元,对每种元,边界二次位移插值函数相同,内部应力场各异,对两类不同选择假定应力场的方法进行了对比研究,以內压力作用下的厚壁筒及厚球为例,对各种应力场进行了数值比较,并选出了一种较好的元,该元给出了远较传统的八节点轴对称假定位移元准确的应力分布。     

Axisymmetric solid-of-revolution elements based on the assumed stress hybrid model

A series of axisymmetric solid-of-revolution elements with 8-node and quadrilateral cross section have been developed based on the assumed stress hybrid model. A quadratic boundary displacement assumption is employed for each element and a variety of interior stress assumptions have been made. Two different kinds of procedure used for developing stress field have been studied. Example problems of a thick-walled cylinder under internal pressure and a thick-walled sphere under internal pressure are utilized to evaluate the various elements and a desirable stress assumption has been identified. Comparisons of present results with those obtained by the use of 8-node element based on the assumed displacement model indicate that this hybrid stress element is far superior in predicting the stress distribution.     

Effect of the order of the finite element and selection of the constrained modes in deformable body dynamics

Finite elements with different orders can be used in the analysis of constrained deformable bodies that undergo large rigid body displacements. The constrained mode shapes resulting from the use of finite elements with different orders differ in the way the stiffness of the body bending and extension are defined. The constrained modes also depend on the selection of the boundary conditions. Using the same type of finite element, different sets of boundary conditions lead to different sets of constrained modes. In this investigation, the effect of the order of the element as well as the selection of the constrained mode shapes is examined numerically. To this end, the constant strain three node triangular element and the quadratic six node triangular element are used. The results obtained using the three node triangular element are compared with the higher order six node triangular element. The equations of motion for the three and six node triangular elements are formulated from assumed linear and quadratic displacement fields, respectively. Both assumed displacement fields can describe large rigid body translational and rotational displacements. Consequently, the dynamic formulation presented in this investigation can also be used in the large deformation analysis. Using the finite element displacement field, the mass, stiffness, and inertia invariants of the three and six-node triangular elements are formulated. Standard finite element assembly techniques are used to formulate the differential equations of motion for mechanical systems consisting of interconnected deformable bodies. Using a multibody four bar mechanism, numerical results of the different elements and their respective performance are presented. These results indicate that the three node triangular element does not perform well in bending modes of deformation.     

基于拟应变法的壳单元大变形分析及在冲击动力问题中的应用

为了简便有效地解决板壳结构的大变形问题，本文针对八节点相对自由度壳单元进行研究。该单元的位移场由壳的中面节点位移和上表面节点的相对位移组成，不带有转动变量。所有的研究都是基于完全的三维位移、应力、应变场。采用拟应变法，对应变场另行假设，能够改善该单元在大变形情况下的计算精度。通过引入Wilson非协调模式，构造了大变形情况下的拟应变场表达式，给出了该单元用于解决非线性动力分析问题的有限元求解方程。通过算例表明，本文针对相对自由度壳单元提出的方法及推导的公式，能够解决冲击动力问题中的大变形问题。     

多参量断裂力学广义杂交/混合裂纹有限元法

通过研究广为人知的断裂力学单变量八节点位移裂纹QPE元和Akin族奇异单元法，本文运用经典局部裂纹解析解，与非协调假设应力杂交-混合元列式方法相结合，提出用于分层各向异性材料的多变量半解析假设应力奇异广义杂交／混合裂纹有限元法，能克服现有位移裂纹元法的域应力分布精度低和高次单元所需计算容量大的局限性，互为补充，更有利于结构裂纹扩展分析和应用研究。文中设计了一个半解析奇异裂纹平面单元，各向同性材料板算例验证了退化二次八节点协调位移裂纹元及六节点非协调奇异应力裂纹元，说明采用稀疏及加密单元网格，两类裂纹单元分别从上下逼近收敛于实验和理论参考解，可得到吻合程度较好的1／√r奇异应变和应力分量以及应力强度因子值，表明了本文奇异裂纹单元理论的优越性。     

一类新的具有无外力圆柱表面的杂交应力元

根据Hellinger-Reissner原理,建立了进一步改进的具有一个无外力圆柱表面三维杂交应力元.元内假定应力场满足以柱坐标表示的平衡方程,及圆柱面上的无外力边界条件.当退化为二维时,也满足协调方程.数值算例表明,当分析带圆弧的槽孔板、块时,在稀疏的有限元网格下,这类单元即可提供较以前各类特殊元、一般假定位移元及一般假定应力元远为准确的三维及二维应力分布.     

THE STRESS SUBSPACE OF HYBRID STRESS ELEMENT AND THE DIAGONALIZATION METHOD FOR FLEXIBILITY MATRIX H

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MODE III 2-D FRACTURE ANALYSIS BY THE SCALED BOUNDARY FINITE ELEMENT METHOD

The scaled boundary finite element method （SBFEM） is a novel semi-analytical technique that combines the advantages of the finite element method and the boundary element method with unique properties of its own. This method has proven very efficient and accurate for determining the stress intensity factors （SIFs） for mode I and mode II two-dimensional crack problems. One main reason is that the SBFEM has a unique capacity of analytically representing the stress singularities at the crack tip. In this paper the SBFEM is developed for mode III （out of plane deformation） two-dimensional fracture anMysis. In addition, cubic B-spline functions are employed in this paper for constructing the shape functions in the circumferential direction so that higher continuity between elements is obtained. Numerical examples are presented at the end to demonstrate the simplicity and accuracy of the present approach for mode Ⅲ two-dimensional fracture analysis.     

A Hybrid Boundary/Finite Element Method for Simulating Viscous Flows and Shapes of Droplets in Electric Fields

A mixed boundary element and finite element numerical algorithm for the simultaneous prediction of the electric fields, viscous flow fields, thermal fields and surface deformation of electrically conducting droplets in an electrostatic field is described in this paper. The boundary element method is used for the computation of the electric potential distribution. This allows the boundary conditions at infinity to be directly incorporated into the boundary integral formulation, thereby obviating the need for discretization at infinity. The surface deformation is determined by solving the normal stress balance equation using the weighted residuals method. The fluid flow and thermal fields are calculated using the mixed finite element method. The computational algorithm for the simultaneous prediction of surface deformation and fluid flow involves two iterative loops, one for the electric field and surface deformation and the other for the surface tension driven viscous flows. The two loops are coupled through the droplet surface shapes for viscous fluid flow calculations and viscous stresses for updating the droplet shapes. Computing the surface deformation in a separate loop permits the freedom of applying different types of elements without complicating procedures for the internal flow and thermal calculations. Tests indicate that the quadratic, cubic spline and spectral boundary elements all give approximately the same accuracy for free surface calculations; however, the quadratic elements are preferred as they are easier to implement and also require less computing time. Linear elements, however, are less accurate. Numerical simulations are carried out for the simultaneous solution of free surface shapes and internal fluid flow and temperature distributions in droplets in electric fields under both microgravity and earthbound conditions. Results show that laser heating may induce a non-uniform temperature distribution in the droplets. This non-uniform thermal field results in a variation of surface tension along the surface of the droplet, which in turn produces a recirculating fluid flow in the droplet. The viscous stresses cause additional surface deformation by squeezing the surface areas above and below the equator plane.     

爆炸载荷作用下加筋板的失效模式分析及结构优化设计

通过对爆炸载荷下具有1根加强筋的固支矩形加筋板的有限元模拟，探讨了抗爆加筋板结构优化设计方法，分析了加筋板的失效模式以及加强筋相对刚度和冲击载荷强度对加筋板失效模式的影响，指出了失效模式Ⅰ下的3种变形模态以及失效模式Ⅱ下的2种子失效模式，得到了失效模式Ⅰ下加强筋和加筋板最大挠度的近似计算公式，提出了单根加筋板的两种失效模式的判别条件，并对具有1根加强筋的固支矩形加筋板抗爆结构进行了优化设计。结果表明，通过数值模拟或模型实验可以求得任意加筋板结构由发生塑性大变形到发生破损的临界条件，从而确定抗爆性能最强时加筋板的质量与加强筋横截面尺寸及间距间的关系，实现对抗爆加筋板结构的优化设计。     

带有沙漏控制的相对自由度壳元非线性动力分析

针对八节点相对自由度壳单元，给出了单元内坐标和位移的插值公式，利用HuWashinzu变分原理，基于拟应变法，在大变形情况下推导了拟应变的表达式，构造了带有沙漏控制的动力问题的有限元求解格式。通过算例表明该文提出的基于相对自由度壳元的沙漏控制算法能够很好地解决非线性动力问题，可改善计算精度和计算效率。     

构造杂交应力单元的柔度矩阵H对角化方法

证明了杂交元柔度矩阵 H非奇异的充分必要条件是假设应力模式线性无关 ;以及等价应力模式形成相同的杂交元。在此基础上建立了假设应力模式的 Hilbert子空间 ,从而可以利用 Schmidt方法简单地得到等价的正交应力模式 ,实现了柔度矩阵 H对角化 ,使得杂交元形成过程中完全避免了繁杂的矩阵求逆运算 ,提高了杂交元分析的计算效率 ,特别在柔度矩阵不容易显式求逆的材料非线性分析中更具有实际意义     

随机缺陷对蜂窝结构动态行为影响的有限元分析

通过对胞壁随机移除的蜂窝结构动态变形过程的有限元模拟,分析了随机缺陷对蜂窝结构变形模式的影响,得到蜂窝结构在两个加载方向上的变形模式图及不同模式间转换的临界速度. 对含缺陷蜂窝结构平台应力的研究发现,当变形模式为过渡模式或动态模式时结构平台应力与冲击速度的平方成线性关系. 相同密度下,低缺陷蜂窝结构的平台应力在由过渡模式向动态模式转变的临界速度附近高于规则蜂窝结构,较高的随机缺陷则使蜂窝结构的平台应力在由准静态模式向过渡模式转变的临界速度附近显著下降.关键词:多孔材料,蜂窝,缺陷,平台应力,有限元分析      

部分应力杂交动力分析模型的混合变分原理

本文通过对具有单重卷积积分的二类变量的Ｇｕｒｔｉｎ型混合变分原理进行修正，采用“部分应力杂交”的概念，建立了线弹性动力分析 部分应力杂交变分格式，并在此基础上构造出时间有限元模型，这种模型对于分层复合板的动力分析尤其适用。     

沉痛悼念许兵先生

通过对胞壁随机移除的蜂窝结构动态变形过程的有限元模拟,分析了随机缺陷对蜂窝 结构变形模式的影响,得到蜂窝结构在两个加载方向上的变形模式图及不同模式间转换的临 界速度. 对含缺陷蜂窝结构平台应力的研究发现,当变形模式为过渡模式或动态模式时结构 平台应力与冲击速度的平方成线性关系. 相同密度下,低缺陷蜂窝结构的平台应力在由过渡 模式向动态模式转变的临界速度附近高于规则蜂窝结构,较高的随机缺陷则使蜂窝结构的平 台应力在由准静态模式向过渡模式转变的临界速度附近显著下降. 关键词：多孔材料,蜂窝,缺陷,平台应力,有限元分析