非连续变形分析的若干问题

基于加法分解及其线性化后的位移增量表达式在更新块体构型时,常导致常规非连续变形分析块体应变计算精度低、块体体积虚假膨胀。根据原始DDA位移模式,分析了该位移模式中因对转?增量1阶近似导致的块体体积自由膨胀;应变分量增量直接叠加导致的块体应变场畸变;以及采用加法分解线性化后的位移增量公式推导块体加速度表达式导致的忽略块体转动时的离心力与科氏力。数值算例表明,原始DDA的位移模式直接导致块体体积自由膨胀、块体内应变场畸变以及忽略了块体转动时离心力与科氏力产生的应变。     

Linear and Geometrically nonlinear analysis of plates and shells by a new refined non-conforming triangular plate/shell element

A refined non-conforming triangular plate/shell element for linear and geometrically nonlinear analysis of plates and shells is developed in this paper based on the refined non-conforming element method (RNEM). A conforming triangle membrane element with drilling degrees of freedom in Cartesian coordinates and the refined non-conforming triangular plate-bending element RT9, in which Kirchhoff kinematic assumption was adopted, are used to construct the present element. The displacement continuity condition along the interelement boundary is satisfied in an average sense for plate analysis, and the coupled displacement continuity requirement at the interelement is satisfied in an average sense, thereby improving the performance of the element for shell analysis. Selectively reduced integration with stabilization scheme is employed in this paper to avoid membrane locking. Numerical examples demonstrate that the present element behaves quite satisfactorily either for the linear analysis of plate bending problems and plane problems or for the geometrically nonlinear analysis of thin plates and shells with large displacement, moderate rotation but small strain.     

A mixed finite element formulation for non-linear analysis of plane problems

Based on the incremental non-linear theory of solid bodies and the Hellinger-Reissncr principle, a mixed updated Lagrangian formulation of the large displacement motion of solid bodies is derived, and an associated mixed finite element model is developed. The model contains the displacements and stresses as the nodal degrees of freedom. The model is used for the large deformation elasto-plastic analysis of plane problems. In solving non-linear problems, the Newton-Raphson method with arc-length control is adopted to trace the post-buckling response. The computational steps to calculate the elasto-plastic stress increments at Gauss points in the elasto-plastic analysis by the present mixed model are described in detail. Numerical results are presented and compared with those of the displacement model and existing solutions to show the accuracy of the present mixed model in the large deformation elasto-plastic analysis of plane problems.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Flexible body dynamics in a local frame with explicitly predicted motion

This paper deals with formulation of dynamics of a moving flexible body in a local frame of reference. In a conventional approach the local frame is normally fixed to the corresponding body and always represents the positions and angles of the body: the positions and angles are represented by Cartesian coordinates and Euler angles or Euler parameters, respectively. The elastic degrees of freedom are expressed by, e.g. nodal coordinates in a finite element analysis, modal coordinates, etc. However, the choice of these variables as the generalized coordinates makes the resulting equations of motion extremely complicated. This is because the representation of the rotation of a body is highly non‐linear and this non‐linearity makes the coefficient matrices dependent on the coordinates themselves. In this paper, we propose an alternative way of treating the issue by explicitly predicting the body motions and regularly updating the local frame. First, the motion of the local frame is assumed to explicitly follow the associated moving body. Then, the equations of motion are derived in a set of generalized coordinates that express both rigid‐body and elastic degrees of freedom in the local frame. These equations are solved by a time integration with a given time interval. The motion of the local frame in the interval is estimated from a prediction of the rigid‐body motions. Then, the gap between the predicted and the actual motions is evaluated. Finally, the predictions are iteratively corrected by the obtained responses in the rigid‐body motions so that the gap should remain within an imposed tolerance. Copyright © 2009 John Wiley & Sons, Ltd.     

Large displacement analysis of three-dimensional beam structures

An updated Lagrangian and a total Lagrangian formulation of a three-dimensional beam element are presented for large displacement and large rotation analysis. It is shown that the two formulations yield identical element stiffness matrices and nodal point force vectors, and that the updated Lagragian formulation is computationally more effective. This formulation has been implemented and the resulted of some sample analyses are given.     

Geometrically nonlinear analysis of shells by quadrilateral flat shell element with drill,shear, and warping

In this paper, a four‐node quadrilateral flat shell element is proposed for geometrically nonlinear analysis based on updated Lagrangian formulation with the co‐rotational kinematics concept. The flat shell element combines the membrane element with drilling degrees of freedom and the plate element with shear deformation. By means of these linearized elements, a simplified nonlinear analysis procedure allowing for warping of the flat shell element and large rotation is proposed. The tangent stiffness matrix and the internal force recovery are formulated in this paper. Several classic benchmark examples are presented to validate the accuracy and efficiency of the proposed new and more proficient element for practical engineering analysis of shell structures. Copyright © 2016 John Wiley & Sons, Ltd.     

A family of simple and robust finite elements for linear and geometrically nonlinear analysis of laminated composite plates

A family of simple, displacement-based and shear-flexible triangular and quadrilateral flat plate/shell elements for linear and geometrically nonlinear analysis of thin to moderately thick laminate composite plates are introduced and summarized in this paper.

The developed elements are based on the first-order shear deformation theory (FSDT) and von-Karman’s large deflection theory, and total Lagrangian approach is employed to formulate the element for geometrically nonlinear analysis. The deflection and rotation functions of the element boundary are obtained from Timoshenko’s laminated composite beam functions, thus convergence can be ensured theoretically for very thin laminates and shear-locking problem is avoided naturally.

The flat triangular plate/shell element is of 3-node, 18-degree-of-freedom, and the plane displacement interpolation functions of the Allman’s triangular membrane element with drilling degrees of freedom are taken as the in-plane displacements of the element. The flat quadrilateral plate/shell element is of 4-node, 24-degree-of-freedom, and the linear displacement interpolation functions of a quadrilateral plane element with drilling degrees of freedom are taken as the in-plane displacements.

The developed elements are simple in formulation, free from shear-locking, and include conventional engineering degrees of freedom. Numerical examples demonstrate that the elements are convergent, not sensitive to mesh distortion, accurate and efficient for linear and geometric nonlinear analysis of thin to moderately thick laminates.     

基于静力凝聚的高精度变截面梁单元及其几何非线性分析研究

基于Euler-Bernoulli梁单元基本假定,通过静力凝聚获得截面特性沿单元轴向连续变化的变截面梁单元高精度刚度矩阵,并提出一种基于随动坐标法求解变截面梁杆结构大位移、大转动、小应变问题的新思路。首先依据插值理论和非线性有限元理论推导出三节点变截面梁单元的切线刚度矩阵,然后使用静力凝聚方法消除中间节点自由度,从而得到一种新型非线性两节点变截面梁单元。结合随动坐标法,在变形后位形上建立随动坐标系,得到变截面梁单元的大位移全量平衡方程。实例计算表明,该新型变截面梁单元具有较高的计算精度,可应用于变截面梁杆系统大位移几何非线性分析。     

一种含有闭环单元的并联机构运动学分析

目的针对自动包装生产线上快速、高精度的分拣,提出一种含有闭环单元的2-UPS/(S+SPR)R并联机构。方法利用螺旋理论建立2-UPS/(S+SPR)R的旋转矩阵,得到机构的自由度。采用闭环矢量法计算出机构的位置逆解。在SolidWorks中,通过Motion分析得到工作空间的所有数据点坐标,然后将全部的数据点导入Matlab,绘制出机构有效的工作空间。结果求解显示该并联机构有绕x轴转动、绕y轴转动和沿y方向移动等3个自由度,得到盾牌形弧状的工作空间。结论该机构具有较大的运动空间,且灵活性好,能够有效地提高包装工作效率。     

Second-order cone programming formulation of discontinuous deformation analysis

In classic discontinuous deformation analysis (DDA), artificial springs must be employed to enforce the contact condition through the open-close iteration. However, improper stiffness parameters might cause numerical problems. The main goal of this paper is to propose a new framework of DDA using second-order cone programming. The complementarity relationship at contacts can be formulated directly; thus, artificial springs are avoided. Stemming from the equations of momentum conservation of each block, the governing equations of DDA can be cast as convex optimization problems. The basic variables in the formulations can be either block displacements or contact forces. The derived optimization problems can be reformulated into a standard second-order cone programming program, which can be solved using standard efficient optimization solvers. The proposed approach is validated by a series of numerical examples.     

A robust quadrilateral membrane finite element with drilling degrees of freedom

A quadrilateral membrane finite element with drilling degrees of freedom is derived from variational principles employing an independent rotation field. Both displacement based and mixed approaches are investigated. The element exhibits excellent accuracy characteristics. When combined with a plate bending element, the element provides an efficient tool for linear analysis of shells.     

修正结构有限元模型的一种方法

本文讨论了一个寻找结构有限元模型的误差源和对它进行修正的方法.为了减少待修正模型的未知参数数目,将质量、刚度矩阵描述为设计变量的函数.在此基础上,先用少量测得的静位移数据修正刚变矩阵;然后用少数几阶实验的固有频率、固有振型数据修正质量矩阵.为了避免测量旋转自由度的静位移和固有振型分量的困难,还讨论了聚缩模型的修正问题.最后,文中附有算例及结果.     

Enhanced rotational Bragg selectivity by use of random phase encoding in volume holographic filter

An out-of-plane angular detection scheme with random phase encoding is proposed. A ground glass is attached on a rod, which is rotated around a center point, so that the rotation of the rod induces the displacement of the ground glass in a circular path. To enhance the rotational sensitivity we adjust the Bragg selectivity of the volume holographic optical element encoded by random phase. Therefore, the rotational sensitivity can be tuned over a large range from several degrees to ten thousandth degrees by changing the radius of rotation. The theoretical calculation, as well as experiment, is demonstrated.     

Rotation‐free triangular shell element using node‐based smoothed finite element method

In this paper, a triangular thin flat shell element without rotation degrees of freedom is proposed. In the Kirchhoff hypothesis, the first derivative of the displacement must be continuous because there are second‐order differential terms of the displacement in the weak form of the governing equations. The displacement is expressed as a linear function and the nodal rotation is defined using node‐based smoothed finite element method. The rotation field is approximated using the nodal rotation and linear shape functions. This rotation field is linear in an element and continuous between elements. The curvature is defined by differentiating the rotation field, and the stiffness is calculated from the curvature. A hybrid stress triangular membrane element was used to construct the shell element. The penalty technique was used to apply the rotation boundary conditions. The proposed element was verified through several numerical examples.     

铆接界面端三维奇异性应力场的研究

针对铆接结构的特点,应用特征函数扩展技术分析柱坐标下接触界面端的应力奇异性问题。建立了柱坐标下圆柱体端面接触边缘附近的三维渐近位移场和应力场渐近表达式,并根据铆钉/被铆接件接触界面端的位移和应力边界条件,建立一个非线性特征方程组。据此方程组可求解界面端邻域的应力奇异性指数、位移和应力角分布函数的数值解。通过与有限元方法计算结果相对比,验证了该方法的有效性。分析了平头、沉头以及半圆头铆钉构成的铆接结构的应力奇异性问题,考察了铆钉材料、几何形式和摩擦系数对接触界面端应力奇异性指数和应力场角分布的影响。     

Development of a finite element contact analysis algorithm for charged‐hydrated soft tissues with large sliding

This paper focuses on a finite element analysis of contact phenomena with large sliding between charged‐hydrated biological soft tissues, such as articular cartilages, based on the triphasic theory. The impenetrability constraint between the contacting bodies and the continuity of the interstitial fluid and ion phases at the contact surfaces are imposed by applying a Lagrange multiplier approach with the contact pressure, chemical potential of the fluid and electrochemical potentials of ions as Lagrange multipliers. A node‐to‐segment one‐pass approach is adopted to cope with large deformations and sliding between the contact surfaces. To pass the contact patch test, contact boundary integrations are performed on both the master and slave contact surfaces. On the other hand, the degrees of freedom of the multipliers at the master nodes are eliminated by projecting the master nodes onto the slave surface to avoid overconstraint. The effectiveness of the proposed algorithm is verified by a couple of numerical examples, in which continuous distributions of displacement, fluid flow, ionic molar flow and Lagrange multipliers on or across the contact surface are confirmed. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical modelling of shear bands by element bands

It is presently a concern and challenge to numerically model shear band localization. Many numerical methods have been developed to take into account the strain and displacement discontinuities across a shear band. In this paper, a contact band element method is proposed to model the shear band with a finite thickness under large shear deformation. The shear band elements, alternatively called contact band elements, are continuously updated based on their current configurations to prevent the large distortions of conventional finite elements and maintain realistic shear band configurations. The contact band element method, with a technique for the special shear band element, consists of the schemes to keep the shear band elements in good shapes, handle the band overlapping, kinking and separation problems. A few examples have shown that the contact band element method is a very efficient way to model the shear bands under large shear deformation. Copyright © 2002 John Wiley & Sons, Ltd.     

Analysis of a rotation‐free 4‐node shell element

In this paper, a shell element for small and large deformations is presented based on the extension of the methodology to derive triangular shell element without rotational degrees of freedom (so‐called rotation‐free). As in our original triangular S3 element, the curvatures are computed resorting to the surrounding elements. However, the extension to a quadrilateral element requires internal curvatures in order to avoid singular bending stiffness. The quadrilateral area co‐ordinates interpolation is used to establish the required expressions between the rigid‐body modes of normal nodal translations and the normal through thickness bending strains at mid‐side. In order to propose an attractive low‐cost shell element, the one‐point quadrature is achieved at the centre for the membrane strains, which are superposed to the bending strains in the centred co‐rotational local frame. The membrane hourglass control is obtained by the perturbation stabilization procedure. Free, simply supported and clamped edges are considered without introducing virtual nodes or elements. Several numerical examples with regular and irregular meshes are performed to show the convergence, accuracy and the reasonable little sensitivity to geometric distortion. Based on an updated Lagrangian formulation and Newton iterations, the large displacements of the pinched hemispherical shell show the effectiveness of the proposed simplified element (S4). Finally, the deep drawing of a square box including large plastic strains with contact and friction completes the ability of the rotation‐free quadrilateral element for sheet‐metal‐forming simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Nanometric position and displacement measurement of the six degrees of freedom by means of a patterned surface element

A method is presented for position and displacement measurements of the six degrees of freedom by use of a patterned surface element observed by a static interferometric vision system. The surface element is made of a regular pattern of circular holes etched on a chromium layer deposited onto a flat glass plate. The in-plane coordinates (x, y, theta(z)) are reconstructed with a subpixel resolution by a vision method based on phase measurements. The out-of-plane coordinates (z, theta(x), theta(y)) are reconstructed by phase-shifting interferometry. Resolutions achieved by the proposed method are in the range of microradians for the measurement of angles and of nanometers for the position and displacement.