On using linear elements in incompressible plane strain problems: a simple edge based approach for triangles

In this paper a simple iterative method is presented for finite element solution of incompressible plane strain problems using linear elements. Instead of using a mixed formulation approach, we use an equivalent displacement/velocity approach in an iterative manner. Control volumes are taken for regions which are to exhibit incompressible behaviour. For triangular elements the control volume is chosen as the area built on the parts of each pair of elements at the sides of an edge. In this case, elements are let to exchange volume. It is shown that the proposed edge based approach removes the deficiency of the linear triangular elements i.e. locking effect. Similar edge based approach is applied to the linear quadrilateral elements. However, if the control volume is chosen as the element volume the formulation gives similar results as the discontinuous mixed formulation using one pressure point without exhibiting instability behaviour. The formulation is based on decomposition of the displacement/velocity field into deviatoric and volumetric parts. The volumetric part is iteratively eliminated without confronting locking or instability phenomenon. The iterative procedure is very cheap and simple to be implemented in any FEM code. Several examples are given to demonstrate the performance of the procedure. Copyright © 2004 John Wiley & Sons, Ltd.     

Polyhedral elements by means of node/edge‐based smoothed finite element method

The node‐based or edge‐based smoothed finite element method is extended to develop polyhedral elements that are allowed to have an arbitrary number of nodes or faces, and so retain a good geometric adaptability. The strain smoothing technique and implicit shape functions based on the linear point interpolation make the element formulation simple and straightforward. The resulting polyhedral elements are free from the excessive zero‐energy modes and yield a robust solution very much insensitive to mesh distortion. Several numerical examples within the framework of linear elasticity demonstrate the accuracy and convergence behavior. The smoothed finite element method‐based polyhedral elements in general yield solutions of better accuracy and faster convergence rate than those of the conventional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.     

On the accuracy of polynomial finite elements for crack problems

A square tensile sheet with symmetric edge cracks is analysed by the finite element method to demonstrate the accuracy obtained with polynomial elements when the mesh patterns are chosen on the basis of the element's interpolation functions. Numerical results are provided for the accuracy of the total strain energy, the stress intensity factor, and the numerical condition of the global stiffness matrix. The stress intensity factor is determined to within 1.0 per cent accuracy using quadratic (linear strain) triangular elements with only 138 degrees-of-freedom.     

On completeness of shape functions for finite element analysis

Some elements commonly used for analysis are examined for examined for completeness of polynomial interpolation and computational efficiency. Extensions to n-dimensional space are shown to be natural consequences of the interpolation, thus all elements considered here allow for finite element approximation in higher than three-dimensional spaces (e.g. space–time interpolations). From the study it is concluded that ‘serendipity’ class elements from the most efficient elements up to third-degree polynomial approximations. The method used here to develop the serendipity shape functions allows for different orders of interpolation along each edge. Thus, in zones where high accuracy is required meshes can now be easily changed from linear to quadratic or higher-order elements. Computations on some simple problems have demonstrated this to be a superior method than using large numbers of low ordered elements.     

Generalized crack closure analysis for elements with arbitrarily-placed side nodes and consistent nodal forces

A new approach was developed for the evaluation of energy release rate by the virtual crack closure technique in quadratic and linear elements. The generalized method allows arbitrary placement of the side nodes for quadratic elements and thus includes both standard elements, with mid-side nodes, and singularity elements, with quarter-point nodes, as special cases of one general equation. It also accounts for traction-loaded cracks. The new derivation revealed that the proper nodal forces needed for crack closure calculations should be the newly-defined “nodal edge forces,” rather than the global or element forces from standard finite element analysis results. A method is derived for calculating nodal edge forces from global forces. These new forces affect energy release rate calculations for singularity elements and for problems with traction-loaded cracks. Several sample calculations show that the new approach gives improved accuracy.     

一种改进的边界轮廓矢量化算法

左右于断层测量的反向建模中，物体断面图像序列的区域边界和实现模型重构的基础，如何从边界轮廓精确提取几何特征是实现重构的关键。本文提出一种改进的矢量化算法，该算法的基本思想是首先提取曲率极值点作为初始分段点。然后判定每两个相邻初始分段点之间轮廓线的形状，并区别直线段与曲线段分别进行拟合和矢量化。实践表明，该算法既能准确提取边界形状的特征关键点，又保证了重构后曲面的光滑规整。     

Evolution of linear edge dislocation in atmospheric turbulence and free space

In accordance with the extended Huygens–Fresnel principle, the evolution of linear edge dislocation propagating through atmospheric turbulence is studied, and the effects of waist width and the slope of linear edge dislocation are investigated. It is shown that when linear edge dislocation beams propagate through atmospheric turbulence, if the waist width is not equal and the slope is not zero, then the linear edge dislocation vanishes and transforms into an optical vortex with a topological charge of ?1 or +1. The optical vortex and an optical vortex that is created annihilate when the transmission distance is far enough. The linear edge dislocation vanishes when the slope is zero. If the waist width is equal, then the linear edge dislocation will vanish regardless of the slope value. For linear edge dislocation beams in free space, when selecting specific parameters, linear edge dislocation always exists.     

A least-squares approach for uniform strain triangular and tetrahedral finite elements

A least-squares approach is presented for implementing uniform strain triangular and tetrahedral finite elements. The basis for the method is a weighted least-squares formulation in which a linear displacement field is fit to an element's nodal displacements. By including a greater number of nodes on the element boundary than is required to define the linear displacement field, it is possible to eliminate volumetric locking common to fully integrated lower-order elements. Such results can also be obtained using selective or reduced integration schemes, but the present approach is fundamentally different from those. The method is computationally efficient and can be used to distribute surface loads on an element edge or face in a continuously varying manner between vertex, mid-edge and mid-face nodes. Example problems in two- and three-dimensional linear elasticity are presented. Element types considered in the examples include a six-node triangle, eight-node tetrahedron, and ten-node tetrahedron. © 1998 John Wiley & Sons, Ltd.     

Application of the control volume mixed finite element method to a triangular discretization

A two‐dimensional control volume mixed finite element method is applied to the elliptic equation. Discretization of the computational domain is based in triangular elements. Shape functions and test functions are formulated on the basis of an equilateral reference triangle with unit edges. A pressure support based on the linear interpolation of elemental edge pressures is used in this formulation. Comparisons are made between results from the standard mixed finite element method and this control volume mixed finite element method. Published 2011. This article is a US Government work and is in the public domain in the USA.     

改进的直线拟合线阵CCD图像边缘检测方法

为了检测经过光学系统成像所得图像的边缘,在直线拟合方法的基础上提出了一种新的边缘检测方法。该方法综合考虑边缘过渡区间所有点的信息,利用直线拟合方法求出一组可能的边缘值,再选择它们的中值作为最终的边缘值。基于该方法的采集系统由线阵 CCD 采集边缘的灰度图像,使用单片机来处理数据,并计算出边缘的位置。实验结果表明,这种改进方法的重复性误差为 0.89 个像素,而且能够在 0.039s 内完成一次边缘检测计算。     

A MODIFIED FINITE-ELEMENT SMEARING TECHNIQUE FOR FREE-EDGE STRESS ANALYSIS

A new finite-element smearing scheme is presented in the present study to provide a practical yet accurate free-edge stress analysis. It is proposed that, in the area away from the free edge of the laminate, all layers are to be smeared into one layers of finite elements, while near the free edge each layer is to be modeled as one layer of finite elements. Compatibility between the smeared elements andthe regular elements is enforcedby the penalty method. The present technique can be applied to general laminates with various geometries, loadings, etc. The results of two example problems can show that the present approach is simple, accurate, and economical in computing free edge stresses of laminates.     

Comparison of different finite elements for 3-D eddy currentanalysis

In order to evaluate the best type of element for the finite-element analysis of 3-D eddy currents, a fundamental model is analyzed using the usual first-order tetrahedral, triangular prism, and brick nodal elements, and also the brick edge element. The effects of the types of elements on the flux and eddy-current distributions are investigated using the A-φ method and the T-Ω method. It is concluded that the brick edge element is best from the viewpoints of accuracy and CPU time     

Extension of the method of auxiliary mapping for three‐dimensional elliptic boundary value problems

In this paper, we extend the method of auxiliary mapping (MAM), introduced by Babuška and Oh, to three dimensions so that the extended MAM (3‐D MAM) can effectively handle three‐dimensional elliptic problems containing the singularities caused by the non‐smooth domains. There are three type of singularities caused by non‐smoothness of domains in R ³: the vertex, the edge, and the vertex–edge combined singularities. To deal with the singularities of these types, we present three auxiliary mappings and formulas for the transformed bilinear forms and the transformed linear functionals by these auxiliary mappings. Then we present 3‐D MAM and constructions of the blending‐type elemental mappings for elements containing singularities. Numerical experiments that show the effectiveness of 3‐D MAM are provided. Copyright © 2001 John Wiley & Sons, Ltd.     

3D Galerkin integration without Stokes' theorem

A direct approach to calculating the hypersingular integral for a three-dimensional Galerkin approximation is presented. The method does not employ either Stokes' theorem or a regularization process to transform the integrand before the evaluation is carried out. Integrating two of the four dimensions analytically, the potentially divergent terms arising from the coincident and adjacent edge integrations are identified and canceled exactly. The method is presented in the simplest possible situation, the hypersingular kernel for the Laplace equation, and linear triangular elements.     

直线拟合方法在一维图像边缘检测中的应用

在线阵CCD测量系统中采用直线拟合方法检测一维图像边缘。首先,测量系统通过线阵CCD获得一维图像,基于阈值比较法,对图像信号设置高、低阈值,截取边缘信号的中间部分作为拟合窗口;然后,利用最小二乘法,对拟合窗口内的边缘信号进行直线拟合;最后,利用图像亮、暗电平的中间电平作为阈值截交拟合直线,以此交点作为图像边缘点的位置。实验结果和分析表明,与阈值比较法相比较,直线拟合方法具有重复精度高、稳定性好的优点,能有效抑制随机噪声的影响,因此可以有效检测一维图像边缘位置。     

基于HSI空间彩色图像多结构元形态边缘检测

构建了一类在HSI 颜色空间基于多结构元彩色形态边缘检测梯度算法实现彩色图像边缘检测新方法,多结构元形态边缘检测有着比单一结构元素形态边缘检测更优越的性能。所定义的算法充分考虑了边缘的方向性和噪声对边缘检测的影响,本算法在噪声抑制方面显示了优良的性能,能满足不同应用的需要。     

Modeling optical properties of liquid-crystal devices by numerical solution of time-harmonic Maxwell equations

We consider numerical modeling of the optical properties of devices typical of beam-steering devices based on liquid-crystal materials: two-dimensional, anisotropic and inhomogeneous dielectric properties, periodic in one dimension. A mathematical formulation of the system of second-order partial differential equations for the components of the time-harmonic electric field is discretized by using a finite-element method based on curl-conforming edge elements. The discrete equations are also interpreted as equivalent finite-difference equations. It is shown how the resulting large sparse complex system of linear algebraic equations can be solved by an iterative method with convergence accelerated by a preconditioner based on fast Fourier transforms. Benchmarking results and the application to a realistic problem are reported. The practical limitations of the approach and its advantages and disadvantages compared with other approaches are discussed.     

基于数学形态学的HSI空间彩色边缘检测方法

针对在RGB空间中很难有效区分颜色相似性的问题,选择更加符合颜色视觉特性的HSI颜色空间进行图像处理,提出一种基于HSI空间的多结构多尺度自适应彩色图像边缘检测方法.首先对H、S、I三个分量采用不同结构和不同尺度的结构元素进行形态学边缘检测,然后对三个边缘分量利用信息熵加权得到融合后的彩色边缘信息.实验结果表明,该方法可以充分利用彩色图像的色度,饱和度和亮度信息,有效地抑制噪声,自适应地提取完整的边缘信息.     

Boundary edge elements and spanning tree technique in three-dimensional electromagnetic field computation

The boundary discretization of the curl-free vector variable such as the magnetic field h and the divergence-free variable such as the surface current density k is discussed. Instead of introducing the scalar variables, the vector variables h and k are discretized by the curl-conform and the div-conform triangular edge elements, respectively. The degrees of freedom are associated with the boundary edges. In order to ensure the null curl of h and the null divergence of k, a spanning tree technique is used to identify the independent edges. The triangular edge elements contain the first-order nodal elements when expressing h or k by the scalar variables. The use of edge elements permits one to solve multiply connected problems if the independent edges are well identified, i.e. the necessary cuts are introduced in multiply connected domains. An automatic tree generation algorithm is presented. It permits one to determine automatically the additional edges on the necessary loops (cuts) of a multiply connected region. Some tree generation examples are illustrated. A numerical application to a three-dimensional multiply connected eddy current problem is reported at the end of the paper.     

Expansion of periodic boundary condition for 3-D FEM analysis usingedge elements

The application of periodic boundary conditions in the analysis of three-dimensional magnetic fields by finite element methods (FEMs) leads to a substantial reduction of computation labor and storage. The expansion of the condition for magnetostatic curl-curl formulation with the magnetic vector potential employing edge tetrahedral elements is discussed. Differences between the definitions of the condition for nodal and edge elements are examined. The vectorial nature of edge elements is emphasized and associated difficulties in the formulation and application of the condition are carefully analyzed and overcome. Details for computer implementation are given and a simple test problem to verify the validity of the software is proposed. Advantages gained when the condition is used for TEAM Workshop problem 13 as an example are shown