Inverse mapping and distortion measures for quadrilaterals with curved boundaries

By using the theory of geodesics in differential geometry, inverse relations of the mapping for 8-node quadrilaterals with curved boundaries are derived and expressed in terms of the element geodesic co-ordinates defined at the local origin. The parameters in the expressions of the geodesic co-ordinates in terms of the isoparametric co-ordinates are suggested to be the distortion measures of the element. The mathematical meanings of these parameters can be discussed and explained by using the coefficient vectors in the equations defining the isoparametric co-ordinate transformation. This method of defining the element distortion measures is consistent and general, and can be applied to any other two- or three-dimensional isoparametric finite elements.     

The reference coordinates and distortion measures for quadrilateral hybrid stress membrane element

The skew Cartesian coordinate system determined by the Jacobian of the isoparametric transformation evaluated at the origin can be shown to be a geodesic coordinate system at the origin. By using a theory in differential geometry, inverse relations of the isoparametric coordinate transformation can be derived and expressed in terms of these geodesic coordinates. In the formulation of hybrid stress finite elements, it is suggested as a new strategy for assumed stresses that such coordinates be used as the reference coordinates. The theory described is exemplified by its applications to the 4-node hybrid stress membrane elements. A set of new distortion-measuring parameters for the quadrilateral element are also proposed based on such theory.     

A new method for finite element transitional mesh generation

In the preparation of finite element meshes, inclusion of transitional blocks is important in order to construct optimal meshes. In this paper, a new method is proposed which is capable of generating quaddominated arbitrary transitional meshes. These meshes are well graded and do not require any mesh smoothening algorithm. The inverse isoparametric mapping combined with an elimination procedure is used to construct transition zones. This new algorithm is described in detail and its efficiency is illustrated with appropriate examples. Different methods available for inverse isoparametric mapping are discussed with their merits and limitations. Many of the existing techniques for inverse isoparametric mapping require the calculation of some special coefficients which may vary with the element type. These techniques lose their clarity and efficiency in the case of three dimensional and higher order two dimensional elements. In this paper, a generalized iterative procedure is proposed to carry out the inverse isoparametric mapping. The computations in this approach are already part of every finite element program based on the isoparametric formulation. Hence implementation of the new algorithm is very simple and straightforward.     

A fully smoothed XFEM for analysis of axisymmetric problems with weak discontinuities

In this paper, we propose a fully smoothed extended finite element method for axisymmetric problems with weak discontinuities. The salient feature of the proposed approach is that all the terms in the stiffness and mass matrices can be computed by smoothing technique. This is accomplished by combining the Gaussian divergence theorem with the evaluation of indefinite integral based on smoothing technique, which is used to transform the domain integral into boundary integral. The proposed technique completely eliminates the need for isoparametric mapping and the computing of Jacobian matrix even for the mass matrix. When employed over the enriched elements, the proposed technique does not require sub‐triangulation for the purpose of numerical integration. The accuracy and convergence properties of the proposed technique are demonstrated with a few problems in elastostatics and elastodynamics with weak discontinuities. It can be seen that the proposed technique yields stable and accurate solutions and is less sensitive to mesh distortion. Copyright © 2016 John Wiley & Sons, Ltd.     

Direct evaluation of singular boundary integrals in two-dimensional biharmonic analysis

Development of techniques to provide rapid and accurate evaluation of the integrations required in boundary element method (BEM) formulations are receiving more attention in the literature. In this work, a series of direct expressions for surface integrals, required for a boundary element solution of the non-homogeneous biharmonic over a general two-dimensional curvilinear surface, are presented. The concept of an isoparametric representation, usually applied to the variation of the field variables and the geometry, is extended to the parametric mapping of the curvilinear geometry. The result renders the typically complicated Jacobian function into a series of polynomial expressions based on the shape function set and several discrete Jacobian values. An application of the isoparametric approximation of the Jacobian for a quadratic element representation is developed. Implementation of this approximation significantly improves the accuracy of the boundary integral solution by eliminating error associated with numerical quadrature. Overall computational efficiency is improved by reducing the time necessary to calculate individual surface integrals and evaluate field variables at internal points. A numerical solution of the boundary integral equations of phenomena governed by the biharmonic equation is presented and compared with an exact analysis.     

A Smoothed Finite Element Method for Mechanics Problems

In the finite element method (FEM), a necessary condition for a four-node isoparametric element is that no interior angle is greater than 180° and the positivity of Jacobian determinant should be ensured in numerical implementation. In this paper, we incorporate cell-wise strain smoothing operations into conventional finite elements and propose the smoothed finite element method (SFEM) for 2D elastic problems. It is found that a quadrilateral element divided into four smoothing cells can avoid spurious modes and gives stable results for integration over the element. Compared with original FEM, the SFEM achieves more accurate results and generally higher convergence rate in energy without increasing computational cost. More importantly, as no mapping or coordinate transformation is involved in the SFEM, its element is allowed to be of arbitrary shape. Hence the restriction on the shape bilinear isoparametric elements can be removed and problem domain can be discretized in more flexible ways, as demonstrated in the example problems.     

Algorithms for determining invertible two- and three-dimensional quadratic isoparametric finite element transformations

This paper presents an algorithm which determines the invertibility of any planar, triangular quadratic isoparametric finite element transformation. Extensions of the algorithm to three-dimensional isoparametric finite element transformations yield conditions which guarantee invertibility of 10-node tetrahedra and 8-node bricks. The mathematical basis for the algorithm focuses on the Jacobian as a continuous function defined over a compact set where the Jacobian attains a maximum and a minimum value. The algorithm then determines whether these values are of opposite sign.     

含两个分量的四边形单元面积坐标理论

为了便于构造抗畸变的四边形单元,建立了一套新的四边形单元面积坐标理论(QAC-2),并给出了相关的积分和微分公式。该坐标系作为自然坐标,具有明确的物理意义,且只含有两个相互独立的坐标分量,因此易于实现与直角坐标和等参坐标的沟通,便于理解和应用;两个坐标分量与直角坐标之间满足线性变换,在构造单元时易于选择完备的多项式序列,且多项式的完备次数不会随着网格的畸变而下降,因此可以保证单元的精度和抗畸变性能。     

六面体单元体积坐标方法

基于二维问题四边形单元面积坐标法的成功思路,建立了三维六面体单元体积坐标的系统方法,包括:1)六面体单元特征参数的定义及单元退化模式研究;2)六面体单元体积坐标定义;3)六面体单元的体积坐标与直角坐标、等参坐标之间的关系;4)六面体体积坐标的微分公式。可以看到,六面体体积坐标保持了局部自然坐标的优点,并且与直角坐标始终保持线性关系。它为构造对网格畸变不敏感的新型六面体有限元模型提供了新工具。     

Eight-node conforming straight-side quadrilateral element with high-order completeness (QH8-C1)

A new eight-node conforming quadrilateral element with high-order completeness, denoted as QH8-C1, is proposed in this article. First, expressions for the interpolation displacement function satisfying the requirements for high-order completeness in the global coordinate system are constructed. Second, the displacement function expression in global coordinates is transformed into isoparametric coordinates, and the relationships between the two series of coefficients for the two kinds of displacement function expressions are found. Third, the displacement function expression is modified to satisfy the requirements of nodal freedom and interelement boundary continuity. The key to the new element construction is the derivation of the linear relationship expressions among 12 coefficients of element displacement interpolation polynomials in the global and isoparametric coordinate systems. As a result, the relationship between quadratic completeness and interelement continuity is explicitly given, and a proof of the completeness and the continuity was conducted to theoretically guarantee the validity of the derivation results. Furthermore, in order to verify the correctness of the theoretical work, nine numerical examples were performed. The computation results from these examples demonstrate that QH8-C1 exhibited excellent performance, including high simulation accuracy, fast convergence, insensitivity to mesh distortion, and monotonic convergence.     

Exterior flow with an isoparametric Hermite cubic element

An isoparametric Hermite cubic element is developed for calculating compressible flows past complex aerofoils in two dimensions. A simple intermediate transformation is used to derive an isoparametric transform whose Jacobian is always non-zero and which attains very high accuracy on the boundary. Application of the method and assessment of the error is illustrated by flow past a circular cylinder.     

Improved co-ordinate transformations for finite elements: The Lagrange cubic case

A modified isoparametric co-ordinate transformation is developed for the case of Lagrange cubic finite elements. This modified transformation produces significant improvements in solution accuracy while eliminating troublesome aspects of the standard isoparametric mapping. In particular, zero Jacobians in the standard transformation, which can occur in elements having non-uniform node spacing, are completely eliminated with the new transformation. Within the analysis, the range of node locations for which the standard transformation is valid (the Jacobian does not vanish) is determined. The treatment of curved sides is discussed and illustrated. Example calculations using two-dimensional serendipity elements show that the new transformation provides solutions that are as much as several orders of magnitude more accurate than those computed using the standard formulation.     

复合材料层合板壳的三维退化热壳元分析

根据壳元的几何形状规定, 推导了壳元几何关系公式; 假设温度沿壳元厚度是二次多项式分布, 在给定边界条件下, 推导了壳元温度函数表达式; 根据传热定理和有限元离散原理, 推导了复合材料热传导系数从材料坐标系到有限元分析的等参元正则坐标系的变换关系, 建立了各向异性复合材料层合板的有限元热分析列式; 通过积分变换, 推导了三维退化热壳元公式。在此工作基础上, 编制了有限元计算程序, 将计算结果和ANSYS 计算结果进行了比较, 结果吻合较好。经证明, 三维退化热壳元可以节省内存和提高计算效率。      

一种双目立体视觉系统的校准方法

基于双目立体视觉系统的图像分析以及人工神经网络的三维空间建模算法,设计了一种针对双目立体视觉相机的校准方法,并可应用于运动目标点的轨迹追踪。将均匀分布目标点的校准平面放置在有效视野内的不同位置,通过双目立体视觉系统来捕获处于不同位置的校准平面图像。在图像处理之后,使用校准点中心的二维坐标作为人工神经网络训练的输入样本集,通过建立人工神经网络模型结构,实现目标点二维平面坐标到三维空间坐标的映射关系。采用这种具有通用性的方法,可以有效修正系统中存在的失真因子,获得目标三维位置信息,而无需进行复杂的相机校准操作。实验表明,提出的方案具有良好的可行性和鲁棒性。     

Multi-Directional Reconstruction Algorithm for Panoramic Camera

of view. It can be applied in virtual reality, smart homes and other fields as well. A multi-directional reconstruction algorithm for panoramic camera is proposed in this paper according to the imaging principle of dome camera, as the distortion inevitably exists in the captured panorama. First, parameters of a panoramic image are calculated. Then, a weighting operator with location information is introduced to solve the problem of rough edges by taking full advantage of pixels. Six directions of the mapping model are built, which include up, down, left, right, front and back, according to the correspondence between cylinder and spherical coordinates. Finally, multi-directional image reconstruction can be realized. Various experiments are performed in panoramas (1024×1024) with 30 different shooting scenes. Results show that the azimuth image can be reconstructed quickly and accurately. The fuzzy edge can be alleviated effectively. The rate of pixel utilization can reach 84%, and it is 33% higher than the direct mapping algorithm. Large scale distortion is also further studied.     

On an isoparametric finite-element for composite laminates with finite rotations

For composite laminates consisting of an arbitrary number of orthotropic laminae first a finite-rotation theory is presented as basis of isoparametric finite-element formulations. The derivation is achieved by a Reissner-Mindlin type kinematic assumption which allows a constant shear deformation distribution across the thickness. The constitutive equations are presented in a general form such that orthotropic material behaviour with material axes varying arbitrarily across the thickness may easily be considered in numerical implementation, also when using curvilinear coordinates. Special attention is taken to predict the force distribution in the deformed shell structure. This theory is then transformed into a four-node isoparametric assumed-strain finite element. Unlike in the degeneration approach, interpolation polynomials are introduced directly for rotation variables determining the deformed position of the unit normal vector. The capability of the finite element developed to deal with strongly nonlinear situations is demonstrated by many examples. Also numerical results are presented permitting a systematical comparison of classical and isoparametric approaches concerning the numerical efficiency.     

Isoparametric Hermite elements

Isoparametric Hermite elements are created using Bogner–Fox–Schmit rectangles on a reference domain and mapping these numerically onto the computational domain. The difficulties involved in devising explicit C¹ shape functions for isoparametric elements are thus avoided, and the resulting elements have all the benefits of full C¹ continuity, the simplicity of the Bogner–Fox–Schmit element and the geometrical flexibility expected from higher-order isoparametric elements. The numerical mapping consists in the finite element solution of a linear boundary value problem, which is inexpensive and is carried out as a preprocessing operation—the required derivatives of the mapping then being supplied to the main analysis as data. Some care is required in defining the differential boundary conditions, and guidance on this is provided. Examples are given showing the success of the mapping procedure, and the use of the resulting elements in the solution of some boundary value problems. The numerical results confirm a convergence analysis provided for the new isoparametric Hermite element.     

Simple cylindrical and spherical finite elements

Finite element approximations are developed for three‐dimensional domains naturally represented in either cylindrical or spherical coordinates. Lines of constant radius, axial length, or angle are used to represent the domain and cast approximations that are natural for these geometries. As opposed to general isoparametric three‐dimensional elements generated in conventional parent space, these elements can be evaluated analytically and do not generate geometric discretization error. They also allow for anisotropic material coefficients that are frequently aligned in either cylindrical or spherical coordinates. Several examples are provided that show convergence properties and comparison with analytical solutions of the Poisson equation.     

面积坐标法构造含转角自由度的四结点膜元

以四边形面积坐标作为工具,构造了两个含转角自由度的广义协调四边形单元AQ4和lAQ4。它们通过强式分片检验,与同类单元相比,具有很高的计算精度,能消除梯形闭锁现象,有很强的抗网格畸变的能力。     

An efficient method for nonlinear analysis of layered shells

Finite element formulation based on explicit through-thickness integration scheme assumes importance when applied to multilayered shells, as it is numerically accurate and computationally efficient. Explicit integration becomes possible on assuming the variation of the inverse Jacobian through the thickness. The element stiffness matrices are discussed for (i) large rotation, and (ii) small rotation. Relative efficiencies of the explicit through-thickness integration schemes are compared with that of the conventional formulation involving numerical integration in three directions in each layer and summation over the layers. The small rotation formulation assuming linear variation of the Jacobian inverse across the thickness and based on further approximation regarding certain submatrices is seen to be computationally efficient. The geometric nonlinear behaviours of laminated composite cylindrical panels subjected to external pressure are discussed. The parameters considered are: number of layers, symmetric/antisymmetric, cross-ply/angleply, boundary conditions and central angle. The strength of shallow panels with longitudinal edges hinged and curved edges free is controlled by the limit point load, while for deep panels it is controlled by the bifurcation load. The boundary conditions have significant influence on load carrying capacities. A list of symbols is given at the end of the paper