绘制二维单元等值线的一种简单方法

等值线是二维标量场可视化的一种主要手段,提出一种画二维单元等值线的简单方法。首先把母元等分成4个子块,然后利用子块顶点的自然坐标和单元的形函数,求出实际单元中对应子块的顶点温度,利用子块顶点温度判断是否有等值线通过。对于有等值线通过的子块,求出其包围盒的尺寸,如果小于等于一个像素,则把对应的像素设置为相应的颜色,否则,在母元中把子块继续4等分,直到实际坐标系中子块的包围盒尺寸小于一个像素为止。应用实践表明,这种方法原理简单,精度高,易于编写程序。     

基于离散方法画高次三角形单元等值线

高次三角形单元在温度场的有限元计算中起着重要作用,该文提出一种画高次三角形单元等值线的离散方法。自然坐标系中的等腰直角三角形母元被均匀划分为N2个子三角形,用子三角形顶点的自然坐标,插值计算温度,若计算出的温度等于等值线的温度,则把屏幕上相应的象素设置成预定的颜色。实践表明,这种方法精度高,速度快,易于编程。     

画应力场云图的一种简单有效的方法

根据屏幕上单元内像素的个数，在局部坐标系中的母元内均匀布置足够的点；然后，利用这些点的坐标及单元的插值函，求出总体坐标系中对应点的坐标及应力；最后，根据预先设定的颜色与应力之间的函数关系，求出屏幕上对应像素的颜色值。     

针对三维有限元数据场的精确后处理算法

为了描述有限元模型中数据场的分布规律，发展与完善了基于单元局部插值的母单元绘制算法，提出了以高斯积分点作为插值基准点的有限元应力云图绘制算法，并给出了具体的基于母单元的等值线绘制算法．与传统三角形算法做系统的比较表明，文中算法使有限元后处理功能充分达到了有限元数值分析的精度，尤其针对高阶插值数据场有更好的效果．     

用VB画二维高阶等参单元彩色云图

介绍一种画二维高阶等参单元彩色云图的新方法,其特点是算法简单,结果精确,易于编程.文中给出了这种方法的完整VB源程序,并给出一个应用实例.     

一种画高阶等参单元彩色云图的新方法

该文介绍一种画高阶等参单元彩色云图的新方法,其特点是算法简单,结果精确,便于编程。文中给出了这种方法的ＶＢ源程序,并给出应用实例。     

绘制等值线的一种离散方法*

介绍了一种绘制等值线的离散方法。母元被均匀划分为矩形的子域,用子域顶点的自然坐标插值计算应力,若该应力接近等值线的应力,则把对应的像素设置成相应的颜色。实践表明,这种方法精度高、速度快,易于实现。     

高次三角形单元的可视化显示

高次三角形单元在有限元计算中起着重要作用，本文提出一种画高次三角形单元云图的新方法。自然坐标系中的等腰直角三角形母元被均匀划分为N^2个等腰直角子三角形，根据映射到屏幕上的子三角形形心处的温度计算颜色，然后进行填充。实践表明，文中方法十分有效。     

一种应力图生成方法

介绍可视化有限元系统VSES中采用的一种应力图生成方法。首先用扫描线转换填充算法判断各象素是否在单元内,根据单元类型用不同插值方法计算单元内各点应力,最后设置颜色函数,计算各模型中各象素点颜色。由颜色函数的不同设置,可得到应力云图和色带图,用轮廓线追踪法获得应力等值线。方法具有速度快、适应性强、显示效果好等特点。     

三维有限元模型的任意剖切及其等值线与彩色云图生成的方法

对三维有限元模型快速有效地生成任意剖面上等值线及彩色云图,是有限元计算后处理中的一个重要技术。该文在建立单元信息描述表的基础上,提出了一种适合于任意三维实体单元类型的通用剖切算法,和在剖切面及外表面生成等值线或高质量彩色云图的方法。     

一种基于二次等参单元的等值线图生成算法

本文提出了一种基于二次等参单元的等值线图生成算法，该算法根据有限元法将整个区域分割成多个互相连续的二次等参单元，通过生成每个等参单元内的等值线，进而生成整个区域内的等值线。     

四结点单元应力云图的一种生成方法

应力云图是形象地理解有限元分析中应力分布的有效手段，提出了四结点单元应力云图的一种生成方法，推导出了四结点四边形单元内任一点的局部坐标的解析表达式，可直接求解单元内任一点的局部坐标。用扫描线法生成单元应力云图时，用一次Bezier曲线描述四边形单元的边，可减少交点的个数。实例表明，该方法是有效的。     

散乱分布数据曲面重构的光顺-有限元方法

提出了一种基于散乱分布的数据点重构三维曲面的有限元方法.根据最佳逼近与数据光顺理论建立正定的目标泛函,采用有限元最佳拟合使泛函极小化,求得最优解.通过八节点等参数有限元插值计算,重新构造出三维曲面.这种光顺-有限元方法有效地抑制了输入数据上误差噪声的影响,与有限元拟合方法相比,所需的输入数据点少,重构的曲面逼近精度高、光顺性好.数值实验表明,该方法简单,便于应用.     

On application of differential geometry to computational mechanics

Developments in the fields of computational science—the finite element method—and mathematical foundations of continuum mechanics result in many new algorithms which give solutions to very complicated, complex, large scaled engineering problems. Recently, the differential geometry, a modern tool of mathematics, has been used more widely in the domain of the finite element method. Its advantage in defining geometry of elements [13–15] or modeling mechanical features of engineering problems under consideration [4–7] is its global character which includes also insight into a local behavior. This fact comes from the nature of a manifold and its bundle structure, which is the main element of the differential geometry.

Manifolds are generalized spaces, topological spaces. By attaching a fiber structure to each base point of a manifold, it locally resembles the usual real vector spaces; e.g. ³. The properties of a differential manifold M are independent of a chosen coordinate system. It is equivalent to say, that there exists smooth or C^r differentiable atlases which are compatible.

In this paper a short survey of applications of differential geometry to engineering problems in the domain of the finite element method is presented together with a few new ideas.

The properties of geodesic curves have been used by Yuan et al. [13–15], in defining distortion measures and inverse mappings for isoparametric quadrilateral hybrid stress four- and eight-node elements in ². The notion of plane or space curves is one of the elementary ones in the theory of differential geometry, because the concept of a manifold comes from the generalization of a curve or a surface in ³.

Further, the real global nature of differential geometry, has been used by Simo et al. [4,6,7]. A geometrically exact beam finite strain formulation is defined. The mechanical basis of such a nonlinear model can be found in the mathematical foundation of elasticity [18]. An abstract infinite dimensional manifold of mappings, a configuration space, is constructed which permits an exact linearization of algorithms, locally. A similar approach is used by Pacoste [5] for beam elements in instability problems.

Special attention is focused on quadrilateral hybrid stress membrane elements with curved boundaries which belong to a series of isoparametric elements developed by Yuan et al. [14]. The distortion measures are redefined for eight-node isoparametric elements in ² for which geodesic coordinates are used as local coordinates.     

Metric Identities and the Discontinuous Spectral Element Method on Curvilinear Meshes

We study how to approximate the metric terms that arise in the discontinuous spectral element (DSEM) approximation of hyperbolic systems of conservation laws when the element boundaries are curved. We first show that the metric terms can be written in three forms: the usual cross product and two curl forms. The first curl form is identical to the “conservative” form presented by Thomas and Lombard [(1979), AIAA J. 17(10), 1030–1037]. The second is a coordinate invariant form. We prove that in two space dimensions, the typical approximation of the cross product form does satisfy a discrete set of metric identities if the boundaries are isoparametric and the quadrature is sufficiently precise. We show that in three dimensions, this cross product form does not satisfy the metric identities, except in exceptional circumstances. Finally, we present approximations of the curl forms of the metric terms that satisfy the discrete metric identities. Two examples are presented to illustrate how the evaluation of the metric terms affects the satisfaction of the discrete metric identities, one in two space dimensions and the other in three.     

A Finite Element Method on Convex Polyhedra

We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping. This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.     

Line art rendering via a coverage of isoparametric curves

A line art nonphotorealistic rendering scheme of scenes composed of freeform surfaces is presented. A freeform surface coverage is constructed using a set of isoparametric curves. The density of the isoparametric curves is set to be a function of the illumination of the surface determined using a simple shading model, or of regions of special importance such as silhouettes. The outcome is one way of achieving an aesthetic and attractive line art rendering that employs isoparametric curve based drawings that is suitable for printing publication