四边形单元退化为三角形单元问题
在编写有限元程序中，采用的是四边形单元，但划分的单元中有三角形单元出现，请问怎么处理四边形单元退化为三角形单元的问题。因为其中涉及Jacobian矩阵，如果退化到三角形，Jacobian矩阵的行列式变成零，导致奇异，请问怎么在程序中处理呢？ 本帖最后由 zzjo 于 2011-10-4 12:31 编辑转一个单元精度问题的帖子：
参看Modeling and Meshing Guide > Chapter 2. Planning Your Approach >2.2. Choosing Between Linear and Higher Order Elements。参看有限元理论书籍，例如《有限元分析及应用》曾攀，清华大学出版社。
有限元理论上，使用三角形单元比使用四边形单元对于应力场的描述能力要差一些。例如，三节点三角形单元是常应变（应力）CST单元，单元内任意一点应变和应力都为常数，因此，采用三节点三角形单元的话，必须在应力变化剧烈的区域加密网格。当然，正因为如此，ANSYS里面并不提供三节点三角形单元，只提供六节点三角形单元PLANE2。4节点4边形单元的应力和应变为一次线性变化的，因此4边形的单元对于应力场的描述能力要强一些。ANSYS提供的四节点四边形单元有PLANE42。
在三节点三角形单元每条边的中点再增加一个内部节点就得到二次函数6节点三角形单元。6节点三角形单元的位移场模式比四节点四边形单元的位移场模式多了x和y的平方项，它们都含有xy项，它们的位移场和应力场描述能力相近 。
高阶单元和低阶单元。很明显，对于同样形状的单元，高阶单元比低阶单元的精度高，位移场应力场的描述能力强。例如，三角形六节点单元比三角形三节点要好。
在ANSYS里面可以把单元分成两类： linear (with or without extra shapes)(No Midside Nodes)（也即低阶单元）和 quadratic（含中间节点的单元，也即高阶单元）.
使用高阶单元的时候要避免其退化为三角形单元和四面体单元，避免出现单元过分扭曲。 在非线性分析中，如果网格划分合理的话用低阶单元可以获得比高阶单元更精确的解，而且花费的计算量更少。///（For structural analyses, these corner node elements with extra shape functions will often yield an accurate solution in a reasonable amount of computer time. When using these elements, it is important to avoid their degenerate forms in critical regions. That is, avoid using the triangular form of 2-D linear elements and the wedge or tetrahedral forms of 3-D linear elements in high results-gradient regions, or other regions of special interest. You should also take care to avoid using excessively distorted linear elements. In nonlinear structural analyses, you will usually obtain better accuracy at less expense if you use a fine mesh of these linear elements rather than a comparable coarse mesh of quadratic elements.）//
建立曲面壳体模型时，一般采用低阶（linear）壳单元就可以了，只是需要注意控制每个单元所跨的角度 //（When modeling a curved shell, you must choose between using curved (that is, quadratic) or flat (linear) shell elements. Each choice has its advantages and disadvantages. For most practical cases, the majority of problems can be solved to a high degree of accuracy in a minimum amount of computer time with flat elements. You must take care, however, to ensure that you use enough flat elements to model the curved surface adequately. Obviously, the smaller the element, the better the accuracy. It is recommended that the 3-D flat shell elements not extend over more than a 15° arc. Conical shell (axisymmetric line) elements should be limited to a 10° arc (or 5° if near the Y axis).）//
非结构分析中，低阶单元和高阶单元的效果一样好，而且耗费更少。而且，退化单元（三角形和四面体）经常在非结构分析中也能得到足够精确的结果。//（For most non-structural analyses (thermal, magnetic, etc.), the linear elements are nearly as good as the higher order elements, and are less expensive to use. Degenerate elements (triangles and tetrahedra) usually produce accurate results in non-structural analyses.）///
线性结构分析中还是一般采用高阶单元。而非线性分析一般采用低阶单元，因为高阶单元积分点多。 //For linear structural analyses with degenerate element shapes (that is, triangular 2-D elements and wedge or tetrahedral 3-D elements), the quadratic elements will usually yield better results at less expense than will the linear elements. However, in order to use these elements correctly, you need to be aware of a few peculiar traits that they exhibit: ///
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