子单元，母单元，分析对象，局部坐标和位移函数之间关系的问题

hanmingde

请问各位大侠：1）坐标变换中，母单元和子单元之间的这个转换，这个子单元和实际分析结构之间有什么关系？是不是严格的按照实际结构的结点处整体坐标下的坐标点进行的？
2）位移函数中的形函数和母单元与子单元之间转换用的形函数应该不同吧？可不可以这么理解：在对同一个单元进行分析的时候，可以得到同样的一个母单元，但求解坐标变换的形函数和位移函数的形函数时，这二者一般不同的？


谢谢！我思路比较乱，还请各位大侠帮小弟指点指点，谢谢

nongda

不是特别明白你的问题
感觉你粘贴上来的这本书不好
实在不喜欢这个母单元/子单元的称呼。
从图形上来看，子单元就是将某个物理模型划分成许多小单元后的其中一个，其中各个点的坐标是(x,y,z)，是其物理空间的真实坐标。母单元就是等参单元的在参数坐标系中的形式，各个点的参数坐标是（ ξ, η, ζ )
子单元坐标x ≈ = ∑Ni(x) * xi = ∑Ni( x(ξ, η, ζ ) ) * xi
其中xi是单元节点的真实物理坐标。
母单元坐标ξ = ∑Ni(ξ) * ξi 这个式子实际没有什么物理意义

tonnyw

1# hanmingde

Let me give 1D illustration. Finite element method involves two approximations.
1. Approximate the exact solution; For example, u = sum^N_i U_i*H_i where U_i is the degree of freedom and H_i is the shape function defined at node i

2. Approximate the geometry. For example, x = sum^N_j X_i*L_j  where X_i is the nodal coordinates.

If H_i and L_j are the same in terms of function type and the number of them, we call it isotropic element.

However, shape functions are easily defined on an element with nice geometry which is called master element. It is also easy to carry out numerical integration on this element. Therefore, in finite element method, we map each physical element to this master element. As a matter of fact, finite element procedure is carried out only on one element: the master element.

In calculating strain, we use the derivative of displacement over the coordinate x.
For instance, du/dx. But as I mentioned before, all the action is taken on the master element whose coordinate is r. Therefore by chain rule, we have du/dx = du/dr*dr/dx

We know that 1 = dx/dx = dx/dr*dr/dx. It can be seen that dr/dx = 1/(dx/dr)

The same philosophy also applies to the 2D and 3D case.

hanmingde

谢谢nongda和tonnyw，我再找找书看看，呵呵