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发表于 2010-12-17 05:27:21
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来自 美国
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. |
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