Читать книгу Finite Element Analysis - Barna Szabó - Страница 48

When the solution is smooth then the most efficient finite element discretization scheme is uniform mesh and high polynomial degree. However, all implementations of finite element analysis software have limitations on how high the polynomial degree is allowed to be and therefore it may not be possible to increase the polynomial degree sufficiently to achieve the desired accuracy. In such cases the mesh has to be refined. Uniform refinement may not be optimal in all cases, however. Consider, for example, the following problem:

(1.118)

where , and f is a smooth function. Intuitively, when is small then the solution will be close to however, because of the boundary condition , has to be satisfied, the function will change sharply over some interval .

Letting and the exact solution of this problem is

(1.119)

which is plotted for various values of on the interval in Fig. 1.11. It is seen that the gradient at rapidly increases with respect to decreasing values of .

Figure 1.11 The solution , given by eq. (1.119), in the neighborhood of for various values of .

This is a simple example of boundary layer problems that arise in models of plates, shells and fluid flow. Despite the fact that is an analytic function, it may require unrealistically high polynomial degrees to obtain a close approximation to the solution when is small.

The optimal discretization scheme for problems with boundary layers is discussed in the context of the ‐version in [85]. The results of analysis indicate that the size of the element at the boundary is proportional to the product of the polynomial degree p and the parameter . Specifically, for the problem discussed here, the optimal mesh consists of two elements with the node points located at , , , where with .

A practical approach to problems like this is to create an element at the boundary (in higher dimensions a layer of elements) the size of which is controlled by a parameter. The optimal value of that parameter is then selected adaptively.