Читать книгу Finite Element Analysis - Barna Szabó - Страница 25

For we have

(1.52)

For we define the shape functions as follows:

(1.53)

Figure 1.3 Lagrange shape functions in one dimension, .

where are the Legendre polynomials. The definition of Legendre polynomials is given in Appendix D. These shape functions have the following important properties:

1 Orthogonality. For :(1.54) This property follows directly from the orthogonality of Legendre polynomials, see eq. (D.13) in the appendix.

2 The set of shape functions of degree p is a subset of the set of shape functions of degree . Shape functions that have this property are called hierarchic shape functions.

3 These shape functions vanish at the endpoints of : for .

The first five hierarchic shape functions are shown in Fig. 1.4. Observe that all roots lie in . Additional shape functions, up to , can be found in the appendix, Section D.1.

Exercise 1.6 Show that for the hierarchic shape functions, defined by eq. (1.53), for .

Figure 1.4 Legendre shape functions in one dimension, .

Exercise 1.7 Show that the hierarchic shape functions defined by eq. (1.53) can be written in the form:

(1.55)

Hint: note that for all n and use equations (D.10) and (D.12) in Appendix D.