Finite Element Analysis. Barna Szabó

of implementation, keeping the condition number of the coefficient matrices small, and personal preferences. For the symmetric positive‐definite matrices considered here the condition number C is the largest eigenvalue divided by the smallest. The number of digits lost in solving a linear problem is roughly equal to . Characterizing the condition number as being large or small should be understood in this context. In the finite element method the condition number depends on the choice of the basis functions and the mesh.

The standard polynomial basis functions, called shape functions, can be defined in various ways. We will consider shape functions based on Lagrange polynomials and Legendre¹² polynomials. We will use the same notation for both types of shape function.

      Lagrange shape functions

      Lagrange shape functions of degree p are constructed by partitioning into p sub‐intervals. The length of the sub‐intervals is typically but the lengths may vary. The node points are , and . The ith shape function is unity in the ith node point and is zero in the other node points:

      (1.50)

      These shape functions have the following important properties:

      (1.51)

For example, for the equally spaced node points are , , . The corresponding Lagrange shape functions are illustrated in Fig. 1.3.

      Exercise 1.5 Sketch the Lagrange shape functions for .

      Legendre shape functions

      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)

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