Finite Element Analysis. Barna Szabó

Finite Element Analysis

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 log Subscript 10 Baseline upper C

. 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 upper I Subscript st into p sub‐intervals. The length of the sub‐intervals is typically 2 slash p but the lengths may vary. The node points are xi 1 equals negative 1 , xi 2 equals 1 and negative 1 less-than xi 3 less-than xi 4 less-than midline-horizontal-ellipsis less-than xi Subscript p plus 1 Baseline less-than 1 . 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) upper N Subscript i Baseline left-parenthesis xi Subscript j Baseline right-parenthesis equals Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column if i equals j 2nd Row 1st Column 0 2nd Column if i not-equals j EndMatrix and sigma-summation Underscript i equals 1 Overscript p plus 1 Endscripts upper N Subscript i Baseline left-parenthesis xi right-parenthesis equals 1 period

For example, for p equals 2 the equally spaced node points are xi 1 equals negative 1 , xi 2 equals 1 , xi 3 equals 0 . The corresponding Lagrange shape functions are illustrated in Fig. 1.3.

Exercise 1.5 Sketch the Lagrange shape functions for p equals 3 .

Legendre shape functions

For p equals 1 we have

(1.52) upper N 1 equals StartFraction 1 minus xi Over 2 EndFraction comma upper N 2 equals StartFraction 1 plus xi Over 2 EndFraction dot

For p greater-than-or-equal-to 2 we define the shape functions as follows:

(1.53) upper N Subscript i Baseline left-parenthesis xi right-parenthesis equals StartRoot StartFraction 2 i minus 3 Over 2 EndFraction EndRoot integral Subscript negative 1 Superscript xi Baseline upper P Subscript i minus 2 Baseline left-parenthesis t right-parenthesis d t i equals 3 comma 4 comma ellipsis comma p plus 1

Graph depicts lagrange shape functions in one dimension, p=2.

Figure 1.3 Lagrange shape functions in one dimension,

where upper P Subscript i Baseline left-parenthesis t right-parenthesis 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 upper I Subscript st . Additional shape functions, up to p equals 8 , can be found in the appendix, Section D.1.

Exercise 1.6 Show that for the hierarchic shape functions, defined by eq. (1.53), upper N Subscript i Baseline left-parenthesis negative 1 right-parenthesis equals upper N Subscript i Baseline left-parenthesis plus 1 right-parenthesis equals 0 for i greater-than-or-equal-to 3 .

Graph depicts legendre shape functions in one dimension, p=4.

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)Скачать книгу