Finite Element Analysis. Barna Szabó

Finite Element Analysis

alt="left-bracket upper M Superscript left-parenthesis k right-parenthesis Baseline right-bracket equals Start 4 By 4 Matrix 1st Row 1st Column m 11 Superscript left-parenthesis k right-parenthesis Baseline 2nd Column m 12 Superscript left-parenthesis k right-parenthesis Baseline 3rd Column midline-horizontal-ellipsis 4th Column m Subscript 1 comma p Sub Subscript k Subscript plus 1 Baseline 2nd Row 1st Column m 21 Superscript left-parenthesis k right-parenthesis Baseline 2nd Column m 22 Superscript left-parenthesis k right-parenthesis Baseline 3rd Column midline-horizontal-ellipsis 4th Column m Subscript 2 comma p Sub Subscript k Subscript plus 1 Baseline 3rd Row 1st Column vertical-ellipsis 2nd Column Blank 3rd Column down-right-diagonal-ellipsis 4th Column vertical-ellipsis 4th Row 1st Column m Subscript p Sub Subscript k Subscript plus 1 comma 1 Superscript left-parenthesis k right-parenthesis Baseline 2nd Column m Subscript p Sub Subscript k Subscript plus 1 comma 2 Superscript left-parenthesis k right-parenthesis Baseline 3rd Column midline-horizontal-ellipsis 4th Column m Subscript p Sub Subscript k Subscript plus 1 comma p Sub Subscript k Subscript plus 1 EndMatrix dot"/>

The terms of the coefficient matrix m Subscript i j Superscript left-parenthesis k right-parenthesis are computable from the mapping, the definition of the shape functions and the function c left-parenthesis x right-parenthesis . The matrix left-bracket upper M Superscript left-parenthesis k right-parenthesis Baseline right-bracket is called the element‐level Gram matrix¹³ or the element‐level mass matrix. Observe that is symmetric. In the important special case where is constant on Ik it is possible to compute once and for all. This is illustrated by the following example.

Example 1.4 When c left-parenthesis x right-parenthesis equals c Subscript k is constant on Ik and the Legendre shape functions are used then the element‐level Gram matrix is strongly diagonal. For example, for p Subscript k Baseline equals 5 the Gram matrix is:

(1.70)

Remark 1.5 For p Subscript k Baseline greater-than-or-equal-to 2 a simple closed form expression can be obtained for the diagonal terms and the off‐diagonal terms. Using eq. (1.55) it can be shown that:

(1.71)

and all off‐diagonal terms are zero for i greater-than-or-equal-to 3 , with the exceptions:

(1.72)

Remark 1.6 It has been proposed to make the Gram matrix perfectly diagonal by using Lagrange shape functions of degree p with the node points coincident with the Lobatto points. Therefore upper N Subscript i Baseline left-parenthesis xi Subscript j Baseline right-parenthesis equals delta Subscript i j where is the Kronecker delta¹⁴. Then, using p plus 1 Lobatto points, we get:

m Subscript i j Superscript left-parenthesis k right-parenthesis Baseline equals StartFraction c Subscript k Baseline script l Subscript k Baseline Over 2 EndFraction integral Subscript negative 1 Superscript 1 Baseline upper N Subscript i Baseline upper N Subscript j Baseline d xi almost-equals StartFraction c Subscript k Baseline script l Subscript k Baseline Over 2 EndFraction w Subscript i Baseline delta Subscript i j

where wi is the weight of the ith Lobatto point. There is an integration error associated with this term because the integrand is a polynomial of degree 2 p . To evaluate this integral exactly n greater-than-or-equal-to left-parenthesis 2 p plus 3 right-parenthesis slash 2 Lobatto points would be required (see Appendix E), whereas only Скачать книгу