Finite Element Analysis. Barna Szabó

Finite Element Analysis - Barna Szabó


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l right-parenthesis minus modifying above u with caret Subscript script l Baseline v prime left-parenthesis script l right-parenthesis period EndLayout"/>

      Numerical example

      Letting c equals 1, f left-parenthesis x right-parenthesis equals 1, script l equals 10 and modifying above u with caret Subscript script l Baseline equals 0.25 we construct the numerical problem using one element and the hierarchic shape functions defined in Section 1.3.1. By definition:

      (1.173)u equals sigma-summation Underscript j equals 1 Overscript p plus 1 Endscripts a Subscript j Baseline upper N Subscript j Baseline left-parenthesis xi right-parenthesis comma v equals sigma-summation Underscript i equals 1 Overscript p plus 1 Endscripts b Subscript i Baseline upper N Subscript i Baseline left-parenthesis xi right-parenthesis

.

γ 10−3 10−6 10−9 10−12 10−15
u() 0.2540348 0.2500004 0.25(0)64 0.25(0)94 0.25(0)124

      where p is the polynomial degree. Therefore u left-parenthesis script l right-parenthesis equals a 2 and v left-parenthesis script l right-parenthesis equals b 2 and, using the Legendre shape functions, for p equals 3 the unconstrained coefficient matrix, without the modifications of Nitsche, is

left-bracket upper M right-bracket equals StartFraction 2 Over script l EndFraction Start 7 By 4 Matrix 1st Row 1st Column 1 slash 2 2nd Column negative 1 slash 2 3rd Column 0 4th Column 0 2nd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 3rd Row 1st Column Blank 2nd Column 1 slash 2 3rd Column 0 4th Column 0 4th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Row 1st Column Blank 2nd Column left-parenthesis s y m period right-parenthesis 3rd Column 1 4th Column 0 6th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 7th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column 1 EndMatrix plus StartFraction c script l Over 2 EndFraction Start 7 By 4 Matrix 1st Row 1st Column 2 slash 3 2nd Column 1 slash 3 3rd Column negative 1 slash StartRoot 6 EndRoot 4th Column 1 slash 3 StartRoot 10 EndRoot 2nd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 3rd Row 1st Column Blank 2nd Column 2 slash 3 3rd Column negative 1 slash StartRoot 6 EndRoot 4th Column negative 1 slash 3 StartRoot 10 EndRoot 4th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Row 1st Column Blank 2nd Column left-parenthesis s y m period right-parenthesis 3rd Column 2 slash 5 4th Column 0 6th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 7th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column 2 slash 21 EndMatrix left-bracket upper N right-bracket equals Start 7 By 4 Matrix 1st Row 1st Column 0 2nd Column 1 slash script l 3rd Column 0 4th Column 0 2nd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 3rd Row 1st Column Blank 2nd Column negative 1 slash script l plus 1 slash left-parenthesis gamma script l right-parenthesis 3rd Column minus StartRoot 12 EndRoot slash script l 4th Column minus StartRoot 18 EndRoot slash script l 4th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Row 1st Column Blank 2nd Column left-parenthesis s y m period right-parenthesis 3rd Column 0 4th Column 0 6th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 7th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column 0 EndMatrix dot

      The unconstrained right hand side vector without the modifications of Nitsche is:

StartSet r EndSet equals left-brace script l slash 2 script l slash 2 minus left-parenthesis script l slash 2 right-parenthesis StartRoot 2 slash 3 EndRoot 0 right-brace Superscript upper T

      and with the modifications of Nitsche it is:

left-brace r Subscript upper N Baseline right-brace equals left-brace modifying above u with caret Subscript script l Baseline slash script l modifying above u with caret Subscript script l Baseline slash left-parenthesis gamma script l right-parenthesis minus modifying above u with caret Subscript script l Baseline slash script l minus modifying above u with caret Subscript script l Baseline StartRoot 12 EndRoot slash script l minus modifying above u with caret Subscript script l Baseline StartRoot 18 EndRoot slash script l right-brace Superscript upper T Baseline period

      1 1 Ludwig Prandtl 1875–1953.

      2 2 The generalized form is also called variational form or weak form.

      3 3 The term “discretization” refers to processes by which approximating functions are defined. The most widely used discretizations will be described and illustrated by examples in this and subsequent chapters.

      4 4 See Definition A.5 in the appendix.

      5 5 Peter Gustav Lejeune Dirichlet 1805–1859.

      6 6 Carl Gottfried Neumann 1832–1925.

      7 7 Victor Gustave Robin (1855–1897).

      8 8 A functional is a real‐valued function defined on a space of functions or vectors.

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