Finite Element Analysis. Barna Szabó
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Название: Finite Element Analysis

Автор: Barna Szabó

Издательство: John Wiley & Sons Limited

Жанр: Физика

Серия:

isbn: 9781119426462

isbn:

СКАЧАТЬ alt="u Superscript black star Baseline equals modifying above u with caret 0 phi Subscript upper K Baseline left-parenthesis x right-parenthesis plus modifying above u with caret Subscript script l Baseline phi Subscript upper L Baseline left-parenthesis x right-parenthesis"/>

integral Subscript 0 Superscript script l Baseline left-parenthesis kappa left-parenthesis u Superscript black star Baseline right-parenthesis prime v prime plus c u Superscript black star Baseline v right-parenthesis d x equals sigma-summation Underscript i equals 1 Overscript upper N Subscript u Baseline Endscripts b Subscript i Baseline left-parenthesis c Subscript i upper K Baseline plus c Subscript i upper L Baseline right-parenthesis

      where Nu is the number of unconstrained equations, that is, the number of equations prior to enforcement of the Dirichlet boundary conditions. (For instance, in Example 1.6 upper N Subscript u Baseline equals 7.) The coefficients c Subscript i upper K, c Subscript i upper L are elements of the assembled coefficient matrix.

Geometric representation of recommended choice of the function u★ in one dimension.
in one dimension.

      Since v element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis, we have b Subscript upper K Baseline equals b Subscript upper L Baseline equals 0 and therefore the Kth and Lth rows of matrix left-bracket upper C right-bracket are multiplied by zero and can be deleted. The Kth and Lth columns of matrix left-bracket upper C right-bracket are multiplied by modifying above u with caret 0 and modifying above u with caret Subscript script l respectively, summed and the resulting vector is transferred to the right‐hand side. The resulting coefficient matrix has the dimension N which is Nu minus the number of Dirichlet boundary conditions. The number N is called the number of degrees of freedom. It is the maximum number of linearly independent functions in upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis.

      Remark 1.7 In order to avoid having to renumber the coefficient matrix once the rows and columns corresponding to φK and φL were eliminated, all elements in the Kth and Lth rows and columns can be set to zero, with the exception of the diagonal elements, which are set to unity. The corresponding elements on the right hand side vector are set to û0 and ûℓ. This is illustrated by the following example.

minus u Superscript double-prime Baseline plus 4 u equals 0 comma u left-parenthesis 0 right-parenthesis equals 1 comma u left-parenthesis 1 right-parenthesis equals 2

      the exact solution of which is

u equals StartFraction exp left-parenthesis 2 right-parenthesis minus 2 Over exp left-parenthesis 2 right-parenthesis minus exp left-parenthesis negative 2 right-parenthesis EndFraction exp left-parenthesis minus 2 x right-parenthesis plus StartFraction 2 minus exp left-parenthesis negative 2 right-parenthesis Over exp left-parenthesis 2 right-parenthesis minus exp left-parenthesis negative 2 right-parenthesis EndFraction exp left-parenthesis 2 x right-parenthesis period

      Using five elements of equal length on the interval upper I equals left-parenthesis 0 comma 1 right-parenthesis and p equals 1 assigned to each element, find the finite element solution for this problem.

left-bracket upper C Superscript left-parenthesis k right-parenthesis Baseline right-bracket equals Start 2 By 2 Matrix 1st Row 1st Column 79 slash 15 2nd Column negative 73 slash 15 2nd Row 1st Column negative 73 slash 15 2nd Column 79 slash 15 EndMatrix comma k equals 1 comma 2 comma ellipsis 5

      where we used kappa Subscript k Baseline equals 1, c Subscript k Baseline equals 4, script l Subscript k Baseline equals 1 slash 5. The assembled unconstrained coefficient matrix is:

left-bracket upper C right-bracket equals Start 6 By 6 Matrix 1st Row 1st Column 79 slash 15 2nd Column negative 73 slash 15 3rd Column 0 4th Column 0 5th Column 0 6th Column 0 2nd Row 1st Column negative 73 slash 15 2nd Column 158 slash 15 3rd Column negative 73 slash 15 4th Column 0 5th Column 0 6th Column 0 3rd Row 1st Column 0 2nd Column negative 73 slash 15 3rd Column 158 slash 15 4th Column negative 73 slash 15 5th Column 0 6th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column negative 73 slash 15 4th Column 158 slash 15 5th Column negative 73 slash 15 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column negative 73 slash 15 5th Column 158 slash 15 6th Column negative 73 slash 15 6th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column negative 73 slash 15 6th Column 79 slash 15 EndMatrix dot

      Upon enforcement of the Dirichlet conditions the system of equations is

left-bracket upper C right-bracket equals Start 4 By 4 Matrix 1st Row 1st Column 158 slash 15 2nd Column negative 73 slash 15 3rd Column 0 4th Column 0 2nd Row 1st Column negative 73 slash 15 2nd Column 158 slash 15 3rd Column negative 73 slash 15 4th Column 0 3rd Row 1st <a href=СКАЧАТЬ