Finite Element Analysis. Barna Szabó

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Column left-brace b 1 b 2 b 5 right-brace Start 3 By 1 Matrix 1st Row r 1 Superscript left-parenthesis 1 right-parenthesis 2nd Row r 2 Superscript left-parenthesis 1 right-parenthesis 3rd Row r 3 Superscript left-parenthesis 1 right-parenthesis EndMatrix plus left-brace b 2 b 3 right-brace StartBinomialOrMatrix r 1 Superscript left-parenthesis 2 right-parenthesis Choose r 2 Superscript left-parenthesis 2 right-parenthesis EndBinomialOrMatrix plus left-brace b 3 b 4 b 6 b 7 right-brace Start 4 By 1 Matrix 1st Row r 1 Superscript left-parenthesis 3 right-parenthesis 2nd Row r 2 Superscript left-parenthesis 3 right-parenthesis 3rd Row r 3 Superscript left-parenthesis 3 right-parenthesis 4th Row r 4 Superscript left-parenthesis 3 right-parenthesis EndMatrix 2nd Row 1st Column equals 2nd Column left-brace b 1 b 2 midline-horizontal-ellipsis b 7 right-brace Start 4 By 1 Matrix 1st Row r 1 2nd Row r 2 3rd Row vertical-ellipsis 4th Row r 7 EndMatrix identical-to StartSet b EndSet Superscript upper T Baseline StartSet r EndSet EndLayout"/>

      where , , , etc.

      1.3.6 Condensation

      Each element has internal basis functions. Those elements of the coefficient matrix which are associated with the internal basis functions can be eliminated at the element level. This process is called condensation.

      Let us partition the coefficient matrix and right hand side vector of a finite element with such that

      where the and . The coefficient matrix is symmetric therefore . Using

(1.77)

      we get

      (1.78)

The condensed stiffness matrices and load vectors are assembled and the Dirichlet boundary conditions are enforced as described in the following section. Upon solving the assembled system of equations the coefficients of the internal basis functions are computed from eq. (1.77) for each element.

      1.3.7 Enforcement of Dirichlet boundary conditions

      When Dirichlet conditions are specified on either or both boundary points then is split into two functions; a function and an arbitrary specific function from , denoted by . We then seek such that

(1.79)

      for all . Observe that the solution is independent of the choice of .

      We denote the global numbers of the basis functions that are unity at and by K and L respectively. For instance, in Example 1.6 and . It is advantageous to define in terms of and :

(1.80)

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