Finite Element Analysis. Barna Szabó

Finite Element Analysis - Barna Szabó


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minus u left-parenthesis script l right-parenthesis right-parenthesis"/>

      (1.166)normal upper Pi left-parenthesis u right-parenthesis equals one half integral Subscript 0 Superscript script l Baseline left-parenthesis left-parenthesis u Superscript prime Baseline right-parenthesis squared plus c u squared right-parenthesis d x plus StartFraction 1 Over 2 epsilon EndFraction left-parenthesis u left-parenthesis script l right-parenthesis minus modifying above u with caret Subscript script l Baseline right-parenthesis squared minus integral Subscript 0 Superscript script l Baseline f left-parenthesis x right-parenthesis u d x period

      Letting epsilon right-arrow 0, the minimizer of the potential energy converges to the solution of the Dirichlet problem; however, the numerical problem becomes ill‐conditioned. Nitsche's method stabilizes the numerical problem making it possible to solve it for the full range of boundary conditions, including epsilon equals 0.

      Stabilization

      and, multiplying eq. (1.165) by v prime left-parenthesis script l right-parenthesis epsilon gamma script l slash left-parenthesis epsilon plus gamma script l right-parenthesis, we have