Finite Element Analysis. Barna Szabó

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method is presented in the following. For additional details we refer to [51].

      Consider the problem:

(1.164)

      with the boundary conditions and . However, at we substitute the natural boundary condition:

(1.165)

where is a small positive number, is called penalty parameter. The role of the penalty parameter becomes clearly visible if we consider the potential energy

      (1.166)

      Letting , 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 .

      Stabilization

On multiplying eq. (1.164) by v and integrating by parts we get

(1.167)

We introduce the stability parameter γ and multiply eq. (1.165) by to get

(1.168)

Adding eq. (1.167) and eq. (1.168) we get

(1.169)

      and, multiplying eq. (1.165) by , we have

(1.170)

Subtracting eq. (1.170) from eq. (1.169) we obtain the generalized formulation:

(1.171) Скачать книгу