Читать книгу Finite Element Analysis - Barna Szabó - Страница 53

Nitsche's method¹⁶ allows the treatment of essential boundary conditions as natural boundary conditions. This has certain advantages in two and three dimensions. An outline of the algorithmic aspects of the 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 .