Finite Element Analysis. Barna Szabó

Finite Element Analysis

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

where epsilon is a small positive number, 1 slash epsilon is called penalty parameter. The role of the penalty parameter becomes clearly visible if we consider the potential energy

(1.166)

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

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

(1.167) minus u prime left-parenthesis script l right-parenthesis v left-parenthesis script l right-parenthesis plus integral Subscript 0 Superscript script l Baseline left-parenthesis u prime v Superscript prime Baseline plus c u v right-parenthesis d x equals integral Subscript 0 Superscript script l Baseline f left-parenthesis x right-parenthesis v d x period

We introduce the stability parameter γ and multiply eq. (1.165) by v left-parenthesis script l right-parenthesis epsilon slash left-parenthesis epsilon plus gamma script l right-parenthesis to get

(1.168)

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

(1.169)

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

(1.170)

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

(1.171)

Letting epsilon equals 0 in eq. (1.171) we get the stabilized method proposed by Nitsche [67]:

(1.172)Скачать книгу