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Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements. (English) Zbl 1197.74178
Summary: A key challenge while employing non-interpolatory basis functions in finite-element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of such conditions for B-spline basis functions, with application to both second- and fourth-order problems. This is achieved using concepts borrowed from Nitsche’s method, which is a stabilized method for imposing constraints on surfaces. Conditions for the stability of the system of equations are derived for each class of problem. Stability parameters in the Nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the Dirichlet boundary. The approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not. Through several numerical examples, the approach is shown to yield optimal rates of convergence.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74K20 Plates
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D07 Numerical computation using splines
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