Imposition of Normal And/or Tangential Boundary Conditions by an Augmented Lagrangian Technique

In order to tackle realistic engineering applications where constrained problems such as the NavierStokes equations are encountered, the trend is to employ Uzawa’s algorithm, which consists of computing the momentum equation separately from the constraints. Examples of constraints include the divergence-free constraint and the zero-normal velocity constraint to a boundary (slip boundary condition). Applications range from the field of solid mechanics [l] to Newtonian and non-Newtonian fluid mechanics [2,3] to name a few. In this work, we present a finite element methodology applied to the solution of the two-dimensional Navier-Stokes. The velocity directional boundary conditions, from which the normal and tangential condition are special cases, are imposed iteratively and simultaneously with the divergence-free constraint by Uzawa’s algorithm. A summary of previous implementation of directional boundary conditions can be found in [4] and [5]. We extend [S] in several aspects, one of which is the capability to handle systems of equations which are not symmetric and positive definite. The present treatment differentiates itself from more standard procedure not only in the novel manner in which the directional constraint is implemented, but also in the simplicity of this approach. The governing equations are presented in Sec. 2, and the description of Uzawa’s algorithm and its implementation to the problem at hand is explained in Sec. 3. In the final sections the implementation is validated on two test problems.