The Weighted Shifted Boundary Method (WSBM) was recently introduced as an enhanced Shifted Boundary Method (SBM) for the simulation of flows with moving boundaries. Earlier work of the authors on no-slip boundary conditions for the two-dimensional Stokes flow is extended here to the more challenging case of the three-dimensional incompressible Navier-Stokes equations at low and moderate Reynolds numbers. The SBM is an immersed finite element method that reformulates an infinite-dimensional boundary value problem over a surrogate (approximate) computational domain – to avoid integrating over cut cells – and modifies the original boundary conditions using Taylor expansions – to maintain accuracy. The WSBM weights the SBM’s variational form with the elemental volume fraction of active fluid, drastically reducing spurious pressure oscillations in time that occur when the total volume of active fluid changes abruptly over a time step. The WSBM induces small mass (i.e., volume) conservation errors, which converge quadratically in the case of piecewise-linear finite element interpolations, as the grid is refined. An extensive set of two- and three-dimensional tests demonstrates the robustness and accuracy of the proposed approach.

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. The surrogate domain is constructed so as to avoid cut cells and the associated problematic implementation and numerical integration issues. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions: hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Stokes flow, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach allows to drastically reduce spurious pressure oscillations in time, which are produced if the total volume of active fluid were to change abruptly over a time step. The proposed Weighted SBM (W-SBM) exactly preserves states of hydrostatic equilibrium, and induces small mass and momentum conservation errors, which converge as the grid is refined. This is in analogy to cutFEMs and related unfitted approaches, which rely on an affine representation of cut boundaries. We demonstrate the robustness and accuracy of the proposed method with an extensive suite of two-dimensional tests.