The Weighted Shifted Boundary Method for Moving-Boundary Flow Simulations

Abstract

This dissertation introduces a novel Weighted Shifted Boundary Method (WSBM) for the simulation of incompressible flows with moving boundaries. The method is built upon the framework of the Shifted Boundary Method (SBM), a class of unfitted finite element methods that reformulate boundary value problems over surrogate computational domains to avoid the challenges associated with cut cells and complex numerical integration. Accuracy is maintained by shifting the boundary conditions using Taylor expansions, both in location and value.The WSBM enhances this framework by incorporating a weighting strategy based on the elemental volume fraction of active fluid within each computational element. This approach significantly improves the temporal stability of the method by reducing spurious pressure oscillations arising from abrupt changes in active fluid volume. It also preserves hydrostatic equilibrium exactly and exhibits only small mass and momentum conservation errors, which are shown to converge optimally under mesh refinement.The method is first formulated and rigorously analyzed in the context of time-dependent Stokes flow. A strong form of the governing equations is established, followed by the derivation of a consistent weak form within appropriately defined functional spaces. Theoretical properties, including stability, convergence, and conservation behavior, are investigated in detail. The WSBM is then applied to a series of two-dimensional benchmark tests, such as the oscillating cylinder and square in both quiescent and cross-flow conditions. Comparisons with reference solutions in the literature demonstrate the method’s accuracy, robustness, and ability to capture subtle flow features.The framework is subsequently extended to the full Navier–Stokes equations and generalized to three-dimensional geometries with complex configurations. Numerical experiments include a rotating sphere in cross-flow, a gyroid-shaped mixer representative of industrial mixing devices, and a rotating Monkey Trefoil structure confined within a pipe. In all cases, the WSBM yields accurate predictions of forces and torques, confirming its capability to simulate challenging moving-boundary flow scenarios.Overall, this work establishes the WSBM as a reliable and efficient computational tool for immersed boundary problems in both academic and industrial contexts. Its flexibility and ease of implementation make it particularly well-suited for integration into existing simulation platforms and for advancing research in numerical methods in computational fluid dynamics.