A weighted shifted boundary method for immersed moving boundary simulations of Stokes' flow

Journal article (2024)

Authors

Danjie Xu Duke University
Oriol Colomés Duke University, Offshore Engineering - CEG
Alex Main Duke University, Ansys
Kangan Li Duke University, The Pennsylvania State University
Nabil M. Atallah Duke University, Lawrence Livermore National Laboratory
Nabil Abboud Ansys
Guglielmo Scovazzi Duke University
Research Group
Offshore Engineering (CEG) (TU Delft)
Computational fluid dynamics Immersed boundary method Approximate boundary Unfitted finite element method Moving boundaries Shifted boundary method
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Published Date

2024

Language

English

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Faculty

Civil Engineering & Geosciences

Department

Hydraulic Engineering

Research Group

Offshore Engineering

Abstract

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.