Abstract

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods and was recently introduced for the Poisson, linear advection/diffusion, Stokes, Navier-Stokes, acoustics, and shallow-water equations, and many additional problems [1, 2]. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. 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. We extend here the SBM to the simulation of incompressible Navier-Stokes flows with moving free-surfaces, by appropriately weighting its variational form with the elemental volume fraction of active fluid [3]. This approach prevents spurious pressure oscillations in time, which would otherwise be produced if the total active fluid volume were to change abruptly over a time step. In fact, the proposed weighted SBM method 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. We present an extensive set of two- and three-dimensional tests to demonstrate the robustness and accuracy of the method. References [1] Alex Main and Guglielmo Scovazzi. The Shifted Boundary Method for embedded domain com- putations. Part I: Poisson and Stokes problems. Journal of Computational Physics, 372:972–995, 2018. [2] Alex Main and Guglielmo Scovazzi. The Shifted Boundary Method for embedded domain computations. Part II: Linear advection–diffusion and incompressible Navier–Stokes equations. Journal of Computational Physics, 372:996–1026, 2018. [3] O Colomés, AG Main, L Nouveau, G Scovazzi. A Weighted Shifted Boundary Method for Free Surface Flow Problems. Journal of Computational Physics, doi: https://doi.org/10.1016/j.jcp.2020.109837. The support of the U.S. Department of Energy, Office of Science, the U.S. Office of Naval Research, and ExxonMobil Upstream Research Company (Houston, TX) are gratefully acknowledged.

Abstract

This talk will present a method for achieving arbitrarily high orders of accuracy when approximating PDE-based moving boundary problems using finite element (and related) methods on moving computational meshes. Its effectiveness will be demonstrated on a nonlinear diffusion problem for which the boundary velocity is only defined implicitly as part of the PDE problem. In this situation high-order approximations can be achieved for both the movement of the boundary and the update of the evolving solution on the moving mesh.