High order treatment of moving curved boundaries: Arbitrary-Lagrangian-Eulerian methods with a shifted boundary polynomials correction

TL;DR

The paper tackles the challenge of enforcing high-order boundary conditions for the Euler equations on moving curved domains within an ALE framework. It extends the shifted boundary polynomial correction (SBPC) to moving meshes and integrates it into the space-time corrector of a direct ALE finite-volume method, enabling accurate treatment of curved boundaries using linear meshes. Numerical experiments in 2D and 3D, including the Kidder problem and fluid-structure interaction with oscillating cylinders, demonstrate that SBPC recovers high-order convergence and reduces boundary-induced errors, outperforming traditional ghost-state approaches. The method significantly simplifies mesh-generation demands while delivering high-fidelity results, with potential extensions to fully viscous flows and topology changes in future work.

Abstract

In this paper we present a novel approach for the prescription of high order boundary conditions when approximating the solution of the Euler equations for compressible gas dynamics on curved moving domains. When dealing with curved boundaries, the consistency of boundary conditions is a real challenge, and it becomes even more challenging in the context of moving domains discretized with high order Arbitrary-Lagrangian-Eulerian (ALE) schemes. The ALE formulation is particularly well-suited for handling moving and deforming domains, thus allowing for the simulation of complex fluid-structure interaction problems. However, if not properly treated, the imposition of boundary conditions can lead to significant errors in the numerical solution, which can spoil the high order discretization of the underlying mathematical model. In order to tackle this issue, we propose a new method based on the recently developed shifted boundary polynomial correction, which was originally proposed on fixed meshes. The new method is integrated into the space-time corrector step of a direct ALE finite volume method to account for the local curvature of the moving boundary by only exploiting the high order reconstruction polynomial of the finite volume control volume. It relies on a correction based on the extrapolated value of the cell polynomial evaluated at the true geometry, thus not requiring the explicit evaluation of high order Taylor series. This greatly simplifies the treatment of moving curved boundaries, as it allows for the use of standard simplicial meshes, which are much easier to generate and move than curvilinear ones, especially for 3D time-dependent problems. Several numerical experiments are presented demonstrating the high order convergence properties of the new method in the context of compressible flows in moving curved domains, which remain approximated by piecewise linear elements.

Figures (12)

• Figure 1: This illustration provides a visual representation of the necessary transformations when working with curved meshes. First, a mapping is required from the standard reference element, $\Omega_{st}$ (left), to the straight-sided element, $\Omega^e_{I}$ (top right), denoted as $\boldsymbol\phi_I:\Omega_{st}\rightarrow\Omega^e_I$. Similarly, a mapping to the curvilinear element (bottom right) is needed, represented as $\boldsymbol\phi_M:\Omega_{st}\rightarrow\Omega^e$. Finally, the deformation mapping $\boldsymbol\phi:\Omega^e_I\rightarrow\Omega^e$ is obtained by composing these transformations as $\boldsymbol\phi=\boldsymbol\phi_M\circ\boldsymbol\phi_I^{-1}$.
• Figure 2: Visual representation of the real and discretized (surrogate) boundaries for curved geometries, with correspondent normals.
• Figure 3: Manufactured solution on 2D moving meshes: initial (left) and final (right) mesh configuration. The mesh is moving radially outward with a velocity field that depends on the distance from the center of the domain. The color map represents the density field.
• Figure 4: Manufactured solution on 3D moving meshes: initial (left) and final (right) mesh configuration. The mesh is moving radially outward. The color map represents the density field.
• Figure 5: Kidder problem in 2D: initial (left) and final (right) mesh configuration. The mesh is moving radially with a velocity field that depends on the Kidder exact solution. The color map represents the pressure field.

Theorems & Definitions (1)

• Remark 1: Polynomial correction evaluation in the ALE framework