Eikonal boundary condition for level set method

TL;DR

The paper addresses robust boundary treatment for level-set evolution on 3D polyhedral domains by introducing an eikonal boundary condition that enforces |∇u| = 1 on ∂Ω. The interior level-set equation is solved in tandem with the boundary eikonal equation, using a semi-implicit linearization and deferred-correction solver to maintain the signed-distance property without reinitialization. Numerical results show EKBC yields accuracy comparable to exact Dirichlet data in test cases and outperforms linearly extended or zero Neumann boundaries in preserving distance, especially under large CFL numbers. This approach enhances the practicality of level-set methods for industrial applications and suggests future integration with extended velocity fields to handle general motions while preserving distance.

Abstract

In this paper, we propose to use the eikonal equation as a boundary condition when advective or normal flow equations in the level set formulation are solved numerically on polyhedral meshes in the three-dimensional domain. Since the level set method can use a signed distance function as an initial condition, the eikonal equation on the boundary is a suitable choice at the initial time. Enforcing the eikonal equation on the boundary for later times can eliminate the need for inflow boundary conditions, which are typically required for transport equations. In selected examples where exact solutions are available, we compare the proposed method with the method using the exact Dirichlet boundary condition. The numerical results confirm that the use of the eikonal boundary condition provides comparable accuracy and robustness in surface evolution compared to the use of the exact Dirichlet boundary condition, which is generally not available. We also present numerical results of evolving a general closed surface.

Figures (8)

• Figure 1: (a) is a polyhedron mesh whose boundary is a cube. (b) shows the inner cross-section of (a). (c) presents a part of the boundary (green) and internal (red) cells. (d) is an illustration of boundary (green) and internal (red) cells with tessellated triangular faces.
• Figure 2: Using the eikonal boundary condition, the equidistant isosurfaces \ref{['eq:eqdist_iso']} of test cases in Table \ref{['tab:test_case']} at the final time $T$ are presented on the mesh $\mathcal{M}_{\text{M}}$ in Table \ref{['tab:meshes']}. The second smallest surface is the evolved surface $\Gamma_{T}(u)$\ref{['eq:zerolevel']}. In \ref{['TS']} and \ref{['TC']}, we diagonally cut to present the isosurfaces for proper visualization.
• Figure 3: Applying $\triangle t_{\text{M}}$\ref{['eq:time_step']}, the characteristic length versus errors $E^{1,\mathcal{Z}}$\ref{['eq:err_L1_zero']}, $E^{\infty,\mathcal{Z}}$\ref{['eq:err_Linf_zero']}, $E^{\text{v}}$\ref{['eq:err_vol']}, $E^1$\ref{['eq:err_L1']} for test cases \ref{['TS']}, \ref{['RS']}, \ref{['ES']}, and \ref{['SS']} are presented by using the Dirichlet (upper row) and the eikonal boundary condition (lower row).
• Figure 4: Applying $\triangle t_{\text{M}}$\ref{['eq:time_step']}, the characteristic length versus errors $E^{1,\mathcal{Z}}$\ref{['eq:err_L1_zero']}, $E^{\infty,\mathcal{Z}}$\ref{['eq:err_Linf_zero']}, $E^{\text{v}}$\ref{['eq:err_vol']}, $E^1$\ref{['eq:err_L1']} for test cases \ref{['TC']}, \ref{['RC']}, \ref{['EC']}, and \ref{['SC']} are presented by using the Dirichlet (upper row) and the eikonal boundary condition (lower row).
• Figure 5: Applying $\triangle t_{\text{M}}$\ref{['eq:time_step']}, the characteristic length versus errors $E^1_{\text{M}}$\ref{['eq:err_L1']} and $E^{1,g}_{\text{M}}$\ref{['eq:err_L1_g']} for test cases \ref{['RSS']} and \ref{['RSC']} in Table \ref{['tab:test_case']} are presented by using EKBC (left column), LEBC (middle column), and ZNBC (right column).

Theorems & Definitions (2)

• Remark 1
• Remark 2