A high-order fully Lagrangian particle level-set method for dynamic surfaces

TL;DR

It is demonstrated that the resulting particle closest-point (PCP) redistancing achieves high-order accuracy for 2D and 3D geometries discretized on highly irregular particle distributions and has better robustness against particle distortion than regression in a monomial basis.

Abstract

We present a fully Lagrangian particle level-set method based on high-order polynomial regression. This enables closest-point redistancing without requiring a regular Cartesian mesh, relaxing the need for particle-mesh interpolation. Instead, we perform level-set redistancing directly on irregularly distributed particles by polynomial regression in a Newton-Lagrange basis on a set of unisolvent nodes. We demonstrate that the resulting particle closest-point (PCP) redistancing achieves high-order accuracy for 2D and 3D geometries discretized on highly irregular particle distributions and has better robustness against particle distortion than regression in a monomial basis. Further, we show convergence in a classic level-set benchmark case involving ill-conditioned particle distributions, and we present an application to an oscillating droplet simulation in multi-phase flow.

Figures (15)

• Figure 1: Discretized 2D domain containing surface.
• Figure 2: Continuum surface model inside the narrow-band. Solid dots denote particles $\mathcal{N}$ in the narrow-band of width $w$, and hollow dots denote particles in the skin outside the narrow-band. Dashed lines represent the narrow-band borders, and the solid lines surrounding the surface $\Gamma_t$ (bold solid line) delimit the space within which particles contribute to the local regression polynomials (light-gray lines) approximating the surface within a neighborhood radius $r_c$.
• Figure 3: Convergence of different particle-mesh interpolation schemes (symbols, inset legend) applied to the ellipse Eq. \ref{['eq:ellipseeq']} discretized on irregularly ($\alpha=0.3$) distributed particles. The star in the legend entries indicates that the respective remeshing formulation is renormalized by the particle volumes.
• Figure 4: Convergence of the maximum error in the SDF after mesh-based CP redistancing of the remeshed level-set values of the 2D ellipse using different remeshing schemes. The star in the legend entries indicates that the respective remeshing formulation is renormalized.
• Figure 5: Convergence of the SDF computed by the present particle closest-point (PCP) method for the 2D ellipse case with irregularly ($\alpha=0.3$) distributed particles and fourth-order minter regression polynomials. The observed convergence order is 4.8 (theoretical: 5) with a numerical solver tolerance of $10^{-14}$.