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Geometric local parameterization for solving Hele-Shaw problems with surface tension

Zengyan Zhang, Wenrui Hao, John Harlim

TL;DR

This work develops a meshfree, geometric local parameterization framework for the two-dimensional Hele-Shaw free boundary problem with surface tension, representing the evolving interface as a point cloud and discretizing the singular boundary integral equation (BIE) using Generalized Moving Least Squares (GMLS). A core contribution is the systematic use of local charts to approximate curvature and normal vectors with high-order accuracy, along with an analytic treatment of the logarithmic singularity in the Green's function, enabling stable, high-order spatial discretization and rigorous convergence analysis. The authors prove probabilistic error bounds for the discretized operators and establish invertibility of the discrete system under coercivity conditions, with convergence rates tied to boundary smoothness and quadrature order. Numerical experiments on circles, perturbed circles, and smooth closed curves confirm high-order spatial convergence and expected temporal accuracy, demonstrating the method's robustness to complex geometries and its potential for extension to 3D and more general source terms.

Abstract

In this work, we introduce a novel computational framework for solving the two-dimensional Hele-Shaw free boundary problem with surface tension. The moving boundary is represented by point clouds, eliminating the need for a global parameterization. Our approach leverages Generalized Moving Least Squares (GMLS) to construct local geometric charts, enabling high-order approximations of geometric quantities such as curvature directly from the point cloud data. This local parameterization is systematically employed to discretize the governing boundary integral equation, including an analytical formula of the singular integrals. We provide a rigorous convergence analysis for the proposed spatial discretization, establishing consistency and stability under certain conditions. The resulting error bound is derived in terms of the size of the uniformly sampled point cloud data on the moving boundary, the smoothness of the boundary, and the order of the numerical quadrature rule. Numerical experiments confirm the theoretical findings, demonstrating high-order spatial convergence and the expected temporal convergence rates. The method's effectiveness is further illustrated through simulations of complex initial shapes, which correctly evolve towards circular equilibrium states under the influence of surface tension.

Geometric local parameterization for solving Hele-Shaw problems with surface tension

TL;DR

This work develops a meshfree, geometric local parameterization framework for the two-dimensional Hele-Shaw free boundary problem with surface tension, representing the evolving interface as a point cloud and discretizing the singular boundary integral equation (BIE) using Generalized Moving Least Squares (GMLS). A core contribution is the systematic use of local charts to approximate curvature and normal vectors with high-order accuracy, along with an analytic treatment of the logarithmic singularity in the Green's function, enabling stable, high-order spatial discretization and rigorous convergence analysis. The authors prove probabilistic error bounds for the discretized operators and establish invertibility of the discrete system under coercivity conditions, with convergence rates tied to boundary smoothness and quadrature order. Numerical experiments on circles, perturbed circles, and smooth closed curves confirm high-order spatial convergence and expected temporal accuracy, demonstrating the method's robustness to complex geometries and its potential for extension to 3D and more general source terms.

Abstract

In this work, we introduce a novel computational framework for solving the two-dimensional Hele-Shaw free boundary problem with surface tension. The moving boundary is represented by point clouds, eliminating the need for a global parameterization. Our approach leverages Generalized Moving Least Squares (GMLS) to construct local geometric charts, enabling high-order approximations of geometric quantities such as curvature directly from the point cloud data. This local parameterization is systematically employed to discretize the governing boundary integral equation, including an analytical formula of the singular integrals. We provide a rigorous convergence analysis for the proposed spatial discretization, establishing consistency and stability under certain conditions. The resulting error bound is derived in terms of the size of the uniformly sampled point cloud data on the moving boundary, the smoothness of the boundary, and the order of the numerical quadrature rule. Numerical experiments confirm the theoretical findings, demonstrating high-order spatial convergence and the expected temporal convergence rates. The method's effectiveness is further illustrated through simulations of complex initial shapes, which correctly evolve towards circular equilibrium states under the influence of surface tension.
Paper Structure (13 sections, 7 theorems, 95 equations, 5 figures, 1 algorithm)

This paper contains 13 sections, 7 theorems, 95 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of the Hele–Shaw problem with surface tension and its boundary integral formulation
  3. Numerical approximation for the free boundary problems
  4. Generalized Moving Least Squares approximation of the curvature
  5. Quadrature rule to approximate the boundary integral equation
  6. Time discretization
  7. Error analysis
  8. Numerical Experiments
  9. Free boundary equation on circle
  10. Free boundary equation on perturbations of a circle
  11. Free boundary equation on smooth closed curves
  12. Conclusion
  13. Proof of Proposition \ref{['prop4.3']}

Key Result

Proposition 3.1

Let $X \subset \Gamma$ be a set of $N$ uniformly sampled i.i.d. data from a $d$-dimensional manifold $\Gamma$. Assume that $g \in C^{\ell+1}(\Gamma)$. With probability higher than $1-\frac{1}{N}$, as $N\to \infty$. Here, $D^\alpha$ denotes the multiindex derivative of order $|\alpha|$, and the constant in the big-$\mathcal{O}$ notation can depend on $d$, but it is independent of $N$.

Figures (5)

  • Figure 1: Root-Mean-Square errors of the curvature as defined in \ref{['rmsecurvature']} as functions of $N$ for the GMLS estimates with $\ell=3$ and 4.
  • Figure 2: Mesh refinement tests.
  • Figure 3: The evolution dynamics for the circular case. In (a), the profiles of $\mathbf x(t)$ at $t=0$, $0.001$, $0.003$ are shown; (b) The errors for radius $R(t)$ are plotted at a time interval $[0,8\times 10^{-3}]$.
  • Figure 4: The evolution dynamics for perturbed circles with different $D_1$ and $D_2$. In (a), (c), and (e), the profiles of $\mathbf x(t)$ at different times and boundary motion $V_n \mathbf n$ at $t=0$ for different $D_1$ and $D_2$ are shown. In (b), (d), and (f), comparisons of the maximum and minimum distances from the boundary points to the center with the steady-state circle radius $r_s$ for different $D_1$ and $D_2$ are shown, where we observe that the perturbed circles with different $D_1$ and $D_2$ evolve into circles with radius $r_s$.
  • Figure 5: The evolution dynamics for smooth closed curves. In (a) and (c), the profiles of $\mathbf x(t)$ at different times are shown. In (b) and (d), the maximum and minimum distances from the boundary points to the center are shown, where we observe that the smooth closed curves evolve into circles.

Theorems & Definitions (11)

  • Proposition 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • Proposition 4.4: McLean mclean1991variation.
  • Proposition 4.5
  • proof
  • Theorem 4.1
  • ...and 1 more