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B-spline-Based ALE-MFS Framework for Evolving Domains

Muhammad Ammad, Leevan Ling, Shu Ma

TL;DR

This paper tackles curvature-driven evolution of 2D domains by marrying a meshless MFS interior velocity with an ALE framework. Boundary data are reconstructed via adaptive B-splines and either prescribed or curvature-driven (mean curvature) velocities are used, while interior motion is obtained from a harmonic extension solved by MFS with sources on a fixed exterior circle. Two a posteriori error indicators, LOOCV and a maximum-principle bound, are developed to assess boundary and interior accuracy across square and zero-padded MFS systems; zero-padding notably stabilizes performance for highly nonconvex geometries. Compared with classical ALE–FEM mesh motion, the B-spline ALE–MFS approach yields equal or superior mesh quality and can be integrated as a drop-in replacement for the harmonic extension step, offering a practical path to robust moving-interface simulations and potential 3D extensions and multiphysics coupling.

Abstract

We develop and analyze a B-spline based arbitrary Lagrangian-Eulerian method of fundamental solutions (ALE-MFS) for curvature-driven motion of two-dimensional evolving domains. Boundary points move with the material to track the geometric flow, while interior points move within an ALE framework via a harmonic extension of the boundary velocity, computed by a meshless MFS with sources on a fixed auxiliary circle, thus avoiding volumetric meshing. Boundary normals and curvature are reconstructed by an adaptive local B-spline scheme that remains robust for strongly nonconvex shapes and large deformations. A posteriori error estimates are obtained from a hatmatrix formulation of leave-one-out cross-validation (LOOCV) for both square collocation and zero-padded least-squares systems, and are complemented by maximum principle indicators for harmonic problems. Numerical experiments on circular, star-shaped, and amoeba-like domains show that square collocation suffices for moderately complex geometries, while zero-padded least-squares significantly improves interior velocity regularity and pointwise transport accuracy for strongly nonconvex shapes, without altering the source or collocation sets. The ALE-MFS algorithm also generates high-quality moving meshes for ALE-finite element methods, with larger minimum angles and slower mesh ratio growth than classical FEM mesh-motion strategies, suggesting a practical and easily integrable alternative for challenging moving-interface simulations.

B-spline-Based ALE-MFS Framework for Evolving Domains

TL;DR

This paper tackles curvature-driven evolution of 2D domains by marrying a meshless MFS interior velocity with an ALE framework. Boundary data are reconstructed via adaptive B-splines and either prescribed or curvature-driven (mean curvature) velocities are used, while interior motion is obtained from a harmonic extension solved by MFS with sources on a fixed exterior circle. Two a posteriori error indicators, LOOCV and a maximum-principle bound, are developed to assess boundary and interior accuracy across square and zero-padded MFS systems; zero-padding notably stabilizes performance for highly nonconvex geometries. Compared with classical ALE–FEM mesh motion, the B-spline ALE–MFS approach yields equal or superior mesh quality and can be integrated as a drop-in replacement for the harmonic extension step, offering a practical path to robust moving-interface simulations and potential 3D extensions and multiphysics coupling.

Abstract

We develop and analyze a B-spline based arbitrary Lagrangian-Eulerian method of fundamental solutions (ALE-MFS) for curvature-driven motion of two-dimensional evolving domains. Boundary points move with the material to track the geometric flow, while interior points move within an ALE framework via a harmonic extension of the boundary velocity, computed by a meshless MFS with sources on a fixed auxiliary circle, thus avoiding volumetric meshing. Boundary normals and curvature are reconstructed by an adaptive local B-spline scheme that remains robust for strongly nonconvex shapes and large deformations. A posteriori error estimates are obtained from a hatmatrix formulation of leave-one-out cross-validation (LOOCV) for both square collocation and zero-padded least-squares systems, and are complemented by maximum principle indicators for harmonic problems. Numerical experiments on circular, star-shaped, and amoeba-like domains show that square collocation suffices for moderately complex geometries, while zero-padded least-squares significantly improves interior velocity regularity and pointwise transport accuracy for strongly nonconvex shapes, without altering the source or collocation sets. The ALE-MFS algorithm also generates high-quality moving meshes for ALE-finite element methods, with larger minimum angles and slower mesh ratio growth than classical FEM mesh-motion strategies, suggesting a practical and easily integrable alternative for challenging moving-interface simulations.
Paper Structure (21 sections, 1 theorem, 26 equations, 11 figures, 1 table)

This paper contains 21 sections, 1 theorem, 26 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Formulation via the ALE Framework
  3. Formulation of the Method of Fundamental Solutions
  4. MFS with Prescribed Boundary Velocities
  5. MFS with Inferred (Unprescribed) Boundary Velocities
  6. Selection of Collocation and Source Points
  7. Error Indicators
  8. LOOCV Indicator
  9. Maximum Principle Error Indicator
  10. Numerical Examples
  11. Comparison of MFS Parameters: Classical Circular Geometry
  12. Assessing LOOCV for MFS in Challenging Domains
  13. Improving MFS Robustness in Complex Domains via Zero-Padded Square Systems
  14. Numerical comparison with a classical ALE–FEM mesh motion
  15. Mesh motion generated by BGN-based ALE--FEM
  16. ...and 6 more sections

Key Result

Lemma 4.1

Let $u \in C^2(\Omega)\cap C^0(\overline{\Omega})$ satisfy $\Delta u = 0$ in a bounded domain $\Omega \subset \mathbb{R}^2$. Then

Figures (11)

  • Figure 1: Illustration of air bubbles rising and deforming in a liquid column, a classic moving boundary problem.
  • Figure 2: Annotated moving domain $\Omega(t)$ and boundary $\Gamma(t)$ in $\mathbb{R}^2$ with boundary velocity $\vec{V}_\Gamma$.
  • Figure 3: Schematic of the MFS setup: the domain $\Omega$ (shaded), boundary collocation points $N_c$ (blue dots), and auxiliary source points $N_s$ (red square) located outside the domain.
  • Figure 4: Flowchart of the adaptive B–Spline procedure used to infer boundary velocities $\vec{V}_\Gamma$ from a discrete interface ammad2025eabe.
  • Figure 5: Circular geometry with circular source points. (a) Initial geometry with uniform distribution of interior points and source points (on a concentric circle) (b) Evolved boundary and interior points at $t=T$ (c) LOOCV indicator $E_{\mathrm{LOO},\infty}$ and maximum principle error $E_{\mathrm{MP}}$ versus $r_s$ (d) $E_{\mathrm{LOO},\infty}$ versus $r_s$ for various fill distances $h$.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Lemma 4.1: Strong maximum principle; e.g., evans2022partial
  • Remark 5.1