Body-fitted tracking of 2d open curves with a level set based mesh evolution method

Technical Summary: Body-Fitted Tracking of 2D Open Curves with a Level Set Based Mesh Evolution Method

Problem Statement
The paper addresses the numerical challenge of tracking the motion of open curves (or collections of disjoint open curves) in two-dimensional space. While significant literature exists for tracking closed domains or boundaries, the treatment of open objects—curves with endpoints strictly contained within the computational domain—remains difficult. Purely Lagrangian methods (particle-based) often struggle with robustness under large deformations and topological changes, requiring frequent resampling. Conversely, purely Eulerian methods (fixed mesh, e.g., standard Level Set) lack an explicit body-fitted discretization, making accurate geometric and mechanical computations (such as curvature evaluation or solving boundary value problems on the curve) difficult without approximations. The goal is to reconcile these requirements: maintaining a precise, meshed representation for calculations while allowing for robust, arbitrary evolution of the curve's shape.

Methodology
The proposed strategy employs a hybrid approach that couples two complementary representations of the open curve $\Gamma$ at each iteration of the evolution process:

1. Explicit Meshed Representation: The curve is discretized as a collection of edges belonging to a conforming triangular mesh of the total computational domain. This allows for high-accuracy geometric and mechanical computations (e.g., Finite Element Method) directly on the curve.
2. Implicit Level Set Representation: The curve is simultaneously captured using two auxiliary scalar functions, $\phi$ and $\psi$, defined on the whole domain.
   • $\phi$ defines a closed curve $\tilde{\Gamma}$ (the zero level set) that extends the open curve $\Gamma$.
   • $\psi$ delimits the open segment $\Gamma$ within $\tilde{\Gamma}$ (specifically, $\Gamma = \{x \mid \phi(x)=0, \psi(x)<0\}$).
   • The endpoints $\Sigma$ are defined where $\phi(x)=0$ and $\psi(x)=0$.
   • An orthogonality condition ($\nabla\phi \cdot \nabla\psi = 0$) is enforced near the endpoints to prevent undesirable collisions of the level sets.

Algorithmic Workflow
The evolution proceeds via an iterative loop (Algorithm 1) that alternates between these representations:

1. Mesh to Level Set: From the current meshed curve, two level set functions are generated. This involves calculating unsigned distances and signed distances using the Fast Marching Method to construct $\phi$ (extending the curve to a closed loop) and $\psi$ (defining the open segment).
2. Velocity Calculation: The velocity field $V$ driving the motion is computed. This step utilizes the explicit mesh for accurate geometric quantities (e.g., curvature) or for solving boundary value problems (e.g., fluid dynamics or elasticity) via the Finite Element Method.
3. Level Set Evolution: The evolution equations for $\phi$ and $\psi$ (advection-like equations) are solved over a time step $\Delta t$ using the Method of Characteristics (Runge-Kutta integration). This updates the curve implicitly, handling large deformations and topological changes robustly.
4. Level Set to Mesh: The new level set functions are converted back into a high-quality, body-fitted triangular mesh. This involves an isosurface extraction step (marching triangles) to create an initial, low-quality mesh, followed by local remeshing operators (edge split, collapse, swap, vertex relocation) to restore element quality while preserving the explicit discretization of the curve.

Key Contributions

• Novel Coupling: The paper introduces a specific algorithmic framework that seamlessly switches between a body-fitted mesh (for accuracy) and a two-level-set representation (for robustness) specifically for open curves.
• Two-Level-Set Generation: It details a method to generate the pair of level set functions ($\phi, \psi$) from a line mesh, ensuring the orthogonality condition at endpoints and the correct extension of the open curve into a closed loop.
• Remeshing Strategy: It outlines a quality-oriented remeshing procedure that extracts the curve from the level sets and regenerates a high-quality mesh, a critical step for maintaining numerical stability in physical simulations.
• Shape Optimization Framework: The method is integrated into a shape optimization context using the Hadamard boundary variation method, deriving shape derivatives for functionals involving both the curve bulk and its endpoints.

Results and Applications
The methodology is validated through several physical simulations and optimization problems:

• Analytical Advection: A benchmark test involving a curve undergoing large deformation and returning to its initial state demonstrates high accuracy (Hausdorff distance error $\approx 10^{-3}$) despite extreme stretching.
• Vortex Sheet Roll-up: The method simulates the dynamics of a 2D vortex sheet, where the velocity is determined by a weakly singular integral over the curve. The results show good agreement with existing Lagrangian benchmarks.
• Anisotropic Perimeter Optimization: The framework optimizes open curves to minimize anisotropic length (Riemannian metric) in the presence of wind fields and topographical constraints, successfully finding optimal paths that detour against wind.
• Additive Manufacturing: The method optimizes the laser path for Electron Beam Melting. The goal is to melt a specific target region in a powder bed while avoiding melting outside it, subject to heat diffusion equations. The optimization successfully shapes the laser path to match the target geometry.
• Fracture Reconstruction: An inverse problem is solved to reconstruct an underground fault line from surface displacement measurements. The method minimizes the least-square difference between observed and predicted displacements, successfully recovering the inclination of a known fault pattern.

Significance and Claims
The paper claims to offer a robust, body-fitted methodology that overcomes the limitations of purely Lagrangian or Eulerian approaches for open curves. By combining the geometric precision of meshed representations with the topological flexibility of Level Set methods, it enables accurate mechanical and geometric computations even during large evolutions.

The authors are modest regarding the scope, noting that:

• The current implementation is 2D, though the methodology is designed with 3D extension in mind (which is ongoing work).
• Topological changes (branching, junctions) are handled naturally by the Level Set aspect but are not the primary focus of the presented examples.
• The 2D implementation may be less efficient than specialized planar solvers, but it serves as a proof-of-concept for a technology intended for 3D applications (e.g., shell optimization, electromagnetic screens).

The work is presented as a preliminary step toward a general 3D framework for open surfaces, with the current 2D results serving to validate the core algorithmic ingredients.