Vertex-Sharing Mesh Parameterization

• The paper introduces a method that maps C² planar curves onto triangulated domains using closest point projection, ensuring a global homeomorphism with C¹ regularity on each edge.
• It employs geometric criteria such as acute conditioning angles and curvature-based mesh quality measures to maintain accurate Jacobian bounds and method robustness.
• The approach facilitates constructing high-order finite elements on immersed boundaries without requiring mesh conformity, thereby automating mesh generation for complex geometries.

Vertex-sharing mesh parameterization refers to a class of methods for mapping planar curves, particularly $C^2$-regular boundaries $\Gamma$, onto structured representations within triangulated ambient spaces. The approach enables the parameterization of $\Gamma$ via closest point projection over a discrete set of mesh edges $\Gamma_h$, utilizing only a straight-edge triangulation of a polygonal superset of the domain. Rigorous geometric criteria guarantee that the resulting piecewise parameterization is a global homeomorphism that is $C^1$ on each edge, with robust control over the Jacobian. This provides a foundation for constructing curved finite elements and high-order methods on immersed boundaries without mesh conformity requirements (Rangarajan et al., 2011).

1. Construction of the Discrete Edge Set $\Gamma_h$

Let $\Gamma \subset \mathbb{R}^2$ be a $C^2$-regular closed curve and $\mathcal{T}_h$ a (possibly nonconforming) straight-edge triangulation containing $\Gamma$. The signed distance function $\phi$ is used to classify mesh vertices: $\phi(v) < 0$ for vertices inside $\Gamma$, $\phi(v) \geq 0$ for vertices outside. A triangle $K=(p,q,r)\in\mathcal{T}_h$ is positively-cut if exactly two vertices satisfy $\phi(\cdot)\geq0$ and the third $\phi(\cdot)<0$. Edges $e_{pq}$ with both endpoints outside ($\phi(p)\geq0$, $\phi(q)\geq0$) and contained in a positively-cut triangle are positive edges. The union of all such positive edges forms $\Gamma_h$: $$\Gamma_h := \bigcup \left\{ e_{pq} \in \mathrm{Edges}(\mathcal{T}_h) \mid \phi(p)\geq 0, \phi(q)\geq 0,\ \exists K=(p,q,r)\in \mathcal{T}_h : \phi(r)<0 \right\}$$ This definition ensures that $\Gamma_h$ consists of mesh edges “sharing” vertices on a single side of $\Gamma$ while lying adjacent to triangles “cut” by the curve.

2. Geometric and Mesh Quality Criteria

Accurate parameterization and injectivity require localized geometric conditions around $\Gamma$. For each positively-cut triangle:

• $h_K$ is the diameter, $\rho_K$ the in-radius, and $\sigma_K=h_K/\rho_K$ the shape parameter.
• The two “positive” vertices $a, b$ are identified, with $a$ termed the proximal vertex if $\phi(a)<\phi(b)$.
• $\theta_K=$ angle at $a$ in $\triangle bac$ (conditioning angle); $\theta_K^{\text{adj}}$ is the minimum of the two adjacent angles from the neighboring triangle sharing $e_{ab}$.
• $M_K:=\max_{x\in\Gamma\cap B(K,h_K)}\kappa(x)$ is the supremum of curvature in the local region.
• $C_K^h = M_K/(1-M_Kh_K)$ provides a condition number for the local curvature and triangle size.

Sufficient conditions for each $K$ are:

1. $h_K < r_n$ (with $r_n$ the regularity radius for $\phi$ and $\pi$),
2. $\theta_K < 90^\circ$,
3. $$\sigma_KC_K^h h_K < \min\{\cos\theta_K, \sin(\theta_K/2)\}$$,
4. $C_K^h h_K < \frac{1}{2}\sin\theta_K^{\text{adj}}$.

These criteria ensure robustness of the closest point projection and the exclusivity of positive edges per triangle.

3. Theoretical Properties: Homeomorphism and Regularity

The main theorem of Rangarajan & Lew (Rangarajan et al., 2011) guarantees that, under the above mesh and geometric restrictions:

• Every positive edge in $\Gamma_h$ belongs to exactly one positively-cut triangle.
• Each positive edge $e_{ab}$, restricted to its relative interior, is mapped by $\pi$ as a $C^1$-diffeomorphism.
• The signed distance along $e_{ab}$ is bounded:

$$-C_K^h h_K^2 < \phi(x) \leq h_K \quad \forall x \in e_{ab}$$

• The Jacobian $J(x) = |\nabla\pi(x) \cdot (b-a)/|b-a||$ satisfies

$$0 < \frac{\sin(\beta_K - \theta_K)}{1 + M_K h_K} \leq J(x) \leq \frac{1}{1 - M_K h_K}$$

where $\cos\beta_K = \sigma_K C_K^h h_K - \eta_K$ and $\eta_K = (\min\{\phi(a),\phi(b)\} - \phi(c))/h_K$.

• Globally, $\pi:\Gamma_h\to\Gamma$ is a homeomorphism and $\Gamma_h$ is a disjoint union of Jordan loops, each covering a connected component of $\Gamma$.

Propositions guarantee that the edgewise $C^1$ regularity and injectivity transfer to global properties via the loop structure of $\Gamma_h$.

4. Algorithmic Implementation and Practical Considerations

A prototypical implementation proceeds through the following steps: A. Preprocessing: Compute the signed-distance $\phi$ and its gradient at each vertex; evaluate the signs on all triangles. B. Positive Edge Detection: Mark positively-cut triangles; identify positive edges by their endpoints and adjacent triangles. C. Mesh Quality Checks: For each positively-cut triangle, evaluate $h_K$, $\rho_K$, $\sigma_K$, proximal vertex, angles, estimate $M_K$ (via local curvature sampling or analytic bound), compute $C_K^h$, and check all geometric conditions. D. Assembly and Parameterization: Construct the (possibly non-simple) graph of positive edges, extract connected components as loops (Jordan curves), and for each edge $e=(a,b)$, parameterize $x(t)=a(1-t)+bt$, $t\in[0,1]$ with $y(t)=\pi(x(t))$. E. Jacobian Diagnostics (Optional): Compute $\nabla\pi$, $J(t)$, confirm alignment with the theoretical bounds on a sampling of each edge.

This workflow is explicitly parallel over triangles/edges, requires only standard mesh data structures, and is independent of mesh conformity to the curve.

5. Mathematical Tools: Closest-Point Projection and Regularity

For a $C^2$-regular curve $\Gamma$ with associated tubular neighborhood $B(\Gamma, r_n)$, both signed-distance $\phi$ and closest point projection $\pi$ are well-defined and $C^1$ within $B(\Gamma, r_n)$. Specifically, $\nabla\phi(x)$ satisfies $|\nabla\phi|=1$, and

$$\nabla\pi(x) = \frac{T(\pi(x))\otimes T(\pi(x))}{1-\phi(x)\kappa_s(\pi(x))}$$

where $T$ is the unit tangent and $\kappa_s$ the signed curvature at $\pi(x)$. On every positive edge $e_{ab}$, $|\phi(x)|\leq h_K<r_n$ ensures $C^1$ regularity. Jacobian lower bounds are maintained via the acute angle restrictions and upper bounds by curvature locally, thereby supporting stable mapping for high-order finite element methods.

6. Applications and Significance in Finite Element Methods

This vertex-sharing parameterization method underpins the construction of high-order curved finite elements for domains defined by $C^2$ (including piecewise $C^2$) planar curves, as demonstrated in (Rangarajan et al., 2011). Its key features are that:

• No mesh vertices are required to lie on $\Gamma$.
• $\Gamma$ can be parameterized (by closest point projection) using only the vertices and edges from a background triangulation.
• The global homeomorphism and $C^1$ regularity ensure accurate trace approximation and quadrature for finite element assembly.

A plausible implication is enhanced automation in mesh generation for complex geometries, as the method’s criteria are compatible with standard meshers and can be checked algorithmically without special meshing around $\Gamma$.

7. Summary Table of Edge Classification

Criterion	Condition	Mesh Element
Positive edge	Both vertices $\phi \geq 0$, adjacent to negatively-signed vertex	Mesh edge
Positively-cut triangle	Two vertices $\phi \geq 0$, one $\phi < 0$	Mesh triangle
Acute conditioning angle	$\theta_K < 90^\circ$	Triangle angle
Size and curvature conditioning	$h_K < r_n$, $$\sigma_KC_K^hh_K < \min\{\cos\theta_K, \sin(\theta_K/2)\}$$	Triangle scalar

The vertex-sharing mesh parameterization is thus a rigorously-defined, computationally efficient approach for geometry-immersed parameterizations with robust theoretical guarantees and direct application to high-order numerical methods (Rangarajan et al., 2011).