Triangular Meshing of Two-Dimensional Circular Domains

Triangular meshing of two-dimensional circular domains is a computational technique that partitions circular areas into finite, non-overlapping triangular elements. This method finds extensive applications in computational geometry, finite element analysis, and computer graphics. MATLAB provides built-in triangulation tools and numerical computation methods to achieve efficient implementation.

Core Methodology Boundary Point Generation: The first step involves generating boundary points using parametric equations (e.g., x = r·cos(θ), y = r·sin(θ)) with uniform sampling along the circumference to ensure even distribution. Internal Point Insertion: To optimize mesh quality, control points can be added inside the circle using strategies like centroid-based distribution or quasi-random sequences (e.g., Halton sequence) to avoid skinny triangles. Triangulation Generation: MATLAB's `delaunayTriangulation` function (or legacy `delaunay`) creates a Delaunay triangulation from point sets, maximizing the minimum angle criterion for improved mesh quality. Constraint Handling: For preserving boundary integrity, constrained triangulation techniques using `alphaShape` or manual edge constraints prevent boundary fragmentation.

Optimization and Extensions Mesh Quality Enhancement: Laplacian smoothing algorithms iteratively adjust vertex positions to reduce poorly-shaped elements. Adaptive Refinement: In physical simulations, mesh density can be increased in high-gradient regions (e.g., stress concentration zones) using error-estimation algorithms. 3D Extension: Similar methodologies apply to spherical surface triangulation with adjustments for spherical coordinates and geodesic distances.

This approach balances computational efficiency with geometric accuracy, serving as a fundamental tool for discretizing complex domains in engineering computations and visualization tasks.