Algorithms for Reliable Mesh Generation and Adaptivity : Longest-Edge Bisection and Largest-Angle Bisection-Trisection

Geometric criteria serve as a cornerstone in adaptive meshing strategies, directly contributing to the enhanced accuracy and computational efficiency of finite element simulations in complex engineering domains. Traditional approaches, most notably the longest-edge bisection, primarily focus on edge-length property without fully incorporating the mesh quality indicating criteria such as aspect ratio, maximum angle, geometric curvature, and minimum triangle area. This thesis presents a novel adaptive mesh refinement framework that combines the largest angle-based bisection and trisection methods with an integrated treatment of key mesh quality indicators. In addition to the traditional approaches, the proposed methodology extends in understanding the challenge of hanging nodes, which arises when adjacent elements are refined inconsistently, leading to non-conforming meshes and potential numerical instability. In the proposed methods, some triangles are first identified for refinement, for example, when their largest interior angle exceeds a user-defined threshold. For moderately distorted elements, the largest-angle bisection method is applied, while highly obtuse triangles are subdivided via a tailored trisection approach. A key innovation in this work is the numerical implementation of largest-angle bisection with explicit handling of hanging nodes, phenomena that occur when adjacent triangles do not share a common refinement along their interfaces. To address this, the thesis develops a novel algorithm that propagates the refined edge information across the mesh, ensuring full conformity while preserving or even enhancing element quality. Furthermore, the largest-angle bisection+trisection strategy is comparatively evaluated against the conventional longest-edge bisection method. Mesh quality indicators are used as benchmarks to assess performance. The findings confirm that the longest-edge bisection method provides a structured and globally uniform refinement, preserving geometric regularity with minimal computational overhead. In contrast, the experimental results demonstrate that our largest-angle bisection+trisection refinement offers a more targeted approach to refine poorly shaped elements, although the challenges related to hanging nodes and mesh conformity remain significant. Therefore, the results provide a promising avenue for advancing finite element methods in a wide range of engineering and scientific applications.