A Comparison of Gradient- and Hessian-Based Optimization Methods for Tetrahedral Mesh Quality Improvement

Discretization methods, such as the finite element method, are commonly used in the solution of partial differential equations (PDEs). The accuracy of the computed solution to the PDE depends on the degree of the approximation scheme, the number of elements in the mesh [1], and the quality of the mesh [2, 3]. More specifically, it is known that as the element dihedral angles become too large, the discretization error in the finite element solution increases [4]. In addition, the stability and convergence of the finite element method is affected by poor quality elements. It is known that as the angles become too small, the condition number of the element matrix increases [5]. Recent research has shown the importance of performing mesh quality improvement before solving PDEs in order to: (1) improve the condition number of the linear systems being solved [6], (2) reduce the time to solution [7], and (3) increase the solution accuracy. Therefore, mesh quality improvement methods are often used as a post-processing step in automatic mesh generation. In this paper, we focus on mesh smoothing methods which relocate mesh vertices, while preserving mesh topology, in order to improve mesh quality. Despite the large number of papers on mesh smoothing methods (e.g., [8, 9, 10, 11, 12, 13, 14]), little is known about the relative merits of using one solver over another in order to smooth a particular unstructured, finite element mesh. For example, it is not known in advance which solver will converge to an optimal mesh faster or which solver will yield a mesh with better quality in a given amount of time. It is also not known which solver will most aptly handle mesh perturbations or graded meshes with elements of heterogeneous volumes. The answers may likely depend on the context. For