Mesh Shape-Quality Optimization Using the Inverse Mean-Ratio Metric: Tetrahedral Proofs

This technical report is a companion to [5] that proves the diagonal blocks of the Hessian matrix for the inverse mean-ratio metric for tetrahedral elements are positive definite. Thus, the block Jacobi preconditioner used in the inexact Newton method to solve the mesh shape-quality optimization problem using the average inverse mean-ratio metric for the objective function is positive definite. Note that [5] only proves these results for triangular elements. We first recall the proposition proved in [5] used to show convexity for fractional functions. Definition 1.1 (Uniform Convexity [6]) Let f : < → <, and let Ω ⊆ < be a convex set. The function f is uniformly convex on Ω with constant κ if there exists a constant κ > 0 such that for all x ∈ Ω, y ∈ Ω, and λ ∈ [0, 1],