A comparison of gradient- and hessian-based optimization methods for tetrahedral mesh quality improvement

Abstract

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.