Mesh quality improvement is an integral part of finite element modeling, as the quality of the solution as well as the speed of the solution procedure are generally tied to the quality of the worst element in the mesh. Generally, a mesh improvement solution includes the introduction of a mesh quality measure followed by an optimization process, with most mesh quality measures being based on geometric parameters such as minimum angle, length, Jacobian, etc. In this work, we introduce an energetic measure of mesh quality given by the deformation energy of a reference element mapped to the current mesh configuration, assuming a hyperelastic constitution. In this way, the approach is analogous to a set of connected, equilateral, elastomeric elements subject to the topology of the given mesh and finding their equilibrium configuration. This formulation allows us to adopt traditional solid mechanics methods and techniques to optimize the mesh quality, significantly reducing the burden of the mesh improvement process. The mesh improvement process we introduce corresponds to solving an energy minimization problem where the optimization variables are the mesh vertices positions as well as the mesh topology. We alternate between smoothing steps using standard gradient-based methods and topological operations to reduce the total pseudo-strain energy of the mesh in an iterative process. This procedure is verified on a mesh with heavily distorted elements that would render a finite element simulation impractical. We show that the combination of both smoothing and topological steps greatly outperforms using only of them and leads to a vastly improved mesh with respect to standard quality metrics. We compare the performance of this method with the mesh improvement procedures available in the software Cubit and show that our method leads to better results. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.