## 1 Introduction  
Mesh quality has great effects on both the accuracy and efficiency of numerical solutions based on the finite element method and the finite volume method [1; 2; 3]. Therefore, meshes with high quality are required for various finite element and finite volume based numerical simulations. For a given geometric model, meshes can be generated using various mesh generation methods [4; 5]. Among them Delaunay-based methods are quite popular, since they are well suited for automation, of reasonable complexity, and produce meshes with high quality. For problems with smooth solutions, a uniform mesh is usually preferred over a nonuniform one because it readily lends itself to efficient solution. Even when adaptivity is required, an isotropic mesh can be preferred for the same reason if it can resolve the solution without using an undue number of mesh points. Anisotropic meshes are favored when there is a need for better alignment of the mesh with certain solution directions, such as those arising due to boundary or interior layers and sharp interfaces. The adaptive moving mesh methods are usually used in these situations, where point locations are dynamically relocated in time while adapting to the evolving structures in the solution [6]. [PERSON] et al. [7; 8; 9; 10; 11; 12; 13] solved a series of time-dependent partial differential equations (PDEs) using the adaptive moving mesh methods. Their works have rigorous mathematical derivations and elegant proofs. The comparison results with the methods based on priori meshes show that the adaptive solution on posteriori generated meshes will converge to the exact solution with higher accuracy and reliability. The adaptive moving mesh methods can effectively improve the accuracy of numerical simulations, and the mesh optimization methods studied in this paper try to ensure the feasibility and improve the accuracy of numerical simulations. However, tangled meshes can occur during mesh generation, mesh optimization, large-scale deformation, moving domains, and mesh morphing. Generally speaking, such meshes cannot be directly used for numerical simulations. Inverted elements should be eliminated before carrying out numerical simulations, since they usually produce invalid solutions [14]. There are many methods for eliminating inverted elements, including remeshing [15], local mesh modification [16], topological transformations [17] and mesh smoothing. Remeshing may not be automatic, especially for models with complex geometries. Local mesh modification can be very effective. However, for a tangled mesh that contains a large number of inverted elements, remeshing the whole computational domain may be a better choice. Topological transformations improve the mesh quality by inserting or removing a node or modifying the node-element connection relationship. Edge flip, edge collapse, and node insertion are three popular topological optimization methods to improve the mesh quality. While remeshing, local mesh modification and topological transformations methods are valuable in mesh untangling, we focus on mesh smoothing or optimization via node-movement in this paper. Mesh smoothing improves the mesh quality by simply relocating or adjusting node positions while preserving the mesh topology. There exists many node movement methods and the majority of them do not specifically address the mesh untangling problem. For example, Laplace smoothing [18] is widely used to improve mesh quality, where node positions are iteratively updated by the arithmetic mean of neighboring node positions. However, it can lead to a tangled mesh from one that initially is untangled. Therefore, a set of methods known as barrier methods are designed to prevent the initial untangled meshes from becoming tangled. However, these methods can not be used when the initial mesh is tangled. To address the above issue, there are several methods specialized to untangle an initial tangled mesh, and they generally do not consider the element shape.