## 1 Introduction
Therefore, a barrier smoothing technique is needed to obtain a valid and high-quality mesh. [PERSON] and [PERSON] tried to untangle the input mesh by maximizing the minimum area/volume of elements within each local patch of the mesh [19]. The system is guaranteed to converge, but not necessarily to an untangled mesh. [PERSON] used a local non-barrier metric that penalized inverted elements [20] to untangle the mesh. Global optimization of the objective function is recommended when untangling larger and more difficult meshes, as all nodes are updated simultaneously. However, the global optimization strategy is very time-consuming in untangling large-scale meshes. As with the other methods, there is no guarantee of obtaining a valid mesh when using this method. [PERSON] and [PERSON] proposed two new metrics for simultaneous mesh untangling and smoothing using the target matrix paradigm [21]. This method can effectively eliminate inverted elements and do not require that one specify a parameter as in Reference [20]. The above methods [20; 21] are adopted for the well-known mesh quality improvement toolkit (Mesquite) [22]. There are also methods developed for simultaneously eliminating inverted elements and improving the mesh quality. [PERSON] et al. [23] proposed a multiobjective optimization method that combined two or more objective functions into a single objective function, and solved it using a nonlinear conjugate gradient method. [PERSON] et al. [24; 25] proposed a method incorporating a modified quality metric that ensured higher penalty for inverted elements. This method is effective but also does not guarantee an untangled output mesh. [PERSON] et al. [26] proposed a method optimizing triangular and quadrilateral meshes on parameterized surfaces. The method expresses the quality metrics for planar elements in terms of the parametric coordinates of the nodes, and then relocates the nodes on the parametric space to improve the mesh quality and eliminate the inverted elements in the physical space. Optimization-based algorithms usually untangle and smooth the input meshes at the expense of higher computational and implementational effort. As a way out of the dilemma between quality improvement effect and computational effort, algorithms that do not need to solve numerical optimization problems have been proposed. [PERSON] et al. (2017) proposed a novel approach for simultaneous mesh untangling and smoothing using a Pointer network. This method predicts the approximate solutions for free nodes using a pre-trained network, and there is no need to solve complex numerical optimization problems. A geometry-based algorithm for smoothing 2D/3D meshes is proposed by [PERSON] et al. (2018, 2020), which can produce high quality meshes comparable to those obtained by the optimization-based methods within shorter runtimes. Although their method is designed for mesh smoothing, the method still shows its potential for mesh untangling. The purpose of this paper is to develop a geometry-based mesh untangling and smoothing approach for 2D meshes. A novel explicit element geometric transformation (EEGT) operation is firstly introduced and analyzed, which can transform an arbitrary initial polygonal element (even inverted) into its regular counterpart if applied iteratively. This is the main driving force behind the proposed approach. In the proposed mesh untangling and smoothing approach, all mesh elements are transformed individually and simultaneously in the first step, then new interior node positions are obtained as weighted means of the scaled transformed element nodes. The proper choice of averaging weights in element stitching ensures that the elements can be transitioned smoothly and uniformly throughout the calculation domain, and the explicit element geometric transformation has strong capability to eliminate inverted elements and improve the mesh quality. To the best of our knowledge, the proposed approach is the first geometry-based framework specially designed for simultaneous mesh untangling and smoothing.