## 5 Conclusions and Future Work  
A novel and efficient geometry-based approach for simultaneously untangling and smoothing isotropic 2D meshes has been proposed for the first time in this paper. It has the following features:  
(1) The proposed approach is based on a new explicit element geometric transformation operation, which can transform an arbitrary polygonal element into its regular counterpart if applied iteratively. It is more reliable and efficient, and its convergence is proved by theoretical analysis.  
(2) The proposed approach involves two main steps, namely independent element geometric transformation and node positions updating. It is very suitable for parallel implementation, especially on GPU.  
(3) The proposed approach has a strong ability to eliminate inverted elements, and the optimized mesh elements can be transitioned smoothly and uniformly throughout the whole computational domain.  
_Limitations_. The proposed approach can fail on meshes with extreme difference in element size. It may also fail when the initial positions of interior nodes are fairly random. Both situations may lead to the failure of eliminating all inverted elements, because the node positions updating may lead to worse node coordinates. As the element geometric transformation aims at transforming a polygonal element into its regular counterpart, the proposed approach based on the regularizing element transformation cannot be used to optimize anisotropic meshes.  
_Future Work_. Further study will be focused on the extension of the proposed approach to volume meshes and anisotropic meshes. Beyond that, it can be combined with topology modification techniques and local mesh regeneration techniques to form a more reliable method for untangling and smoothing of 2D meshes.  
\begin{table}
\begin{tabular}{c c} \hline \hline Compute Capability & 5.0 \\ \hline Number of stream multiprocessor & 5 \\ \hline Cores per stream multiprocessor & 128 \\ \hline Warp Size & 32 \\ \hline Clock Rate & 1.12 GHz \\ \hline Global Memory & 4 GB \\ \hline Peak FP32 Performance & 1.3 Tflops \\ \hline \hline \end{tabular}
\end{table}
Table 4: The detailed parameters of Quadro K2200.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Meshes** & **Platform** & **Number of Elements** & **Number of Inverted Elements** & **Time(s)** \\ \hline Quadrilateral mesh example & CPU & 140 & 23 & 0.006 \\  & GPU & & 4.184 \(\times\) 10\({}^{-4}\) \\ \hline Mesh deformation example & CPU & 11,363 & 459 & 29.849 \\  & GPU & 10,085 & 4914 & 37.488 \\ \hline Triangular mesh example & CPU & 10,085 & 4914 & 1.199 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of optimization results on CPU and GPU.  
**Author Contributions:** Conceptualization, S.S. and Z.G.; methodology, S.S. and Z.G.; software, Z.G.; theoretical analysis, Z.G. and M.G.; writing--original draft preparation, Z.G.; writing--review and editing, S.S., Z.G. and M.G. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by National Key R&D Program of China grant number 2017 YFC0803300 and 2018 YFC0809700.  
**Conflicts of Interest:** The authors declare no conflict of interest.