# Designing Parallel Adaptive Laplacian Smoothing for Improving Tetrahedral Mesh Quality on the GPU  
[PERSON]  
1 School of Engineering and Technology, China University of Geosciences (Beijing), Beijing 100083, China; [EMAIL_ADDRESS] (N.X. ); [EMAIL_ADDRESS] (L.X. )1 School of Engineering and Technology, China University of Geosciences (Beijing), Beijing 100083, China; [EMAIL_ADDRESS] (N.X. )1 School of Engineering and Technology, China University of Geosciences (Beijing), Beijing 100083, China; [EMAIL_ADDRESS] (N.X. )1 School of Engineering and Technology, China University of Geosciences (Beijing), Beijing 100083, China; xixing@cugb. is called mesh smoothing [12; 13; 14; 15]. Mesh smoothing is more widely used because it does not change the connectivity of the mesh; one of the popular approaches is Laplacian mesh smoothing [16; 17; 18]. The process of Laplacian mesh smoothing is straightforward, requiring only that the locations of mesh nodes be updated to the geometric center of their neighbors during each iteration. In this case, the mesh topology will not be changed. However, in large-scale and complicated application scenarios, the computational models consist of a large number of nodes and elements, which means that in tetrahedral or hexahedral mesh models, a large number of tetrahedrons or hexahedrons need to be involved in the iterative computations. In this case, the complex iteration process will result in expensive computational costs when using Laplacian mesh smoothing. The use of parallel computing is an effective strategy to improve the efficiency of the Laplacian mesh smoothing algorithm; the powerful parallelism features of a modern graphics processing unit (GPU) can be utilized. Currently, parallel computing on GPUs has been widely used in numerical computing, artificial intelligence, and other applications [19; 20; 21; 22]. In Laplacian mesh smoothing, when the input mesh changes from a simple triangular mesh to a complex tetrahedral or hexahedral mesh, there are obviously larger number of tetrahedrons need to be calculated, Laplacian mesh smoothing process in serial calculation will become extremely long, and the experimental cost will be expensive. Therefore, it is necessary to accelerate Laplacian mesh smoothing algorithm on a GPU. Several feasible algorithms accelerated on the GPU have been proposed to optimize mesh quality. For example, [PERSON] et al. [23] proposed an ordinary Laplacian mesh smoothing algorithm accelerated on the GPU; [PERSON] and [PERSON] [24] proposed three efficient parallel algorithms to improve finite element meshes based on the Laplacian smoothing. In addition, other parallel optimization strategies have been proposed for various types of meshes. For example, [PERSON], [PERSON] et al. [25] presented a parallel approach for optimizing surface meshes by redistributing vertices on a feature-aware higher-order reconstruction of a triangulated surface. [PERSON], [PERSON] et al. [26] described a parallel adaptive mesh refinement strategy for two-phase flows using tetrahedral meshes. [PERSON] [27] proposed a multi-threaded parallel version of a sequential quality improvement algorithm for tetrahedral meshes, which combined mesh smoothing operations and local reconnection operations. In our previous research, we developed parallel Laplacian mesh smoothing algorithms for triangular meshes accelerated on the GPU [23; 28]. Moreover, a parallel ordinary Laplacian mesh smoothing for tetrahedral mesh accelerated on the GPU [29] has been presented. However, there is a shortcoming in the previously proposed ordinary Laplacian mesh smoothing algorithm. Specifically, the potential invalid nodes will be created in the concave area of the mesh. In FEM, the creation of invalid nodes will lead to distorted elements, which will strongly reduce the computational accuracy [30; 31; 32; 33].