###### Abstract
Meanwhile, some problems still need to be solved in practical applications, such as generating triangles outside the domain and determining the insertion position of new nodes. Some scholars have studied the problem of determining the insertion position of a new node and proposed corresponding schemes, such as utilizing triangular topological relations and multiple methods to reduce the computational burden in the localization procedure [3; 4]. [PERSON] utilized circle-dynamic movement to speed up searching for a new point [5]. [PERSON] and [PERSON] utilized dividing the dataset and a growing algorithm for searching for a third point to limit the determination of the search range, respectively [6; 7]. [PERSON] used the directional line connecting the current and subsequent insertion points to determine the target triangle [8]. [PERSON] and [PERSON] reduced the search time for the third point by subdividing and ordering point sets into regions [9]. [PERSON] controls the insertion order of nodes in different cells based on the triangle center of gravity method [10], which optimizes the position and process of the inserted nodes to locate the target triangle, which locates the insertion point quickly and improves the algorithm's efficiency. Scholars studied the improvement of delta mesh division, with [PERSON] and [PERSON] proposing an improved Delaunay refinement algorithm based on the original Delaunay triangular mesh [11; 12]. [PERSON] proposed an improved algorithm for Delaunay triangular dissections  
Figure 1: Voronoi diagram and a Delaunay triangular mesh. to discretize 2D domains using a multi-constraint insertion algorithm [13]. [PERSON] proposed a hybrid computational approach that partially uses GPUs and CPUs to speed up the computation [14]. [PERSON] combined Delaunay's advanced frontier technology to generate the boundary initial mesh [15]. [PERSON] and [PERSON] proposed an improved parallel Delaunay mesh generation algorithm based on the geometrical domain decomposition strategy and the incremental inserting algorithm, respectively, and partitioning algorithm based on their respective advantages to propose an improved parallel Delaunay mesh generation algorithm [16; 17]. [PERSON] and [PERSON] based the combination of the partition method, the incremental inserting algorithm, and the triangular mesh growth algorithm to generate a high-quality triangular mesh with improvement [18; 19]. [PERSON] proposed an effective automatic generation of ideal triangular divisions, ensuring the generated triangles have a range of angles [20]. In recent years, most innovative research on Delaunay triangular meshes has been directed towards the point localization problem of insertion nodes and improving the triangular mesh algorithm. Nevertheless, the placement of the insertion node is arbitrary, and the decision-making process is laborious. Despite the enhanced triangular mesh algorithm enhancing the efficiency of mesh construction, it does not eliminate the occurrence of deformed meshes. The Delaunay triangular mesh generation concept allows the immediate determination of the inserted node's position. Consequently, this reduces the need for extensive node position evaluations, minimizes the occurrence of deformed meshes, and ultimately enhances the overall mesh quality.