Study on Delaunay Triangular Mesh Delineation for Complex Terrain Based on the Improved Center of Gravity Interpolation Method

Abstract

Wind energy resources in complex terrain are abundant. However, the default mesh division of various terrains often needs more specificity, particularly in wind resource analysis. The mesh division method can diminish computational efficiency and quality in intricate topographical conditions. This article presents a combined algorithm for generating Delaunay triangular meshes in mountainous terrains with significant variations in terrain. The algorithm considers the uncertainty of inner nodes and mesh quality, addressing both the advantages and drawbacks of the Delaunay triangular mesh. The proposed method combines the triangular center of gravity insertion algorithm with an incremental inserting algorithm. Its main goal is to enhance the quality and efficiency of mesh generation, specifically tailored for this type of complex terrain. The process involves discretizing boundary edges and contour lines to obtain point sets, screening boundary triangles, and comparing the triangle area to the average boundary triangle area, combining with the incremental inserting algorithm to generate a triangular mesh of complex terrain. After an initial debugging of the mesh, it is determined whether increasing the internal nodes is necessary to insert the triangle centers of gravity. Upon implementing actual mountainous terrain in the simulation software, a comparison of the resulting meshing demonstrates that the proposed method is highly suitable for complex mountainous terrain with significant variations in elevation. Additionally, it effectively improves the quality of the Delaunay triangular mesh and reduces the occurrence of deformed cells during the meshing process.

1. Introduction

China’s inland terrain is complex, and most of the wind energy gathering areas are mountains, plateaus, hills, basins, and other complex terrain areas, which are rich in wind energy resources due to irregular landforms, complex terrain, and other factors. As China’s wind power resource field continues to extend, the construction of wind farms is mainly in wind energy-rich areas. From the perspective of wind resource development, wind energy resources in complex terrain areas are more abundant than in flat terrain areas. Among the many factors affecting the distribution of wind resources, the height difference significantly influences wind speed distribution, especially in mountainous areas where the terrain difference is noticeable, where the wind speed will increase with an increase in height difference, and there is also an abundant availability of wind resources. Improving the efficiency and accuracy of analysis and calculation through the mesh division is even more prominent for the problem of complex terrain, which is in the study of the distribution of wind resources. If the mesh division is rough, the calculation results may be unreliable; if the mesh division is fine, it is possible to enhance the accuracy of the calculation results, but it will significantly extend the calculation time. Wan Y, when analyzing the wind resources of wind farms by applying the wind resource software Meteodyn WT, mentions that the default mesh division of the WT software is simple in the defined mapping area. It has the universality of all-terrain, but this mesh division will significantly reduce the complex terrain’s calculation efficiency and mesh quality. So, it is necessary to research the more applicable mesh division method for complex terrain [1].

Among many types of meshing, a Delaunay triangulated irregular network (D-TIN) has a simple data structure and miniature storage space, is easy to update, and adapts to multiple point distribution methods. It is commonly employed to characterize the topographical features of the Earth’s surface. It has found extensive application in mesh demarcation, the development of digital elevation models, and three-dimensional visualization. In 1908, G. Voronoi first mathematically limited each discrete point’s effective range of action and defined the Voronoi diagram on the two-dimensional plane. In 1934, B. Delaunay evolved the Delaunay triangular network from the Voronoi diagram, which is easier to analyze and apply [2], as shown in Figure 1.

In recent years, the formation of the Delaunay triangular mesh has garnered significant study interest and attention from scholars both domestically and internationally and classified this research into three primary algorithms: the partition method, the incremental inserting algorithm, and the growth algorithm. Among them, the triangular network growth algorithm could be more efficient, with less room for improvement. The partition algorithm is highly efficient, but the operation process occupies a large amount of memory due to recursion. It must deal with a manageable amount of data. The incremental inserting algorithm approach consumes less memory and is suitable for dealing with extensive data. However, when the number of triangles grows more and more, the process of point localization and local optimization procedure (LOP) optimization becomes slower. Operation efficiency cannot be guaranteed. 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]. Yang, B. utilized circle-dynamic movement to speed up searching for a new point [5]. He, J. and Han, L. 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]. Zhang, S. used the directional line connecting the current and subsequent insertion points to determine the target triangle [8]. McCullagh, M. J. and Ross, C. G. reduced the search time for the third point by subdividing and ordering point sets into regions [9]. Zhang, J. 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 Shewchuk, R. and Chen, X. proposing an improved Delaunay refinement algorithm based on the original Delaunay triangular mesh [11,12]. Secchi S. proposed an improved algorithm for Delaunay triangular dissections to discretize 2D domains using a multi-constraint insertion algorithm [13]. Qi, M. proposed a hybrid computational approach that partially uses GPUs and CPUs to speed up the computation [14]. Sun, D. combined Delaunay’s advanced frontier technology to generate the boundary initial mesh [15]. Zeng, Y. and Huang, Q. 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]. Qing, W. and Angelo, L.D. 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]. Chew, L.P. 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.

In this paper, through the analysis of the existing Delaunay triangular mesh generation algorithm and its improvement method, based on the type of mountainous terrain with high terrain difference in complex terrain, for the uncertainty of the internal nodes in the Delaunay triangular section and the mesh quality advantages and disadvantages of the problem, we put forward a kind of triangular center of gravity insertion method (TCOGIM) suitable for the improvement of complex terrain in mountainous terrain: discretizing boundary edges and contour lines to obtain point sets, the area ratio of the average value of the boundary triangle, and determining whether the triangle needs to interpolate the center of gravity to increase the internal nodes of the process to deter-mine the location of the nodes, combined with the incremental inserting algorithm to generate a triangular mesh of complex terrain.

2. Delaunay Triangular Mesh

2.1. Delaunay Theory

In the context of a given set of discrete points, as p = {p1, p2, …, pn}, a Delaunay triangular section can be obtained by connecting any two points with a straight line, ensuring that line segments connect all points and that no two line segments intersect at any location other than the vertex. The three vertices of a triangle can determine its circumcircle. If this circumcircle contains no additional points, considering it is empty, it is classified as a Delaunay triangle. Conversely, if it locates the remaining vertices of the triangle within the circumcircle, it is considered non-empty and classified as a non-Delaunay triangle.

The determination of whether a triangle is a Delaunay triangle requires the aid of the Delaunay criterion: (1) The empty circle property: the Delaunay triangulation net is unique (no four points can be co-circular), and no other points can exist within the outer circle of any triangle in the Delaunay triangulation net; (2) Maximizing the minimum angle property: the minimum angle is the largest among the possible triangles formed by the collection of scatter points’ triangular dissections. During the dissection process, these two criteria minimize the likelihood of narrow and irregular triangles appearing, and the mesh pattern generated is unique for the same set of discrete points.

2.2. Bowyer–Watson Algorithm

Improvements in triangular sectioning based on Delaunay’s algorithm are proliferating. This article focuses on the Bowyer–Watson (B–W) algorithm’s incremental inserting algorithm. The method shown in Figure 2 is as follows: If we add a point P to the triangular section, all the vertices of the outer circle containing the vertex P form a “star polygon”, as shown in the middle of the shaded part of Figure 2b, and we delete all the triangles in its interior, making a “cavity”. Then, the inner triangles are all deleted to form a “cavity”, as shown in Figure 2c. The vertices at the boundary of the cavity are connected to the vertex P to obtain a new Delaunay triangular section, which replaces the triangles deleted in the original section. Then, a new Delaunay triangular section that contains the vertex P is obtained.

3. Delaunay–TCOGIM Algorithm

3.1. Delaunay–TCOGIM Algorithm Design Ideas

Due to the randomness of the insertion of the discrete nodes in the Delaunay method, the number of malformed triangular meshes will increase. As a result, we have implemented TCOGIM to determine the position of the nodes. We insert the center of gravity to construct the triangular mesh, with the triangular area and the average size of the boundary triangles as the determining conditions. The thought behind the Delaunay–TCOGIM algorithm is mainly to discretize the points on the boundary and contour lines, screen the internal mesh of the region, discretize the point sets in clockwise or anticlockwise order, judge the center of gravity of the boundary triangles according to the right-handed helix rule for the initial debugging of the mesh, and compare it with the average value of the boundary triangles’ area to judge whether it is necessary to interpolate the center of gravity to increase the internal nodes, and then combine it with the B–W algorithm to reconstruct the triangles to cycle judgment on the newly-generated triangle unit area until all triangles meet the conditions to satisfy the Delaunay criterion to generate a Delaunay triangular mesh.

The basic steps of the Delaunay–TCOGIM algorithm are boundary discretization, building the initial triangle mesh, building the initial Delaunay mesh, finding the inner region of the boundary, building the mesh for the inner region, and finding the Delaunay mesh. During the construction of the internal mesh within the region, the Delaunay method differs from the combined Delaunay–TCOGIM algorithm. The Delaunay method selects triangle cells with a triangle area more significant than the average value of the boundary triangles. They identify these selected triangle cells as inserting the set of triangle cells into the point and inserting a specific number of discrete nodes as internal nodes within this set of triangle cells. This process subsequently updates the set of triangles.

3.2. Steps of the Delaunay–TCOGIM Algorithm

The specific steps of the Delaunay–TCOGIM algorithm are as follows:

(1)

Boundary discretization: The initial input data of the algorithm is a series of point sets obtained by discretizing the boundary edges and contour lines after topographic map recognition. According to the basic idea of the algorithm in this article, in the first one-dimensional mesh division, the mesh boundaries and contour lines are divided into discrete points and line segments and then transformed into boundary nodes, boundary edges, and discrete point sets. The obtained data includes all the necessary mesh data needed for subsequent mesh division.

(2)

Construct the initial triangle mesh: Based on the preceding mesh data, determine the center point of the target area, the maximum height (y-axis direction), and the maximum width (x-axis direction), multiply the maximum value of the two by a coefficient greater than 1, and then use the center point of the target area as the center and this length as the length of the side of the box to construct a square, the Delaunay box. The coordinates of the box’s four vertices serve as the initial nodes of the Delaunay mesh; the rectangle and any diagonal serve as the initial mesh boundary; and the two triangles serve as the initial triangle mesh.

(3)

Construction of the initial Delaunay mesh: After constructing a sufficiently large enclosing box, the B–W algorithm described in Section II divides the mesh into its initial subdivisions. Use this enclosing box to turn the meshing problem into a convex envelopment problem with discrete points of the curves in the mesh boundary region and the four vertices of the enclosing box. To do this, start the enclosing box as two triangles with divisions at all discrete points. Then, the discrete points are used as insertion points for the B–W algorithm to build the Delaunay triangle.

(4)

Identify the inner region of the boundary: The Delaunay mesh obtained through the above description is a triangular mesh containing the Delaunay enveloping box. However, the boundary of the delineated region cannot be shown very intuitively in the mesh of the Delaunay enveloping box, and only the triangular mesh of the inner region need to be constructed for practical application. For this problem, we are identifying the region boundaries by using a coloring algorithm to differentiate between the internal and external components of the region [21] and removing the triangles and Delaunay enclosing boxes outside the mesh region.

The following steps are the precise sequence of the coloring algorithm:

• Step 1. Assign v = −1 to each triangle mesh cell.
• Step 2. Find the triangle mesh cell containing vertices, assign it a color, i.e., v = c (define c = 0), and store it in stack S.
• Step 3. If stack S is not empty, one of the triangle mesh cell e is taken out of the stack.
• Step 4. Traverse the three adjacent cells of the triangle mesh cell e for the following reasons:

  (1)

  If the value of the visited triangle cell is not −1, then this triangle cell contains colored vertices. Return to Step 4.

  (2)

  If the value of the visited triangle cell is −1, proceed directly to Step 5.

• Step 5. Determine the common edge of the visited triangle cell and cell e using the following method:

  (1)

  Assign a color to the visited triangle mesh, i.e., v = c, and store it in stack S if the joint edge of the visited triangle cell and cell e is not a region boundary edge.

  (2)

  If the joint edge of the visited triangle cell and cell e is a region boundary edge, proceed to Step 4 to traverse the neighboring cells.

• Step 6. Make c = c + 1 and clear stack S. If there is still a cell with v = −1, proceed to Step 3. Otherwise, exit.

The value of c is the number of region components. In this instance, the number of region components is 2. The region interior is a class, and the triangle mesh cells of the interior components with the color 0 are to be deleted, whereas the triangle mesh cells with the color 1 are outside the region and should be left intact.

(5)

Constructing a regional internal mesh: For the mesh delineation after region identification, the nodes are all mesh boundary points, and there are no regional internal points in the mesh delineation. Therefore, creating some regional internal points is necessary to complete the accurate mesh delineation of the terrain region. In Delaunay triangulation, in combination with the improved TCOGIM algorithm with the B–W algorithm, the discrete points of the boundary are discretized in either a clockwise or anticlockwise order to generate the triangular cell set D and to determine whether the triangular cells therein are boundary triangles. The boundary triangles are then added to the triangular cell set TE, and these indicate the boundary edges b_ib_j. Calculate the center of gravity G of the boundary triangular cells and employ the right-handed spiral rule to judge b_ib_jG. Retain the triangle if the direction is positive and reject it if the direction is negative. Calculate the area of all boundary triangles and calculate the average. Compare the area of each triangle in the triangular cell set TE with the average comparison value. If the triangle’s area is less than the average, proceed to the next triangle; if greater than that, insert the center of gravity to apply the B–W algorithm to re-cluster the triangles. The average measure of the newly generated triangle cell mesh area and the average value of the boundary triangle area are evaluated in a cyclical manner until all generated triangles satisfy the necessary conditions. Figure 3 shows the exact procedure for constructing the regional internal mesh in this process.

(6)

Evaluate the Delaunay mesh: The Delaunay criterion mentioned in Section II is the empty circle property and the maximum minimal angle property. It is necessary to evaluate the Delaunay criterion for the triangular mesh generated by the combination of the Delaunay–TCOGIM algorithm, and only the triangular mesh that satisfies the Delaunay criterion is a Delaunay triangular mesh. If the determination procedure yields a triangular geometry that is not Delaunay, the diagonal exchange method must be used to obtain Delaunay triangles.

Figure 4 illustrates the diagonal exchange technique: Remove the common side BD of the non-Delaunay triangles to obtain the quadrilateral ABCD, then connect the other diagonal AC to obtain two new triangles.

The flowchart of the specific Delaunay–TCOGIM algorithm is shown in Figure 5.

4. Example Analysis

Yunnan Province is an interior region located at a low latitude. The Tropic of Cancer intersects the southern region, with the topography being elevated in the northwest and somewhat lower in the southeast. There is a gradual decrease in height from the north to the south, with a significant change in altitude due to the mountainous plateau terrain. The province’s mountainous region comprises 88.64% of its entire area, mostly due to its topography and monsoon climate. This region is abundant in wind energy resources. This section provides an example of a wind farm located in a mountainous, complicated terrain with significant variations in elevation. The specific wind farm in Yunnan Province is used as a case study, and the topographic features of this wind farm are illustrated in Figure 6.

4.1. Validation Results of the Combined Delaunay–TCOGIM Algorithm

Applying the Delaunay–TCOGIM algorithm to a specific example: take 1/4 of the number of discrete point sets of the boundary and contour lines, as depicted in Figure 7.

Figure 8 shows the construction results of applying the B–W algorithm to build the initial Delaunay mesh.

As shown in Figure 9, use the coloring algorithm for region identification to obtain the initial triangular mesh. It is evident from the mesh that all discrete points lie only on the boundary. The resulting triangular mesh is irregular, with many deformed triangles and poor quality.

It compares the area of the triangles and the average value of the scope of the boundary triangles to determine whether to insert the center of gravity to delineate the internal area further. It repeats the judgment cycle after the triangular mesh generation by applying the Delaunay criterion to determine the final age of tetrahedrons. Figure 10 depicts the Delaunay triangular mesh after one process using twice the average value of the boundary triangles as its base.

4.2. Comparison with the Delaunay Method

Based on the initial Delaunay triangles in Figure 9, the Delaunay method randomly adds point sets to the interior. It uses the incremental inserting algorithm to make the Delaunay triangle mesh. Figure 11 shows a comparison between the Delaunay triangular mesh made by the combined Delaunay–TCOGIM algorithm and the Delaunay triangular mesh made by the Delaunay method after cycling with 1.52 times the mean value of the boundary triangles.

4.3. Mesh Quality Analysis and Runtime Comparison

After five cycles of the Delaunay–TCOGIM algorithm with 1.52 times the average value of the boundary triangles,

Table 1. Variations in mesh quality differentiation.

Quality Factor	Multiples of Average Value	Initial Value	Cycle Times
			First	Second	Third	Fourth	Fifth
$q_M$	1.52	0.7208	0.8462	0.8757	0.8762	0.8766	0.8766
$q_J$		0.6220	0.8066	0.8532	0.8546	0.8560	0.8558

compares the mesh average quality coefficient q_M and the mesh association quality coefficient q_J (see the Appendix A for specific formulas). After one cycle, there was a notable enhancement in the mesh quality and reduced aberration. After three runs, the average and association quality coefficients tended to reach a stable state. The combined Delaunay–TCOGIM algorithm produced a Delaunay triangular mesh with an average quality of 0.8766 and a degree of distortion of 0.8558.

Next, we will appropriately adjust the selection of contour discrete point sets by reducing the number of sets for the higher three contours to one-fifth of their original number while keeping the number of sets for the other contours unchanged. It compares the triangular mesh generated by cycling five times with the average value of the boundary triangles, 1.52 times, with that caused by the Delaunay method. It reaches the quality of the Delaunay triangular mesh generated by the two ways in

Table 2. Delaunay triangular network quality assessment.

Arithmetic	Total Points	Number of Triangular Mesh Cells	${\mathit{q}}_{\mathit{M}}$	${\mathit{q}}_{\mathit{J}}$
Delaunay–TCOGIM algorithm	457	837	0.8766	0.8558
Delaunay method			0.6682	0.5044

. Refining the triangular grid cells improved the grid quality, allowing it to better meet the wind resource study requirements. In contrast, the triangular grid generated by the Delaunay method produces more deformed grid cells, which degrades the grid quality.

Mesh cell regularity p quantifies the proximity of individual triangles to positive triangles. In contrast, mesh geometric irregularity p_G quantifies the irregularity of the mesh geometry (see the Appendix A for specific formulas). As shown in

Table 3. Delaunay triangle mesh comparison.

Arithmetic	Total Meshes	Meshes								${\mathit{p}}_{\mathit{G}}$
		p = 0~0.3	p = 0.3~0.4	p = 0.4~0.42	p = 0.42~0.44	p = 0.44~0.46	p = 0.46~0.48	p = 0.48~0.5	p = 0.5
Delaunay–TCOGIM algorithm	837	30	136	68	99	116	168	220	0	0.0617
Delaunay method		305	206	55	51	77	60	83	0	0.1659

, a significant proportion of the triangular meshes generated by this algorithm are close to or reach the positive triangles (p ≥ 0.46), and the number of meshes that deviate from the positive triangles to a greater extent decreases (p ≤ 0.4). In this algorithm, the percentage of meshes close to or reaching the positive triangles is 46.36%, and the portion of meshes deviating from the positive triangles is 19.83%. In contrast, in the Delaunay method, the percentage of meshes close to or reaching the positive triangles is 17.08%. The portion of meshes deviating from the positive triangles is 61.05%, a significant decrease from the part of meshes varying from the positive triangles. This algorithm has significantly enhanced the quality of the triangular Delaunay mesh.

The CPU time is an essential indicator of the efficiency of the algorithms. With the area-to-boundary triangle ratio set at 1.5 in both algorithms, the discrete points in the final mesh are the same or similar. The comparison of CPU running times between the two algorithms is shown in

Table 4. CPU runtime comparison.

Number of Cycles/Times	Number of Discrete Points of This Algorithm	Number of Discrete Points of Delaunay Method	Running Time of This Algorithm/s	Running Time of Delaunay Method/s
1	268		8.86	8.94
2	427		14.06	14.69
3	450	449	18.70	21.67
4	457	455	24.09	23.19
5	459	462	32.61	29.78

. The CPU time comparison in Table 4 shows that the algorithm’s running time in this paper is smaller than that of the Delaunay method when the number of loops is ≤3. However, this trend reverses when the loop count exceeds 3. It can be inferred that when the final mesh’s discrete points remain constant, the algorithm in this paper gradually inserts fewer discrete points as the loop count increases. In contrast, the Delaunay method inserts the same number of points in each loop, which leads to the need to constantly judge the size of the average of the triangle and the boundary triangle area in the algorithm, and so the CPU running time will also increase.

In this section, we applied the proposed Delaunay–TCOGIM algorithm and the Delaunay method to a specific typical mountainous terrain. By comparing the mesh quality metrics and CPU running time discussed in Section C, we saw that the Delaunay–TCOGIM algorithm produces a higher-quality mesh compared to the Delaunay method. The CPU running time of the Delaunay–TCOGIM algorithm is lower than the CPU running time of the Delaunay method for a certain number of cycles with the same discrete points. Therefore, under a reasonable selection of the number of cycles and multiplier, the Delaunay–TCOGIM algorithm demonstrates better applicability to the mountainous complex terrains with high terrain differences and can effectively improve the quality and efficiency of the Delaunay triangular mesh.

5. Conclusions

In this article, through the analysis of existing algorithms and their improvement methods based on the type of mountainous terrain with high relief in complex landscapes, the uncertainty of internal nodes in Delaunay triangulation, and the problem of good and bad mesh quality, the TCOGIM algorithm is proposed to be applied to assist in the improvement of mountainous terrain with high relief in complex landscapes, combined with the B–W algorithm, which provides a method to efficiently and accurately analyzing and calculating the mesh in studying the distribution of wind resources in mountainous terrain. The method employed for mesh delineation involves comparing it with the average value of the boundary triangles’ area to determine the necessity of interpolating the center of gravity for increased internal nodes. This process is then coupled with the B–W algorithm to reconstruct the triangles to cycle judgment on the newly generated triangle unit area. This cycle continues until all triangles meet the conditions, satisfying the Delaunay criterion and generating a Delaunay triangular mesh. Through simulation combined with an actual wind farm application in Yunnan Province, we compare the triangular mesh in the same region generated by the Delaunay–TCOGIM algorithm with that produced by the Delaunay method in terms of metrics, such as mesh quality coefficient and mesh regularity, as well as the CPU running time of the two algorithms. The comparison results show that the Delaunay–TCOGIM algorithm exhibits better applicability to complex terrain with high elevation differences. It effectively enhances the quality of the Delaunay triangular mesh, mitigates the generation of distorted grid cells, and improves the efficiency of grid generation. However, when choosing the number of loops and the multiplier, we should choose a reasonable value to avoid the calculation time being too long, which affects the efficiency. Moreover, the results of making changes to this wind farm’s contour discrete point set show that after reducing the selection of part of the contour discrete points, the triangular mesh can be made more efficiently. The results show that after reducing the selection of some contour discrete points, the more refined the triangular grid cells are, the better the grid quality is.