## 1. Introduction

Numeral simulation has been applied to solve many problems in various fields, and mesh generation always is a vital topic in finite element analysis. Among the mesh generation algorithms, Delaunay triangulation [

1,

2,

3,

4], the advancing front technique [

5], the mapping method [

6] have been widely developed and applied due to the high quality of the generated mesh and easy implementation of the algorithms. Many researchers have focused on tetrahedral mesh generation methods [

7,

8]. For example, Meng [

9,

10] proposed a novel link-Delaunay-triangulation method to achieve geometric and topological consistency. Indirect methods [

11,

12] were proposed to convert Delaunay triangular meshes into quadrilateral meshes by combining adjacent pairs of triangles. Constrained triangulated surfaces can be used to mesh a three-dimensional geological model by using a series of tetrahedral meshes [

13,

14]. Other methods [

15,

16] for generating three-dimensional meshes were also proposed.

Moreover, Shimada et al. [

17] and Yamakawa [

18] first proposed the bubble method for mesh generation of the non-manifold geometry. This method generates point sets and the Delaunay triangulation technique is used for mesh generation. Yokoyama [

19] proposed an algorithm for three-dimensional bubble mesh generation. In this method, many points were inserted by the mapping-based method. Zhang [

20] used bubble method to discretize two-parameter surfaces into a high-quality mesh. Guo [

21,

22] proposed a fast local mesh generation algorithm based on the node-set generated by this method and proposed a subdivided dynamic bubble system to shorten the calculation time. Wang [

23] presented an automatic triangular mesh generation method, based on bubble dynamics simulation and a modified Delaunay method to generate high-quality mesh on complex surfaces efficiently. Jeong-Hun Kim [

24] implemented bubble method for adaptive mesh generation in two and three dimensions.

The geological model is a three-dimensional model representing the ground surface, strata, and faults, which consists of multiple geological blocks describing the structure and geometry of the stratum. Each block in the model has a different shape. The thin layer and the fault are very flat geometry, and the thick layer is a very thick geometry. Therefore, when the geological model is transformed into a numerical model, the geological model needs to be divided into the high-quality mesh to meet a balance between calculation accuracy and efficiency.

In this paper, we propose a clustering-based bubble method for generating high-quality tetrahedral meshes of geological models. The proposed bubble method is conducted based on the spatial distribution of point set of given surface meshes using the clustering method. There are four main procedures in the method. First, the inputted geological models consisting of triangulated surface meshes are divided into several parts based on spatial distribution of point set, which can be used for the determination of the positions and radii of initial points. Second, a procedure based on distance of nearby bubbles is used to obtain the initial size of bubbles. Third, by enforcing the forces acting on bubbles, all bubbles inside the 3D domain reach an equilibrium state by the motion control equations. Finally, the center node of the bubble can form a high-quality node distribution in the domain, and then the required tetrahedral mesh is generated.

The novelty of the proposed method can be explained as follows. As mentioned above, most of the above bubble methods focus on two-dimensional cases and three-dimensional parameter surface. However, since geological structures and conditions are generally quite complex, a three-dimensional geological model cannot be well represented by parametric surfaces but can be well represented of triangulated surfaces. The proposed bubble method is specifically proposed to generate the high-quality meshes of geological models consisting of triangulated meshes.

The main contributions of this paper can be summarized as follows.

- (1)
We apply the clustering algorithm based on the spatial distribution to divide the surface points for determining the initial bubbles.

- (2)
A procedure based on the distance of nearby bubbles to determine the initial size of bubbles.

- (3)
We evaluate the quality of the tetrahedral mesh in the geological model using three indicators.

The structure of this paper is organized as follows.

Section 1 introduces related research about mesh generation for surfaces and three-dimensional domains.

Section 2 introduces the background of bubble meshing method.

Section 3 introduces the method for geological model discretization based on the physical analogy between nodes and bubbles. In

Section 4, mesh quality indices are introduced to evaluate the resultant meshes and applications to demonstrate the effectiveness are provided. In

Section 5, the advantages and shortcomings are analyzed. Finally, the presented work is concluded in

Section 6.