The realistic representation of an irregular geological body is essential to the construction of a particle simulation model. A three-dimensional (3D) sphere generator for an irregular model (SGIM), which is based on the platform of Microsoft Foundation Classes (MFC) in VC++, is developed to accurately simulate the inherent discontinuities in geological bodies. OpenGL is employed to visualize the modeling in the SGIM. Three key functions, namely, the basic-model-setup function, the excavating function, and the cutting function, are implemented. An open-pit slope is simulated using the proposed model. The results demonstrate that an extremely irregular 3D model of a geological body can be generated using the SGIM and that various types of discontinuities can be inserted to cut the model. The data structure of the model that is generated by the SGIM is versatile and can be easily modified to match various numerical calculation tools. This can be helpful in the application of particle simulation methods to large-scale geoengineering projects.
Although particle-based methods such as the discrete element method (DEM) [
The first step in applying particle simulation methods to solve any practical engineering problem involves the generation of a particle model of discrete objects, which are packed in a form that realistically represents the physical model of the problem. Extensive research has been conducted on this topic [
The majority of studies on sphere generation algorithms focus on ways to efficiently generate spheres and pack them in a container [
Few studies focus on the implementation of an irregular body into spherical particle modeling for complex engineering, especially large-scale geological body models with extremely irregular surfaces. The majority of existing methods are predominantly applied to the secondary development of certain existing programs [
To overcome these difficulties, the platform of Microsoft Foundation Classes (MFC) in VC++ is used to develop a program, which is referred to as a 3D sphere generator for irregular models (SGIM) in this paper. Step-by-step visualization modeling operations can be implemented using this program. Different types of discontinuities can be inserted to cut the model to obtain a more realistic rock mass, which is significant to the application of particle simulation methods in numerical simulation of large-scale geoengineering problems.
The structure of the SGIM is illustrated in Figure
Program structure of the SGIM.
The first function, which is referred to as the basic-model-setup function, facilitates the development of the basic model. It comprises a cuboid with edges parallel to the coordinate axes. In the current version of the SGIM, the equal spheres packing method is applied. All balls in the model contain the same radii and are packed by row, column, and layer. Note that selection of the particle size is a fundamental problem in modeling with spherical particles. An arrangement of spheres with equivalent sizes may result in a lattice effect on the mechanical behavior of the particle assembly. For this reason, the random packing method is usually recommended. However, implementation of this method is time consuming for a large-scale geological body. In the proposed model, a small particle size was adopted, which can reduce the lattice effect. The optimal size was based on a trial-and-error method. The number of rows, columns, and layers is obtained as
The center coordinates of a sphere are defined as
The second function is used to excavate the basic model. Excavating faces are composed of a series of interconnected triangular facets. These facets are obtained by inputting the coordinates of the points located along the edges of the excavating faces. The principle of an excavating function is to delete the spheres of the basic model if their centers are located in excavating spaces.
As shown in Figure
Excavating space of a triangular facet.
Let the coordinates of the three points be
(1) The equation of the plane
(2) The coordinates of the three points
(3) The equations of the planes that correspond to the three sides of the prism
(5) The coordinates of the three points
(6) The equations for the planes that correspond to the three sides of the prism
(7)
Note that, when
The excavating space of one triangular facet can be calculated with these seven steps. The total excavating spaces are the union of the excavating spaces of all triangular facets.
The cutting function, which is employed to cut the basic model with discontinuities, is based on certain assumptions. Several studies indicate that nearly equivalent lengths of discontinuities exist in the strike direction and the dip direction. As a result, the hypothesis that the discontinuity in 3D space can be described as a thin disc is often adopted [
Figure The center of the sphere must be located inside the infinite-length cylinder with one of its normal sections as the discontinuity. This is guaranteed by ( When When When
A disc plane that represents discontinuity in 3D space.
The simulated geological body and its discontinuities are illustrated in this example. This model tries to visualize the body in three dimensions using the proposed algorithm. The spherical particle calculation model of an open-pit slope in the Zijinshan gold-copper mine, which is the largest gold mine of the Zijin Mining Group in China, was constructed. The assumption that the geological body contains two sets of joints and a fault was applied. The dip and strike directions of these discontinuities are arbitrary in this study.
Figure
The basic model.
The basic model, which is excavated by adding excavating faces, is shown in Figure
The model after excavating.
The site picture
The spherical particle calculation model
Two groups of discontinuities with zero thickness and one group of discontinuity with a thickness of 4.0 are inserted into the model, as shown in Figure
Model after insertion of discontinuities.
Resulting spheres with different colors that adhere to discontinuities.
The final spherical particle model is shown in Figure
Spherical particle model for numerical calculation.
The SGIM exhibits the following advantages: the SGIM does not rely on existing software, and the data structure of the model generated by the SGIM can be easily modified to match numerical simulation tools, such as PFC3D. By utilizing the SGIM, spheres can be generated for extremely irregular 3D models to reasonably consider discontinuities in complex engineering problems.
Because the focus of this study is to develop a sphere generator for the 3D model for irregular geological bodies, some aspects are not considered in the present version of the SGIM and should be addressed in future studies. First, random packing of equal and unequal spheres should be adopted. In certain numerical simulation tools, such as PFC3D, ideal numerical simulation results can be obtained when the ratio of particle diameters is located within a certain range. Second, an algorithm to calculate the inner excavating space is required. Numerous excavating spaces frequently exist inside a geological body in practical engineering problems. However, the excavating function applied in this study is only suitable for excavation on the upper surface of the model. Third, the joint in a geological body is usually described as a thin disc. However, certain large-scale faults with complex shapes may exist in a geological body. A curved surface may be more suitable for this situation. Because some distribution laws of joints exist for a jointed rock mass, individually inputting all discontinuities is time consuming and impractical. Therefore, the incorporation of alternate ways to describe discontinuities, such as the Monte Carlo method, is required to simulate the network of discontinuities.
This work was supported by the Key Research Program of the Chinese Academy of Sciences (no. KZZD-EW-05-03), the National Basic Research Program (973 Program) of China (no. 2011CB710602), and the China National Natural Science Foundation (51139004, 40972201).