# Generation of Numerical Models of Anisotropic Columnar Jointed Rock Mass Using Modified Centroidal Voronoi Diagrams

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## Abstract

**:**

## 1. Introduction

## 2. Typical Shapes of Columnar Jointed Rock Mass

## 3. Constrained Centroidal Voronoi and Implementation Method

#### 3.1. Voronoi Diagram Algorithm and Its Constraints

#### 3.1.1. Classical Voronoi Tessellation

_{i}is generated. It can be seen that the domain is divided into 10 patches and each patch has a single seed. For the generation of a Voronoi diagram, a Delaunay triangulation is first implemented. Then the perpendicular bisector is plotted to partition the domain into different Voronoi cells. With the development of computational geometry, Voronoi tessellation is included in many software and packages like MATLAB, Mathematica, and SciPy. However, there are two obvious shortcomings of the classical Voronoi diagram:

- (a)
- The Voronoi cell is not closed. For the Voronoi diagram in Figure 4, only one cell is closed and the other nine cells are open. This brings inconvenience for the analysis.
- (b)
- The shape of the Voronoi cell is random and it is very hard to generate a Voronoi diagram with a specified statistical distribution.

#### 3.1.2. Constrained Voronoi Diagram Generation

_{i}and domain D. Taking a model with 16 points (x

_{i}, y

_{i}) and a square domain in Figure 5a as example, this algorithm can be described as follows [22]:

- (a)
- A Voronoi diagram is generated using a classical tessellation method (Figure 5b).
- (b)
- All the open cells with a vertex outside the domain D are identified (shown in Figure 5c).
- (c)
- A set of new seeds symmetric to the seeds of open cells with respect to the domain boundary are created (Figure 5d).
- (d)
- New Voronoi diagram is generated with a classical tessellation (Figure 5e).
- (e)
- By removing the open cells as well as the related seeds, the final Voronoi diagram is shown in Figure 5f.

#### 3.2. Centroidal Voronoi Algorithm

#### 3.2.1. Random and Centroidal Voronoi Diagram

#### 3.2.2. Lloyd’s Algorithm

- (1)
- For an initial seeds y
_{i}, generate a Voronoi diagram using constrained Voronoi tessellation; - (2)
- Compute the centroid z
_{i}of the Voronoi diagram of y_{i}; - (3)
- Move the generating point y
_{i}to its centroid z_{i}; - (4)
- Repeat Steps 1 to 3 until all generating points converge to the centroids.

#### 3.2.3. Estimation of the Centroid

_{i}is the region area, and ρ(χ) is the density function with ρ(χ) = 1 being the default.

**(1) Integration method on triangle partitions**

_{i}(x

_{i}, y

_{i}) (i = 1, 2, 3), the coordinates of centroid are given by:

_{i}(x

_{gi}, y

_{gi}) and S

_{i}, the coordinates of the centroid of the polygon can be obtained as:

**(2) Sampling method**

_{i}(x

_{i}, y

_{i}) in this area, then the centroid of this cell can be approximated as the average of these points:

#### 3.3. Numerical Implementation and Discussion

## 4. Modeling of Columnar Jointed Rock Mass

- (1)
- For a Voronoi diagram with CV larger than the specified value, calculate the centroid P
_{C}. - (2)
- The new generator P
_{new,g}is set at the midpoint of the old generator P_{old,g}and the centroid P_{C}. Calculate the coefficient of variation CV of the new Voronoi diagram. - (3)
- Repeat Steps 1 and 2 to obtain the generator until the coefficient of variation CV converges to the specified value with a prescribed accuracy.

## 5. Columnar Joints Generation: A Case Study of the Baihetan Hydropower Station

#### 5.1. Engineering Geological Investigation

_{2}β). In the construction of the plant’s hydraulic structures, columnar jointed basalt is found to be widely distributed at the arch dam foundation, underground caverns, abutment slopes, and other water conservancy tunnels (Figure 11).

#### 5.2. Columnar Jointing Model Generation

## 6. Conclusions and Discussion

- (1)
- The coefficient of variation is an effective parameter for representing the deviation between the generator and the centroid of Voronoi cell, which has the same effect as energy in a centroidal Voronoi tessellation. Furthermore, it can reflect the heterogeneity of the cells forming the columnar jointed rock mass.
- (2)
- A modified Lloyd’s algorithm is proposed to generate the Voronoi diagram with a specified coefficient of variation. Two algorithms for estimating the centroid were presented and discussed.
- (3)
- This work proposed the description of columnar jointed rock mass with six parameters and a detailed procedure for modelling columnar jointed rock mass. Taking the columnar basalt in the Baihetan hydropower station as an example, numerical models for columnar jointed rock mass with the specified geological properties were generated. The numerical results indicated that the method was effective and efficient.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Typical columnar jointed rock masses: (

**a**) Devil’s Postpile, Yosemite in California; (

**b**) Fingal’s Cave, Staffa in Scotland; and (

**c**) Giant’s Causeway, Antrim in Northern Ireland.

**Figure 3.**Diagrammatic drawing of the joints in columnar basalt: (

**a**) vertical columnar joint, and (

**b**) horizontal transverse joint.

**Figure 4.**An illustration of Voronoi tessellation with 10 generators: (

**a**) a set of generators, and (

**b**) the resulting Voronoi diagram.

**Figure 5.**Illustration of the stages of the constrained Voronoi tessellation algorithm: (

**a**) initial seeds, (

**b**) classical Voronoi tessellation, (

**c**) open Voronoi cells, (

**d**) symmetry seeds, (

**e**) new Voronoi tessellation, (

**f**) constrained Voronoi diagram.

**Figure 6.**Random and centroidal Voronoi diagram: generators (white dots) and centroids (black dots): (

**a**) random Voronoi diagram, and (

**b**) centroidal Voronoi diagram.

**Figure 7.**Discretization of n-polygon: (

**a**) polygon with n vertices, and (

**b**) n − 2 discrete triangles.

**Figure 8.**Illustration of the sampling method: (

**a**) Voronoi tessellation, and (

**b**) random sampling points.

**Figure 9.**Implementation of the constrained centroidal Voronoi algorithm: (

**a**) initial condition, (

**b**) step 5, (

**c**) step 20, (

**d**) step 50, (

**e**) change of energy, and (

**f**) change of coefficient of variation.

**Figure 11.**Site conditions of the Baihetan Hydropower Station: (

**a**) construction site, and (

**b**) typical columnar jointed basalt.

**Figure 12.**Typical P

_{2}β

_{3}columnar basalt at Baihetan Hydropower Station: (

**a**) geological photo, and (

**b**) joints skeleton.

**Figure 13.**Numerical specimen generation: (

**a**) Voronoi diagram with CV = 56.45%, (

**b**) Voronoi diagram with CV = 44.18%, (

**c**) extrude 2-D Voronoi diagram with direction, (

**d**) cut columnar rock with transverse joint, (

**e**) block model of columnar rock, and (

**f**) particle model of columnar rock.

**Figure 14.**Columnar joints with different coefficient of variation: (

**a**) block model with CV = 40%, (

**b**) block model with CV = 30%, (

**c**) block model with CV = 20%, (

**d**) block model with C = 10%, (

**e**) particle model with CV = 40%, (

**f**) particle model with CV = 30%, (

**g**) particle model with CV = 20%, and (

**h**) particle model with CV = 10%.

**Figure 15.**Columnar joints with different joint dip angles: (

**a**) block model with dip = 0°, (

**b**) block model with dip = 30°, (

**c**) block model with dip = 60°, (

**d**) block model with dip = 90°, (

**e**) particle model with dip = 0°, (

**f**) particle model with dip = 30°, (

**g**) particle model with dip = 60°, and (

**h**) particle model with dip = 90°.

Parameter | Dip | DD | CD | CV | TD | TP |
---|---|---|---|---|---|---|

Values | 72° | 145° | 25/m^{2} | 44.18% | 1.5 m | 0.3 |

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**MDPI and ACS Style**

Meng, Q.; Yan, L.; Chen, Y.; Zhang, Q.
Generation of Numerical Models of Anisotropic Columnar Jointed Rock Mass Using Modified Centroidal Voronoi Diagrams. *Symmetry* **2018**, *10*, 618.
https://doi.org/10.3390/sym10110618

**AMA Style**

Meng Q, Yan L, Chen Y, Zhang Q.
Generation of Numerical Models of Anisotropic Columnar Jointed Rock Mass Using Modified Centroidal Voronoi Diagrams. *Symmetry*. 2018; 10(11):618.
https://doi.org/10.3390/sym10110618

**Chicago/Turabian Style**

Meng, Qingxiang, Long Yan, Yulong Chen, and Qiang Zhang.
2018. "Generation of Numerical Models of Anisotropic Columnar Jointed Rock Mass Using Modified Centroidal Voronoi Diagrams" *Symmetry* 10, no. 11: 618.
https://doi.org/10.3390/sym10110618