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A Multidimensional Clustering Analysis Method for Dividing Rock Mass Homogeneous Regions Based on the Shape Dissimilarity of Trace Maps

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Abstract

Dividing rock mass homogeneous regions according to discontinuities has important engineering significance. This study proposes a multidimensional clustering analysis method for dividing rock mass homogeneous regions based on shape dissimilarities, which reflect the comprehensive properties of discontinuities. The method mainly includes four procedures: (a) division of small trace maps, (b) calculation of shape dissimilarity based on the box-counting dimension and trace angle, (c) construction of the feature vector of a small trace map, (d) multidimensional clustering analysis. The attributes of clustering samples in this method are all shape dissimilarities with dimension = 1, so there is no need to assign weights to the attributes in the clustering analysis, i.e., it is a multiple-factor division method for rock mass homogeneous regions without manual setting of weights. Simulation experiments are used to validate the method, and the results show that this method is effective. In addition, some limiting conditions and test method about division results are discussed. Finally, a rock exposure at a hydropower station dam is used as a case study to illustrate the utility of the proposed method. The proposed method is believed to be a potentially useful tool for studying the spatial variability of geometric parameters of rock masses.

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Abbreviations

b :

Long side of a rectangular trace map

h :

Short side of a rectangular trace map

n h :

Number of divided parts of the short side of a trace map

n b :

Number of divided parts of the long side of a trace map

d i :

Equivalent side length of the rectangular box

N(di):

Number of valid boxes when the equivalent side of the rectangular boxes is di

SD ij :

Shape dissimilarity from the ith small trace map to the jth small trace map

BD i :

Box-counting dimensions of the ith small trace map

BD j :

Box-counting dimensions of the jth small trace map

MA i :

Mean value of the trace angles of the ith small trace map

SA i :

Standard deviation of the trace angles of the ith small trace map

MA j :

Mean value of the trace angles of the jth small trace map

SA j :

Standard deviation of the trace angles of the jth small trace map

FV i :

Feature vector of the ith small trace map

FV j :

Feature vector of the jth small trace map

cos(A, B):

Cosine of vectors A and B

arccos(cos(A, B)):

Angle of intersection between vectors A and B

MI :

Mcclain index

S w :

Within-cluster proximities

S b :

Between-cluster proximities

N w :

Numbers of Sw

N b :

Numbers of Sb

S w m :

Mean values of Sw

S b m :

Mean values of Sb

q :

Number of clusters

C k :

Serial number set of the samples in the kth cluster

x i :

Feature vectors of the ith small trace map

x j :

Feature vectors of the jth small trace map

n k :

Number of samples in the kth cluster

AH :

Accuracy of the homogeneous region results

N as :

Number of small trace maps

N cs :

Number of correct small trace maps

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Acknowledgements

This study was funded by the National Key R&D Program of China (2018YFC1505005), the National Natural Science Foundation of China (41972264, 41772287, 41772322), and the Zhejiang Provincial Natural Science Foundation Project (LY18E090002). The authors would like to thank Dr. Wen Zhang from Jilin University for providing the joint data of the engineering application.

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Authors and Affiliations

  1. Department of Civil Engineering, Zhejiang University, Hangzhou, 310058, China

    Jiongchao Wang, Jun Zheng, Qing Lü & Jichao Guo

  2. State Key Laboratory of Eco-Hydraulics in Northwest Arid Region, Xi’an University of Technology, Xi’an, 710048, China

    Mingming He

  3. Shaanxi Key Laboratory of Loess Mechanics and Engineering, Xi’an University of Technology, Xi’an, 710048, China

    Mingming He

  4. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu, 610065, China

    Jianhui Deng

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  1. Jiongchao Wang
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Correspondence to Jun Zheng.

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Wang, J., Zheng, J., Lü, Q. et al. A Multidimensional Clustering Analysis Method for Dividing Rock Mass Homogeneous Regions Based on the Shape Dissimilarity of Trace Maps. Rock Mech Rock Eng 53, 3937–3952 (2020). https://doi.org/10.1007/s00603-020-02145-9

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