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

S S symmetry

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

Qingxiang Meng 1,2,* , Long Yan 2,3,*, Yulong Chen 4,* and Qiang Zhang 5,6

1 College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China 2 Research Institute of Geotechnical Engineering, Hohai University, Nanjing 210098, China 3 Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing 210098, China 4 Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China 5 China Institute of Water Resources and Hydropower Research, Beijing 100048, China; [email protected] 6 State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China * Correspondence: [email protected] (Q.M.); [email protected] (L.Y.); [email protected] (Y.C.)

Received: 21 October 2018; Accepted: 7 November 2018; Published: 9 November 2018

Abstract: A columnar joint network is a natural fracture pattern with high symmetry, which leads to the anisotropy mechanical property of columnar basalt. For a better understanding the mechanical behavior, a novel modeling method for columnar jointed rock mass through field investigation is proposed in this paper. Natural columnar jointed networks lies between random and centroidal Voronoi tessellations. This heterogeneity of columnar cells in shape and area can be represented using the coefficient of variation, which can be easily estimated. Using the bisection method, a modified Lloyd’s algorithm is proposed to generate a Voronoi diagram with a specified coefficient of variation. Modelling of the columnar jointed rock mass using six parameters is then presented. A case study of columnar basalt at Baihetan Dam is performed to demonstrate the feasibility of this method. The results show that this method is applicable in the modeling of columnar jointed rock mass as well as similar polycrystalline materials.

Keywords: anisotropic columnar jointed rock; numerical model; centroidal Voronoi diagram; coefﬁcient of variation

1. Introduction Columnar jointed rock is a typical geological structure formed by a lot of ordered colonnades like the world famous natural wonders of Fingal’s Cave and Giant’s Causeway [1,2]. Being a miraculous natural phenomenon, the study on the geological structure of columnar jointed rock can be dated back to the 17th century [3]. Nowadays, the formation of columnar jointing is reasonably understood as a result of cracks propagation in cooling lava ﬂows [4–7]. In situ observations and laboratory tests have also been employed in the study of columnar jointing [8–10]. With the increase of human activities, some engineering projects have encountered columnar jointing, and a large hydropower station is even founded on it [11]. Because of its adverse geologic conditions, study on the mechanical properties of columnar jointed rock mass is very important in engineering projects. King et al. studied elastic-wave propagation in columnar joints with a series of cross-hole acoustic measurements made between four horizontal boreholes drilled from a near-surface underground opening situated in a basaltic rock mass [12]. In connection with the Baihetan

Symmetry 2018, 10, 618; doi:10.3390/sym10110618 www.mdpi.com/journal/symmetry Symmetry 2018,, 10,, 618x FOR PEER REVIEW 2 of 15

Hydropower Station project, several studies have been implemented for the mechanical property of Hydropowercolumnar jointed Station rock. project, Meng severalsystematically studies analyzed have been the implemented anisotropic forproperties the mechanical of columnar property jointed of columnarbasalt using jointed analytical rock. and Meng numerical systematically methods analyzed [13]. Zheng the anisotropic et al. proposed properties a 3-D of modeling columnar method jointed basaltfor columnar using analytical basalt using and numericalrandom Voronoi methods tessellation [13]. Zheng [14]. et al.In proposedReferences a 3-D[15–17], modeling 2-D and method 3-D fordiscrete columnar element basalt simulation using random methods Voronoi of column tessellationar are [14 implemented,]. In References and [15 the–17 ],representative 2-D and 3-D discreteelementary element volume simulation scale and methods equivalent of columnar mechan areical implemented, parameters and are the obtained. representative Basedelementary on meso- volumestructural scale analysis, and equivalent macro-anisotropic mechanical constitutive parameters model are obtained. is developed Based and on 3-D meso-structural numerical simulation analysis, macro-anisotropicis performed for the constitutive dam foundation model considering is developed the and effect 3-D of numerical columnar simulation joints [18]. isLaboratory performed tests for theare damcarried foundation out using considering experimental the effectanalogs of columnarof jointing joints under [18 ].uniaxial Laboratory compression tests are [19]. carried These out usingresearch experimental results indicate analogs that the of jointingcharacteristics under of uniaxial columnar compression jointed rock [ 19mass]. These are very research complex results with indicateanisotropic that and the nonlinear characteristics properties. of columnar jointed rock mass are very complex with anisotropic and nonlinearMost properties.of the modelling methods of columnar jointed rock mass use the program developed by ZhengMost et al. of the[14]; modelling however, methodsthis method of columnar has some jointed shortcomings, rock mass such use as the a programlow efficiency developed and byimmutable Zheng et columnar al. [14]; however,shapes, as this pointed method out has in someReference shortcomings, [20]. The suchobjective as a of low this efficiency paper is and to immutableovercome these columnar shortcomings shapes, asin the pointed generation out in of Reference columnar [20 joints]. The in objectiverock mass. of An this efficient paper isand to overcomecontrollable these method shortcomings for the generation in the generation of column of columnarar joints is joints proposed in rock based mass. on An a efficientconstrained and controllablecentroidal Voronoi method fortessellation the generation diagram. of columnar In Section joints 2, the is proposed characteristics based onof atypical constrained columnar centroidal rock Voronoimass are tessellation briefly introduced. diagram. The In Section properties2, the of characteristics Voronoi diagrams of typical and columnar the latest rock related mass research are briefly on introduced.this topic are Thediscussed properties in Section of Voronoi 3. Based diagrams on these and discussion the latest and related algorithms, research a detailed on this procedure topic are discussedfor the generation in Section of columnar3. Based jointed on these rock discussion mass is developed and algorithms, in Section a detailed4. A case procedurestudy on columnar for the generationjointed basalt of columnargeneration jointed for the rock Baihetan mass isHydropower developed in Station Section is4 .presented A case study in Section on columnar 5. Finally, jointed the basaltconclusion generation is given for in the Section Baihetan 6. Hydropower Station is presented in Section5. Finally, the conclusion is given in Section6. 2. Typical Shapes of Columnar Jointed Rock Mass 2. Typical Shapes of Columnar Jointed Rock Mass With the development of hydropower station, columnar jointing is often encountered in southwesternWith the China. development As a result of hydropowerof cooling lava station, flows and columnar ash-flow jointing tuffs, it is occurs often in encountered many types inof southwesternvolcanic rocks China.in the formation As a result of ofa regular cooling array lava of flows polygonal and ash-flow prisms tuffs,or columns it occurs [21]. in Several many types famous of volcanictypical columnar rocks inthe joints formation are shown of a in regular Figure array 1. of polygonal prisms or columns [21]. Several famous typical columnar joints are shown in Figure1.

(a) (b) (c)

Figure 1.1. TypicalTypical columnar columnar jointed jointed rock rock masses: masses: (a) Devil’s (a) Devil’s Postpile, Postpile, Yosemite Yosemite in California; in California; (b) Fingal’s (b) Cave,Fingal’s Staffa Cave, in Staffa Scotland; in Scotland; and (c) Giant’s and (c) Causeway, Giant’s Causeway, Antrim in Antrim Northern in Northern Ireland. Ireland.

Although the characteristics of columnar jointed rock is typical, they they vary vary for for different different locations. locations. The column in in some some areas areas is is regular regular and and straight straight like like Giant’s Giant’s Causeway, Causeway, while while some some other other places places like likeBaihetan Baihetan Dam, Dam, it is it irregular. is irregular. Besides, Besides, the the diamet diametersers of ofcolumns columns vary vary from from meters meters to to centimeters. centimeters. Columns are usually parallel and straight, and and the length can can be as much as several meters. Furthermore, the number number of of sides sides of of an an individual individual column column also also varies varies from from three three to toeight. eight. In Inorder order to togive give a more a more specific speciﬁc illustration illustration of the of the columns, columns, a colorized a colorized O’Reilly O’Reilly map map is presented is presented in Figure in Figure 2. In2. this figure, hexagons are green, heptagons are blue, octagons are purple, pentagons are orange, and

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Symmetry 2018, 10, x FOR PEER REVIEW 3 of 15 In this ﬁgure, hexagons are green, heptagons are blue, octagons are purple, pentagons are orange, and Symmetry 2018, 10, x FOR PEER REVIEW 3 of 15 squaressquares are are red. red. From From the the introduction introduction of of columnarcolumnar jointed rock rock mass mass above, above, a adiagrammatic diagrammatic drawing drawing is proposed in Figure3. It is appropriate to model columnar joints using Voronoi diagrams. is proposedsquares are in red. Figure From 3. the It isintroduction appropriate of tocolumnar model columnarjointed rock joints mass using above, Voronoi a diagrammatic diagrams. drawing is proposed in Figure 3. It is appropriate to model columnar joints using Voronoi diagrams.

FigureFigureFigure 2. 2.A 2.A colorizedAcolorized colorized map map of of about about about 200200 200 columnscolumns columns atat thethe at Giant’s theGiant’s Giant’s Causeway Causeway Causeway made made by madeby O’Reilly O’Reilly by in O’Reilly 1879in 1879 in 1879[1,21].[1,21]. [1,21 ].

(a) (b) (a) (b) FigureFigure 3. 3.Diagrammatic Diagrammatic drawing ofof the the joints joints in columnar in columnar basalt: ( basalt:a) vertical (a columnar) vertical joint, columnar and (b) joint, Figurehorizontal 3. Diagrammatic transverse joint. drawing of the joints in columnar basalt: (a) vertical columnar joint, and (b) and (b) horizontal transverse joint. horizontal transverse joint. 3. Constrained3. Constrained Centroidal Centroidal Voronoi Voronoi and and Implementation Implementation Method Method 3. Constrained Centroidal Voronoi and Implementation Method 3.1. Voronoi3.1. Voronoi Diagram Diagram Algorithm Algorithm and and Its Its Constraints Constraints 3.1. Voronoi Diagram Algorithm and Its Constraints 3.1.1.3.1.1. Classical Classical Voronoi Voronoi Tessellation Tessellation 3.1.1. ClassicalA Voronoi Voronoi diagram Tessellation is a classical domain partitioning method named after Georgy Voronoi, a A Voronoi diagram is a classical domain partitioning method named after Georgy Voronoi, a Russian mathematician. Voronoi generation is the dual algorithm of Delaunay triangulation. A RussianA mathematician. Voronoi diagram Voronoi is a classical generation domain is the partitioning dual algorithm method of Delaunaynamed after triangulation. Georgy Voronoi, A typical a typical Voronoi figure is generated using the perpendicular bisector of the lines composed by a set of VoronoiRussian ﬁgure mathematician. is generated Voronoi using the generation perpendicular is the bisectordual algorithm of the lines of Delaunay composed triangulation. by a set of points A typicalpoints Voronoi called seeds.figure Theis generated Voronoi usingdiagram the is perpendicular a topic in computational bisector of the geometry lines composed and has alreadyby a set of called seeds. The Voronoi diagram is a topic in computational geometry and has already widely pointswidely called applied seeds. in some The otherVoronoi research diagram areas islike a hydrologytopic in computational and crystal mechanics. geometry and has already applied inAs some shown other in Figure research 4, a areasVoronoi like diagram hydrology with and10 random crystal seeds mechanics. Pi is generated. It can be seen widely applied in some other research areas like hydrology and crystal mechanics. Asthat shown the domain in Figure is divided4, a Voronoi into 10 patches diagram and with each 10patch random has a single seeds seed.Pi is For generated. the generation It can of be a seen As shown in Figure 4, a Voronoi diagram with 10 random seeds Pi 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 that the domain is divided into 10 patches and each patch has a single seed. For the generation of a

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VoronoiSymmetry diagram, 2018, 10, x FORa Delaunay PEER REVIEW triangulation is first implemented. Then the perpendicular bisector4 of 15 is plotted to partition the domain into different Voronoi cells. With the development of computational Voronoi diagram, a Delaunay triangulation is first implemented. Then the perpendicular bisector is geometry,Voronoi diagram,Voronoi a tessellationDelaunay triangulation is included is firstin implemented.many software Then and the perpendicularpackages like bisector MATLAB, is plotted to partition the domain into different Voronoi cells. With the development of computational Mathematica,plotted to partition and SciPy. the domain However, into differentthere are Voronoi two obvious cells. With shortcomings the development of the of classicalcomputational Voronoi geometry, Voronoi tessellation is included in many software and packages like MATLAB, Mathematica, diagram:geometry, Voronoi tessellation is included in many software and packages like MATLAB, andMathematica, SciPy. However, and SciPy. there areHowever, two obvious there are shortcomings two obvious of theshortcomings classical Voronoi of the classical diagram: Voronoi (a) The Voronoi cell is not closed. For the Voronoi diagram in Figure 4, only one cell is closed and (a)diagram: The Voronoi cell is not closed. For the Voronoi diagram in Figure4, only one cell is closed and the the other nine cells are open. This brings inconvenience for the analysis. (a) otherThe nineVoronoi cells cell are is open. not closed. This bringsFor the inconvenience Voronoi diagram for in the Figure analysis. 4, only one cell is closed and (b) The shape of the Voronoi cell is random and it is very hard to generate a Voronoi diagram with (b) Thethe shape other ofnine the cells Voronoi are open. cell isThis random bringsand inconvenience it is very hard for the to generateanalysis. a Voronoi diagram with a a specified statistical distribution. (b) specifiedThe shape statistical of the Voronoi distribution. 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 3.1.2. Constrained Voronoi Diagram Generation 3.1.2.To Constrainedovercome the Voronoi first drawback Diagram ofGeneration classical Voronoi tessellation, a constrained Voronoi diagram To overcome the first drawback of classical Voronoi tessellation, a constrained Voronoi diagram algorithmTo overcome is employed the firstto generate drawback a ofclos classicaled Voronoi Voronoi tessellation tessellation, with a constrained seed points Voronoi Pi and diagram domain D. algorithm is employed to generate a closed Voronoi tessellation with seed points Pi and domain D. Takingalgorithm a model is employed with 16 topoints generate (xi, yai )clos anded a Voronoisquare domaintessellation in Figurewith seed 5a aspoints example, Pi and thisdomain algorithm D. Taking a model with 16 points (xi, yi) and a square domain in Figure5a as example, this algorithm can canTaking be described a model as with follows 16 points [22]: (xi, yi) and a square domain in Figure 5a as example, this algorithm be described as follows [22]: can be described as follows [22]: (a) A Voronoi diagram is generated using a classical tessellation method (Figure 5b). (a) A Voronoi diagram is generated using a classical tessellation method (Figure5b). (b)(a) All A the Voronoi open diagramcells with is agenerated vertex outside using a the classical domain tessellation D are identified method (Figure(shown 5b). in Figure 5c). (b) All the open cells with a vertex outside the domain D are identified (shown in Figure5c). (c)(b) A Allset ofthe new open seeds cells symmetricwith a vertex to outside the seeds the of domain open cells D are with identified respect (shown to the indomain Figure boundary 5c). are (c) A set of new seeds symmetric to the seeds of open cells with respect to the domain boundary are (c) createdA set of(Figure new seeds 5d). symmetric to the seeds of open cells with respect to the domain boundary are created (Figure5d). (d) Newcreated Voronoi (Figure diagram 5d). is generated with a classical tessellation (Figure 5e). (d)(d) NewNew Voronoi Voronoi diagram diagram is is generatedgenerated with a a classical classical tessellation tessellation (Figure (Figure 5e).5e). (e) By removing the open cells as well as the related seeds, the final Voronoi diagram is shown in (e)(e)By By removing removing the the openopen cellscells as well as as the the rela relatedted seeds, seeds, the the final final Voronoi Voronoi diagram diagram is shown is shown in in Figure 5f. FigureFigure5f. 5f.

((aa)) (b()b ) Figure 4. An illustration of Voronoi tessellation with 10 generators: (a) a set of generators, and (b) Figure 4. An illustration of Voronoi tessellationtessellation withwith 1010 generators:generators: ((aa)) aa set set of of generators, generators, and and (b) ( theb) the resulting Voronoi diagram. resultingthe resulting Voronoi Voronoi diagram. diagram.

(a) (b) (c) (a) (b) (c) Figure 5. Cont.

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(d) (e) (f) (d) (e) (f) Figure 5. Illustration of the stages of the constrained Voronoi tessellation algorithm: (a) initial seeds, (Figureb)Figure classical 5. 5.Illustration Illustration Voronoi of tessellation,of the the stages stages ofof(c the)the open constrainedcons Voronoitrained Voronoi Voronoicells, (tessellationd tessellation) symmetry algorithm: algorithm: seeds, ((ae)) (initial anew) initial seeds,Voronoi seeds, tessellation,(b)( classicalb) classical Voronoi(f) constrainedVoronoi tessellation, tessellation, Voronoi (c) open (diagram.c) open Voronoi Voronoi cells, (cells,d) symmetry (d) symmetry seeds, (seeds,e) new ( Voronoie) new tessellation,Voronoi (f)tessellation, constrained (f Voronoi) constrained diagram. Voronoi diagram. 3.2. Centroidal Voronoi Algorithm 3.2.3.2. Centroidal Centroidal Voronoi Voronoi Algorithm Algorithm 3.2.1. Random and Centroidal Voronoi Diagram 3.2.1.3.2.1. Random Random and and Centroidal Centroidal Voronoi Voronoi Diagram Diagram The Voronoi diagram created using a constrained Voronoi tessellation method with randomly TheThe Voronoi Voronoi diagram diagram created created usingusing aa constrainedconstrained Voronoi Voronoi tessellation tessellation method method with with randomly randomly generated seeds is shown in Figure 6a. It can be seen that the Voronoi cell is irregular and there is a generatedgenerated seeds seeds is is shown shown in in Figure Figure6 6a.a. ItIt cancan bebe seen that the the Voronoi Voronoi cell cell is is irregular irregular and and there there is a is a clear deviation between the Voronoi seeds and the centroids. In order to obtain more regular clearclear deviation deviation between between the the Voronoi Voronoi seeds seeds and theand centroids.the centroids. In order In order to obtain to obtain more regularmore regular columnar columnar as shown in Figure 2, an algorithm for centroidal Voronoi diagrams (Figure 6b) is proposed as showncolumnar in as Figure shown2, anin Figure algorithm 2, an foralgorithm centroidal for centroidal Voronoi diagramsVoronoi diagrams (Figure6 (Figureb) is proposed 6b) is proposed to tackle to tackle this problem. thisto problem. tackle this problem.

(a) (b) (a) (b) Figure 6. Random and centroidal Voronoi diagram: generators (white dots) and centroids (black Figuredots): 6. ( aRandomRandom) random and andVoronoi centroidal centroidal diagram, Voronoi Voronoi and (b diagram:) centroidaldiagram: generators Voronoigenerators (whitediagram. (white dots) dots) and centroidsand centroids (black (black dots): dots):(a) random (a) random Voronoi Voronoi diagram, diagram, and (b )and centroidal (b) centroidal Voronoi Voronoi diagram. diagram. Compared with the random Voronoi diagram, the seed of a cell is coincident with its centroid in centroidalCompared Voronoi with the tessellation random (CVT).Voronoi It diagram,is widely theused seed in related of a cell fields is coincident like mesh with generation its centroid and in centroidaldata compression. VoronoiVoronoi tessellationtessellation Previous study (CVT). (CVT). has It It isindicated is widely widely usedth atused some in relatedin naturalrelated ﬁelds patternsfields like like mesh like mesh Giant’s generation generation Causeway and dataand datacompression.can compression. be represented Previous Previous by studythe centro study hasidal indicated has Voronoi indicated that tessellation some that naturalsome [5]. natural patterns patterns like Giant’s like Giant’s Causeway Causeway can be canrepresented be representedThere by are the a number centroidalby the centroof methods Voronoiidal Voronoi for tessellation the generation tessellation [5]. of centroidal[5]. Voronoi diagram, among which areThere the algorithms are a number by ofLloyd methods and by for for MacQueen the the generation generation [23,24], of of centroidalwhich centroidal are Voronoiwidely Voronoi used, diagram, and someamong other which aremethods thethe algorithmsalgorithms are variations by by Lloyd Lloyd of andthese and by two MacQueenby methods.MacQueen [ 23Co,24 nsidering[23,24],], which which that are widelyMacQueen’s are widely used, algorithm and used, some and needs other some methodstwice other methodsareas variations many are Monte variations of these Carlo two ofsimulations these methods. two in methods. Considering each iterat Coion,nsidering that which MacQueen’s consumesthat MacQueen’s algorithm a large amount algorithm needs computation twice needs as manytwice asMonte manytime, Carlo Lloyd’s Monte simulations centroidal Carlo simulations inmethod each iteration, is employedin each which iterat for modelling consumesion, which columnar a consumes large amount jointed a large computation rock amount mass in thiscomputation time, study. Lloyd’s time,centroidal Lloyd’s method centroidal is employed method foris employed modelling for columnar modelling jointed columnar rock mass jointed in thisrock study. mass in this study.

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3.2.2. Lloyd’s Algorithm In this work, Lloyd’s algorithm, named after Stuart P. Lloyd, is employed. It is an iteration method to partition the domain into well-shaped and uniformly-sized convex cells [25]. The convergence of Lloyd’s algorithm to a centroidal Voronoi diagram has been proven. The random Voronoi diagram can be iterated to a centroidal Voronoi diagram using Lloyd’s algorithm and the algorithm can be simpliﬁed as follows:

(1) For an initial seeds yi, generate a Voronoi diagram using constrained Voronoi tessellation; (2) Compute the centroid zi of the Voronoi diagram of yi; (3) Move the generating point yi to its centroid zi; (4) Repeat Steps 1 to 3 until all generating points converge to the centroids.

3.2.3. Estimation of the Centroid The centroidal Voronoi algorithm is very simple but the calculation of the centroid of each Voronoi cell is time-consuming. In each iteration of Lloyd’s method, the centroidal positions of all shapes of the Voronoi diagram are calculated. Thus, the estimation of the centroid is the most time-consuming task. For determining the centroid of a Voronoi region, a simple formula is: R V χρ(χ)dVi z = i (1) i R ρ(χ)dV Vi i where χ is the position, Vi is the region area, and ρ(χ) is the density function with ρ(χ) = 1 being the default. In order to calculate the position of centroid fast, two efﬁcient methods for evaluating the integrals are used in this work. One is based on an integration algorithm on triangle partitions and the other is a sampling method. An integration algorithm can calculate the coordinates of centroids exactly but not as fast as a sampling method; whereas a sampling method is fast but the results are not as accurate as that of an integration algorithm. (1) Integration method on triangle partitions For an arbitrary polygon with n vertices, it can be divided into n − 2 triangles with a clockwise or anti-clockwise direction (Figure7). For each triangle with vertex Ai (xi, yi)(i = 1, 2, 3), the coordinates of centroid are given by: xg = (x1 + x2 + x3)/3 (2)

yg = (y1 + y2 + y3)/3 (3) The area of the triangle is calculated as:

S = ((x2 − x1) × (y3 − y1) − (x3 − x1) × (y2 − y1))/2 (4)

Considering that the polygon can be discretized into n − 2 triangles and the centroid and area of each triangle can be expressed as Gi(xgi, ygi) and Si, the coordinates of the centroid of the polygon can be obtained as: n−2 ∑ xgiSi = X = i 1 (5) g n−2 ∑ Si i=1 Symmetry 2018, 10, x FOR PEER REVIEW 7 of 15

Symmetry 2018, 10, 618 n−2 7 of 15  ygiiS = i=1 Yg n−2n−2 (6) ∑ ygiSSi i=1 i Y = i=1 (6) g n−2 ∑ Si i=1

(a) (b)

FigureFigure 7.7. Discretization ofof n-polygon:n-polygon: (a)) polygonpolygon withwith nn vertices,vertices, andand ((bb)) nn −− 22 discrete discrete triangles. triangles.

(2) Sampling method Although thethe integration integration on on triangle triangle partitions partitions is a goodis a good solution solution for the for estimation the estimation of the centroid of the centroidof a polygon, of a polygon, it takes a longit takes time a tolong complete time to eachcomplete step ineach Lloyd’s step iteration.in Lloyd’s For iteration. a quick For calculation, a quick calculation,a simple and a efﬁcientsimple and sampling efficient method sampling is proposed method asis proposed an alternative. as an alternative. For a certain area with different Voronoi tessellations,tessellations, a set of random points are distributed in this area (Figure 8). For each Voronoi cell, if there are n points Pi(xi, yi) in this area, then the centroid this area (Figure8). For each Voronoi cell, if there are n points Pi(xi, yi) in this area, then the centroid of ofthis this cell cell can can be be approximated approximated as theas the average average of theseof these points: points: n n Xxn= / (7) Xg gi= xi/n (7) ∑= i=i1 1

nn Y = y n Yyng =∑ i/ / (8) gii=1 i=1 In this way, the determination of centroids changes to the partition of areas into different Voronoi In this way, the determination of centroids changes to the partition of areas into different cells, i.e., the “N-D nearest point search” problem. Luckily, the nearest point search problem has Voronoi cells, i.e., the “N-D nearest point search” problem. Luckily, the nearest point search problem a mature mathematical solution [26] and this algorithm has been written in a number of software has a mature mathematical solution [26] and this algorithm has been written in a number of software libraries, such as Python and MATLAB. Using these libraries, all the random points can be divided libraries, such as Python and MATLAB. Using these libraries, all the random points can be divided efﬁciently and the centroid of Voronoi diagram can be computed quickly. With more sampling points, efficiently and the centroid of Voronoi diagram can be computed quickly. With more sampling the calculation of centroid will be more accurate. If the number of random points is selected properly, points, the calculation of centroid will be more accurate. If the number of random points is selected this algorithm can be both fast and relatively accurate. properly, this algorithm can be both fast and relatively accurate.

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(a) (b)

FigureFigure 8.8.Illustration Illustration of of the the sampling sampling method: method: (a) Voronoi(a) Voronoi tessellation, tessellation, and (b and) random (b) random sampling sampling points. points. 3.3. Numerical Implementation and Discussion 3.3. NumericalIn the iterations Implementation of Lloyd’s and algorithm, Discussion the Voronoi generators and centroids are known, and energy n can beIn employed the iterations to describe of Lloyd’s the Voronoi algorithm, diagram the [Vo23].ronoi For thegenerators Voronoi tessellationand centroids{Ω iare}i= 1known,, the energy and is defined as: n n {Ω } energy can be employed to describe the VoroZ noi diagram [23]. For the Voronoi tessellation i i=1 , E = ρ(χ)k χ − χ k2dσ (9) the energy is defined as: ∑ g i=1 Ωi where χ is the position of the Voronoi generator,n χ represents the position of the centroid, and ρ(χ) = 1 =−ρχg() χ χ 2 σ Ed||g || (9) is the density function with ρ(χ) = 1 being theΩi default. i=1 However, if the generator of Voronoi diagram is not known beforehand, energy cannot be χ χ estimated.where For is the example, position it isof notthe easyVoronoi to calculate generator, the energyg represents of Voronoi the cells position in Figure of the2. Indirectly, centroid, theand coefficent of variation, which has been proven as an effective parameter in the estimation of Voronoi ρχ() is the density function with ρχ()=1 being the default. diagram [27,28], is employed for the description of the properties of Voronoi cells. In the process of However, if the generator of Voronoi diagram is not known beforehand, energy cannot be smoothing energy from a randomly-generated Voronoi diagram to a centroidal Voronoi diagram, the estimated. For example, it is not easy to calculate the energy of Voronoi cells in Figure 2. Indirectly, coefficient of variation value is recorded with the following formula: the coefficent of variation, which has been proven as an effective parameter in the estimation of Voronoi diagram [27,28], is employed for the descriSDption of the properties of Voronoi cells. In the CV = (10) process of smoothing energy from a randomly-genermated Voronoi diagram to a centroidal Voronoi diagram, the coefficient of variation value is recorded with the following formula: where SD is the standard deviation and m is the mean of Voronoi cell areas. The coefficient of variation gives a quantitative indication on the spatial distribution:SD the higher the coefficient of variation, the CV = (10) higher the tendency of cells to aggregate into clusters. m Based on the related algorithms described, numerical implementation is done in MATLAB, inwhere which SD some is the good standard programming deviation techniquesand m is the from mean the of open-source Voronoi cell program areas. The of Burkardtcoefficient are of referencedvariation gives [29]. Thea quantitative 50 seeds for indication Voronoi diagramon the sp areatial generated distribution: randomly. the higher After the 50 iterations,coefficient the of pointsvariation, are well-spacedthe higher the and tendency Lloyd’s algorithmof cells to aggregate for computing into centroidalclusters. Voronoi tessellation converged overallBased (Figure on 9the). Inrelated this way,algorithms the coefficient described, of variationnumerical has implementation the same evolution is done trend in MATLAB, as energy in (E),which and some it is reasonablegood programming to use coefficient techniques of variation from the to describeopen-sou therce distribution program of property Burkardt of theare Voronoireferenced tessellation. [29]. The 50 seeds for Voronoi diagram are generated randomly. After 50 iterations, the points are well-spaced and Lloyd’s algorithm for computing centroidal Voronoi tessellation converged overall (Figure 9). In this way, the coefficient of variation has the same evolution trend as energy (E), and it is reasonable to use coefficient of variation to describe the distribution property of the Voronoi tessellation.

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(a) (b) (c)

-4.9 -0.5 -5 -1 -5.1

-5.2 -1.5 -5.3

-5.4 Log(CV) -2 Log(Energy)

-5.5 -2.5 -5.6

-5.7 -3 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 Step Step

(d) (e) (f)

FigureFigure 9. Implementation 9. Implementation of theof the constrained constraine centroidald centroidal Voronoi Voronoi algorithm: algorithm: ((aa)) initialinitial condition, ( (bb) ) step 5, (c)step step 5, 20, (c) (stepd) step 20, ( 50,d) step (e) change50, (e) change of energy, of energy, and (andf) change (f) change of coefﬁcient of coefficient of of variation. variation.

4. Modeling4. Modeling of Columnarof Columnar Jointed Jointed Rock Rock MassMass AnalysisAnalysis of typicalof typical shapes shapes of of columnar columnar jointingjointing shows shows that that a columnar a columnar jointed jointed rock rock mass mass can be can be modelled through the extrusion of a 2-D Voronoi diagram. Hence, the generation of 2-D columnar modelled through the extrusion of a 2-D Voronoi diagram. Hence, the generation of 2-D columnar jointing, which is consistent with the site condition is a key task. In Section 3, it is shown that jointing, which is consistent with the site condition is a key task. In Section3, it is shown that coefficient of variation can describe the distribution property in the iterations of Lloyd’s algorithm. coefficientThe centroidal of variation Voronoi can diagram describe is theapplicable distribution in the propertydescription in of the a natural iterations phenomenon of Lloyd’s such algorithm. as The centroidalGiant’s Causeway. Voronoi In diagramfact, natural is applicablecolumnar jointing in the is description not an exactly of acentroidal natural phenomenonVoronoi diagram, such as Giant’sand Causeway. it lies between In fact, random natural and columnar centroidal jointing Voronoi is notdiagrams. an exactly In the centroidal process of Voronoi transforming diagram, a and it liesVoronoi between diagram random from and totally centroidal random Voronoito centroidal, diagrams. the value In of the CV process indicates of that transforming the Voronoi acells Voronoi diagramchange from from totally heterogeneous random to to centroidal, homogeneous. the value Therefore, of CV it indicatesis feasible that to describe the Voronoi the characteristics cells change from heterogeneousof columnar to joints homogeneous. by employing Therefore, two main parame it is feasibleters: the to columnar describe density the characteristics representing the of columnarscale jointsand by coefficient employing of twovariation main reflecting parameters: the variation the columnar property. density representing the scale and coefficient A procedure for the generation of columnar jointed rock mass is proposed (Figure 10). Based on of variation reflecting the variation property. field investigation of geological conditions and site images, the columnar jointing at the site of interest isA characterized procedure for by thesix parameters: generation columnar of columnar density jointed (CD), rock coefficient mass isof proposedvariation (CV), (Figure dip direction10). Based on field(DD), investigation dip angle of (DIP), geological transverse conditions joint distance and site (TD), images, and probability the columnar (TP). jointing The number at the of site random of interest is characterizedseeds for generating by six parameters: a Voronoi diagram columnar is estimated density (CD),according coefficient to the calculated of variation density. (CV), Based dip direction on (DD),the dip idea angle of the (DIP), bisection transverse method, jointa novel distance modified (TD), constrained and probability centroidal (TP).Voronoi The smooth number algorithm of random seedsisfor implemented generating to a make Voronoi the CV diagram conver isge estimatedto the specified according value as to follows: the calculated density. Based on the idea(1) of theFor bisection a Voronoi method, diagram awith novel CV modifiedlarger than constrained the specified centroidalvalue, calculate Voronoi the centroid smooth PC algorithm. is implemented(2) The new to make generator the CVPnew,g converge is set at tothe the midpoint specified of the value old as generator follows: Pold,g and the centroid PC. Calculate the coefficient of variation CV of the new Voronoi diagram. P (1) (3)For Repeat a Voronoi Steps diagram 1 and 2 withto obtain CV largerthe generator than the unti specifiedl the coefficient value, calculateof variation the CV centroid convergesC .to (2) Thethe new specified generator valueP new,gwith ais prescribed set at the accuracy. midpoint of the old generator Pold,g and the centroid PC. 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. Symmetry 2018, 10, 618 10 of 15 Symmetry 2018, 10, x FOR PEER REVIEW 10 of 15 Symmetry 2018, 10, x FOR PEER REVIEW 10 of 15 Having obtained a 2-D Voronoi diagram with specified CD and CV, it can be extruded to a 3-D HavingHaving obtained obtained a a 2-D 2-D Voronoi Voronoi diagram diagram with with specified specified CD CD and and CV, CV, it it can can be be extruded extruded to to a a 3-D 3-D columnar shape in accordance with DD and DIP. Then for each extruding Voronoi cell and its length, columnarcolumnar shape shape in in accordance accordance with with DD DD and and DIP. DIP. Then Then for for each each extruding extruding Voronoi Voronoi cell cell and and its its length, length, a transverse joint can be generated by the determination of the intersection between the extruding aa transverse transverse joint joint can can be be generated generated by by the the determination determination of of the the intersection intersection between between the the extruding extruding Voronoi cell and the transverse plane. Sometimes, the transverse joint is non-penetrating, and the VoronoiVoronoi cell cell and and the the transverse transverse plane. plane. Sometimes, Sometimes, the the transverse transverse joint joint is is non-penetrating, non-penetrating, and and the the transverse joint can be selected with the specified TD and TP. In this way, columnar jointed rock mass transversetransverse joint joint can can be be selected selected with with the the specified specified TD TD and and TP. TP. In In this this way, way, columnar columnar jointed jointed rock rock mass mass with the site properties can be generated. withwith the the site site properties properties can can be be generated. generated.

Figure 10. Procedure for modelling columnar joints. Figure 10. Procedure for modelling columnar joints. Figure 10. Procedure for modelling columnar joints. 5. Columnar Joints Generation: A Case Study of the Baihetan Hydropower Station 5.5. Columnar Columnar Joints Joints Generation: Generation: A A Case Case Study Study of of the the Baihetan Baihetan Hydropower Hydropower Station Station 5.1. Engineering Geological Investigation 5.1.5.1. Engineering Engineering Geological Geological Investigation Investigation The Baihetan Hydropower Station is a multi-purpose project for harnessing the Yangtze River, TheThe Baihetan Baihetan Hydropower Hydropower Station Station is is a a multi-purpose multi-purpose project project for for harnessing harnessing the the Yangtze Yangtze River, River, developed mainly for power generation, flood control, sediment flushing, and improving the developeddeveloped mainly mainly for for power power generation, generation, ﬂood flood control, control, sediment sediment ﬂushing, flushing, and and improving improving the the navigation conditions in the reservoir and downstream. The Baihetan Hydropower Station is located navigationnavigation conditions conditions in in the the reservoir reservoir and and downstream. downstream. The The Baihetan Baihetan Hydropower Hydropower Station Station is is located located on the downstream reaches of the Jinsha River. At the dam site, the most apparent stratigraphic onon the the downstream downstream reaches reaches of of the the Jinsha Jinsha River. River. At At the the dam dam site, site, the the most most apparent apparent stratigraphic stratigraphic lithology is basalt that belongs to the Emeishan (Emei Mt.) formation of the Permain System (P2β). In lithologylithology is is basalt basalt that that belongs belongs to to the the EmeishanEmeishan (Emei(Emei Mt.)Mt.) formation of of the the Permain Permain System System (P (P2β).β ).In the construction of the plant’s hydraulic structures, columnar jointed basalt is found to be widely2 Inthe the construction construction of of the the plant’s plant’s hydraulic hydraulic structures,structures, columnarcolumnar jointedjointed basaltbasalt isisfound found to to be be widely widely distributed at the arch dam foundation, underground caverns, abutment slopes, and other water distributeddistributed at at the the arch arch dam dam foundation, foundation, underground underground caverns, caverns, abutment abutment slopes, slopes, and and other other water water conservancy tunnels (Figure 11). conservancyconservancy tunnels tunnels (Figure (Figure 11 11).).

(a) (b) (a) (b) Figure 11. Site conditions of the Baihetan Hydropower Station: ( a) construction site, and (b) typical Figure 11. Site conditions of the Baihetan Hydropower Station: (a) construction site, and (b) typical columnar jointed basalt. columnar jointed basalt.

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A typical figure of columnar jointed basalt is drawn by the Huadong Engineering Cooperation LimitedA typical (ECIDI), figure as shown of columnar in Figure jointed 12 with basalt relate is drawnd parameters by the Huadong listed in Table Engineering 1. The orientation Cooperation of Limitedcolumnar (ECIDI), is regular as shownwith Dip in Figure= 72° and 12 withDD = related145°, and parameters the length listed of columnar in Table 1is. Theabout orientation 1.5 m. Based of ◦ ◦ columnaron the skeleton is regular figure, with a Dipdigital = 72 imageand DDprocessing = 145 , andtechnique the length is employed of columnar in the is about analysis. 1.5 m. The Based area on of theeach skeleton columnar figure, is calculated a digital and image the processingcoefficient of technique variation is is employed obtained. inUsing the analysis.a sample of The 10 areatypical of eachcolumnar columnar basalt is drawings, calculated andthe statistical the coefficient mean of of variation coefficient is obtained. of variation Using was a sampleabout 44.18% of 10 typical and it columnarimplied that basalt thedrawings, discreteness the of statistical columnar mean basalt of coefficientwas quite oflarge. variation Furthermore, was about the 44.18% density and of itcolumnar implied cells that could the discreteness also be obtained of columnar from skeleton basalt drawings, was quite which large. gave Furthermore, a density of the about density 20 in of a columnar1 m × 1 m cellssquare. could also be obtained from skeleton drawings, which gave a density of about 20 in a 1 m × 1 m square.

(a) (b)

FigureFigure 12.12. TypicalP P22β3 columnarcolumnar basalt basalt at Baihetan Baihetan Hydropower Station: (a) geological photo,photo, andand (b)) jointsjoints skeleton.skeleton.

Table 1. Parameters of columnar jointed rock mass in dam foundation of Baihetan Hydropower Station. Table 1. Parameters of columnar jointed rock mass in dam foundation of Baihetan Hydropower Station. Parameter Dip DD CD CV TD TP Parameter ValuesDip 72◦ DD145 ◦ 25/m CD 2 44.18% CV 1.5 m TD 0.3 TP Values 72° 145° 25/m2 44.18% 1.5 m 0.3 5.2. Columnar Jointing Model Generation 5.2. Columnar Jointing Model Generation Considering the scale effect of heterogeneous materials, a proper size has to be selected. For convenience,Considering the scale size ofeffect the numericalof heterogeneous model mate wasrials, taken a as proper 1 m ×size1 m.has Into orderbe selected. to ensure For thatconvenience, the 3-D model the size could of the be generated,numerical anmodel initial was area take ofn 5 as m 1× m5 × m 1 wasm. In chosen order forto ensure the generation that the of3-D a 2-Dmodel Voronoi could diagram. be generated, From an the initial density area of of site 5 m conditions, × 5 m was itchosen was estimated for the generation that about of 500 a 2-D seeds Voronoi were needed.diagram. A From random the Voronoi density diagramof site conditions, was generated it was in Figureestimated 13a that with about a CV of500 56.45%. seeds Withwere aneeded. speciﬁed A CVrandom value Voronoi of 44.18%, diagram a Voronoi was diagramgenerated could in Figure be generated 13a with (Figure a CV 13of b)56.45%. by applying With a the specified algorithm CV proposedvalue of 44.18%, in Section a 4Voronoi. Then an diagram extrusion could at a certain be gene directionrated (Figure with parameter 13b) by applying DIP and DDthe wasalgorithm made toproposed extend the in Section rock from 4. Then 2-D to an 3-D extrusion (Figure at13 ac). certain Afterwards, direction horizontal with parameter hidden joints DIP and was DD implemented was made toto cutextend the rock the (Figurerock from 13d). 2-D Using to these3-D (Figure operations, 13c). the Afterwards, coordinates horizontal information hidden of all rockjoints blocks was couldimplemented be calculated. to cut the By rock trimming (Figure the 13d). solid Using with th aese certain operations, boundary, the coordinates the block information information of of the all columnarrock blocks jointed could rock be calculated. is shown inBy Figure trimming 13e. the It was solid easier with for a certain the particle boundary, model the in block Figure information 13f, where theof the model columnar could jointed be generated rock is byshown importing in Figure the 13e. DFN It (discretewas easier fracture for the networks) particle model in Figure in Figure 13d to13f, a cubicwhere particle the model model, could which be generated could be doneby importing easily in the DEM DFN packages (discrete like fracture PFC (Particle networks) Flow in Code),Figure YADE13d to (Yeta cubic another particle dynamic model, engine), which andcould LIGGGHTS be done easily (LAMMPS in DEM improved packages for like general PFC (Particle granular Flow and granularCode), YADE heat transfer (Yet another simulations). dynamic engine), and LIGGGHTS (LAMMPS improved for general granular and granular heat transfer simulations).

SymmetrySymmetry2018 2018,,10 10,, 618 x FOR PEER REVIEW 1212 ofof 1515

(a) (b)

(e) (f)

FigureFigure 13.13. NumericalNumerical specimen generation: ( a)) Voronoi diagramdiagram withwith CVCV == 56.45%,56.45%, ((bb)) VoronoiVoronoi diagramdiagram withwith CV CV = = 44.18%, 44.18%, (c )(c extrude) extrude 2-D 2-D Voronoi Voronoi diagram diagram with with direction, direction, (d) cut (d) columnar cut columnar rock withrock transversewith transverse joint, (joint,e) block (e) block model model of columnar of columnar rock, androck, (f and) particle (f) particle model model of columnar of columnar rock. rock.

ColumnarColumnar jointedjointed rockrock massmass withwith differentdifferent CVCV isis shownshown inin FigureFigure 1414.. ItIt cancan bebe seenseen thatthat thethe volumevolume ofof columnarcolumnar becamebecame moremore andand moremore uniformuniform withwith thethe decreasedecrease ofof coefﬁcientcoefficient ofof variation.variation. ColumnarColumnar jointedjointed rockrock massesmasses withwith differentdifferent directionsdirections are are shown shown in in FigureFigure 15 15.. TheThe resultresult showsshows thatthat thethe directiondirection waswas alsoalso anan importantimportant parameterparameter affectingaffecting thethe morphologymorphology ofof columnarcolumnar jointedjointed rockrock masses. From all these cases, it can be concluded that this method is applicable and effective in the

Symmetry 20182018,, 1010,, 618x FOR PEER REVIEW 13 of 15 Symmetry 2018, 10, x FOR PEER REVIEW 13 of 15 modeling of columnar jointed rock mass with complex structures, which is extremely helpful in the masses. From all these cases, it can be concluded that this method is applicable and effective in the modelinganalysis of of physical columnar and jointed mechanical rock mass properties with complex of such materials.structures, which is extremely helpful in the analysismodeling of of physical columnar and jointed mechanical rock massproperties with complexof such materials. structures, which is extremely helpful in the analysis of physical and mechanical properties of such materials.

(a) (b) (c) (d) (a) (b) (c) (d)

(e) (f) (g) (h) (e) (f) (g) (h) Figure 14. Columnar joints with different coefficient of variation: (a) block model with CV = 40%, (b) Figureblock model 14. ColumnarColumnar with CV joints= joints 30%, with ( withc) block different different model coefficient coefﬁcient with CV of= of20%, variation: variation: (d) block (a)( a blockmodel) block model with model Cwith = with10%, CV CV(=e )40%, particle = 40%, (b) (blockmodelb) block model with model CVwith = with CV40%, = CV 30%,(f) particle = ( 30%,c) block (modelc) model block with with model CV CV = with 30%,= 20%, CV (g ()d =particle) 20%,block (model dmodel) block with with model C CV = 10%, with= 20%, (e C) particleand = 10%, (h) (modelparticlee) particle with model model CV with = with40%, CV CV( =f) 10%.particle = 40%, ( fmodel) particle with model CV = with 30%, CV (g) = particle 30%, (g )model particle with model CV with= 20%, CV and = 20%, (h) particleand (h) particlemodel with model CV with = 10%. CV = 10%.

(a) (b) (c) (d) (a) (b) (c) (d)

(e) (f) (g) (h) (e) (f) (g) (h) Figure 15.15. Columnar joints with different joint dip angles:angles: (a) blockblock modelmodel withwith dipdip == 00°,◦,( (b) blockblock Figuremodel withwith15. Columnar dipdip == 3030°,◦,( (jointsc) block with model different withwith dipdipjoint == 6060°,dip◦,( (angles:d) block (a model) block with model dipdip with == 90°, 90◦ dip,( (e) =particle 0°, (b) modelblock withmodel dip with = = 0°, 0dip◦ ,((f )f= particle) 30°, particle (c) modelblock model model with with dip with dip = 30°, dip = 30 (=g◦ 60°,),( particleg )(d particle) block model model with with withdip =dip dip 60°, = = and90°, 60◦ (,he and)) particleparticle (h) particle modelmodel withmodel dipdip with == 0°,90°. dip (f ) =particle 90◦. model with dip = 30°, (g) particle model with dip = 60°, and (h) particle model with dip = 90°. 6. Conclusions and Discussion 6. Conclusions and Discussion Aimed atat the the analysis analysis of columnarof columnar jointed jointed rock massrock withmass complex with complex structures, structures, this paper this proposed paper aproposed novelAimed method a novelat the for methodanalysis the generation for of thecolumnar generation of numerical jointed of model.numericalrock mass From model. with the complex resultsFrom the obtained, structures, results theobtained, this following paper the proposedconclusionsfollowing conclusionsa cannovel be method drawn: can be for drawn: the generation of numerical model. From the results obtained, the following conclusions can be drawn: (1) The coefficient of variation is an effective parameter for representing the deviation between the (1) The coefficient of variation is an effective parameter for representing the deviation between the

Symmetry 2018, 10, 618 14 of 15

(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: Q.M contributed in formulating the research question; Y.C., L.Y. and Q.Z. prepared the original manuscript. All authors contributed in addressing reviewers comments during the review process and proofreading of the accepted manuscript. Funding: This research was funded by National Key R&D Program of China (2017YFC1501100), the National Natural Science Foundation of China (Grant Nos. 51709089, 11572110, 51479049, 11772116), and Open fund of state key laboratory of hydraulics and mountain river engineering (Sichuan University: SKHL1725) are gratefully acknowledged. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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