Random generation of natural sand assembly using micro x-raytomography and spherical harmonics

B. ZHOU*,{ and J. WANG{

Particle morphology is an essential characteristic in determining the mechanical properties of naturalsands. Based on micro x-ray computed tomography data, this paper introduces a mathematicalapproach using spherical harmonics (SHs) to characterise and reconstruct particle morphology inthree dimensions. The basic geometric properties of natural sand particles (volume and surface area)and two empirical engineering indices (sphericity and convexity) are the main focus of theinvestigation on validating the efficiency of SH analysis. This approach is shown to be a robusttechnique for reproducing particle morphology in terms of shape irregularity and surface texturewhen the maximum harmonic degree is greater than ten. By using principal component analysis forthe obtained SH descriptors of the scanned particles, a virtual sand assembly, consisting ofstatistically reconstructed particles with random shapes but major morphological features, issuccessfully generated. This approach can be used in efficient discrete element modelling to furtherstudy the micromechanics of natural sands.

KEYWORDS: microscopy; particle-scale behaviour; sands; statistical analysis

ICE Publishing: all rights reserved

NOTATION

aa reconstructed spherical harmonic (SH) coefficientsam

n , a0mn SH coefficients and normalised SH coefficients

a0mn mean of all a0mnM 6 d dimensions of the principal components (PCs)m, n SH order and degreeN(0, 12) standard normal distributionnmax maximum SH degree usedP, l PCs and the corresponding eigenvalues after princi-

pal component analysis (PCA)Pm

n (x) associated Legendre function

r(h, Q) polar radius of the particle surface at h and Qr(h,Q) SH-reconstructed polar radius of the particle surface

at h and QV(x, y, z) Cartesian coordinates of the particle surface verticesxi random real variableY m

n (h,Q) SH seriesy scores of the PCsh, Q polar angle and azimuthal angle of the spherical

coordinatesm, s mean value and standard deviation of the normal

distribution

INTRODUCTIONDiscrete element method (DEM) analysis taking intoaccount particle morphology plays an important role inunderstanding the micro- and macromechanical propertiesof natural sand, which has been of interest to geologicaland geotechnical engineers for many years. In this context,a large number of studies have been devoted to the

DEM-orientated reconstruction of particle morphology.For example, by using overlapping disc/sphere clump logic,DEM-aided particle assemblies have been successfullygenerated with different irregular shapes, including ellipses(Mahmood & Iwashita, 2010), rods (Azema & Radjai,2012), polygons (Zhou et al., 2013) in a two-dimensional(2D) scenario and realistic three-dimensional (3D) shapesbased on x-ray micro computed tomography (mCT)(Katagiri et al., 2010; Cil & Alshibli, 2012, 2014).

When carefully reviewing all of the modelling methods ofa sand assembly, it is important to notice that the conceptof ‘random’ or ‘natural’ particle shape has rarely beenmentioned. Based on mCT information, the one-to-oneDEM-aided particle morphology can be accurately recon-structed using the method of Ferellec & McDowell (2010).However, the limitation that a large amount of repeatedparticles would be generated within a large assembly due tothe finite number of scanned particle templates is obvious.In fact, for a natural sand, the micromorphology of theconstituent particles is totally random and distinct fromeach other while containing some statistical features. Toovercome this limitation, Mollon & Zhao (2013, 2014)proposed a statistical method to generate a virtual packingof particles with realistically complex yet controllableshapes based on the theory of random fields for sphericaltopology combined with a Fourier shape descriptor. Thecore objective of the current study was to reconstruct alarge particle assembly with random shapes, while repre-senting the major morphological characteristics of naturalsands.

The key issue in achieving this objective is to introduce aquantitative and global descriptor that not only charac-terises but also reconstructs the sand particle morphology.A large number of morphological parameters, such asaspect ratio, roundness, sphericity, convexity and fractaldimensions, are commonly used in engineering (Krumbein& Sloss, 1963; Kolay & Kayabali, 2006). As a singleparameter gives only limited information about the particlefeatures, a large set of parameters is always needed to plot arelatively comprehensive figure of the particle morphology.

Manuscript received 29 September 2014; first decision 23October 2014; accepted 9 December 2014.Published online at www.geotechniqueletters.com on 12January 2015.*Department of Civil Engineering and Mechanics, HuazhongUniversity of Science and Technology, Wuhan, China{Department of Architecture and Civil Engineering, CityUniversity of Hong Kong, Kowloon, Hong Kong

Zhou, B. and Wang, J. (2015) Geotechnique Letters 5, 6–11, http://dx.doi.org/10.1680/geolett.14.00082

6

Even so, no matter how many parameters are used as theinput, they are still incapable of reconstructing a realparticle. In order to overcome this limitation, a mathema-tical approach called the spherical harmonic (SH) series isintroduced for the characterisation and reconstruction ofparticle morphology. Recently, this method has beenwidely used in biomedical image analysis (Chung et al.,2008) and computer graphics (Shen et al., 2009). Earlierapplications of SH analysis in engineering fields weresuccessfully used to reconstruct the morphology of star-likeconcrete aggregates (Garboczi, 2002) and convex rockfragments (Saadatfar et al., 2005).

The current paper first validates the efficiency of thisapproach by analysing scanned particles of LeightonBuzzard sand (LBS) and highly decomposed granite(HDG). All of the SH descriptors of the real sand particleswere obtained based on their mCT images. Based onprincipal component analysis (PCA) of the obtained SHdescriptors, a virtual LBS assembly was successfullyreconstructed containing a large number of randomlyshaped particles while representing the major morphologi-cal features of the real sand particles.

VALIDATION OF SPHERICAL HARMONIC ANALYSISLeighton Buzzard sand is a typical quartz sand charac-terised by chemical inertness and considerable hardness dueto its major constituent minerals (e.g. silicon dioxide andfeldspar). The major geological transportation processesmake LBS particles sub-rounded and smooth, as demon-strated by the typical LBS particles shown in Fig. 1(a).HDG is mainly derived from weathering and erosion ofgranitic rock outcrops, and its particle morphology isalways characterised by angularity and roughness, asshown in Fig. 1(b). Twenty LBS particles and four HDGparticles were randomly picked from the screening packing,with sizes ranging from 1?18 to 2?36 mm. These selectedparticles were then scanned using a Phoenix nanome|x with apixel size of 10 mm at the Advanced Engineering MaterialFacility of Hong Kong University of Science and Tech-nology. As the input for SH analysis, three major steps areneeded for image processing of the initial mCT data – imagesegmentation, boundary identification and noise removal.

Matlab (Mathworks, 2010) was used in this work; the imageprocessing techniques are discussed in detail by Fonsecaet al. (2012). After these steps, a set of surface vertices withCartesian coordinates V(x, y, z) of one given particle can beobtained.

As expressed in equation (1), the major goal of the SHanalysis is to expand the polar radius of the real particlefrom a standard sphere and calculate the associatedcoefficients of the SH series, which is called the SHdescriptor in this study

r(h,Q)~X?n~0

Xn

m~{n

amn Y m

n (h,Q) (1)

In equation (1), r(h, Q) is the polar radius from the particlecentre with the corresponding spherical coordinates h [ ½0,p�and Q [ ½0,2p�, which can be obtained by the coordinatetransformation of the surface vertices V(x, y, z), Y m

n (h,Q) isthe SH series given by equation (2) and am

n are thecorresponding SH coefficients that need to be determined.

Y mn (h,Q)~

(2nz1)(n{m)!

4p(nzm)!

� �1=2

Pmn (cos h)ei m Q (2)

where Pmn (x) are the associated Legendre functions, which

can be easily obtained by using the built-in function‘legendre’ in Matlab, and n and m are the degree and orderof Pm

n (x), respectively. Note that n is a non-negative integerfrom zero to infinity according to the required fittingprecision and thus the total number of one set of am

n is(n + 1)2.

Taking r(h,Q) as the input on the left-hand side ofequation (1), a linear system of equations can be obtainedwith (n + 1)2 unknowns. Generally, in the mCT image of ascanned particle, the number of surface vertices is largeenough to meet the precision requirement. Additionally,nmax 5 12 has been shown to be adequate for mostengineering applications by Garboczi (2002). In this study,nmax was set to 15 for all of the cases. According to thenecessary condition of the system of linear equations,equation (1) will have an optimal solution if the total numberof input surface vertices is greater than the (nmax + 1)2

unknowns. Adopting the standard least-squares estimation,

1

–1

–1 –1

–1

–1 –1

–1

–1 –1

(a)

(b)

–11 1

1

11

1

1

11

0

0 0 0 0

0

1

0

0 0–1 –1

10 0

0

1

–1

–1 –1

–1

–1 –1

–1

–1 –1

–1

1 1

1

11 1

1

11

0

0 0 0 0

0

1

0

0 0 –1 –110 0

0

Fig. 1. Three-dimensional images of scanned sand particles: (a) four typical LBS particles; (b) four typical HDG particles

Random generation of natural sand assembly using micro x-ray tomography and spherical harmonics 7

it is easy to solve these linear equations and determine theoptimised coefficients of am

n .By using the SH descriptor am

n obtained previously, it ispossible to reconstruct the surface morphology usingequation (3), and calculate the particle volume and surfacearea for validation

r(h,Q)~X?n~0

Xn

m~{n

amn Y m

n (h,Q) (3)

Figure 2 visualises the SH reconstructions of the particlemorphology of LBS01 (upper row) and HDG01 (lowerrow). Compared with their mCT images, it is clear that theSH reconstruction has an increasing level of resemblance tothe actual particle morphology with the increasing degree nof the SH series. In the current study, nmax was set to 15,which was found to be sufficient for the morphologicalreconstruction of both shape irregularity and surfacetexture, even for the highly irregular HDG particles.However, a more sophisticated morphological reconstruc-tion representing higher levels of texture and roughnessdetails might be anticipated, entailing higher resolutionmCT scanning and a higher SH degree. Furthermore, basedon the reconstructed vertices and meshes, it is easy tocalculate the particle volume, surface area, sphericity andconvexity for engineering applications. Taking LBS01 forvalidation, Fig. 3 shows the evolution of the particlevolume and surface area of the SH reconstructionscompared with the results from the mCT image analysis.

It was found that both particle volume and surface areaconverge rapidly to a stable value that matches the imageprocessing result after n becomes greater than ten.Conclusively, it was found that the SH descriptor iscapable of characterising and reconstructing the particlemorphology of natural sands such as LBS and HDG.However, a more quantitative approach by investigatingthe topological mapping error between the SH recon-structed and scanned particle is necessary to calibrate amore reasonable SH degree n in future work, such as usingsigned distance functions (Osher & Fedkiw, 2003).

RANDOM GENERATION OF NATURAL SANDASSEMBLYBecause only four HDG particles were scanned, thesecannot fully represent the statistical information of thissand. Therefore, in this section, the detailed process of therandom generation of a natural sand assembly is illustratedmainly based on the obtained SH descriptors of the LBSparticles. It is evident from the SH analysis that a totalnumber of (nmax + 1)2 variables of a complete SH descriptoram

n is needed to reconstruct a particle. However, it is quite adifficult task to establish the quantitative correlationbetween these variables and the target particle morphologybased on the limited information of the finite number ofscanned particles. For this reason, the most important stephere was to apply PCA (Jolliffe, 2005) for the obtained SHdescriptors to reduce the dimensionality of the variables inthe SH space.

To implement the PCA, two major transformations forthe normalisation of the particle morphology were neces-sary, as illustrated in Fig. 4.

N The original surface vertices were rotated until theprincipal directions of the particle inertia tensor wereparallel to the global coordinate axes.

N The particle volume was scaled to the unit sphere.

The updated SH descriptor a0mn of the normalised particlemorphology can still be calculated by SH analysis. In thisstudy, PCA was performed on the covariance matrix of theSH coefficients a0mn of all the LBS particles. The majorcalculation was readily completed by invoking the intrinsicbuilt-in function ‘princomp’ in Matlab, yielding twoimportant outputs – the principal components (PCs) Pand the corresponding eigenvalues l. Mathematically, PCsand their eigenvalues describe the major axes andcontributions of the total morphological variance in thehigh-dimensional SH space. As shown in Fig. 5, thevariance contribution of different PCs attenuates rapidly

mCT image n = 1

1–1–1

1

1

–1

00

0

1–1–1

1

1

–1

00

0

n = 5 n = 10 n = 15

Fig. 2. Spherical harmonic reconstructions of particle morphology with different SH degrees (n); the upper and lower rows are LBSand HDG reconstructions, respectively

12

10

8

6

4

2

14 VolumeSurface area Image processing result

Image processing result

00

Volu

me:

mm

3 an

d su

rface

are

a: m

m2

12108SH reconstruction degree, n

642 14

Fig. 3. Particle volume and surface area of LBS01 as afunction of SH degree

8 Zhou and Wang

with the increase of the sequence number, and the firstthree PCs accounted for 52?3% of the total morphologicalvariation. Each set of a0mn can be reconstructed by these PCswith a score vector y, calculated by

y~PT a0mn {a0mn� �

(4)

where P is the PC matrix of dimensions M 6 d (M 5(nmax + 1)2 is the number of all variables and d is the numberof PCs used for reconstructing a0mn ) and a0mn is the meanvector of all a0mn . Each set of a0mn was then expressed by

a0mn ~a0mn zP:y (5)

The resulting PCs scores of equation (4) were complexnumbers. However, the imaginary parts of scores of thefirst three PCs were infinitesimal relative to the real parts.For these reasons, the first three PCs were selected for theparticle morphology reconstruction in the current study.Figure 6 shows the correlation of the scores of the firstthree PCs normalised by (l(i))1/2 (i 5 1, 2, 3). It is clear thatthe data points are randomly distributed in the whole area,lacking of any distinct correlation pattern. Additionally,the correlation coefficients between PC1 and PC2(Fig. 6(a)) and PC1 and PC3 (Fig. 6(b)) were infinitelysmall. These results clearly indicate that the scores of thefirst three PCs are largely independent of each other.

It is desirable now to investigate the probability densityfunctions (PDFs) of the normalised scores of the first three

PCs separately. Considering the finite number of particles,the cumulative distribution functions (CDFs) of thenormalised PCs scores can be easily obtained, as shownin Fig. 7. It is interesting to find that these three CDFs canbe nicely fitted by the accumulation of a same standardnormal distribution with m 5 0 and s 5 1. By using theindependent PDFs of the normalised scores of the firstthree PCs, it is easy to reconstruct a random SH descriptoraa for a virtual particle developed by equation (5), asexpressed by

aa~a0mn zX3

i~1

xi(l(i))1=2P(i) (6)

where P(i) is a vector of the ith PC and xi , N(0,12) is arandom real variable following the standard normaldistribution.

Figure 8 shows the reconstructed sand assembly contain-ing 50 virtual LBS particles by using equation (6).Compared with the mCT images of the typical LBS particlesin Fig. 1(a), it can be observed that all of the virtualparticles resemble the mother particles in the way that theyretain the major morphological features of the motherparticles although each particle has a unique and randomshape. Figure 9 further shows the correlation betweensphericity and convexity for all of the real and virtual LBSparticles. All the virtual particle data were found todistribute around the scanned particle data, with the virtual

1.0

0.5

0

–0.5

–1.0

1.5

–1.51.5

1.0

0.5

0

–0.5

–1.0

1.5

–1.51.5

1.0 1.51.00.5

0.5–0.5 –0.5–1.0 –1.0

–1.5 –1.5 –1.5 –1.5

0

1.00.5

–0.5–1.0

00

1.51.0

0.5–0.5

–1.00

Fig. 4. Illustration of the normalisation of particle morphology for LBS01

0.20

0.15

0.10

0.05

Mor

phol

ogic

al v

aria

nce

cont

ribut

ion

0.25

00 18161412

Principal component108642 20

Fig. 5. Morphological variance contribution of different PCs as a function of sequence number

Random generation of natural sand assembly using micro x-ray tomography and spherical harmonics 9

particle data showing a little more scatter than the scannedparticle data. As a whole, both kinds of data can be nicelyfitted by the same trend line. These results demonstrate the

effectiveness of PCA in the statistical reconstruction ofirregularly shaped particles that contains the essentialmorphological features of the target sand particles.

In the future, more PCs will be selected and the PDFs oftheir scores will be investigated considering both real andimaginary parts. It is expected that the morphologicalvariation of higher level texture details can be fullyrepresented by the virtual particles. In addition, moremorphological parameters, including angularity, roundnessand aspect ratios, of different sands will be investigated tofurther verify that the generated particles represent themajor morphological features of real sand particles.

CONCLUSIONSBased on x-ray micro computed tomography (mCT) data ofa finite number of real sand particles, spherical harmonic(SH) descriptors could be obtained after image processingand SH analysis. These SH descriptors could be furtherused in characterising and reconstructing the 3D particlemorphology and random generation of natural sandassemblies. Two major conclusions can be drawn fromthis study.

N SH analysis is a robust technique for representingparticle morphology in terms of shape irregularity andsurface textures. The precision of the SH reconstructionrelies strongly on the mCT resolution and the maximumSH degree used. For natural sand particles such asLeighton Buzzard sand and highly decomposed granite,geometrical characteristics such as volume and surface

2.0

2.0

–2.0

–2.0

1.5

1.5

–1.5

–1.5

1.0

1.0

–1.0

–1.0

0.5

0.5–0.5

–0.5

2.5

2.5

–2.5

(a)

(b)

Normalised score of PC1

Correlation: 3.17×10–12N

orm

alis

ed s

core

of P

C2

–2.50

0

2.0

2.0

–2.0

–2.0

1.5

1.5

–1.5

–1.5

1.0

1.0

–1.0

–1.0

0.5

0.5–0.5

–0.5

2.5

2.5

–2.5

Normalised score of PC1

Correlation: 3.36×10–11

Nor

mal

ised

sco

re o

f PC

3

–2.50

0

Fig. 6. Correlation of the normalised scores of the first threePCs: (a) between PC1 and PC2; (b) between PC1 and PC3

0.8

0.6

0.4

0.2

1.0PC1 data

PC1 fitting (μ = 0, σ = 1.0)PC2 fitting (μ = 0, σ = 1.0)PC3 fitting (μ = 0, σ = 1.0)

PC2 dataPC3 data

02.01.51.00.50

Normalised PC score–0.5–1.0–1.5–2.0

Cum

ulat

ive

dist

ribut

ion

func

tion

–2.5 2.5

Fig. 7. Cumulative distribution functions of the normalisedscores of the first three PCs

Fig. 8. Generated LBS assembly with 50 virtual particles

0.96

0.92

0.88Con

vexi

ty

0.84

1.00

0.960.92

Scanned LBS particlesVirtual LBS particlesFitting line

Sphericity0.880.84 1.00

0.800.80

Fig. 9. Correlation between sphericity and convexity of realand virtual LBS particles

10 Zhou and Wang

area can be approximatively determined when themaximum SH degree is greater than ten.

N Statistically, by using PCA for the normalised SHdescriptors of all scanned particles, a virtual sandassembly can be reconstructed with a large number ofrandomly shaped particles. Moreover, these virtualparticles successfully reproduce the major morphologi-cal features of the mother particles in terms of sphericityand convexity.

Further development of the current study will focus onthe DEM-aided reconstruction of natural sand assemblies,which will gain new insights into the micromechanicalbehaviour (e.g. kinematics and breakage) of sand particlescompared with in situ experimental observations using mCTscanning.

ACKNOWLEDGEMENTSThis study was supported by General Research FundCityU 120512 from the Research Grants Council of theHong Kong SAR and Research Grant 51379180 from theNational Science Foundation of China.

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Random generation of natural sand assembly using micro x-ray tomography and spherical harmonics 11