- Access to this full-text is provided by Springer Nature.
- Learn more

Download available

Content available from Granular Matter

This content is subject to copyright. Terms and conditions apply.

Vol.:(0123456789)

1 3

Granular Matter (2021) 23:41

https://doi.org/10.1007/s10035-021-01105-6

ORIGINAL PAPER

The spherical primitive andperlin noise method torecreate realistic

aggregate shapes

S.Michot‑Roberto1· A.Garcia‑Hernández1 · S.Dopazo‑Hilario2· A.Dawson1

Received: 4 October 2020 / Accepted: 24 February 2021 / Published online: 1 April 2021

© The Author(s) 2021

Abstract

An algorithm to re-create virtual aggregates with realistic shapes is presented in this paper. The algorithm has been imple-

mented in the Unity 3D platform. The idea is to re-create realistically the virtual coarse and crushed aggregates that are

normally used as a material for the construction of roads. This method consists of two major procedures: (i) to combine a

spherical density function with a noise matrix based on the Perlin noise to obtain shapes of appropriate angularity and, (ii)

deform the shapes until their minor ferret, aspect ratio and, thickness are equivalent to those wanted. The eﬃciency of the

algorithm has been tested by reproducing nine types of aggregates from diﬀerent sources. The results obtained indicate that

the method proposed can be used to realistically re-create in 3D coarse aggregates.

Keywords Aggregates· 3D virtual modelling· Morphological properties· Physics engines· Concavities

1 Introduction

In order to design civil engineering materials, such as

asphalt, with optimised workability, durability and mechani-

cal properties, it is of the foremost importance to understand

how size distribution and shape of aggregates inﬂuence their

porosity, stone interlock and amounts of asphalt binder or

mortar required for an eﬀective binding action [1, 2].

In order achieve a proper design of aggregate-based

materials, the specifications allow ranges of gradations

and shapes of the particles. There is no standard to deter-

mine how particle changes inﬂuence the overall material

behaviour [3]. This is because laboratory studies of material

behaviour are expensive and time consuming. On the other

hand, most of the computer simulations of granular materials

at the scale levels, still have drawbacks such as generating

realistic randomized 3D particles, which are diﬃcult to gen-

erate [4–6] and the computational cost involved in quantify-

ing the interaction between numerous particles [7, 8].

The ﬁrst step to create computer simulations of aggregate-

based materials is to reproduce realistic aggregates virtually.

For example, [9] employed a spherical harmonic function to

reproduce over ten thousand of real-shaped aggregates based

on the reconstructed aggregate data by X-ray CT technology,

and further quantiﬁed their shape and size [10, 11]. [13]

created realistic particles by assembling spheres, which is

could be very costly computationally. [14] created angular

particles which, although appear realistic do not have mor-

phological properties, such as angularity or sphericity con-

trolled. [15] simulated aggregates using revolution solids.

[16] deformed aggregate projections obtained from a CT

Scan gradually, untilmatching with a target distribution.

Garcia etal. similarly, [13, 14, 17] were able to generate

aggregates with concavities by creating the particles based

on surface coordinates generated by a spherical harmonic

series.

After the virtual aggregates are available, these need to be

packed. A very extended methodology to pack virtual aggre-

gates is the Discrete Element Modelling (DEM) method,

which considers a granular material’s discrete nature, has

limitations when it comes to incorporating particles with

realistic shapes. The reason is because it is computation-

ally expensive, since a granular assemblage is composed of

a multitude of particles interacting [5]. Many studies have

modelled packings of aggregates with realistic shapes, using

diﬀerent Discrete Element Methods [19–21]. However, the

* A. Garcia-Hernández

alvaro.garcia@nottingham.ac.uk

1 Nottingham Transportation Engineering Centre, Department

ofCivil Engineering, University ofNottingham,

NottinghamNG72RD, UK

2 XR-Project, Carrer Josep Vilaseca, 23. Cardedeu,

08440Barcelona, Spain

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

S.Michot-Roberto et al.

1 3

41 Page 2 of 11

computational cost of DEM is also very high. For that rea-

son, these authors propose to develop new techniques that

allow a realistic modelling of particle size and morphology,

for a better and deeper study on how they inﬂuence diﬀerent

materials behaviour.

An alternative to DEM methods are Physics Engines,

which are mainly used in the videogame and animation

industry. These serve to perform simulations based on real

physics parameters. Examples of previous related works are

those from Peng Cao etal., who modelled convex polyhe-

drons as coarse aggregates [6] and created their convex hull

using the Quickhull algorithm [7]. Aggregate conglomerates

were created by importing the particles to the Physic Engine

SWIG-Bullet multi-body, based on the Monte Carlo method

[8]. Izadi etal. simulate pluviation phenomenon using the

Bullet physics engine as well [22]. The gravel was repre-

sented through randomly shaped polyhedrons, created with

the Voronoi’s tessellation [23]. Thus, it can be concluded

that a physics engine can be employed as a geotechnical

engineering simulation tool. In addition, Garboczi etal. [12]

and Zhu etal. [13] developed algorithms that work similarly

to those in a physics motor to realise random pacing of real-

istically shaped aggregates. From these papers, the sections

about the overlapping between particles will be relevant for

the adoption of Physical Engines to pack virtual aggregates.

Finally, [16, 24] have also used statistically-based methods

to generate packs of particles from planar 2D surfaces.

The objective of this paper is to show a novel algorithm

for the creation of virtual realistic 3D aggregates. The

algorithm uses inputs from 2D aggregates to deﬁne the

3D geometries. Previous research has shown thatthere are

linear correlations between 3 and 2D size descriptors [28].

The algorithm consists of a spherical density function and

a Perlin noise to deform the spherical density function. By

combining both density functions, a scalar ﬁeld is created,

which generates a cloud of points that re-creates the shape

of the aggregates in a realistic way. Then, the points are tri-

angulated by using the Marching Cube algorithm [25–27].

Finally, the particles are deformed until they match the

aspect ratio and minor Feret of the aggregates that we want

to deform. The model proposed also oﬀers the possibility

of changing the particle sizes and shapes by considering the

statistical parameters of real aggregate samples.

2 Materials andmethods

2.1 Experimental measurement ofaggregates’

morphological properties

Nine types of coarse aggregates from diﬀerent sources and

with diﬀerent shape characteristics have been used in this

study, namely glass spheres, crushed glass, round gravel,

limestone of maximum size 14, 10, and 6mm and two types

of granite of 14mm. Each type of stone had its morphologi-

cal information extracted in order to implement the proba-

bilistic parameters in the virtual aggregates model. The

shape factors used to measure the aggregates are described

in Table1.

Pictures have been obtained from the aggregates rest-

ing over their most stable position and, the analysis of their

geometries has been carried out by the digital image pro-

cessing software ImageJ, see examples of the aggregates in

Fig.1. The image analysis included (i) establishing the scale,

(ii) highlighting the borders of the particles and correcting

the shading following the procedure mentioned in [32] and,

(iii) measuring and calculation the properties from Table1,

using the BioVoxel plugin and its functions for extended

particle analysis [33]. The distribution of the properties from

Table1 has been analysed in the Minitab software using the

Weibull distribution, which is a commonly accepted one to

create aggregate distributions. The morphological informa-

tion of the aggregate types has been quantiﬁed in terms of

the 50th percentile, P50.

Finally, the height of the aggregates has also to be

adjusted to that of the aggregates. It has been deﬁned by a

new parameter,

𝛽

, which we will call thickness from now

on, and is the ratio between the minor Feret and the average

height of 50 stones, measured with a calliper. The height has

been measured when the stone is lying in the most stable

position, in the direction perpendicular to the surface.

Table 1 Morphological

parameters to characterize

particles

Parameters Formula Reference Equation

Major and Minor Feret diameter Meassured with ImageJ [29]

Perimeter,

P

Area,

A

Aspect ratio, AR

AR

=

MinFeret

MaxFeret

ISO9276-6, [30] (1)

Sphericity/Cox’s circularity,

C

C

=

4𝜋A

P

2(2)

Roundness/Pentland’s circularity,

R

R

=

4A

𝜋MaxFeret

2[31] (3)

Thickness,

𝛽

𝛽

=

MinFeret

Averageheight

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

The spherical primitive andperlin noise method torecreate realistic aggregate shapes

1 3

Page 3 of 11 41

2.2 Physics engine: Unity 3D

The platform chosen to do this research is Unity3D, which

utilises NVIDIA´s PhysX physics engine, and C# as the

development language. PhysX is a multi-threaded physics

simulation, which supports real-time simulation of rigid and

soft bodies, ﬂuids and springs, with a focus on application to

the gaming industry [34]. This physics engine is well docu-

mented and allows to detect collisions between objects with

complex shape thanks to its built-in collision detection and

response algorithms and mesh collide [35].

2.3 Create complex virtual particles withdents

similar tothatofaggregates

In order to re-create dented virtual aggregates, this sec-

tion presents the spherical primitive + Perlin noise method,

will deforms a spherical primitive using a Perlin noise

matrix, until its angularity and circularity, their denting

level, becomes like that of an aggregate. The Perlin noise is

widely used in the animation and movies industry in order

to simulate natural-looking landscapes. In general terms,

it consists of a generator of pseudo-random numbers,

which are softened by a weighted average of the points

[36]. Figure2 summarises the process to produce a virtual

aggregate.

Firstly, a spherical primitive with radius r, is inscribed

in a grid of N x N x N voxels, see Fig.2a, where N is the

number of voxels and d, the distance between the nodes

of the voxels. The radius of the sphere is deﬁned by the

following equation:

An increase in the number of voxels, N, implies an

improvement of the resolution of the 3D model but has

the inconvenience of a higher computational cost. In

this paper, a grid of 7 × 7 × 7 voxels with 1 unit distance

between them has been chosen by trial and error, since it

has been found that this permits a good resolution while

avoiding large memory requirements [37].

Then, a 3D density function

f(x,y,z)

that determines

which lattice nodes, with coordinates,

x,y,z

, are inside

or outside the sphere is deﬁned. In this case, the nodes

outside the sphere are positive, the nodes inside the sphere

are negative and, the nodes at the surface have a value

of 0. The formula that deﬁnes the weights in the density

function is (5). In addition, a weight parameter (

Sc

), has

been added that increases the density values of the spheri-

cal functions. This factor is used to control the inﬂuence

of the spherical density function in the ﬁnal shape of the

aggregates. In other words, it controls the sphericity of

the virtual aggregates. As the numbers of voxels is 7, and

the distance between them 1, the radius, from the centre

of the voxel, is 3 and the density function values inside the

sphere range between 0 and 3.

(4)

r

=d×

N

2

Fig. 1 Examples of the aggre-

gates analysed

Granite 1, 14 mm (G1) Granite 2, 14mm (G2) Round Gravel, 20 mm (RG)

Limestone, 14 mm (L1) Limestone, 10 mm (L2) Limestone, 6 mm (L3)

Glass Sphere, 15 mm (GS) Crushed Glass, 6 mm, (CG) Slag, 6 mm (S)

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

S.Michot-Roberto et al.

1 3

41 Page 4 of 11

where

v

is the distance from the voxel’s centre to the

centre of the sphere.

In order to introduce dents in the particles that are like

those in crushed aggregates, a second grid with 7 × 7 × 7

voxels is generated and combined with the spherical one,

see Fig.2b. The node values, with coordinates

x,y,z

,

(

R(x,y,z)=Pv

), of the second grid are obtained from a

Perlin noise generator, which values may range between

0 and a maximum value of 4, depending on the degree

of deformation desired, we have called the highest value

of the Perlin’s noise range, f. The advantage of using a

Perlin noise is that it produces a gradient, natural looking

surfaces [39]. If the value is 0, there is no deformation. If

the value is 4, the deformation is the highest. This value

has been determined by the trial and error method and, a

higher value could produce unrealistic shapes (see Fig.3),

with holes. Each aggregate has a diﬀerent matrix. To con-

trol the angularity of the aggregates,

Pv

can be multiplied

(5)

f(x,y,z)=(−r+v)∗Sc

by a factor that we have named Intensity of deformation,

Idef

, which weights

Pv

.

Finally, the density function of the aggregates,

F(x,z,y)

,

is created by multiplying the spherical density function

and the noise density function as follows, see also Fig.2c:

Finally, a Marching Cube algorithm is used in order to

triangulate the shape of the virtual saggregate. The points

in the voxels’ edges that cross the density values equal to

0 are identiﬁed by the Marching Cube and triangles are

generated that transverse them and form the shape of the

virtual aggregate.

In occasions, the method presented above may produce

virtual aggregates with topological inconsistencies, such

as holes in the mesh, see Fig.3 as an example. In this

research, it could happen for two reasons: (i) due to the

intrinsic nature of the Marching Cube algorithm, which

in occasions may interpret that some faces correspond

to holes [17] and (ii) the random density matrix which

can deform the particles excessively. These ambiguities

occur when there is more than one possible way to trian-

gulate the points at the voxels’ corners and these can be

interpreted as holes. A solution for this could simply be

to increase the number of voxels in the matrix, N; other

researchers have also identiﬁed this problem and proposed

solutions that minimise but, they do not correct it com-

pletely [38].

Such virtual stones are discarded by a particle control

routine that uses an operation from Unity that “sends a

virtual laser ray” that can detect surfaces. In this way the

surface of aggregates is mapped and aggregates with holes

are detected, depending on whether some rays go through

the stone, and when any are found, the stone is discarded.

(6)

F

(x,z,y)=f(x,y,z)+R(x,y,z)=

[

(−r+v)∗S

c]

∗

[

P

v

∗I

def ]

(a) Spherical density matrix (b) Perlin noise matrix (c) Particle with complex shape

*=

Fig. 2 Process to create a dented particle

Hole

Un-realistic aggregate

(excessive irregularity) Realistic aggregate

Fig. 3 Example of un-realistic and realistic virtual aggregates

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

The spherical primitive andperlin noise method torecreate realistic aggregate shapes

1 3

Page 5 of 11 41

2.4 Deform theparticles toadjust their minor Feret

andaspect ratio inthemost stable position

tothatofaggregate projections

The minor Feret and aspect ratio are enough to characterise

the dimensions of each particle projection [18]. The algo-

rithm described in the previous section produces angular

particles which minor Ferets and aspect ratios do not ﬁt

those of the actual aggregates. For that reason, a method has

been created that rescales the angular particles, to achieve

the desired minor Feret and aspect ratio.

As it will be shown in the Results section, the aspect ratio

and minor Feret of each aggregate type have been character-

ised using the Weibull distribution function, as it was done

in [18]. This distribution is deﬁned by two parameters, one

of shape,

k

, and another one of scale,

𝜆

, which have been

measured experimentally, see also Table2.

To select the aspect ratio and minor Feret of each parti-

cle, a pseudorandom function, with range between 0.05 and

0.95 generates numbers, which are imputed as probabilities

in Weibull distribution functions of aspect ratio and minor

Feret that uses as scale and shape factors measured experi-

mentally. The results are values of minor Feret an aspect

ratio which fall within the same distribution of aggregates

measured experimentally.

A virtual laser is used to scan every virtual particle and

select the diﬀerent views of the particle, see Fig.4, based

on the position of the points on the particle’s surface “hit”

by the ray. Then, an array with the coordinates of the out-

line plan view is created, which enables deﬁnition of the

perimeter. From this array, the minor Feret and the major

Feret are identiﬁed and measured, see Fig.4d. The area is

measured based on the triangular division formed by the

segments between the perimeter array and the centre of

the particle’s projection, Fig.4d, following the procedure

described in [40]. Note that these particles are still approxi-

mately spherical.

Finally, one of the views is selected and, the virtual aggre-

gates are rescaled so that their planar projections have the

minor Feret and aspect ratios within the Weibull probability

range measured experimentally. In addition, the thickness

of the aggregates is adjusted so that their thickness ﬁts that

determined experimentally.

2.5 Select input parameters thatdeﬁne thesize

andshape ofvirtual aggregates

A summary of all the input parameters required to reproduce

the aggregates can be found in Table2. In order to calculate

automatically the input combinations, we have used a DE

algorithm [41]. The outputs that the algorithm has used are

P50

, which is the median value of the perimeter and

A50

, the

median value of the area.

2.6 Explanation oftheuse ofvirtual aggregates

throughpacking.

G2 materials were used to prepare an aggregate packing

to explain the use of the virtual aggregates. The packing

weighted approximately 500 ± 5g. The aggregates were

placed in a cylindrical container to compact the aggre-

gates, whose dimensions are 9.4cm in diameter and 8cm

in height. The compaction test was conducted using a

vibratory table with flexible springs and an engine speed

of 3600rpm, making the table vibrated in a horizontal

sine wave. The frequency was 60Hz, and the three axes’

displacement ranged between 0 and 1.58mm. The stones

were firstly vibrated for 2min and secondly, a metal

cylinder, with weight 2.7kg and 8cm of diameter was

dropped from a 5cm height above the aggregates and, the

Table 2 Input parameters to control size and shape of virtual aggre-

gates in the model

Input parameters

Angularity and sphericity Controls sphericity,

Sc

Intensity of deformation,

Idef

Perlin’s noise range, f

Size and shape Minor Feret, shape factor,

kMF

Minor Feret, scale factor,

𝜆MF

Aspect ratio, shape factor,

kAR

Aspect ratio, scale factor,

𝜆AR

Thickness Β

Perimeter

Minor Feret

Major

Feret

(c) Plan view of a

virtual stone

(a)

Side view of a

virtual stone

(b) Front view of a

virtual stone

(d) Shape factors

Fig. 4 Orthogonal views of virtual aggregates and shape factors

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

S.Michot-Roberto et al.

1 3

41 Page 6 of 11

vibration continued for five additional minutes. Finally,

the pack of G2 aggregates was CT scanned as described in

[42], and the images were treated with the image process-

ing software ImageJ following the procedure described

in [43].

An equivalent mass of G2 virtual aggregates was

compacted in the Physics Engine by pouring them in a

cylindrical container with the same dimensions as the

real one. The total number of particles in the simulation

was defined by the total mass of the sample and the den-

sity, as Unity 3D measures each particle’s volume. The

vibrations over the virtual cylinder mimicked those of the

experiment and, compaction was stopped when the height

of the virtual specimen reached that of the experiment.

Parameters used in the simulation were concave collider,

default contact onset 0.0001m, bounciness 0.315, static

friction 0, 30 iterations per frame and a fixed timestep

of 0.002.

3 Results anddiscussion

3.1 Experimental values oftheaggregates studied

The minor Feret and Aspect Ratio data obtained from

ImageJ have been analysed using the Anderson–Darling

adjustment, which indicates the goodness-of-ﬁt to a distri-

bution curve; the closer the result is to zero, the better the

ﬁt of the curve to the data points [21]. Based on [18], a two-

parameter Weibull distribution functions has been used to ﬁt

the minor Feret and Aspect Ratio. An example can be seen

in Fig.5, which shows the cumulative probabilities for the

minor Feret of G1 aggregates. The Anderson–Darling test

statistic value, A2, is 0.479.

Table3 shows the values that characterise the aggregates

studied. See for example the glass spheres, which show

aspect ratio approximately equal to 1, minor Feret approxi-

mately 15mm, and area and perimeter like those of the cir-

cle with equivalent diameter.

The aggregates with the largest shape parameter values

are RG and GS, whereas the aggregates with the smallest

shape parameter values are S and L3. The aggregates with

a more uniform size distribution are L1 and G1, with shape

values greater than 10; followed by S, CG and G2, with

shape values between 5 and 10. In addition, the aggregates

with highest AR are GS, RG, and L2, while the aggregates

with lowest AR are CG, L3 and S. Those are also the aggre-

gates with higher and lower angularity and circularity. Fur-

thermore, the thicker aggregates, with β greater than 0.5

are RG, G2 and L1, whilst the thinner aggregates are L3,

S, and CG.

Finally, Fig.6 shows the aggregate classiﬁcation based

on their circularity and sphericity, using the Krumbein chart

Fig. 5 Weibull plot of Minor Feret

Table 3 Experimentally

measured shape factors that

characterise the aggregates

studied

Min Feret50

is the median value of minor feret,

AR50

is the median value of aspect ratio,

Max Feret50

is is

the median value of major feret,

P50

is the median value of the perimeter,

A50

is the median value of the

area,

R50

is the median value of the radius,

C50

is the median value of circularity

G1 G2 RG L1 L2 L3 GS CG S

kMF

12.57 5.69 4.45 13.84 3.76 3.66 379 5.12 8.54

𝜆MF

(mm) 15.07 13.27 19.18 15.22 7.86 5.23 14.9 8.7 5.34

MinFeret50

(mm) 14.62 12.36 16.82 14.68 7.19 4.76 15.12 8.04 4.92

kAR

7.15 7.9 7.82 9.59 6.73 6.59 385 7.84 6.63

𝜆AR

0.75 0.73 0.8 0.79 0.71 0.67 0.99 0.66 0.68

AR50

0.71 0.69 0.76 0.78 0.68 0.65 0.99 0.64 0.66

MaxFeret50

(mm) 20.42 17.57 23.25 19.19 10.61 7.14 15.26 11.2 7.43

P50

(cm) 6.03 5.5 6.72 5.81 3.15 2.13 4.79 3.25 2.11

A50

(cm2) 2.08 1.56 2.73 2.02 0.52 0.24 1.83 0.64 0.25

R50

0.65 0.7 0.71 0.69 0.61 0.61 1.00 0.65 0.61

C50

0.71 0.67 0.73 0.78 0.67 0.68 1.00 0.70 0.68

𝛽

0.52 0.6 0.53 0.54 0.44 0.39 1.00 0.49 0.42

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

The spherical primitive andperlin noise method torecreate realistic aggregate shapes

1 3

Page 7 of 11 41

used for visual evaluation of aggregates [44]. From results

from Table3, it can be concluded that

R50

and

C50

are lin-

early related:

In addition, a linear trend can be also observed between

these two parameters,

AR50

and

𝛽

, which implies that, for the

aggregates used in construction materials such as asphalt or

concrete, its value does not need to be measured:

𝛽

does not show any clear relationship with

MinFeret50

and

MaxFeret50

.

3.2 Input parameters togenerate theaggregates

studied

The input parameters that control the scalar density func-

tions are

Sc

,

Idef

, f. Varying these parameters controls the

degree of particle deformation and contributes to achieve

various levels of angularity and sphericity. Their combi-

nation has been found by trial and error through a visual

inspection using Krumbein’s chart classiﬁcation, which can

be found in [44]. Figure7 shows the input values used to

ﬁnd a wide range of morphologies that corresponds to those

commonly found in the aggregates used in civil engineering.

Figure8, shows the virtual aggregates in a Krumbein

chart, based on the circularity and sphericity of the aggre-

gates. This Figure has been prepared to give the reader an

idea of the general shape of the virtual aggregates and the

parameters used to create them. The objective is to show that

the methodology shown in this paper can be used to re-create

any aggregate from those commonly used in civil engineer-

ing. It is remarkable that this methodology has allowed to

(7)

R50

=

1.10

𝛽−

0.12;R2

=

0.91

(8)

AR50

=0.56𝛽+0.42;R

2

=

0.86

precisely control the angularity and circularity of the virtual

aggregates.

To compare the real vs virtual aggregates, these have been

simply represented one versus the other. Figure9 shows the

distribution of Minor Feret of G2 aggregates as an example.

In general, it was found that the main diﬀerence corresponds

to the extremes of the distribution. This can induce errors

in the representations of aggregates, such as aggregates that

are too elongated, or bigger than normal. If these aggregates

would be used to make virtual asphalt mixtures or other

types of aggregate-based materials, the most extreme virtual

aggregates should be eliminated in order to produce correct

mixtures. Obviously, the simulation shows a high level of

similarity because the virtual aggregates have been made to

ﬁt the experimental data.

The algorithm has been validated by comparing the real

and virtual morphological information of each type of aggre-

gate. The validation has been done by a correlation parame-

ter named reliability index, which compare the median of the

virtual stone´s samples to the experimental results, in order

to know the quality of ﬁt of the virtual aggregate model with

the real data; see Eq.(9). A reliability index value equal to

one indicates a perfect adjustment.

The reliability indexes are shown in Table 4. As the

Weibull’s distribution parameters are inputs imposed by the

user, those generated by the algorithms are identical or very

close to them.

It is worth mentioning that the perimeter, area, circu-

larity and roundness show the lowest accuracy. This is

due to (i) errors induced from the selection of the high

number of variables involved in the creation of parti-

cles, which were chosen by the trial and error method;

(9)

Reliability index = P50 virtual

P

50 real

Fig. 6 Aggregate chart classiﬁ-

cation of the aggregates studied

Angular

0.50-0.65

Sub-Angular

0.65-0.70

Sub-Angular/Sub-Rounded

0.70-0.85

Rounded

> 0.85

High

Sphericity

> 0.80

Medium

Sp

hericity

0.70

– 0.85

Low

Sphericity

< 0.70

L1 RG

L2 L3

G2

G1

S

GS

CG

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

S.Michot-Roberto et al.

1 3

41 Page 8 of 11

for example, f,

Idef

, and

Sc

and, (ii) the measurements of

area and perimeter by ImageJ, which may be influenced

by shadowing and thresholding. In the future, an optimi-

sation algorithm can be used to identify the aggregates

more precisely.

3.3 Comparison ofasimulated aggregate packing

toanexperiment

In order to give an idea of the capabilities of using Phys-

ics Engines to produce aggregate packings, Fig.10 shows

a comparison between a simulation and a packing of

High

Sphericity

S= 1

Medium

Sphericity

S= 0.8

Low

Sphericity

S= 0.70

Very Angular Sub-AngularSub-Rounded Rounded

Angular

Sc3.0

Idef 0.015

f4

Sc3.0

Idef 0.020

f3

Sc3.0

Idef 0.030

f3

Sc3.0

Idef 0.090

f1.7

Sc3.0

Idef 1.000

f0.5

Sc1.4

Idef 0.02

f4

Sc1.4

Idef 0.045

f3

Sc1.4

Idef 0.055

f3

Sc1.4

Idef 1.000

f1.7

Sc1.4

Idef 1.200

f0.5

Sc1.2

Idef 0.015

f4

Sc1.2

Idef 0.020

f3

Sc1.2

Idef 0.030

f3

Sc1.2

Idef 0.070

f1.7

Sc1.2

Idef 1.400

f0.5

Fig. 7 Aggregate chart classiﬁcation of the aggregates studied

Fig. 8 Aggregate chart classiﬁ-

cation of the virtual aggregates

studied

Angular

0.50-0.65

Sub-Angular

0.65-0.70

Sub-Angular/Sub-Rounded

0.70-0.85

Rounded

> 0.85

High

Sphericity

> 0.80

Medium

Sp

hericity

0.7

0–0.85

Low

Sphericity

< 0.70

L1 RG

L2

L3

G2

S

GS

CG

S

c

3.0

I

def

1.000

f0.5

S

c

1.2

I

def

0.020

F4

S

c

1.4

I

def

0.035

f4

S

c

1.3

I

def

0.055

f3

S

c

1.6

I

def

0.048

f3

S

c

1.7

I

def

0.090

f2

S

c

1.4

I

def

0.045

f3

G1

S

c

1.6

I

def

0.030

f3

S

c

1.2

I

def

0.050

f3

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

The spherical primitive andperlin noise method torecreate realistic aggregate shapes

1 3

Page 9 of 11 41

aggregates G2, obtained through X-ray CT scans. The

goodness of ﬁt of the vertical distribution of pores has

been found via the Kolmogorov–Smirnov Test [45], using

the critical value for a conﬁdence level at 95% (

Dcrit,0.05

),

using:

where

nx

and

ny

are the population size of the samples.

Dcrit,0.05

= 0.028; if

Dcrit,0.05

is smaller than the maximum

absolute distance between the expected and observed distri-

bution functions (

Ds

), we can conclude that the values from

the experiments and simulations are equivalent. In this case

Ds=0.013

, hence, we can conﬁrm that the virtual aggre-

gate’s packing has a very close AV distribution function than

the experimental one. Finally note that for the same equiva-

lent mass, the experiment comprised 104 particles, while the

simulation 110 particles.

4 Conclusions

This article has shown a novel methodology to re-create

commonly used aggregates in civil engineering. It is based

on a spherical primitive that is deformed by means of a

Perlin noise. Then, the shape and size of the virtual aggre-

gates is adjusted by deforming the shapes until they fall

(10)

D

crit,0.05 =1.36

√

1

nx

+1

ny

,

0

10

20

30

40

50

60

70

80

90

100

0102030

Cumulative frequency (%)

Minor Feret (mm)

Unity 3D

ImageJ

Simulation

Experiment

Fig. 9 Distribution of Minor Ferets for G2 aggregates, calculated

from simulations and experiments

Table 4 Reliability indexes

resultant from comparing the

simulations with experiments

G1 G2 RG L1 L2 L3 L4 L5 GS CG S

MinFeret50

0.997 0.989 0.964 0.978 1.037 1.017 1.024 0.969 0.999 0.969 1.027

AR50

0.997 0.989 0.964 0.978 1.037 1.017 1.024 0.969 0.999 0.969 1.027

MaxFeret50

1.017 0.999 0.924 1.062 0.907 1.024 0.998 0.938 1.007 1.103 0.934

P50

0.831 0.879 0.895 0.854 1.165 0.830 0.854 0.845 0.976 0.809 0.806

A50

0.887 0.860 0.971 1.155 1.194 1.113 1.194 1.205 1.036 1.235 1.245

R50

1.120 1.008 1.033 1.199 1.146 1.183 1.156 1.161 0.975 1.132 1.161

C50

1.051 1.043 1.035 0.929 0.949 0.954 0.957 1.008 0.983 1.095 1.143

Fig. 10 Vertical distribution

of air voids in a simulated

and experimental packs of G2

aggregates

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

01020304050

Height (%)

Air Vo id Content (%)

Simulation

Experiment

Simulation

Experiment

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

S.Michot-Roberto et al.

1 3

41 Page 10 of 11

in the same statistical range of the real aggregates. The

algorithm has been implemented in the NVIDIA PhysX

physical engine, with the objective of being able to use

these aggregates in future research that will involve creat-

ing virtual aggregate gradations.

The following conclusions have been obtained:

• This methodology allows to re-create the morphologies

of aggregates virtually. Based on the experience of the

researchers, hundreds of aggregates were generated in

a few seconds.

• This algorithm has allowed to re-create realistic virtual

aggregate shapes that include concavities. The authors

are convinced that the precision of these virtual aggre-

gates will be enough to analyse the inﬂuence of the

aggregate shapes on the packing of aggregates.

• The angularity and sphericity of the particles are con-

trolled by three inputs that weight the spherical primi-

tive and Perlin noise. In the current research, these

inputs have been adjusted by trial and error, until

the shape of the virtual shapes was found. In future

research, this will be updated by using an optimisation

algorithm, which will allow the creation of aggregates

in a fast and eﬃcient way.

• The comparison between the virtual and the real geo-

metrical properties’ data indicates that the algorithm to

create virtual aggregates can achieve minor and major

Ferets like those measured in real aggregate samples.

In addition, this methodology allows to create virtual

particles with area, perimeter, sphericity, and angular-

ity similar to those of the real materials.

• In occasions, the methodology proposed in this article

produced aggregates that presented unnatural shapes,

including holes and bottlenecks. The reasons for this

are still unsolved. The unnatural particles appeared

very seldomly. However, if hundreds of particles are

produced to create an aggregate packing and some

unnatural particles appear results could be aﬀected. For

this reason, the authors have developed a new method-

ology to produce virtual aggregates that is presented in

[18].

• It has been found that the methodology allows to re-

create full gradations by using the Weibull distribu-

tion to decide the minor Feret of the virtual aggregates.

However, in future research the extreme aggregates

need to be eliminated, to allow re-creating unrealistic

aggregates.

Acknowledgements The authors want to acknowledge the sponsorship

of Tarmac Ltd, which is the UK’s leading sustainable building materi-

als & construction solutions businesses. Speciﬁcally, the contribution

of Ms Kerry Nadel measuring the shape of the aggregates is very much

appreciated.

Declarations

Conflict of interest The authors declare that they have no conﬂict of

interest.

Open Access This article is licensed under a Creative Commons Attri-

bution 4.0 International License, which permits use, sharing, adapta-

tion, distribution and reproduction in any medium or format, as long

as you give appropriate credit to the original author(s) and the source,

provide a link to the Creative Commons licence, and indicate if changes

were made. The images or other third party material in this article are

included in the article’s Creative Commons licence, unless indicated

otherwise in a credit line to the material. If material is not included in

the article’s Creative Commons licence and your intended use is not

permitted by statutory regulation or exceeds the permitted use, you will

need to obtain permission directly from the copyright holder. To view a

copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.

References

1. Aragão, F.T.S., Pazos, A.R.G., da Motta, L.M.G., Kim, Y.R., do

Nascimento, L.A.H.: Eﬀects of morphological characteristics of

aggregate particles on the mechanical behavior of bituminous

paving mixtures. Constr. Build. Mater. 123, 444–453 (2016)

2. Okonkwo, O., Arinze, E.: Eﬀects of aggregate gradation on

the properties of concrete made from granite chippings. J Steel

Struct Constr (2016). https:// doi. org/ 10. 4172/ 2472- 0437. 10001

09

3. Huang, H.: Discrete element modeling of railroad ballast using

imaging based aggregate morphology characterization. In: PhD

Thesis. University of Illinois at Urbana-Champaign (2010)

4. Xu, R., Yang, X.H., Yin, A.Y., Yang, S.F., Ye, Y.: A three-dimen-

sional aggregate generation and packing algorithm for modeling

asphalt mixture with graded aggregates. J. Mech. 26(2), 165–171

(2010)

5. Movshovitz, E.I., Asphaug, N.: Discrete element modeling of

complex granular ﬂows. Am. Geophys. Union Meeting, San Fran-

cisco, USA (2010)

6. Cao, P., Jin, F., Feng, D., Zhou, C., Hu, W.: Prediction on dynamic

modulus of asphalt concrete with random aggregate modeling

methods and virtual physics engine. Constr. Build. Mater. 125,

987–997 (2016)

7. Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algo-

rithm for convex hulls. ACM Trans. Math. Softw. 22(4), 469–483

(1996)

8. Paltani, S.: Monte Carlo methods. Stat. Course Astrophys. 2010–

2011, 2010–2011 (2011)

9. Application to aggregates used in concrete: Cem. Concr. Res.

31(10), 1621–1638 (2002)

10. Erdogan, S.T., etal.: Shape and size of microﬁne aggregates:

X-ray microcomputed tomography vs. laser diﬀraction. Powder

Technol. 177(2), 53–63 (2007)

11. Bullard, J.W., etal.: Deﬁning shape measures for 3D star-shaped

particles: sphericity, roundness, and dimensions. Powder Technol.

249, 241–252 (2013)

12. Garboczi, E.J., etal.: Contact function, uniform-thickness shell

volume, and convexity measure for 3D star-shaped random parti-

cles. Powder Technol. 237, 191–201 (2013)

13. Zhu, Z.G., etal.: Parking simulation of three-dimensional multi-

sized star-shaped particle. Model. Simul. Mat. Sci. Eng. 22,

035008 (2014)

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

The spherical primitive andperlin noise method torecreate realistic aggregate shapes

1 3

Page 11 of 11 41

14. He, H.: Computational modelling of particle packing in concrete,

PhD Thesis, TU Delft, (2010)

15. Mollon, G., Zhao, J.: Generating realistic 3D sand particles using

Fourier descriptors. Granular Matter 15, 95–108 (2013)

16. Tahmasebi, P.: Packing of discrete and irregular particles. Comp.

Geotech. 100, 52–61 (2018)

17. Zhang, H., Sheng, P., Zhang, J., Ji, Z.: Realistic 3D modeling

of concrete composites with randomly distributed aggregates by

using aggregate expansion method. Constr. Build. Mater. 225,

927–940 (2019)

18. Garcia, A., Michot-Roberto, S., Dopazo-Hilario, S., Chiarelli,

A., Dawson, A.: Creation of realistic virtual aggregate avatars.

Powder Technol. (2020). https:// doi. org/ 10. 1016/j. powtec. 2020.

10. 036

19. Zhao, D., Nezami, E.G., Hashash, Y.M.A., Ghaboussi, J.: Three-

dimensional discrete element simulation for granular materials.

Eng. Comput. 23(7), 749–770 (2006)

20. Burtseva, L., Salas, B.V., Romero, R., Werner, F.: Multi-sized

sphere packings: models and recent approaches. Preprint 02/15.

(2015). https://doi.org/https:// doi. org/ 10. 13140/2. 1. 4515. 6169

21. Mollon, G., Zhao, J.: Fourier–Voronoi-based generation of realis-

tic samples for discrete modelling of granular materials. Granular

Matter 14, 621–638 (2012)

22. Izadi, E., Bezuijen, A.: Simulation of granular soil behaviour

using the Bullet physics library. Geomech. Micro Macro I–II,

1565–1570 (2013)

23. Eliáš, J.: Simulation of railway ballast using crushable polyhedral

particles. Powder Technol. 264, 458–465 (2014)

24. Tahmasebi, P., Sahimi, M., Andrade, J.E.: Image-based modeling

of granular porous media. Geophys. Res. Lett. 44, 4738–4746

(2017)

25. Congote, J., Moreno, A., Barandiaran, I., Barandiaran, J., Posada,

J., Ruiz, O.: Marching cubes in an unsigned distance ﬁeld for sur-

face reconstruction from unorganized point sets. GRAPP 2010—

Proc. Int. Conf. Comput. Graph. Theory Appl. 143–147 (2010)

26. Newman, T.S., Yi, H.: A survey of the marching cubes algorithm.

Comput. Graph. 30(5), 854–879 (2006)

27. Miltenberger, K.: Polygonising a scalar ﬁeld. http:// paulb ourke.

net/ geome try/ polyg onise/

28. Su, D., Yan, W.M.: Prediction of 3D size and shape descriptors

of irregular granular particles from projected 2D images. Acta

Geotech. 15, 1533–1555 (2020)

29. Psimadas, D., Georgoulias, P., Valotassiou, V., Loudos, G.:

Molecular nanomedicine towards cancer. J. Pharm. Sci. 101(7),

2271–2280 (2012)

30. Olson, E.: Particle shape factors and their use in image analysis—

part 1: theory. J. GXP Compliance 15(3), 85–96 (2011)

31. Ribeiro, S., Bonetti, C.: Variabilidade morfométrica de sedimentos

arenosos: revisão de métodos e uso do software ImageJ na difer-

enciação de ambientes deposicionais na Ilha de Santa Catarina e

região continental, SC, Brasil. Gravel 11(1), 37–47 (2013)

32. Portal ImageJ Information and Documentation. FFT (Fast Fourier

Transform). 1–4 (2011)

33. Brocher, J.: BioVoxxel Toolbox (2018) [Online]. Available:

https:// imagej. net/ BioVo xxel_ Toolb ox

34. Maciel, A., Halic, T., Lu, Z., Nedel, L.P., De, S.: Using the PhysX

engine for physics-based virtual surgery with force feedback. Int

J Med Robot. 5(3), 341–353 (2009)

35. Seugling, A., Rolin, M.: Evaluation of physics engines and imple-

mentation of a physics module in a 3d-authoring tool. Umea Univ.

Master Thesis (2006)

36. Perlin, K.: An image synthesizer. ACMSIGGRAPH Comput.

Graph. 19(97–8930), 287–296 (1985)

37. Kanzler, M., Rautenhaus, M., Westermann, R.: A voxel-based ren-

dering pipeline for large 3D line sets. IEEE Trans. Vis. Comput.

Graph 25(7), 2378–2391 (2019)

38. Nielson, G.M., Hamann, B.: The asymptotic decider: resolving

the ambiguity in marching cubes. In: Proceedings of Visualization

83–91 (1991)

39. Lagae, A., etal.: A survey of procedural noise functions. Comput.

Graph. Forum 29(8), 2579–2600 (2010)

40. Wang, D., Ding, X., Ma, T., Zhang, W., Zhang, D.: Algorithm for

virtual aggregates’ reconstitution based on image processing and

discrete-element modeling. Appl. Sci. 8(5), 1–16 (2018)

41. Abou-foul, M., Chiarelli, A., Triguero, I., Garcia, A.: Virtual

porous materials to predict the air void topology and hydraulic

conductivity of asphalt roads. Powder Technol. 352, 294–304

(2019)

42. Aboufoul, M., Chiarelli, A., Triguero, I., Garcia, A.: Virtual

porous materials to predict the air void topology and hydraulic

conductivity of asphalt roads. Powder Technol. 352, 294–304

(2019)

43. Aboufoul, M., Garcia, A.: Factors aﬀecting hydraulic conductivity

of asphalt mixture. Mater. Struct. 50(1–16), 2017 (2017)

44. Su, D., Yan, W.M.: Quantiﬁcation of angularity of general-shape

particles by using Fourier series and a gradient-based approach.

Constr. Build. Mater. 161, 547–554 (2018)

45. Massa, S.: Lecture 13: kolmogorov smirnov test & power of tests.

(2016). http:// www. stats. ox. ac. uk/ ~massa/ Lectu re% 2013. pdf

Publisher’s Note Springer Nature remains neutral with regard to

jurisdictional claims in published maps and institutional aﬃliations.

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1.

2.

3.

4.

5.

6.

Terms and Conditions

Springer Nature journal content, brought to you courtesy of Springer Nature Customer Service Center GmbH (“Springer Nature”).

Springer Nature supports a reasonable amount of sharing of research papers by authors, subscribers and authorised users (“Users”), for small-

scale personal, non-commercial use provided that all copyright, trade and service marks and other proprietary notices are maintained. By

accessing, sharing, receiving or otherwise using the Springer Nature journal content you agree to these terms of use (“Terms”). For these

purposes, Springer Nature considers academic use (by researchers and students) to be non-commercial.

These Terms are supplementary and will apply in addition to any applicable website terms and conditions, a relevant site licence or a personal

subscription. These Terms will prevail over any conflict or ambiguity with regards to the relevant terms, a site licence or a personal subscription

(to the extent of the conflict or ambiguity only). For Creative Commons-licensed articles, the terms of the Creative Commons license used will

apply.

We collect and use personal data to provide access to the Springer Nature journal content. We may also use these personal data internally within

ResearchGate and Springer Nature and as agreed share it, in an anonymised way, for purposes of tracking, analysis and reporting. We will not

otherwise disclose your personal data outside the ResearchGate or the Springer Nature group of companies unless we have your permission as

detailed in the Privacy Policy.

While Users may use the Springer Nature journal content for small scale, personal non-commercial use, it is important to note that Users may

not:

use such content for the purpose of providing other users with access on a regular or large scale basis or as a means to circumvent access

control;

use such content where to do so would be considered a criminal or statutory offence in any jurisdiction, or gives rise to civil liability, or is

otherwise unlawful;

falsely or misleadingly imply or suggest endorsement, approval , sponsorship, or association unless explicitly agreed to by Springer Nature in

writing;

use bots or other automated methods to access the content or redirect messages

override any security feature or exclusionary protocol; or

share the content in order to create substitute for Springer Nature products or services or a systematic database of Springer Nature journal

content.

In line with the restriction against commercial use, Springer Nature does not permit the creation of a product or service that creates revenue,

royalties, rent or income from our content or its inclusion as part of a paid for service or for other commercial gain. Springer Nature journal

content cannot be used for inter-library loans and librarians may not upload Springer Nature journal content on a large scale into their, or any

other, institutional repository.

These terms of use are reviewed regularly and may be amended at any time. Springer Nature is not obligated to publish any information or

content on this website and may remove it or features or functionality at our sole discretion, at any time with or without notice. Springer Nature

may revoke this licence to you at any time and remove access to any copies of the Springer Nature journal content which have been saved.

To the fullest extent permitted by law, Springer Nature makes no warranties, representations or guarantees to Users, either express or implied

with respect to the Springer nature journal content and all parties disclaim and waive any implied warranties or warranties imposed by law,

including merchantability or fitness for any particular purpose.

Please note that these rights do not automatically extend to content, data or other material published by Springer Nature that may be licensed

from third parties.

If you would like to use or distribute our Springer Nature journal content to a wider audience or on a regular basis or in any other manner not

expressly permitted by these Terms, please contact Springer Nature at

onlineservice@springernature.com