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Computational packing of aggregates for the study of virtual

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asphalt samples

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Andrea Chiarelli1,a, Andrew R. Dawson2,b, A. García3,c

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1Nottingham Transportation Engineering Centre (NTEC), Faculty of Engineering, The University

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of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

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achiarelli.andrea@gmail.com

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bandrew.dawson@nottingham.ac.uk

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calvaro.garcia@nottingham.ac.uk

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Abstract

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In this paper, a proof of concept of a method is presented for the study of granular materials, such

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as asphalt, based on the use of a physics engine. To begin with, virtual aggregates are generated

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with randomized 3D shapes and a size distribution based on a chosen gradation curve. Then, the

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aggregates are placed in a constrained volume and subjected to a simulated vibration until

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satisfactory compaction is reached. Finally, the packed stone assembly obtained is saved as a 3D

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model, so that the virtual aggregates can be used for further studies such as the analysis of the void

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space in the material. All the steps in the method are described and discussed, along with the

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approximations made. Furthermore, an analysis of the void space is performed to determine if the

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method is able to generate air pores with realistic features. The analysis is performed by comparing

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the void space of a computationally packed aggregate assembly to that of a real asphalt core with

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the same aggregate gradation. The preliminary results obtained show that the modelling approach

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is able to represent effectively the air pores, thus, suggesting that further studies to advance this

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proof of concept should be conducted.

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Keywords: packing, modelling, aggregates, asphalt

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1. Introduction

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In civil and structural engineering, hydrogeology, and petroleum engineering, many materials are

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granular materials, thus, their microscopic structure is characterized by the presence of air voids

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with various sizes [1]. For this reason, the development of computational methods that are able to

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reproduce the shape of real granular materials is of great interest.

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Numerous applications can be envisioned for realistic 3D models of granular materials. First of

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all, a 3D model of the porous space can be used for the design of novel materials with controlled

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air void content and air void size distribution. Second, virtual models can be used to study

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properties of permeable civil engineering materials such as pavements, as this kind of material

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affects e.g. the traffic conditions in rainy weather [2] and soil pollution [3]. Third, 3D models of

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the pore space in granular materials can be used to study air and vapor movement which are of

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great importance in soil science [4] and natural air conditioning.

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In the field of computational methods, the generation of domains for the analysis of materials with

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air voids is usually approached (i) with mathematical or geometrical models, (ii) with the discrete

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elements method (DEM), or (iii) with the use of X-ray CT scans.

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Mathematical models are generally used for fluid-dynamics, as their aim is the generation of the

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pore space. Examples of mathematical models are the percolation method [5, 6, 7], or the use of

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Fourier transforms coupled with Voronoi tessellation [8]. Models for the generation of the pore

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space based only on geometrical principles have been studied by the authors [9, 10].

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DEM simulations are used for mechanical analyses where the particles composing the materials,

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i.e., the grains, are organized together in a confined space and the relationship between them is

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defined by using physical laws considering friction or energy [11, 12]

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X-ray CT scan imaging is a more flexible means to generate virtual domains for computational

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simulations, as the virtual models can be used for both mechanical and fluid-dynamics analyses.

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In fact, from X-ray CT scans it is possible to isolate both the grains and the pore space, thus, a

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number of different analyses can be performed [1, 13, 14, 15]. In addition, X-ray CT scans can be

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used to computationally characterize a material [16]. The use of X-ray CT scans, however, comes

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with a serious limitation, namely, the availability of a CT scanner.

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Currently, only X-ray CT scan imaging is able to describe effectively the stones and the void space

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at the same time. Mathematical and geometrical methods either neglect the stones or consider

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idealized stone shapes, while DEM methods typically use spherical virtual stones that inevitably

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yield an unrealistic shape of the void space due to the regular solids considered. The authors [17]

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were able to use a discrete element simulation to pack stones with realistic shapes in 3D, however,

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only one stone shape was chosen and therefore the stone size distribution was neglected.

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For these reasons, it is important to develop a method that allows a study of the aggregates and the

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void space based on realistic (or customized) aggregate gradations and shapes. Such a method is

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expected to allow engineers and researchers to better design granular materials, because the study

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of the physical properties could be performed based on known (and, potentially, controlled)

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assemblies of stones and pores rather than on a separate analysis of either the stones or the pores.

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In this paper, a proof of concept of a method for packing aggregates with realistic shapes and

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gradation and based on the use of a physics engine is presented. In order to achieve a preliminary

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confirmation that the development of the proof of concept should be continued, a sample virtual

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model created with the approach proposed by the authors is compared to real X-ray CT scans of

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an asphalt mixture. The comparison performed here is focused only on the study of the air pores

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in the virtual material because the 3D shapes of the real stones used were not available.

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Figure 1. Flowchart of the method for packing of aggregates and for the generation of

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virtual air pores.

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2. Development of the packing method

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In Fig. 1, a flowchart of the whole method is shown. In the next subsections, every step is

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described in detail to allow a clear understanding of the whole process

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2.1. Generation of the virtual aggregate

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Before the generation of the assembly of packed virtual stones is started, a number of parameters

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need to be chosen. To begin with, since virtual stones with realistic shapes are used, it is necessary

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to choose their number, size, and aspect ratio. Each stone is created as a virtual 3D object by

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computing the convex hull of a cloud of random points (see Fig. 2).

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Figure 2. Generation of a virtual stone (cloud of points and corresponding convex hull).

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In particular, 50 random points are seeded in the 3D space shown in Fig. 2, then the most “external”

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points in the cloud are used to define the surface of the stone (this enveloping surface is the

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abovementioned convex hull). The number of randomly seeded points can vary based on the user

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needs to create simpler or more complex 3D shapes. It is relevant to mention that too high or too

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low numbers of points could lead to unrealistic stones, e.g., a cube or oddly-shaped flake,

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respectively. The generation of random points in this paper was based on the use of the uniform

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distribution, however, any other distribution (even a custom one) could be followed.

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The points can be generated in any arbitrary interval in each direction, thus, allowing the

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generation of virtual stones of any chosen shape, aspect ratio, or faces. In this paper, the cloud of

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points that originates each virtual stone is created with a 0.6:0.7:1 aspect ratio between the x,y, and

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z axes. It is important to give the virtual stones an aspect ratio that is not 1:1:1 (i.e., a sphere),

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otherwise the maximum degree of packing will be constrained to the maximum level of packing

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that can be obtained for a set of polydisperse spheres, which corresponds to a rather high air void

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content (more than 35% [18]).

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In addition, the stones are seeded into a 3D space which is sized according to a granulometric

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curve that can be chosen by the user, thus, any kind of grading can potentially be simulated (e.g.,

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open grading or dense grading). The seeding and stone construction is executed with MATLAB

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and the stones are saved in the .obj file format to allow easy handling in the next steps.

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2.2 Compaction and packing of the virtual aggregate

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Once virtual aggregates have been created, they can be loaded in the 3D compaction environment.

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The virtual compaction method proposed in this paper exploits the physics engine of a videogame

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development software to vibrate the mixing container where the stones are placed based on Eq. 1-

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3.

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Some input values need to be chosen for the compaction process, which is based on the vibration

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of the volume where the stones are mixed. Therefore, the size of the mixing container has to be

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picked based on the amount of material that is simulated, while its movement needs to be defined

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by the means of some equations. In this paper, sinusoidal equations were chosen, i.e.:

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𝑝(𝑡) = 𝑝0+ 𝑎 ∙ sin(𝜔𝑡) (1)

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𝑞(𝑡) = 𝑞0+ 𝑏 ∙ sin(𝜔𝑡) (2)

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𝑟(𝑡) = 𝑟0+ 𝑐 ∙ sin(𝜔𝑡) (3)

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where 𝑝, 𝑞, and 𝑟 are the coordinates of the mixing container on the x,y, and z axes at time 𝑡; the

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subscript 0 describes the starting position of the container; and 𝑎, 𝑏, 𝑐, and 𝜔 are arbitrary

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constants. For the preliminary results shown in this paper, the vertical oscillation (𝑦 axis, described

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by Eq. 2) was set to 0, as it would have added a further degree of freedom that is not discussed in

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the present analysis. Finally, the values of 𝑎 and 𝑐 were set to 1 and 𝜔 was set to 5. These values

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are arbitrary and were picked after a number of trial simulations in order to achieve a level of

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compaction that showed realistic features.

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When the stones are loaded in the mixing environment they are assigned a position, a rotation, a

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mesh and a corresponding mesh collider, which is used to detect when items with complex shapes

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touch one another. The use of a mesh collider in a physics engine is an advancement from previous

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methods of aggregate packing, because it allows the use of a range of realistic shapes for

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aggregates in the place of simple spheres (see, e.g., [19]) and prevents the need of implementing

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the specific physics that rule the contact between particles as done in [17] because they are included

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in the physics engine. When the virtual stones are imported in the mixing environment, they are

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also assigned their material properties, i.e., mass and friction factors. This is because these

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parameters rule how the particles interact with each other during the virtual compaction process.

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Bitumen and filler were not explicitly included in the simulations performed as done in recent

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studies on virtual aggregate compaction [19]. In [19], each stone is considered as if it was coated

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by a very thin layer of bitumen, thus, using modified contact laws to take bitumen and filler into

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account during the compaction and mixing process. In this paper, contact laws are not manually

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implemented and the contact between particles is completely handled by the physics engine. This

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is an approximation, however, the approach proposed can be justified by the high quality of the

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results that it yields.

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In Fig. 3, a mixing volume with virtual stones is shown. It can be observed that compaction is

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achieved only by vibration and that no load is applied to the upper surface of the virtual stones. A

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surface load could be applied, however, further studies should assess whether it would be

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beneficial for the realism of the result or not.

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(a) Mixing container and directions of the oscillation.(b) Detail of particles being compacted.

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Figure 3. Mixing container with virtual stones during the packing process.

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The mixing container can be vibrated for an arbitrary amount of time based on any chosen criteria.

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In this paper, the vibration was interrupted when the thickness of the stone assembly stopped

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decreasing for at least 1000 time steps, i.e., when no further compaction could be achieved by

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vibration in a practical amount of time. Finally, it is relevant to mention that in the packing process

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the aggregates were free to move and rotate with no constraints. Therefore, 6 degrees of freedom

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are allowed for each particle during the simulation.

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2.3 Extraction of the aggregate assembly and analysis of the pore space

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At the end of the virtual compaction process the aggregates can be saved in a 3D file format for

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further elaboration. The most direct way to extract the pores from the stone assembly is the use of

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a Boolean operation. In fact, if a solid box is created the stone assembly can be subtracted from it

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(Boolean subtraction), thus, generating a 3D representation of the pores. An example of this can

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be seen in Fig. 4, where the pores are now a solid as opposed to the previous situation (see Fig. 3)

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where the aggregates were the solid part. As a matter of fact the volume representations of both

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the aggregates and the pores may be combined and represented at the same time, so that the

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material could be studied as a whole. In Fig. 4, it can be appreciated that the position of the virtual

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stones was unconstrained during the mixing process. In addition, a number of stone sizes can be

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observed in the virtual sample.

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Another option to analyze the packed aggregates is the creation of virtual tomographies of the

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stone assembly, as done e.g. in [9] and [10]. This method is obviously less straightforward and

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requires further elaboration of the 3D volume, however, it allows a more precise analysis of the

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data that is obtained. When the virtual tomographies are created, the distance between slices of the

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virtual material is conceptually similar to the vertical spacing in X-ray CT scanners. For this

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reason, a stack of virtual tomography images can be created to pursue the study of the pore space

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in terms of a number of properties.

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For the study of the aggregate packing method presented in this paper the authors developed an

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automatic slicing algorithm that is able to read the 3D files generated by the virtual compaction

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system and then analyze them. In particular, the algorithm developed calculates the intersection

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between arbitrary horizontal slicing planes with the 3D model and returns the profile of the stones

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in the layer corresponding to the vertical position of the slicing plane.

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Figure 4. Pores as a 3D structure and cross sections in an open-graded mixture.

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The implementation of the slicing algorithm was made in MATLAB and a visual explanation of

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its working mechanism is shown in Fig. 5. The blue plane in Fig. 5 is the slicing plane at a height

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equal to 50% in the virtual sample shown, while the red profiles represent the intersection between

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the 3D geometry and the slicing plane. While the slicing plane is shifted from the bottom to the

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top of the packed virtual stone assembly the red profiles can be exported as images or image stacks.

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The algorithm developed offers a high flexibility, as it allows the automatic generation of an image

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stack very similar to those usually obtained with X-ray CT scanners quickly and with a very high

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accuracy. An alternative method for creating virtual tomographies is the use of the void tracking

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method studied by the authors in [9] and [10], however, the newly developed algorithm is several

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times faster and, more importantly, treats each stone as a set of surfaces, thus, allowing an exact

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projection of the stone profiles on the slicing planes. As a result, the new slicing method does not

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cause any concern about precision or grid spacing as seen in [10].

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Figure 5. Visual explanation of the slicing method used to generate virtual tomographies.

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3. Examples of outputs and comparison with a real asphalt sample

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In order to provide confirmation of the effectiveness of this proof-of-concept method a virtual

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porous asphalt core was built. The virtual asphalt core was built with the stone size distribution

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shown in Table 1, which was previously used to build a porous asphalt core with and air void

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content of 25%. Note that the percentages shown in Table 1 are rescaled, as in the construction of

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the virtual model the dust used for the manufacture of the real sample was not considered.

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Table 1. Sample size distribution used to build a porous virtual asphalt sample.

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Aggregate size (mm)

Percent of total mass (%)

20 mm

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14 mm

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10 mm

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6.3 mm

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The stone shapes (aspect ratios) were defined as explained in section 2.1 because the real aspect

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ratios of the stones used for the asphalt core were not known.

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Virtual tomographies were performed on the asphalt sample created computationally in order to

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show the high degree of realism reached with the packing method used. In Fig. 6, a virtual

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tomography is shown along with a real CT scan obtained from the above-mentioned asphalt core.

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(a) Virtual asphalt sample (~23% air voids). (b) Real asphalt sample (~25% air voids).

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Figure 6. Comparison between a virtual tomography and a real X-ray CT scan (thresholded

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2D images).

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It can be observed that the shapes of the stones in the virtual tomography (Fig. 6a) resemble the

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real shapes and that the planar void fraction is, visually, very similar. In addition, the virtual stones

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seem to be arranged in a realistic way when compared to the real ones. From a first visual

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examination, also the air voids on both sides of Fig. 6 seem to have similar sizes and shapes, thus,

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suggesting that there is an acceptable degree of similarity. In order to confirm what can be visually

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assessed three analyses were run to compare the morphology of the pore space in both the virtual

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and the real samples. The analyses considered in this paper measure the fractal dimension, the

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connectivity density, and the volume fraction. In Table 2, the properties calculated with the BoneJ

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plugin for ImageJ (connectivity density and volume fraction) and with MATLAB (fractal

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dimension) for the two samples are shown.

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Table 2. Morphological properties of the air pores in the virtual and in the real asphalt

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samples.

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Property

Pores in real sample

Pores in virtual sample

Fractal dimension (-)

1.80

1.76

Connectivity density (pixel-3)

0.000346

0.000392

Volume fraction in 3D (%)

25.5

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Tortuosity (-)

1.37

1.30

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The data in Table 2 suggests that the samples are comparable and their degree of similarity is likely

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to be acceptable, especially considering that the shape of the real stones was unknown. At this

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preliminary stage, the desirable degree of similarity is not known.

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To begin with, it is interesting to notice that the fractal dimension (Minkowski–Bouligand

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dimension or box-counting dimension [9]) is similar in the samples and falls in the interval 1.63-

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1.82 previously reported in [20] for real asphalt samples. The fractal dimension is important

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because it was shown to provide information on the homogeneity of asphalt [20], which in turn is

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related to the quality of a pavement [21].

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Figure 7. Cross sections of a composite model made by joining virtual and real air pore

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structures.

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Furthermore, a successful comparison of the connectivity density between the samples is a rather

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strong indication of the potential of the approach. In fact, the connectivity density is calculated by

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the means of the Euler characteristic of the pore space and it is a volume weighted measure of the

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number of connected elements in a network [22]. As a consequence, similar connectivity densities

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mean that there is a similar number of connected elements per unit volume in the samples (real

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and virtual) under analysis.

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The volume fraction of the samples can be assessed either in 2D or in 3D, thus, explaining why

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the values of air void content apparent as areas in Fig. 6 don’t match the volume fractions seen in

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Table 2. For example, the measures of air void content shown in Figure 6 are obtained by image

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analysis, while with different algorithms this parameter can be computed for a 3D volume. The

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latter option is that used by BoneJ, and the results are very satisfactory due to the fact that the

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volume fraction of the virtual sample is very similar to that of the real sample using the same

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aggregate gradation. This suggests that by packing virtual stones with a specific gradation curve it

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is possible to obtain realistic air void contents. This result could be helpful for mixture design once

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the method is more thoroughly tested.

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It is also important to consider the tortuosity of the voids in the samples under analysis. In this

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paper, the tortuosity of the pores was calculated by skeletonizing the pore space and by dividing

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the length of the maximum path found by the Euclidean distance between its start and end points.

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The similar values of tortuosity obtained suggest that fluids should behave similarly in the virtual

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and the real materials. The lower tortuosity obtained for the virtual sample is reasonable, because

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in the virtual sample there are no binder and filler material, thus, the paths in the pore space are

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expected to be less tortuous than in the real sample. Tortuosity is of interest when studying

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phenomena such as water evaporation [4], drainage [23], or the use of air convection in asphalt

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pavements for energy harvesting purposes [24]. Currently, it is very complex to obtain large pore

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networks for computational studies due to the fact that the computational domains used for the

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study of fluid dynamics are usually obtained from X-ray CT scans. As a result, the size is

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constrained by the maximum size and discrimination allowed by the scanning equipment. On the

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contrary, when a computational method such as the one used in this paper is used, any arbitrary

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size and thickness can be obtained for the study of fluid flow in the pores. In principle, the

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discrimination of particle shape can also be obtained to any desired degree of complexity. In these

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cases, though, the available computational power is the limiting factor.

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Since the numerical results shown in Table 2 do not offer visual evidence of the goodness of the

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approach, the real and the virtual samples studied were joined in a composite model (see Fig. 7).

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In Fig. 7, two perpendicular cross sections of such composite model are shown to let the reader

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see that without the labels in the figure it would be very hard, if not impossible, to tell the real and

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virtual scans from one another. The 3D models that were joined were cut to the same size and the

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number of faces in each of them was made equal so that the data would be compatible.

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Finally, it is relevant to add that due to the lack of bitumen and filler in the model it is not possible

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to simulate very dense mixtures at the moment. The main reason for this is that with no bitumen

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and filler there will always be some relatively large (based on the mixture under analysis) portions

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of space that cannot be filled.

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4. Possible applications

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The proof-of-concept method described in this paper could potentially be used for a number of

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applications due to its apparent effectiveness and efficiency, e.g.:

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Design and analysis of asphalt mixtures: the authors expect that further developments of

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the approach presented in this paper could allow the study of a wide variety of stone

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assemblies with a range of stone shapes and sizes. Once the method is refined and

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thoroughly tested, the analysis of asphalt samples could be performed not only about the

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air pores but also about the aggregates. This is because when the stone assemblies are saved

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after the packing simulation each stone is exported as a separate entity with its position and

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rotation, thus, allowing further studies on the packing process. As a result, the method

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could assist in the creation of new and optimized asphalt mixtures or other granular

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materials.

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Computational fluid dynamics simulations: the modelling approach chosen allows the

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creation of 3D domains for computational analyses such as simulations of water, air, or

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multi-phase flow in the pores of granular materials.

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Improved study of contact forces between aggregates: it is expected that the modelling

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approach could lead to the study of contact forces between stones with realistic shapes as

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opposed to current studies that consider ideal spherical aggregates.

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Packing models: due to the flexibility of the approach chosen, the authors expect that the

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method could be used to study how objects with different shapes reach a random packed

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condition when placed inside a constrained volume.

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5. Conclusions

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In this paper, a proof of concept of a new method for packing aggregates was introduced and

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discussed. The following conclusions can be drawn:

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The method provides packed stone assemblies that visually resemble real stone

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assemblies.

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The morphological analysis of a virtual porous sample and of a real porous sample

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provided a first demonstration of the feasibility of the method.

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The use of a real gradation curve to generate a packing of virtual stones allowed the

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generation of an air void content very similar to that obtained in a real asphalt core.

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Stones with arbitrary shapes can be used to achieve a high degree of realism as opposed

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to stones with idealized shapes such as spheres. It is, however, important to pursue the

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study of real aggregate shapes to increase the verisimilitude of the results.

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For future work, further to the preliminary observations made on a single virtual sample, a

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possible validation of the modelling approach could be pursued by comparing a high number of

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virtual samples with different gradations and stone shapes and sizes to a high number of real X-

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ray CT scans.

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Acknowledgements

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The authors thank the University of Nottingham for the financial support provided for the Ph.D.

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of Andrea Chiarelli. New Lecturer's Award to Alvaro Garcia is also acknowledged.

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