Asphalt concrete reconstruction and computational

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Asphalt concrete reconstruction and

computational mechanical modelling using 3D

grid division

By:

Simón Espinosa León

Thesis advisor:

Silvia Caro Spinel, PhD.

Thesis Co-advisor:

Daniel Castillo, PhD.

Faculty of Engineering

Department of Civil Engineering

Bogotá, Colombia

August 2020

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Table of Contents Table of Contents .......................................................................................................................................... 2

Figure list ....................................................................................................................................................... 3

Table list ........................................................................................................................................................ 4

Abstract ......................................................................................................................................................... 5

Resumen ........................................................................................................................................................ 6

1. Introduction ........................................................................................................................................... 7

1.1. Past work and motivation ............................................................................................................. 8

1.2. AC specimen description ............................................................................................................... 8

1.3. General Objective ........................................................................................................................ 10

1.4. Specific Objectives ....................................................................................................................... 10

2. Methodology ....................................................................................................................................... 11

2.1. X-Ray CT scans reconstruction .................................................................................................... 11

2.2. Grid generation and volume fraction .......................................................................................... 13

2.3. Cutting planes .............................................................................................................................. 15

2.4. FE model partitions and material assignment from interpolation rule ...................................... 15

2.5. FE simulation under axial compression ....................................................................................... 18

2.6. The results of the response force of the AC specimen are analyzed and compared to those

obtained using the different grid divisions and material assignation rules. ........................................... 19

3. Results and analysis ............................................................................................................................. 20

3.1. Stress-Strain curve ....................................................................................................................... 20

3.2. Force and elastic modulus at the end of simulation ................................................................... 22

3.2.1. Force at the end of simulation ............................................................................................ 22

3.2.2. Elastic modulus at the end of simulation ............................................................................ 23

3.3. Computational cost ..................................................................................................................... 24

4. Conclusions .......................................................................................................................................... 26

5. Acknowledgements ............................................................................................................................. 28

6. References ........................................................................................................................................... 29

Appendix 1. Partition plane generator (.m) ............................................................................................ 31

Appendix 2. Specimen reconstruction and volume fraction calculation (.m) ......................................... 32

Appendix 3. Abaqus simulation script (.py) ............................................................................................. 33

Appendix 4. Abaqus results processor (.py) ............................................................................................ 35

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Figure list Figure 1. X-ray CT scans a) Img 75 b) Img 90 c) Img 110 ............................................................................. 10

Figure 2. Average volume fraction distribution .......................................................................................... 12

Figure 3. 2D Image grey-level histogram..................................................................................................... 12

Figure 4. Image reconstruction, a) Original, b) Color fix, c) Phase division ................................................. 13

Figure 5. Sample grid division n = 2 ............................................................................................................. 13

Figure 6. Group of cell reconstruction view with different number of divisions a) n=2, b) n=4, c) n=6, d) n=8,

e) n=15, f) n=30, g) n=40, h) n=70 ............................................................................................................... 14

Figure 7. 3 points for cutting plane ............................................................................................................. 15

Figure 8. FE model a) Partitions, b) Material assignment ........................................................................... 16

Figure 9. Volume fraction for a specific image (1 means aggregate, 0 means asphalt mortar) ................. 17

Figure 10. Interpolation rules to assign the mechanical properties of a cell a) 50/50 rule, b) 75/25 rule . 18

Figure 11. FE model constraints a) n=2, b) n=10, c) n=40 ........................................................................... 18

Figure 12. Stress-Strain curve for the specimen a) 50/50 rule, b) 72/25 rule ............................................. 21

Figure 13. Rule Stress-Strain comparison.................................................................................................... 21

Figure 14. Force at the end of the simulation a) 50/50 rule b) 75/25 rule ................................................. 22

Figure 15. Force at end of the simulation combined .................................................................................. 23

Figure 16. Young’s modulus at end of simulation ....................................................................................... 23

Figure 17. Computational cost to create the models a) Normal scale, b) Log scale ................................... 24

Figure 18. Computational cost to run the models ...................................................................................... 25

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Table list Table 1. Asphalt mixture design data ............................................................................................................ 8

Table 2. Coarse aggregate angularity and texture properties ...................................................................... 9

Table 3. Aggregate physical properties ......................................................................................................... 9

Table 4. Cube dimensions respective to n................................................................................................... 14

Table 5. Elements of the Prony series (normalized values), mortar viscoelastic mechanical properties. .. 17

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Abstract This study introduces a three-dimensional (3D) grid division approach to account for the internal

composition of asphalt concrete (AC) when performing finite element (FE) computational mechanics

simulations. This methodology, which is extension of an existing two-dimensional (2D) technique, initiates

by processing Computed Tomography (CT) scan data, to create a simplified reconstruction of an asphalt

concrete (AC) specimen. Once the sample has been reconstructed, it is divided into two main phases: i)

the coarse aggregate, and ii) the asphalt mortar. Then, the specimen is partitioned in homogenized cubic

cells to approximate the mechanical response of the sample. In this study, 70 consecutive images were

used to reconstruct a 70x70x70 mm AC sample. A grid was used to divide the sample into cubic cells and

approximate the aggregate fraction within the cell. Using two binary interpolation rules (50/50 and 75/25),

each cell was assigned a phase: either aggregate or asphalt mortar, depending on the limit value of each

rule. This procedure was done for 10 different partitions; (2, 4, 6, 8, 10, 15, 20, 25, 30, 40); the number of

cells corresponded to the number of partitions cubed, so there was a maximum of 64,000 cells, each

2.8x2.8x2.8 mm in size. To estimate a mechanical response from the simplified reconstructed specimen,

the AC model was then subjected to a constant displacement using the FE software Abaqus®. It was found

that the 50/50 rule converges with 8 divisions, and the overall stress reported in the specimen tends to

decrease as the number of divisions increase. For the 75/25 rule, the opposite was found, as the number

of divisions increase the recorded overall stress increased; also, the results for this rule did not converge

and the end measured stress where one order of magnitude smaller than those of the 50/50 rule. Finally,

the computational cost to build the finite element models follows an exponential increase in time and

required a very significant RAM capacity, while the computational cost and resources to run the AC model

was not significant.

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Resumen Esta investigación desarrolla una estrategia simple de particiones en tres dimensiones (3D) para describir

la composición interna de una mezcla asfáltica para simulaciones computacionales de elementos finitos.

Esta metodología proviene de un estudio en dos dimensiones. Inicia con la reconstrucción de tomografía

computarizada (CT) para obtener un espécimen de mezcla asfáltica. Posterior a la reconstrucción, se divide

en dos fases principales: i) el agregado grueso y ii) la matriz asfáltica. El espécimen se divide en celdas

cúbicas homogenizadas para aproximar la respuesta mecánica de la muestra. Para esto, se tienen 70

imágenes consecutivas utilizadas para recrear un espécimen de 70x70x70 mm de mezcla asfáltica. Este

cubo se pasa por una grilla cúbica y se calcula la fracción de agregado dentro de cada grilla. Para la

asignación de la fase a cada celda específica se tienen dos reglas de interpolación binaria, la regla 50/50 y

la regla 75/25. A partir de la regla de interpolación binaria seleccionada se le asigna a cada celda las

propiedades de un material que representa una de las dos fases. Este procedimiento se realiza para 10

particiones: (2, 4, 6, 8, 10, 15, 20, 25, 30, 40) y el número de celdas generadas es el número de particiones

al cubo para un máximo de 64,000 celdas creando cada celda de 2.8x2.8x2.8mm. La respuesta mecánica

se calcula tras simular la muestra bajo un desplazamiento constante en el programa de elementos finitos

Abaqus®. Se encontró que la regla de interpolación de 50/50 converge rápidamente, a partir de las 8

divisiones los cambios no son significativos. Adicionalmente, a medida que el número de divisiones

aumenta, el esfuerzo disminuye. Lo contrario sucede con la regla de 75/25, esta no converge y a medida

que el número de divisiones aumenta el esfuerzo aumenta también. Finalmente, el costo computacional

asociado a la creación de los modelos tiene un aumento exponencial en el tiempo requerido.

Adicionalmente, requiere de una capacidad de RAM muy alta para general los modelos mientras el costo

computacional para correr el modelo no es significativo.

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1. Introduction Asphalt concrete (AC) is a highly complex heterogeneous material composed of three main phases;

the coarse aggregate, the asphalt mortar or asphalt matrix (i.e., blend of asphalt binder and the fine

portion of the aggregates) and air voids. For the study, only the first two phases will be considered, to

reduce the complexity of the multiphase material. The properties of the AC are dependent on the

proportion of each phase and the disposition of the microstructure. Therefore, analyzing an asphaltic

sample at a microstructural level in three dimensions carries an extremely high computational cost mainly

due to all the information it contains. Therefore, we seek to find a binary method of interpolation between

both phases that reduces the computational cost and efficiently analyze the overall response of a three-

dimensional asphalt sample.

For some years, there has been specialized software for the computational representation of

composite materials. An example of this work is the one developed by Castillo et al. (2015), where the

effect of particle size and air voids on the initiation and propagation of the fracture is investigated. In other

application, through the generation of probable microstructures, Castillo and Al-Qadi (2019) presented

the computational model of a two-dimensional asphalt sample, exploring potential interpolations of the

properties of the asphalt matrix and aggregates to achieve an acceptable computational result in the

mechanical response of the mix at a lower computational cost than when individually modeling the

aggregate particles and the asphalt matrix through grid divisions. Although this model did not include air

voids, the results were very good compared with the expected results. Furthermore, since the generation

of the samples is digital, this model allows various simulations to be done at a very low cost.

Another work in this direction is that conducted by Chen et al (2017). The authors generated random

structures of the aggregates in two and three dimensions in which they accounted for a fine matrix, the

particle size distribution of the aggregates, the air voids, among other properties. This method has the

advantage of giving high flexibility in numerical simulation as it is of digital origin. In addition, it allows

relationships to be identified by varying one of the components of the mixture. Additionally, Chen et al.

(2017) investigated the dynamic modulus of the asphalt sample while Castillo and Al-Qadi (2019)

investigated the mechanical response of a progressive load.

However, although the mentioned studies are based on the digital generation of an AC sample, the

idea in this project is to study a real sample, as Dai (2011) did. The benefit of using an actual three-

dimensional (3D) sample is the possibility of characterizing the actual internal structure of an asphalt mix

and not only random predictions, as in the cases mentioned. However, a disadvantage of having a real

sample is that, unless the work includes multiple different specimens, the model will be limited to the

response of the analyzed sample.

Another disadvantage of using an actual sample is the reliance on the sample 3D reconstruction

process based on individual two-dimensional (2D) images obtained through X-Ray Computed Tomography

(CT) techniques, because the final quality of the computational simulations is based on what it is

reconstructed. Once the reconstruction of the sample has been done, it is assumed that it is the best

representation of the scanned AC specimen. Everything done after the reconstruction is done on the

reconstructed information, so if this lacks any data or has incorrect information the error will be carried

forward during the study.

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1.1. Past work and motivation

This project seeks to improve and complement the research carried out by Castillo and Al-Qadi (2019),

through the generation of 3D models (not 2D as the reference) with binary interpolated properties and

based on CT scans of a real AC sample. The objective is to find a reasonable trade-off between a more

accurate representation of the internal distribution of phases of the AC sample and a lower computational

cost. During the study, the computational cost will be considered to be proportional to the number of grid

divisions (n) used to represent the internal phases of the AC.

During the research, only two phases will be taken into account: i) the coarse aggregate, and ii) the

asphalt mortar, also called the fine aggregate matrix; i.e., combination of asphalt binder and aggregates

passing the sieve No. 16 (1.18 mm). The way to differentiate the phases is a pixel threshold. By dividing

the model using a 3D grid and estimating a unique property within every cell/grid element, the complexity

of the model is reduced. Finite Element (FE) methods will be used to process the reconstructed specimen

with the assigned properties. To do so, the commercial FE software Abaqus® was used to re-construct the

model and run the simulations. The simulation will be done with a controlled displacement and the mesh

size was be the same as the grid which is a compromise between accuracy and computational cost that

Castillo and Al-Qadi (2019) did not take. The decision to have the mesh size the same size as the grid was

because it was thought that the computational cost would be too high otherwise, Castillo and Al-Qadi

(2019) had a 2D sample so they did not have to compromise accuracy for computational cost.

1.2. AC specimen description

The original specimen used for this study was a fine-graded soft limestone superpave type C. The

asphalt mixture design data may be observed in Table 1. From the percent passing it is observed that the

maximum aggregate size is 1 in or 25mm. Table 2 shows the angularity and texture properties for the

coarse aggregate used. Finally, Table 3 identifies the aggregates physical properties.

Table 1. Asphalt mixture design data

Parameter Sieve Size

Sieve No. (mm)

Percent

Passing (%)

Pb (%) 5.2 1 in (25) 100

Va (%) 6.7 3/4 in (19) 99

VMA (%) 13.7 1/2 in (12.5) 95

VFA (%) 70.9 3/8 in (9.5) 92.5

Gmm 2.515 No. 4 (4.75) 77.5

Gsb 2.653 No. 8 (2.36) 43

Binder

Grade PG 76-22

No. 16 (1.18) 30

No. 30 (0.600) -

Binder

Specific

Gravity

1.02

No. 50 (0.300) -

No. 200 (0.075) 6

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Table 2. Coarse aggregate angularity and texture properties

Aggregate Type Angularity Texture Remark

Soft Limestone (SL) 2195 80 Sub-rounded and polished aggregate

Angularity Index Range:

Rounded (0-2100), Sub-rounded (2100-4000),

Sub-angular (4000-5400), and Angular (5400-10,000)

Texture Index Range:

Polished (0-165), Smooth (165-275), Low Roughness (275-350),

Moderate Roughness (350-460), and High Roughness (460-1,000)

Table 3. Aggregate physical properties

Aggregate Tests Soft Limestone

(SL)

Test

Method

Los Angeles % Wt.

Loss-Bituminous 34 Tex-410-A

Mg Soundness-Bituminous1 41 Tex-411-A

Mg Soundness-Stone2 29 Tex-411-A

Polish Value 25 Tex-438-A

Micro-Deval % Wt. Loss-

Bituminous 19.7 Tex-461-A

Fine Aggregate Acid

Insolubility 2 Tex-612-J

Micro-Deval %Wt. Loss 20.4 Tex-461-A

1Using HMAC Application Sample Size Fractions 2Using Other Applications Sample Size Fractions

The actual dimensions of the scanned specimen are: 148mm high and 100mm diameter.

Additionally, it contains 7% of air voids, 25% of mastic and 68% of aggregate. The CT scanned image

resolution was 512 x 512 pixels with an equivalent of 100 x 100mm image and a 1mm interval between

images. Figure 1 shows an example of the CT scans on three different locations and how the actual

specimen looks when digitally sliced.

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a) b) c)

Figure 1. X-ray CT scans a) Img 75 b) Img 90 c) Img 110

1.3. General Objective

Explore the use of grid divisions on a 3D AC specimen reconstructed from a real sample using X-ray CT

scans and obtain a reasonable trade-off between accuracy and computational cost.

1.4. Specific Objectives

1. Find the most appropriate pixel threshold to reconstruct the asphalt concrete specimen using

a grid division technique.

2. Understand the behavior on the mechanical response of the computational models when

using a binary interpolation rule for assigning the type of material within each cell of the

specimen.

3. Find the most appropriate range of divisions (n) where the trade-off between result accuracy

and computational cost is the lowest.

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2. Methodology For the study, five steps were followed to accomplish the result and obtain a response of the AC specimen.

The steps are the following;

1. The sample is reconstructed from 2D images obtained through CT-Scans. Its individual pixels

are analyzed and the phase is assigned depending on the pixel value. This step was done in

MatLab and the code was developed inhouse and can be found in Appendix 2.

2. A grid is superimposed on the reconstructed sample. The sample is divided into cubic cells and

the volume fraction between phases is calculated. This was done in the same MatLab code

explained in the previous step. This algorithm generates an output text file with a coordinate

for each cell and the respective volume fraction.

3. Cutting planes are generated to form the cell partitions for the FE model prior to the material

assignation. Each cutting plane is defined by three points, this is done with a MatLab algorithm

that can be found in Appendix 1. A text file is generated with all three coordinates for each

plane.

4. Create an FE model, generate the partitions using the cutting planes, and assign a material to

each cell depending on the volume fraction and the specific material interpolation rule chosen.

This is can be seen in a Python code in Appendix 3.

5. The FE AC model is subjected to axial compression loading, and the reaction force on top of

the specimen is calculated. This is within the same code as the previous step.

6. The results of the response force of the AC specimen are analyzed and compared to those

obtained using the different grid divisions and material assignation rules.

The mechanical properties for each cell are assigned using a binary interpolation rule in Step 4. The

reason the interpolation rule was chosen to be binary was to reduce computational cost and simplify the

material property to one of the two selected phases (i.e., a grid is considered to be aggregate or mortar,

depending on the aggregate fraction and the assignation rule). Castillo and Al-Qadi (2019) showed

promising results with a binary rule but it required some adjustments to improve the results. Each step

and interpolation rules will be explained in depth in this section.

2.1. X-Ray CT scans reconstruction

To accomplish the sample reconstruction, it was important to find the most appropriate way to

reconstruct the images. The decision was to reconstruct the scanned AC specimen using a threshold value

of pixel color. The criteria to decide the best threshold range was qualitative, because to the naked eye it

is very easy to differentiate between phases. Therefore, the threshold range was slowly changed and the

image reconstructions where compared. Once a range was decided, a total volume fraction of aggregates

within the specimen was calculated, and it was expected to be between in an interval of 60 and 70% (i.e.,

60 to 70% of the total volume of the specimen consists of aggregates). In a previous study, Zelelew and

Papagiannakis (2011) found that the threshold for coarse aggregate should be values above 158 in a grey

scale (0-255). However, such a value should not be taken as an exact indication. Many variables will affect

the pixel values and frequencies (i.e. the histogram) of a given image, including the technique of image

acquisition, quality of lightning, resolution, and previous image manipulation. The overall volume fraction

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calculated for the analyzed AC specimen was 61.67%, and the distribution between the scanned 2D slices

can be observed in Figure 2.

Figure 2. Average volume fraction distribution

The selected threshold range for aggregate was between 165 and 215. Figure 3 shows the complete

histogram for a 2D sample image. The upper limit was set to avoid white values that represent glares

within the scans. In Figure 4 the original image is observed, followed by a color equalization to better

observe the phases and the reconstruction with two material phases. The pixels in yellow are considered

to be part of an aggregate, and assigned the value 1, while the rest of the pixels have the value 0 to

facilitate the future steps of calculating the volume fraction and reduce complexity. Appendix 2 contains

the full written code in MatLab to adjust and reconstruct the images.

Figure 3. 2D Image grey-level histogram

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a) b) c)

Figure 4. Image reconstruction, a) Original, b) Color fix, c) Phase division

2.2. Grid generation and volume fraction

For this step, the idea is to impose a grid into the reconstructed volume and obtain an even number

of cells. It was decided to use cubic cells, to simplify the problem and reduce the computational cost. Each

cell with the same number of pixels in average. The grids where generated depending on the number of

partitions and the selected number of partitions (n) was (2, 4, 6, 8, 10, 15, 20, 25, 30, 40). The number of

cells is calculated as n3. Notice that as n increases, the n values are more distanced to reduce the

computational cost and because it is expected to have a lower change in the results with higher n’s while

with lower n’s the change is expected to be more significant.

The size of each cell is calculated as the total number of pixels (359 on each edge) divided by the

number of partitions. That result is rounded down and considered the standard width of the cell, then, the

remainder of the division is calculated and those are considered additional pixels. Additional pixels are

divided between the first cells creating slightly wider cells. The same procedure is done with the number

of 2D images to give a thickness and the initial cells will be slightly thicker, each image is assumed to be 1

mm thick. The procedure is considered appropriate as the width of a pixel is less than 0.2 mm. Once all

the cells have been generated, the values are added and then divided by the total number of pixels. This

will generate the fraction of the cell that corresponds to the aggregate phase as those pixels had a 1

assigned. A coordinate to the center of the cell is calculated and those values are recorded. Figure 5 shows

an example of how the sample is observed after it is reconstructed and a grid is generated (n=2, in this

case).

Figure 5. Sample grid division n = 2

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A specific image may not be compared with a reconstructed group of cells, as they do not represent a

2D image but a 3D stack of images. When the number of divisions is 2, a 2D view will show a stack of 35

original 2D scanned images, when n is 30, a 2D view will represent a stack of two scanned 2D images.

Figure 6 shows how the 2D view of groups of cells change as n changes. The initial images, before Figure

6f, do not have a proper representation of the microstructure. Figure 6h shows a better representation of

the microstructure as it only has one image and it is not a stack of consecutive images.

a) b) c) d)

e) f) g) h)

Figure 6. Group of cell reconstruction view with different number of divisions a) n=2, b) n=4, c) n=6, d) n=8, e) n=15, f) n=30, g) n=40, h) n=70

Table 4 presents the average length of each edge of a cell for the divisions presented in Figure 6. It is

evident that as the number of divisions increases, the reconstructed model is clearer as the cell is also a

better representation of the internal microstructure of the AC sample. The maximum size for a coarse

aggregate for the specimen used is 1 in or 25mm, so as cells are smaller the shape is better described and

not only the tendency to show the general location of the aggregates. An additional challenge is that the

cells contain more than one image, so it is not possible to make a proper comparison between

reconstructed images with different number of divisions or with the actual scan.

Table 4. Cube dimensions respective to n

n Average cube length [mm]

2 35

3 23.33

6 11.67

8 8.75

15 4.67

30 2.33

40 1.75

70 1

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2.3. Cutting planes

The FE model will create a LxLxL cube that must be partitioned. To do so, it requires 3 points that

reference a partition plane. Partition planes are used to create the cubic cells in the FE model previous to

the material assignment. The reference points are defined as;

1. The first point is on one of the axes at a multiple of the distance L/n where L is the length of the

cube (70mm) and n is the number of partitions. i.e. (0, i*L/n, 0)

2. The second point is directly above the first point. It is on the same axes but it has an offset of L on

another axis. i.e. (0, i*L/n, L)

3. Finally, the third point has an offset from the second point on the other perpendicular axes and it

marks the third coplanar point of the partition plane. i.e. (L, i*L/n, L)

All three points may be observed on Figure 7 where ‘i’ is a number between 1 and n-1. The generation of

these points is within a cycle that can be seen in Appendix 1. Each time the cycle ends, i increases its value

creating the next set of reference points with an offset of L/n from the previous set, until it reaches the

desired number of sets (n-1). Then, it starts generating reference points on a perpendicular plane until all

three reference plains (xy, xz and yz) have n-1 sets of reference points. A minimum of 3 partition planes

are needed to make cubic cells, in Figure 5 3 planes are used each placed perpendicular to a reference

plane (xy, xz, yz).

Figure 7. 3 points for cutting plane

2.4. FE model partitions and material assignment from interpolation rule

The FE model is a 3D 70x70x70mm deformable body cube in Abaqus that will be assigned a material

(aggregate or asphalt mortar) on each partition, depending on the aggregate volume fraction of each cell

(calculated on Step 2). The partitions are creating by cutting the cube through n-1 number of planes on

each axis. The cutting planes are defined by three points that where calculated on the previous step. Figure

8 shows the FE model where Figure 8a has the raw partitions that generate 8 cells, followed by Figure 8b

that has the material assigned to all 8 cells (the color represents a material has been assigned). It is

important to take into account that all the cells are the same size, they don’t have the problem associated

with the additional pixels mentioned on Step 2.

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a) b)

Figure 8. FE model a) Partitions, b) Material assignment

The cell material assignment depends on the chosen interpolation rule. For the study, two binary rules

were chosen, the 50/50 rule and the 75/25 rule. The idea behind each rule is to set a volume fraction limit

(50 and 75 respectively): a cell with aggregate fraction below that limit will be assigned as asphalt mortar,

and if the value is higher than the limit it will be taken as aggregate. The aggregate volume fractions for

each cell are known at this point as they were calculated previously on Step 2. For the 50/50 rule, the limit

of the aggregate fraction per cell is 0.50, while for the 75/25 rule the limit is 0.75. In the first case, the rule

means that if the aggregate volume fraction is lower than 0.5, then asphalt mortar is assigned, while if it

is higher than 0.5, the cell is assigned aggregate properties. In the second case, cells with aggregate fraction

values lower than 0.75 are assigned asphalt mortar properties, while cells with fraction values larger than

0.75 are assigned aggregate properties. These rules were chosen due to the promising results published

by Castillo and Al-Qadi (2019). They found that this type of true-false criterion shows good results when

modeling a low-density AC and proper results when modeling a high-density AC. This study seeks the most

appropriate limit, so it evaluates two values.

The materials used for the FE simulation are the same used by Castillo and Al-Qadi (2019). The coarse

aggregate is an elastic material with a Young’s modulus of 25 GPa and a Poisson’s coefficient of 0.16. The

asphalt mortar has an elastic instantaneous axial Young’s modulus of 112 Pa and a Poisson’s coefficient of

0.4. It also has a viscoelastic behavior, described using a Prony series, some of which parameters are

normalized for their use in Abaqus, as presented in Table 5. The viscoelastic properties were obtained at

25°. Details on the testing and further information can be found in Kim et al (2005).

Eq 1 shows the Prony series, nevertheless, Abaqus uses 𝑔𝑖, that is defined as 𝐺𝑖/𝐺0. More information

about viscoelastic materials and its modeling in Abaqus can be found at in the Abaqus manual (Abaqus

documentation, 2020).

𝐺(𝑡) = 𝐺0 − ∑ 𝐺𝑖 [1 − exp (−𝑡

𝜏𝑖)]

𝑛

𝑖=1

, and 𝐺∞ = 𝐺0 − ∑ 𝐺𝑖

𝑛

𝑖=1

(Eq 1)

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Table 5. Elements of the Prony series (normalized values), mortar viscoelastic mechanical properties.

i g1 Prony ki Prony1 τi Prony

1 0.653333 0 0.0015716

2 0.2277331 0 0.0163883

3 0.0968486 0 0.0994745

4 0.0185536 0 0.7358037

5 0.0029643 0 6.7740964

6 0.0004768 0 65.786517

7 7.54E-05 0 643.36493

8 1.19E-05 0 6514.2857

9 2.25E-06 0 80976.19

Figure 9 displays the aggregate fraction with n = 70 for the image shown in Figure 4. The color bar on

the right describes the color code used, where 1 corresponds to complete aggregate and 0 is no aggregate

(asphalt mortar). Figure 10 shows how the interpolation rules work. Figure 10a represents the 50/50 rule

while Figure 10b has the 75/35 rule. In addition, both plots have an example with a volume fraction of 0.25

and another volume fraction of 0.6. It may be appreciated that for both cases the 0.25 volume fraction will

be assigned the mortar phase while the 0.6 volume fraction will have an aggregate phase in the first rule

and a mortar phase with the second rule. Appendix 3 has the full code to generate the partitions from the

points calculated in the previous step and then to assign the material to each individual cell depending on

the volume fraction calculated on Step 2 and the chosen binary rule.

Figure 9. Volume fraction for a specific image (1 means aggregate, 0 means asphalt mortar)

1 For 3D simulations it is recommended to use bulk properties (𝑘𝑖) identical to 𝑔𝑖. Nevertheless, as all the results where

calculated with no bulk properties. It was tested for 8 divisions and the results showed a maximum difference under 0.0005%.

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a) b)

Figure 10. Interpolation rules to assign the mechanical properties of a cell a) 50/50 rule, b) 75/25 rule

2.5. FE simulation under axial compression

Once all the previous steps have been done, the model is created in Abaqus® and ready to enter the

simulation. Two simulations are carried out for each n: one for each material assignation rule. Therefore,

a total of 20 simulations were carried out. The computation mechanical test corresponds to the application

of an axial compression load, under displacement-controlled conditions. The compression is at constant

vertical displacement during 0.5s with a total displacement of 0.1mm. The base has a full displacement

constraint and the top is only free to move vertically. The superior face is free to move vertically as the

controlled displacement is carried out in this face. A rigid body was not used to generate a distributed load

as the FE software did not allow the assembly of a rigid 3D plate with the 3D deformable body, for that

reason it was used a constraint instead and it gives the same results. In that way, the movement is

controlled, and the simulation is comparable with an axial load test with the superior face glued to the top

and the inferior face glued at the bottom. Figure 11 shows the mentioned boundary conditions for the

model with n = 2. The mesh size is the same as the cell size for each case.

a) b) c)

Figure 11. FE model constraints a) n=2, b) n=10, c) n=40

All the results are measured on each node of the superior face. The displacement is linear and identical

for all nodes as that is the controlled variable, then the reaction force differs on each node.

Page 19 de 35

2.6. The results of the response force of the AC specimen are analyzed and compared

to those obtained using the different grid divisions and material assignation

rules.

Using the code in Appendix 4 the results are compiled and then opened in MatLab to be processed.

The reaction force for every node are added for each time of the time interval displacement to end up

with only one reaction force result. That force is then divided by the area of the superior face (70x70 mm)

to calculate the stress in the superior face at each interval of the controlled displacement. This was done

for each grid division and binary rule. During this step, it would have been important to compare the

calculated stress results with a laboratory testing of the specimen stress.

Page 20 de 35

3. Results and analysis The results and analysis will be divided into three sections, the first section is the Stress-Strain curves

where the two rules are compared, and the tendencies are described. The second section is the force and

elastic modulus at the end of the simulation for both rules, here it will be observed the convergence of the

end results and the change in the elastic modulus. Finally, the third section will show the time

computational cost by comparing the time to create the model and to process it. All the measured results

were taken on the nodes of the superior face.

As explained before, the mechanical simulation was done in Abaqus CAE®, and it consisted performed

by applying a controlled compressive displacement of 0.1 mm during a 0.5 s interval, the reaction force

was calculated by adding the force on each node within the superior face. The reported stress was

calculated by adding all the reaction forces from the superior face and dividing it by the total area of that

face. The strain is the displacement divided by the initial height of the specimen.

3.1. Stress-Strain curve

The Stress-Strain curves were constructed using as input the applied displacement and the sum of the

reaction force registered at all nodes of the superior face. The resulting curves for the first rule (50/50)

may be seen in Figure 12a, while the results for the second rule (75/25) are observed in Figure 12b.

Opposite tendencies are observed in Figure 12. Each time n increases on the 50/50 rule, the stress line

tends to decrease, as seen in Figure 12a. On the contrary, while n increases on the 75/25 rule, the stress

line tends to increase, as appreciated on Figure 12b. The first trend was expected because when n is low

the number of cells is low, and each cell should have the average aggregate fraction (61%), making all of

the cells be simulated as aggregate. Then, as the number of divisions increase, more cells start to take the

properties of the asphalt mortar, generating the stress to fall. The contrary happens for the second rule,

low number of divisions makes all the cells take the properties of the mortar and, as the divisions increase,

the stress increases as more cells take the properties of aggregate.

It is clear that the stress in the 50/50 rule converges when the number of divisions reaches 8, with 512

cells. After this value, the stress results start to have less significant changes (i.e., less than 3% between

consecutive n values); this is evident as the curves get closer together. This is different to what seems to

happen with the 75/25 material assignation rule, where results seem to converge near 6 or 8 divisions but

then it disperses again, and the stress results are significantly different from the next one (i.e. more than

34% between consecutive curves). The reason for the 75/25 rule to have such an unexpected behavior

could be due to the uneven relationship between phases, as the number of divisions increases and the

cells are divided, the individual cell can change the phase and cause the mentioned behavior.

Page 21 de 35

a) b) Figure 12. Stress-Strain curve for the specimen a) 50/50 rule, b) 72/25 rule

When comparing the results between the two rules, a significant discrepancy may be observed.

Figure 13 presents the results from Figure 12 within the same figure, and it is clear that the order of

magnitudes differ and the trends are different, they don’t seem to converge. It may also be seen that as

the number of divisions increase, the results of both rules tend towards each other, the second rule faster

than the first rule as the first rule is converging. Also, it is clear that the behavior observed seems more

elastic than viscoelastic, Figure 12a have all the curves seem completely elastic while Figure 12b have a

more viscoelastic behavior but still very elastic.

Figure 13. Rule Stress-Strain comparison

If the binary interpolation rules are defined as X/(100-X), where X is the limit aggregate fraction, it

could be hypothesized that when chosen an X near the mean aggregate fraction (61% for this study) the

variability would be very low and it would converge with a small number of divisions. This does not mean

it is the correct interpolation rule, to decide which is the most appropriate rule it is important to test inside

Page 22 de 35

a laboratory the real specimen after it is scanned. This was not possible for the study as the CT images

were provided by Michigan State University.

3.2. Force and elastic modulus at the end of simulation

3.2.1. Force at the end of simulation

The convergence and trend of a simulation can be analyzed by taking a look at the results at the end

of the simulation. The computational trade off will be examined as well as it can be seen how results

change with the number of divisions. The results seen in Figure 14a show how the change in the force is

very low for the material assignation 50/50 rule after n is 8, and almost insignificant between 30 and 40

divisions (i.e., less than 4%). The behavior is very similar to that in Figure 8 from Castillo and Al-Qadi (2019)

study. The results have converged and an increase in divisions does not have a significant change in the

force.

Figure 14b shows a different tendency. First, it does not show a clear convergence as it did on the

previous figure. Second, the force tends to increase and not to decrease, also there is a peak around 8

divisions and then the force decreases until 15 divisions, and then the force starts increasing again. More

simulations are required to find the convergence and analyze the behavior. It is interesting to see how on

Figure 15 the end force for both interpolation rules would seem to be between them, as n increases even

when the convergence has not been met for the second interpolation rule.

a) b)

Figure 14. Force at the end of the simulation a) 50/50 rule b) 75/25 rule

Page 23 de 35

Figure 15. Force at end of the simulation combined

To get an actual comparison it is important to test the real specimen under the same simulated

conditions to have an understanding of what the behavior should look like. The test must be done at 250

and with a constant displacement with the bottom and top face glued to the testing machine to get the

same constraints as the simulated ones. The results from the simulations will be slightly more rigid as the

simulation does not include air voids and the real specimen does.

3.2.2. Elastic modulus at the end of simulation

The elastic modulus is calculated from the values observed in Figure 15. As expected, the modulus

results follow the same trend as the force at the end of the simulation from Figure 15 as they are all divided

by the same strain. Conventional AC have Young’s Modulus between 2,500 and 4,000 MPa and as observed

in Figure 16 when the number of divisions is 40 the modulus is around 5,000 MPa. The fact that no air

voids are considered could cause de increase in the rigidity of the sample. Also, it shows the same results

as section 3.1 where the modulus with n = 2 is that of an aggregate. The modulus was calculated as the

response was almost linear as seen in Figure 13.

Figure 16. Young’s modulus at end of simulation

Page 24 de 35

3.3. Computational cost

It is expected to see an increase in computational time as the number of divisions increase, that is

what is observed on Figure 17. It is clear that the increase in time is not significant until the number of

divisions reaches 20; as appreciated in Figure 17a, the increase from 20 to 25 is very substantial (i.e. more

than 300%) and each increase after that is more considerable. Thus, Figure 17b shows the same values

using a logarithmic scale and it is now evident that it describes the tendency much better. For that reason,

is important to find the most appropriate number of divisions because the computational time and

resources needed increase exponentially as the number of divisions increase.

For this study, it was planned to have 3 additional divisions, 50, 60 and 70. However, it was not possible

to generate the model for those divisions and in the process, the RAM (Random Access Memory) in use

for the generation of the model without the use of a GUI (Graphical User Interphase) was more than 80Gb.

The resources needed for the generation of the models are beyond those of a conventional computer and

the virtual machine used had not been able to complete the missing models for the time of this document.

This is similar to the computational challenge described by You et al (2012).

a) b)

Figure 17. Computational cost to create the models a) Normal scale, b) Log scale

The computational cost to run the models was not significant compared to the time required to

create the models. It is seen in Figure 18 that the run time is under 20 minutes for all but one simulation.

The last simulation for the first rule had an unexpected peak with a measured time slightly over one hour,

while the corresponding simulation with the second rule did not exceed 20 minutes. That increase in time

can be attributed to a wide variety of factors, like a parallel process or an internal error of the virtual

machine or other reasons that will not be discussed here. However, it is important to mention that it is an

outlier (over 87% difference) and not the expected value or trend (around 16%) between the two rules for

the same number of divisions. This is expected, as the procedure is the same and the model should be

identical with different material assignment rules. Once the model has been created, the computational

cost is not significant, and the resources required are those of a conventional computer.

Page 25 de 35

Figure 18. Computational cost to run the models

The research vice-president of Universidad de los Andes conceded a virtual machine with 256 Gb of

RAM and 24 Cores of CPU. The model for 50 and 60 divisions has been running for over 14 days and it has

not finished, while the model with 70 division crashes after 5 days. The CPU required is not significant

compared with the RAM to create the models. It would be interesting to see if the results change if the

mesh used is not the same size as the cells but much smaller, theoretically the precision should increase

and the computational cost to run as well and as mentioned, the computational cost to run is not

significant as initially hypothesized.

Page 26 de 35

4. Conclusions An AC specimen scanned using X-Ray CT techniques was digitized using a threshold for the pixels and

then divided into grids to determine the mechanical response of each cell or division. Only two main

phases where considered for the AC specimen: coarse aggregate and asphalt mortar. The reconstructed

cube was divided into cubic cells and each cell was assigned a phase depending on the aggregate fraction

measured during the digitalization and the interpolation rule. The number of divisions to create the cells

where (2, 4, 6, 8, 10, 15, 20, 25, 30, 40) and the number of cells is the number of divisions to the cube. The

interpolation rules chosen followed a Boolean logic, a different limit was chosen for each rule, 50% for the

first rule and 75% for the second rule.

Binary interpolation rules were chosen as they require low computational cost to run due to the

simplification in the material properties. Therefore, the computational challenge is associated with the

creation of the model and with the generation of all the required partitions. This challenge is associated

with all grid division methodology. The increase in the number of divisions produces as a result an

exponential increase in the time to create the model. Thus, a powerful machine is required to generate

the models due to the requirement of over 80Gb of RAM for the program to run. Consequently, the

compromise between the results and computational cost is very important, as the computational cost

increases rapidly with small changes in the number of divisions.

After the cells were assigned with a material property, the model was implemented in FE model and

a displacement-controlled axial compression load was applied on the surface of the specimen. The stress

measured on the top face as a result of the applied loading condition was analyzed, as well as the reaction

force measured on the top face at the end of the simulation and the computational time to create and run

each model.

The results show that 50/50 material assignation rule converges rapidly with increase in the number

of grid divisions, opening the possibility to make decisions on the trade-off between the results and the

computational cost. On the contrary, the 75/25 rule did not converge and show a trend of rapid increase

in the reaction force. Due to this behavior it is not possible to make a decision of compromise between

results and computational cost, thus, more results are required to find a convergence. It was observed

that the end reaction force of the actual specimen should be between 7 and 40kN as those are the limits

imposed by the two rules.

Although the two interpolation rules did not converge to the same stress or cross each other, they

showed a tendency in the way the curves move as the number of divisions increase and the microstructure

is better represented. It was found that the 50/50 interpolation rule does not give enough weight to the

aggregate, as the stress decreases with the increase in divisions, contrary to what happened with the 75/25

rule that gave too much weight to the aggregate and not enough to the mortar phase, that made the stress

increase as the divisions increased. The results show that the correct interpolation rule must be one in

between the two analyzed rules to best describe the microstructure. Nevertheless, it is necessary to have

the real response of the specimen obtained from the laboratory or from 3D simulations using actual

aggregate particles, to accurately choose the interpolation rule.

Even when the results only show an approximation to the actual response, it is important to see the

capacity of the binary interpolations to converge and reduce its computational cost, not only by describing

the multiphase material using grid divisions but also buy using a low number of divisions to get an

Page 27 de 35

appropriate result. The benefits of using binary interpolation rules facilitated the model creation as only

two materials and two sections had to be created and not a specific material for each cell.

Even though grid divisions and binary interpolation rules reduce computational time, the creation of

the models can take several tens of thousands of minutes to generate and require highly capable computer

with an important capacity to successfully accomplish the job. The two most important aspects of grid

divisions are:

7. A small n can be identified which represents the real/expected behavior of a specimen.

8. Better understand the aggregate distribution to then generate random specimens with a small

number of divisions.

Recommendations for future studies on this topic include:

1. Get the response of a real specimen in the laboratory.

2. Find the best binary interpolation rule to assign the properties.

3. Understand the aggregate distribution

4. Implement a program that can position the material stochastically in 3D

Page 28 de 35

5. Acknowledgements The author would like to thank Dr Silvia Caro and Dr Daniel Castillo for the immense support towards

the development of this study and the generosity sharing their knowledge and experience. The author

would like to thank the Universidad de Los Andes research vice-president for the conceded resources and

all the computational power required for the study. Finally, to Dr. Emin Kutay from Michigan State

University for providing the raw CT-images and allowing its use.

Page 29 de 35

6. References Abaqus documentation. (2020). Time domain viscoelasticity.

Caro, S., Castillo, D., & Masad, E. (2015). Incorporating the heterogeneity of asphalt mixtures in flexible

pavements subjected to moisture diffusion . International Journal of Pavement Engineering .

Caro, S., Castillo, D., & Silva, M. S. (2014). Methodology for Modeling the Uncertainty of Material

Properties in Asphalt Pavements. Journal of Materials in Civil Engineering.

Caro, S., Castillo, D., Darabi, M., & Masad, E. (2016). Influence of different sources of microstructural

heterogeneity on the degradation of asphalt mixtures. International Journal of Pavement

Engineering.

Caro, S., Masad, E., Bhasin, A., & Little, D. (2010). Coupled Micromechanical Model of Moisture-Induced

Damage in Asphalt Mixtures. Journal of Materials in Civil Engineering.

Caro, S., Masad, E., Silva, M. S., & Little, D. (2010). Stochastic micromechanical model of thed eterioration

of asphalt mixtures subject to moisture diffusion processes. International Journal for Numerical

and Analytical Methods in Geomechanics.

Castillo, D., & Al-Qadi, I. L. (2019). Mechanical modelling of asphalt concrete using grid division.

International Journal of Pavement Engineering.

Castillo, D., & Caro, S. (2014). Effects of air voids variability on the thermomechanical response of asphalt

mixtures . International Journal of Pavement Engineering.

Castillo, D., & Caro, S. (2014). Probabilistic modeling of air void variability of asphalt mixtures in flexible

pavements. Construction and Building Materials.

Castillo, D., Caro, S., Darabi, M., & Masad, E. (2015). Studying the effect of microstructural properties on

the mechanical degradation of asphalt mixtures. Construction and Building Materials.

Castillo, D., Caro, S., Darabi, M., & Masad, E. (2016). Modelling moisture-mechanical damage in asphalt

mixtures using random microstructures and a continuum damage formulation. Road Materials

and Pavement Design .

Castillo, D., Caro, S., Darabi, M., & Masad, E. (2017). Influence of aggregate morphology on the mechanical

performance of asphalt mixtures. Road Materials and Pavement Design.

Chen, J., Wang, H., & Li, L. (2017). Virtual testing of asphalt mixture with two-dimensional and three-

dimensional random aggregate structures. International Journal of Pavement Engineering .

Chen, J., Wang, H., Dan, H., & Xie, Y. (2018). Random Modeling of Three-Dimensional Heterogeneous

Microstructure of Asphalt Concrete for Mechanical Analysis. Journal of Engineering Mechanics.

Dai, Q. (2011). Two- and three-dimensional micromechanical viscoelastic finite element modeling of stone-

based materials with X-ray computed tomography images. Construction and Building Materials.

Fenton, G. A., & Griffiths, D. (2008). Risk Assessment in Geotechnical Engineering. Hoboken, N.J.: John

Wiley & Sons.

Kim, Y.-R., Allen, D. H., & Little, D. N. (2005). Damage-induced modeling of asphalt mixtures through

computational micromechanics and cohesive zone fracture. Journal of Materials in Civil

Engineering.

Page 30 de 35

Manrique-Sanchez, L., & Caro, S. (2019). Numerical assessment of the structural contribution of porous

friction courses (PFC). Construction and Building Materials.

Masad, E., Muhunthan, B., Shashidhar, N., & Harman, T. (1999). Internal Structure Characterization of

Asphalt Concrete Using Image Analysis. Journal of Computing in Civil Engineering.

Wang, H., Huang, Z., You, Z., & Chen, Y. (2014). Three-dimensional modeling and simulation of asphalt

concrete mixtures based on X-ray CT microstructure images. Journal of Traffic and Transportation

Engineering.

Wills, J., Caro, S., & Braham, A. (2017). Influence of material heterogeneity in the fracture of asphalt

mixtures. International Journal of Pavement Engineering.

You, T., Al-Rub, R. K., Darabi, M. K., Masad, E. A., & Little, D. N. (2012). Three-dimensional microstructural

modeling of asphalt concrete using a unified viscoelastic–viscoplastic–viscodamage model.

Construction and Building Materials.

Zelelew, H., & Papagiannakis, A. (2011). A volumetrics thresholding algorithm for processing asphalt

concrete X-ray CT images. International Journal of Pavement Engineering.

Page 31 de 35

Appendix 1. Partition plane generator (.m)clc

clear all

l=70; %Cube length

ni=[2,4,6,8,10,15,20,25,30,40,50,60,70]; %Number of cuts

for i=1:13

B=[];

n= ni(i)

% Planes in z

for z=1:n-1

p1=[0,0,z*l/n]; %First coordinate on axis

p2=[0,l,z*l/n]; %Second coordinate on axis at the

end of y

p3=[l,l,z*l/n]; %Thrid coordinate on axis at the end

of x and y

B=[B;p1;p2;p3]; %Planes

end

% Planes in x

for x=1:n-1

p1=[x*l/n,0,0];

p2=[x*l/n,l,0];

p3=[x*l/n,l,l];

B=[B;p1;p2;p3];

end

% Planes in y

for y=1:n-1

p1=[0,y*l/n,0];

p2=[0,y*l/n,l];

p3=[l,y*l/n,l];

B=[B;p1;p2;p3];

End

fileName = strcat('cc',num2str(n),'.txt')

dlmwrite(fileName,B)

end

Page 32 de 35

Appendix 2. Specimen reconstruction and volume fraction calculation

(.m) clc clear all nii=[2,4,6,8,10,15,20,25,30,40,50,60,70]; %Cuts per axis for ii=1:13 n= nii(ii); ni=30; %First image nf=100; %Last image j=0; %Counts registered images THsup=215; %Upper Threshold THinf=165; %Lower Threshold for i=1:150 D = strcat(num2str(i),'.jpg'); %Image name if exist(D, 'file') j = j+1; I = imread(D); %Import image I = I(77:435,77:435); %Cut image into square for x=1:length(I) for y=1:length(I) if I(x,y)>=THinf && I(x,y)<=THsup I(x,y)=1; %Aggregate takes number 1 else I(x,y)=0; %Not aggregate number 0 end end end S{j} = I; %Cell with each image end end [volFrac,gridCoord] =VolFrac(S,n); %3D image is sent and

number of cuts, I get the % volume fraction

and coordinate

vf = [gridCoord volFrac]; %Output matrix sum(volFrac)/n^3 figure imagesc(ima{50}) figure imagesc(S{50}) fileName = strcat('cm',num2str(n),'.txt') %dlmwrite(fileName,vf) end

function [volFrac,gridCoord,imagenn] = VolFrac(mat,grids) %Function to calculate the volume fraction between two

material phases within a grid l = length(mat{1}); %Pixels wide n = floor(l/grids); %Pixels per grid nr = rem(l,grids); %Additional pixels ni = 25; %First image of interest nf = ni+69; %Last image of interest (70mm) k = 0; %Image counter p = l/70; %Pixel size (≈5.12) nn = 70/grids;

% Reduce sample to 70 images for j=ni:nf k=k+1; %Images Im{k}=mat{j}; %Save image end ns = floor(k/grids); %Grid thickness nsr = rem(k,grids); %Additional thickness g=0; %Grid counter gz=0; %Grid counter in z for i=1:grids if i==1 nss = 1; %Value initial image else nss = z+1; %First image end

gz = gz+1; %Grid z +1 gx = 0; %Restart grid x counter xf = 0; %Restart final image in x nrx = nr; %Restart additional pixels in x % Iterate in x for x=1:grids gx = gx+1; %Grid in x +1 gy = 0; %Restart grid in y xi = xf+1; %Lower value in grid %Add additional pixels if necessary if nrx > 0 xf = xi+n; %Upper grid value nrx = nrx-1; %One additional pixel less else xf = xi+n-1; %Upper pixel in the grid end xt = xf-xi; %Grid with yf = 0; %Restart final image y nry = nr; %Restart additional pixel y %Iterate in y for y=1:grids gy = gy+1; %Grid count y +1 yi = yf+1; %Lower grid value %Add additional pixels if necessary if nry > 0 yf = yi+n; %Upper grid value +1 nry = nry-1; %Additional pixels less else yf = yi+n-1; %Upper grid value end yt = yf-yi; %Grid with g = g+1; %Grid counte vol{g}=0; %Add additional pixel if necessary if i<= nsr %Pass through image for z=nss:nss+ns vol{g} = vol{g}+sum(Im{z}(xi:xf,yi:yf));

%Accumulate pixels within grid end else %Pass through images for z=nss:nss+ns-1 vol{g} = vol{g}+sum(Im{z}(xi:xf,yi:yf));

%Accumulate pixels within grid end end zt = z-nss+1; %Images within grid t = (xt+1)*(yt+1)*zt; %Total pixels in grid volS = sum(vol{g}); %Pixels with aggregate volF{g} = volS/t; %Volume fraction c = [(2*x-1)*nn/2,(2*y-1)*nn/2,(2*i-

1)*nn/2];%Coordinate in center of grid gridC{g} = c; %Save grid coordinates imagen{i}(x,y)=volF{g}; end end end %Output format as matrix volFrac = cell2mat(volF)'; gridC = cell2mat(gridC); gridC = reshape(gridC,3,[])'; gridCoord = gridC; imagenn = imagen; end

Page 33 de 35

Appendix 3. Abaqus simulation script (.py) """

@author: Simon Espinosa

"""

from abaqus import *

from abaqusConstants import *

import __main__

import section

import regionToolset

import displayGroupMdbToolset as dgm

import part

import material

import assembly

import step

import interaction

import load

import mesh

import optimization

import job

import sketch

import visualization

import xyPlot

import displayGroupOdbToolset as dgo

import connectorBehavior

import os

import time

from numpy import array # for python

# display function

def disp(line): # print in terminal and ABAQUS python

console

print line

print >> sys.__stdout__, line

l = 70.0 # Dim of cube

ni = array([2,4,6,8,10,15,20,25,30,40,50,60,70]) #Num of

grids

for i in xrange(0,1): #13

tStart = time.clock()

#Creat model

Mdb()

#Model name

n = ni[i]

nn = str(n)

modelName = 'Model-'+nn

mdb.models.changeKey(fromName='Model-1',

toName=modelName)

print >> sys.__stdout__, nn

print(modelName)

# Creat cube

s =

mdb.models[modelName].ConstrainedSketch(name='__profile__',

sheetSize=400.0)

g, v, d, c = s.geometry, s.vertices, s.dimensions,

s.constraints

s.setPrimaryObject(option=STANDALONE)

s.rectangle(point1=(0.0, 0.0), point2=(l, l))

session.viewports['Viewport:

1'].view.setValues(nearPlane=462.133,

farPlane=679.652, width=455.141, height=248.429,

cameraPosition=(

20.3919, 101.652, 570.892), cameraTarget=(20.3919,

101.652, 0))

p = mdb.models[modelName].Part(name='Cubo',

dimensionality=THREE_D,

type=DEFORMABLE_BODY)

p = mdb.models[modelName].parts['Cubo']

p.BaseSolidExtrude(sketch=s, depth=l)

s.unsetPrimaryObject()



1'].setValues(displayedObject=p)

del mdb.models[modelName].sketches['__profile__']

# Path to data points

path = 'C:\Users\...\cc'+nn+'.txt'

datalist = []

# Import data into a matrix

with open(path, "rb") as fp:

for row in fp.readlines():

tmp = row.split(",")

try:

datalist.append((float(tmp[0]),

float(tmp[1]), float(tmp[2])))

except:pass

numcor = len(datalist) # Number of data points

for i in xrange(0,numcor):


# Creat datumPoint from coordinates importet fro

cc.txt

p.DatumPointByCoordinate(coords=datalist[i])

for ii in xrange(2,numcor,3):


c = p.cells

pickedCells = c

v1, e1, d2 = p.vertices, p.edges, p.datums

# Creat partitions using points from cc.txt

p.PartitionCellByPlaneThreePoints(cells=pickedCells,

point1=d2[ii],

point2=d2[ii+1],

point3=d2[ii+2])

# Create materials

p = mdb.models[modelName]

p.Material(name='Agg')

mt1 = p.materials['Agg']

mt1.Elastic(table=((25000.0, 0.16), ))

p.Material(name='Mat')

mt2 = p.materials['Mat']

mt2.Elastic(moduli=INSTANTANEOUS, table=((112.0, 0.4),

))

mt2.Viscoelastic(domain=TIME,

time=PRONY, table=((0.653333017, 0.0, 0.001571618),

(0.22773307, 0.0,

0.016388301), (0.096848609, 0.0, 0.099474509),

(0.018553647, 0.0,

0.735803657), (0.002964298, 0.0, 6.774096386),

(0.000476788, 0.0,

65.78651685), (7.53574e-05, 0.0, 643.3649289),

(1.1875e-05, 0.0,

6514.285714), (2.25001e-06, 0.0, 80976.19048)))

p.HomogeneousSolidSection(name='Agg', material='Agg',

thickness=None)

p.HomogeneousSolidSection(name='Mat', material='Mat',

thickness=None)

#Save

mdb.saveAs( pathName='C:/Users/.../'+modelName+'.cae')

elapsed = (time.clock()-tStart)

disp('Total time for creat'+modelName+',

'+str(elapsed/60)+' minutes.')

#Interpolation rule

vfi = array([0.5,0.75])

for ivf in xrange(0,2):

#Creat model

Mdb()

# Open model

openMdb(pathName='C:/Users/.../'+modelName+'.cae')

#Model name

vfrule = vfi[ivf] # The volume fraction rule ej 50-

50

# Path to data points

pathcm = 'C:\Users\...\cm'+nn+'.txt'

datalistcm = []

datalistccm = []

# Import data into a matrix

with open(pathcm, "rb") as fp:

for row in fp.readlines():

tmp = row.split(",")

try:

datalistcm.append((float(tmp[0]),

float(tmp[1]), float(tmp[2])))

datalistccm.append(float(tmp[3]))

except:pass

numcorcm = len(datalistcm) # Number of data points

for i in xrange(numcorcm):

# Part, cells


c = p.cells

# Select cell and apply material

cells = c.findAt(((datalistcm[i]), ))

names = 'Set-' + str(i)

region = p.Set(cells=cells, name=names)

# Check what material it is

if datalistccm[i] >= vfrule:

p.SectionAssignment(region=region,

sectionName='Agg')

sectionName='Agg'

else:

Page 34 de 35

p.SectionAssignment(region=region,

sectionName='Mat')

sectionName='Mat'

# Create amplitud for load

mdb.models[modelName].TabularAmplitude(name='Amp',

timeSpan=STEP,

smooth=SOLVER_DEFAULT, data=((0.0, 0.0), (0.5, -

0.1)))

# Create the instance for the cube

a = mdb.models[modelName].rootAssembly


1'].setValues(displayedObject=a)


1'].assemblyDisplay.setValues(

optimizationTasks=OFF,

geometricRestrictions=OFF, stopConditions=OFF)


a.DatumCsysByDefault(CARTESIAN)


a.Instance(name='Cubo', part=p, dependent=OFF)

# Mesh


partInstances =(a.instances['Cubo'], )

a.seedPartInstance(regions=partInstances, size=l/n,

minSizeValue=l/n/10)


partInstances =(a.instances['Cubo'], )

a.generateMesh(regions=partInstances)

# Step


1'].assemblyDisplay.setValues(

adaptiveMeshConstraints=ON)

mdb.models[modelName].ViscoStep(name='Step-1',

previous='Initial',

timePeriod=0.5, maxNumInc=10000,

timeIncrementationMethod=FIXED,

initialInc=0.01, cetol=0.0)


1'].assemblyDisplay.setValues(step='Step-1')

# Constraint

mdb.models[modelName].FieldOutputRequest(name='F-

Output-2',

createStepName='Step-1', variables=('E', 'S',

'U','RF'))


1'].assemblyDisplay.setValues(step='Initial')


1'].assemblyDisplay.setValues(loads=ON, bcs=ON,

predefinedFields=ON, connectors=ON,

adaptiveMeshConstraints=OFF)


1'].assemblyDisplay.setValues(step='Step-1')


f1 = a.instances['Cubo'].faces

for i in xrange(0,n):

for j in xrange(0,n):

datalist = ((2*i+1)*l/n/2 ,l ,(2*j+1)*l/n/2

)

if i==0 and j==0 :

faces1 = f1.findAt((datalist,),)

else:

faces1 += f1.findAt((datalist,),)

region = regionToolset.Region(faces=faces1)

mdb.models[modelName].DisplacementBC(name='BC-3',

createStepName='Step-1',

region=region, u1=UNSET, u2=1.0, u3=UNSET,

ur1=UNSET, ur2=UNSET,

ur3=UNSET, amplitude='Amp', fixed=OFF,

distributionType=UNIFORM,

fieldName='', localCsys=None)

#BCSup


a.regenerate()







datalist = ((2*i+1)*l/n/2 ,l ,(2*j+1)*l/n/2

)

if i==0 and j==0 :


else:



mdb.models[modelName].DisplacementBC(name='BC-Sup',

createStepName='Initial',

region=region, u1=SET, u2=UNSET, u3=SET,


ur3=UNSET, amplitude=UNSET,

distributionType=UNIFORM, fieldName='',

localCsys=None)

#BCInf


a.regenerate()







datalist = ((2*i+1)*l/n/2 ,0 ,(2*j+1)*l/n/2

)

if i==0 and j==0 :


else:



mdb.models[modelName].DisplacementBC(name='BC-Inf',

createStepName='Initial',

region=region, u1=SET, u2=SET, u3=SET,


ur3=UNSET, amplitude=UNSET,

distributionType=UNIFORM, fieldName='',

localCsys=None)

#Save

mdb.saveAs(pathName='C:/Users/.../'+modelName+'-

'+str(vfrule)+'.cae')

disp(modelName+'-'+str(vfrule))


#Display time to creat model with material

disp('Total time for '+modelName+'-'+str(vfrule)+',


#Job

jobName = 'Job-'+nn

mdb.Job(name=jobName, model=modelName, description='',

type=ANALYSIS,

atTime=None, waitMinutes=0, waitHours=0, queue=None,

memory=90,

memoryUnits=PERCENTAGE, getMemoryFromAnalysis=True,

explicitPrecision=SINGLE,

nodalOutputPrecision=SINGLE, echoPrint=OFF,

modelPrint=OFF, contactPrint=OFF, historyPrint=OFF,

userSubroutine='',

scratch='', resultsFormat=ODB,

multiprocessingMode=DEFAULT, numCpus=1,

numGPUs=0)

# alt A - Write inp

#disp('Submitting...')

#mdb.jobs[jobName].writeInput(consistencyChecking=OFF)

# alt B - Create odb

disp('Submitting...')

mdb.jobs[jobName].submit(consistencyChecking=OFF)

disp('Waiting...')

mdb.jobs[jobName].waitForCompletion()

# print elapsed time


disp('Total time for '+modelName+', '+str(elapsed/60)+'

minutes.')

#Finish

Mdb()

Page 35 de 35

Appendix 4. Abaqus results processor (.py) from abaqus import * # for abaqus

from abaqusConstants import *

import __main__

import section

import regionToolset

import displayGroupMdbToolset as dgm

import part

import material

import assembly

import optimization

import step

import interaction

import load

import mesh

import job

import sketch

import visualization

import xyPlot

import displayGroupOdbToolset as dgo

import connectorBehavior

#import abq_ExcelUtilities.excelUtilities

from numpy import array # for python

import time

# display function

def disp(line): # print in terminal and ABAQUS python

console

print line

print >> sys.__stdout__, line

### Extract and report vertical reaction force and

displacement of rigid body

#nStart = 1

#nEnd = 10

#nRange = xrange(nStart,nEnd+1)

ni = array([2,4,6,8,10,15,20,25,30,40]) #Num of grids

# 1. Create xy data from surface

print >> sys.__stdout__, ''

for i in xrange(0,10):

vfi = array([0.5,0.75])

for ivf in xrange(0,2):

tStart = time.clock()

### ODB name

n = ni[i]

nn = vfi[ivf]

jobName = 'Job-'+str(n)+'-'+str(int(nn*100))

odbName = jobName+'.odb'

disp(odbName)

odb = session.openOdb(name=odbName, readOnly=False)


1'].setValues(displayedObject=odb)

xydata = session.xyDataListFromField(odb=odb,

outputPosition=NODAL, variable=(('RF', NODAL, ((COMPONENT,

'RF2'), )), ('S', INTEGRATION_POINT, ((COMPONENT, 'S22'),

)), ), nodeSets=('_PICKEDSET5', ))

z = session.xyDataObjects.keys()

disp(z)

lz = len(z)

lz2 = len(z)/2

disp(lz2)

reportName = 'rf'+str(n)+'-'+str(int(nn*100))+'.txt'

report2Name = 's'+str(n)+'-'+str(int(nn*100))+'.txt'

x = []

y = []

for j in xrange(0,lz2):

disp(j)

x.append(session.xyDataObjects[z[j]])

y.append(session.xyDataObjects[z[j+lz2]])

x = tuple(x)

y = tuple(y)

session.writeXYReport(fileName=reportName, xyData=x)

session.writeXYReport(fileName=report2Name,

xyData=y)

for k in xrange(0,lz2):

disp(k)

del session.xyDataObjects[z[k]]

del session.xyDataObjects[z[k+lz2]]

odb.close()

x = []

y = []

z = []

# print elapsed time


disp('Total time for '+jobName+',


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