# 2D Digital Reconstruction of Asphalt Concrete Microstructure for Numerical Modeling Purposes

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Asphalt Concrete Microstructure Characterization

#### 1.2. Numerical Representation of AC Microstructure

^{2}homogenization, as proposed by Feyel and Chaboche [22]. The whole methodology was illustrated first with a 2D academic problem of a tapered bar with a periodic microstructure. The aggregate particles were modeled as linear elastic and their geometry was idealized as hexagons of equal size. In the second example, the 2D representative volume element (RVE) accounted for much more realistic AC microstructure. The aggregate particles were modeled as irregular polygons. There was no information on the procedure of their generation, however.

**A**, effective parameters are assessed in terms of the lower scale quantities. The Mori-Tanaka scheme [23] is developed for this purpose. Aggregate particles of the idealized shape (ellipse) were used due to the known analytical formulas for the localization tensor

**A**. At each step of the upscaling, the effective matrix parameters were assessed in terms of the underlying matrix-inclusion material. For the viscoelastic parameters, the corresponding principle was used. Laplace-Carson transform was used to compute effective creep functions, and its inverse was employed to express them in the time domain.

^{3}required 1000

^{3}voxels. In terms of the FEA, it would result in a problem solved using the same number of finite elements, which is the overkill mesh.

## 2. Materials and Methods

#### 2.1. Samples and Images Preparation

#### 2.2. Image Processing

#### 2.2.1. Conversion RGB to Grayscale Image

#### 2.2.2. Image Binarization

#### 2.2.3. Enhanced Binarization

- An image is processed as described in Section 2.2.1 and Section 2.2.2 using the human visual inspection to control the process;
- A small number of inclusions is manually reconstructed;
- Those manually reconstructed inclusions are used as references;
- A threshold value described in Section 2.2.2 is iteratively updated to provide the improved agreement between the reference inclusion shapes and those reconstructed by the algorithm. So far, we have used the area of the inclusion as the quantity to be compared.

#### 2.3. Controlled Geometry Simplification

#### 2.3.1. Reference AC Microstructure

#### 2.3.2. Shortest Edge Elimination

- Reduce the initial number of vertices by simple removal of their specified percentage (10%, 20%, etc.). The vertices are removed with regular interval, i.e., the 1st, 11th, 21st, etc. (when the percentage equal to 10% is specified). This simplification is justified by the high resolution of the processed image.
- Remove iteratively the shortest edge along the inclusion boundary.

#### 2.3.3. Local Geometry Enhancement

- Find the extreme (outermost) pixels along the inclusion boundary in all four directions, i.e., top, bottom, left and right (see Figure 8a). In the case of multiple values, we selected higher (for left and right extrema) and located more to the left pixels (for top and bottom extrema). Typically, a quadrilateral is generated.
- Iteratively, we looped over all the approximated geometry edges. The inclusion boundary pixel along the respective segment (see Figure 8b,d) with the largest distance from the approximated edge was searched. Consequently, a new vertex for the approximated geometry was added and the number of its edges increases. At this step, one can introduce additional requirements on the newly created edges. For instance, a minimum edge length can be verified before the current edge splitting.

#### 2.3.4. Convex Subdomain Approach

## 3. Results

#### 3.1. Heat Flow Problem

- in Figure 6b, which was generated on the basis of the initial microstructure geometry without any simplification,
- in Figure 7b, which was generated using the shortest edge elimination algorithm with the percentage of eliminated boundary pixels equal to about 90%,
- in Figure 8d, which was obtained after 2 local geometry enhancements, and
- in Figure 9e, after geometry correction.

_{REF}− NDOF

_{H})/NDOFREF × 100%

_{REF}stands for the overkill mesh number degrees of freedom and NDOF

_{H}is the number of degrees of freedom of the solution obtained using a coarser mesh resulting from the simplified geometry. Consequently, the solution error is computed as

_{MAX,REF}− U

_{MAX,H}|/|U

_{MAX,REF}| × 100%

_{MAX,REF}denotes the maximum solution value in the analyzed domain (herein, it is the temperature) obtained using the overkill mesh. U

_{MAX,H}is the maximum solution value obtained using a respective coarser mesh.

#### 3.2. Linear Elasticity (Plane Strain) Problem

_{x}

^{2}+ u

_{y}

^{2}, where u

_{x}and u

_{y}stand for the horizontal and vertical displacement component, respectively).

#### 3.3. Discussion

## 4. Conclusions

- Image processing can be used in order to reconstruct the AC microstructure geometry.
- The initial inclusion boundaries can be effectively simplified using the algorithms presented in this paper.
- A large NDOF reduction can be obtained due to the user-controlled microstructure geometry simplification with a small solution error introduced.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kim, Y.-R.; Souza, F.V.; Teixeira, J.E. A two-way coupled multiscale model for predicting damage-associated performance of asphaltic roadways. Comput. Mech.
**2013**, 51, 187–201. [Google Scholar] [CrossRef] - Kim, Y.-R. Modeling of Asphalt Concrete, 1st ed.; McGraw Hill: New York, NY, USA, 2009. [Google Scholar]
- Islam, M.-R.; Tarefder, R.-A. Pavement Design: Materials, Analysis, and Highways; McGraw Hill: New York, NY, USA, 2020. [Google Scholar]
- Liu, Y. Discrete Element Methods for Asphalt Concrete: Development and Application of User-Defined Microstructural Models and a Viscoelastic Micromechanical Model. Ph.D. Thesis, Michigan Technological University, Houghton, MI, USA, 2011. [Google Scholar]
- Wollny, I.; Hartung, F.; Kaliske, M.; Liu, P.; Oeser, M.; Wang, D.; Falla, G.C.; Leischner, S.; Wellner, F. Coupling of microstructural and macrostructural computational approaches for asphalt pavements under rolling tire load. Comput. Aided. Civ. Inf.
**2020**, 35, 1178–1193. [Google Scholar] [CrossRef] - Neumann, J.; Simon, J.-W.; Mollenhauer, K.; Reese, S. A framework for 3D synthetic mesoscale models of hot mix asphalt for the finite element method. Build. Mater.
**2017**, 148, 857–873. [Google Scholar] [CrossRef] - Wimmer, J.; Stier, B.; Simon, J.W.; Reese, S. Computational homogenisation from a 3D finite element model of asphalt concrete-linear elastic computations. Finite Elem. Anal. Des.
**2016**, 110, 43–57. [Google Scholar] [CrossRef] - Onifade, I.; Jelagin, D.; Birgisson, B.; Kringos, N. Towards asphalt mixture morphology evaluation with the virtual specimen approach. Road Mater. Pavement Des.
**2015**, 17, 579–599. [Google Scholar] [CrossRef] - Fakhari Tehrani, F.; Absi, J.; Allou, F.; Petit, C. Heterogeneous numerical modeling of asphalt concrete through use of a biphasic approach: Porous matrix/inclusions. Comput. Mater. Sci.
**2013**, 69, 186–196. [Google Scholar] [CrossRef] - Sepehr, K.; Svec, O.J.; Yue, Z.Q.; El Hussein, H.M. Finite element modelling of asphalt concrete microstructure. WIT Trans. Eng. Sci.
**1994**, 6, 225–232. [Google Scholar] [CrossRef] - Mo, L.; Huurman, M.; Wu, S.; Molenaar, A. 2D and 3D meso-scale finite element models for ravelling analysis of porous asphalt concrete. Finite Elem. Anal. Des.
**2008**, 44, 186–196. [Google Scholar] [CrossRef] - Mitra, K.; Das, A.; Basu, S. Mechanical behavior of asphalt mix: An experimental and numerical study. Constr. Build. Mater.
**2012**, 27, 545–552. [Google Scholar] [CrossRef] - Sadd, M.H.; Dai, Q.; Parameswaran, V.; Shukla, A. Microstructural Simulation of Asphalt Materials: Modeling and Experimental Studies. J. Mater. Civ. Eng.
**2004**, 16, 107–115. [Google Scholar] [CrossRef] - Huang, W.; Wang, H.; Yin, Y.; Zhang, X.; Yuan, J. Microstructural Modeling of Rheological Mechanical Response for Asphalt Mixture Using an Image-Based Finite Element Approach. Materials
**2019**, 12, 2041. [Google Scholar] [CrossRef] - Schüller, T.; Jänicke, R.; Steeb, H. Nonlinear modeling and computational homogenization of asphalt concrete on the basis of XRCT scans. Constr. Build. Mater.
**2016**, 109, 96–108. [Google Scholar] [CrossRef] - Nejad, F.M.; Motekhases, F.Z.; Zakeri, H.; Mehrabi, A. An Image Processing Approach to Asphalt Concrete Feature Extraction. J. Ind. Intell. Inf.
**2015**, 3, 54–60. [Google Scholar] [CrossRef] - Liu, P.; Hu, J.; Wang, D.; Oeser, M.; Alber, S.; Ressel, W.; Fala, G. Modelling and evaluation of aggregate morphology on asphalt compression behavior. Constr. Build. Mater.
**2017**, 133, 196–208. [Google Scholar] [CrossRef] - Ziaei-Rad, V.; Nouri, N.; Ziaei-Rad, S.; Abtahi, M. A numerical study on mechanical performance of asphalt mixture using a meso-scale finite element model. Finite Elem. Anal. Des.
**2012**, 57, 81–91. [Google Scholar] [CrossRef] - You, T.; Al-Rub, R.; Darabi, M.; Masad, E.; Little, D. A thermo-viscoelastic–viscoplastic–viscodamage constitutive model for asphaltic materials. Int. J. Solids Struct.
**2011**, 1, 191–207. [Google Scholar] [CrossRef] - Qu, X.; Wang, D.; Wang, L.; Huang, Y.; Hou, Y.; Oeser, M. The State-of-the-Art Review on Molecular Dynamics Simulation of Asphalt Binder. Adv. Civ. Eng.
**2018**, 2018, 4546191. [Google Scholar] [CrossRef] - Aigner, E.; Lackner, R.; Pichler, C. Multiscale prediction of viscoelastic properties of asphalt concrete. J. Mater. Civ. Eng.
**2009**, 21, 771–780. [Google Scholar] [CrossRef] - Feyel, F.; Chaboche, L. FE
^{2}multiscale approach for modelling the elasto-viscoplastic behaviour of long fibre SiC/Ti composite materials. Comput. Methods Appl. Mech. Eng.**2000**, 183, 309–330. [Google Scholar] [CrossRef] - Mori, T.; Tanaka, K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall.
**1973**, 21, 571–574. [Google Scholar] [CrossRef] - Jaworska, I. An effective contour plotting method for presentation of the postprocessed results. Comput. Vis. Graph. Comput. Imaging Vis.
**2006**, 32, 1112–1117. [Google Scholar] - Preparata, F.P.; Shamos, M.I. Computational Geometry; Springer: Berlin/Heidelberg, Germany, 1985. [Google Scholar]
- Klimczak, M.; Cecot, W. MsFEM upscaling for coupled thermo-mechanical problem. In Lecture Notes in Computer Science, Computational Science, Proceedings of the International Conference on Computational Science, Krakow, Poland, 16–18 June 2021; Maciej, P., Kranzlmüller, D., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A., Eds.; Springer: Berlin/Heidelberg, Germany, 2021; Volume 12746, pp. 562–575. [Google Scholar]

**Figure 2.**Test stand with DSLR camera (1), light source (2), tripod (3), uniform background (4), intervalometer (5) and tested specimen (6).

**Figure 4.**Truecolor AC specimen image with a length scale marked (

**a**) and the corresponding grayscale image (

**b**). The processed images are presented at the same length scale.

**Figure 5.**Binarized AC specimen image (

**a**), processed image with filled inclusions (

**b**), processed image after erosion and filtering operations (

**c**), and final image after dilation operation (

**d**).

**Figure 6.**Boundaries of the captured inclusions overlaid on the original image (

**a**) and the corresponding fine mesh (

**b**).

**Figure 7.**Boundaries of the simplified inclusions overlaid on the original image (

**a**) and the corresponding fine mesh (

**b**).

**Figure 8.**Specification of the outermost boundary pixels and the corresponding initial approximated inclusion geometry (

**a**). Results of 2 subsequent iterations of the algorithm (

**b**–

**d**) the fine mesh obtained for the last approximated geometry. In (

**a**,

**c**), the light blue curve denotes the initial inclusion boundary and the red one shows its approximated counterpart. A single small red circle denotes the pixel from the boundary with the largest distance from the corresponding edge of the approximated geometry. In the next iteration, this point becomes a new vertex.

**Figure 9.**Boundaries of the approximated inclusions (initial set of 11,000 pixels) obtained by convex polygons (

**a**) and their 860 vertices (

**b**). Badly fitted convex area (

**c**) corrected by inserting the additional vertex (

**d**) and the fine mesh (

**e**) corresponding to (

**b**)—including correction shown in (

**d**).

**Figure 10.**Map of temperature [°C] obtained using the overkill mesh (

**a**) and meshes corresponding to the approximated geometry using shortest edge elimination (

**b**), local geometry enhancement (

**c**), and convex subdomain approach (

**d**).

**Figure 11.**Horizontal (

**a**,

**c**,

**e**,

**g**) and vertical (

**b**,

**d**,

**f**,

**h**) displacement components [m]. The first row presents the reference solution and the remaining ones refer to the geometries obtained using the shortest edge elimination, the local geometry enhancement and the convex subdomain approach, consecutively.

Algorithm | NDOF | Maximum Temperature [°C] | NDOF Reduction [%] | Relative Error [%] |
---|---|---|---|---|

Reference Solution | 110,557 | 43.03 | - | - |

Shortest Edge Elimination | 5524 | 42.72 | 95.00 | 0.72 |

Local Geometry Enhancement | 10,445 | 43.01 | 90.55 | 0.05 |

Convex Subdomain Approach | 16,251 | 44.09 | 85.30 | 2.46 |

Algorithm | NDOF | Maximum Displacement Magnitude [m] | NDOF Reduction [%] | Relative Error [%] |
---|---|---|---|---|

Reference Solution | 221,114 | 3.37 × 10^{−6} | - | - |

Shortest Edge Elimination | 11,048 | 3.41 × 10^{−6} | 95.00 | 1.19 |

Local Geometry Enhancement | 20,890 | 3.36 × 10^{−6} | 90.55 | 0.30 |

Convex Subdomain Approach | 32,502 | 3.24 × 10^{−6} | 85.30 | 3.86 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Klimczak, M.; Jaworska, I.; Tekieli, M.
2D Digital Reconstruction of Asphalt Concrete Microstructure for Numerical Modeling Purposes. *Materials* **2022**, *15*, 5553.
https://doi.org/10.3390/ma15165553

**AMA Style**

Klimczak M, Jaworska I, Tekieli M.
2D Digital Reconstruction of Asphalt Concrete Microstructure for Numerical Modeling Purposes. *Materials*. 2022; 15(16):5553.
https://doi.org/10.3390/ma15165553

**Chicago/Turabian Style**

Klimczak, Marek, Irena Jaworska, and Marcin Tekieli.
2022. "2D Digital Reconstruction of Asphalt Concrete Microstructure for Numerical Modeling Purposes" *Materials* 15, no. 16: 5553.
https://doi.org/10.3390/ma15165553