# Computational Generation of Virtual Concrete Mesostructures

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## Abstract

**:**

## 1. Introduction

## 2. Concrete Mesostructure Generator (CMG)

#### 2.1. Modeling a Generic Aggregate

#### 2.2. Generating a Concrete Mesostructure

#### 2.3. Data for the Calibration of the CMG

#### 2.4. Comparison of Simulations vs. Laboratory Measurements

## 3. Estimation of the Elastic Properties Using Computational Homogenisation

#### 3.1. Data for Model Validation

#### 3.2. Computational Modeling

## 4. Direct Computation of Elastic Properties

#### 4.1. Data Generation and Pre-Processing

#### 4.2. 3D-CNN Architecture

#### 4.3. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CMG | Concrete Mesostructure Generator |

CT | Computed Tomography |

ANN | Artificial Neural Network |

ITZ | Interfacial-Transition Zone |

RSA | Random Sequential Adsorption |

SRSA | Semi-Random Sequential Adsorption |

LS-FFT | Lippmann–Schwinger Fast Fourier Transform-based homogenisation |

FCH | Finite Cell Homogenisation |

MT | Mori–Tanaka homogenisation |

3D-CNN | 3 Dimensional Convolutional Neural Network |

ReLU | Rectified Linear Unit |

SGD | Stochastic Gradient Descent |

MSE | Mean Squared Error |

ML | Machine Learning |

## Appendix A. Analytical Homogenisation of Mortar Matrix

## References

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**Figure 1.**Concrete Mesostructure Generator (CMG): 2D section of a 3D polyhedron enclosing an ellipsoid in CMG (

**Left**), calculation of tangent points between the ellipsoid surface and polyhedron planes with random angles (

**Right**).

**Figure 2.**Concrete Mesostructure Generator: 2D section of a 3D polyhedron with concave depressions (

**Left**), calculation of basis points on the ellipsoid surface for the Gaussian surface generation (

**Right**).

**Figure 4.**(

**a**,

**b**) Visualisation of the polyhedrons with maximum length of 2.5 cm (50 voxels), elongation ratio of 2.5 with 30 number of cuts, with and without coating. (

**c**) Selected section featuring concave regions of the polyhedron.

**Figure 5.**Influence of the number of cuts (N) and the elongation on the aggregate shape. The elongation corresponds to the value $1/\xi $.

**Figure 6.**(

**Left**) Virtual concrete mesostructure RVEs generated by the CMG. Visualisation of concrete mesostructures with aggregates only (

**Center**) and visualisation of the mortar matrix (

**Right**).

**Figure 7.**Visualisation of virtual concrete samples of size 5, 10, and 20 cm (

**Top**), and their associated statistical data, namely particle size distribution curve (

**Center**). Absolute volume fraction and particle counts with respect to size (

**Bottom**).

**Figure 8.**(

**Left**,

**Center**) Qualitative comparison of an actual concrete slice and the statistically equivalent virtual concrete generated by the CMG. (

**Right**) Statistical data of the cumulative volume fraction of aggregates (laboratory measurements of AB16 vs. virtual AB16).

**Figure 9.**(

**Left**) Virtual concrete specimens of size $10\times 10\times 40$ cm discretised by $200\times 200\times 800$ voxels. (

**Right**) Specimen of size $60\times 60\times 30$ cm, constructed by stacking eight identical periodic blocks of virtual concrete mesostructures of size $30\times 30\times 15$ cm.

**Figure 10.**Time required to generate a virtual mesostructure using PyCMG [37] as a function of the volume fraction of aggregates on a 6-year-old Intel(R) Core(TM) i5-4210U CPU @ 1.7 GHz laptop.

**Figure 11.**Visualisation of virtual concrete samples of size 5 cm used in the homogenisation procedure.

**Figure 12.**The 2D volume fraction of samples (5, 10, and 20 cm) at a different slicing position and the corresponding standard deviation of 2D volume fraction.

**Figure 13.**3D-CNN architecture used for training and prediction of an ML model for predicting the elastic concrete properties directly from the mesostructure of the material.

**Figure 14.**Visualisation of input slice of mesostructure with outputs from each 3D-CNN layer (

**Left**); evolution of the mean squared error (MSE) vs. epochs (

**Right**).

**Figure 15.**Comparison of homogenised results from FCH simulations (high-fidelity) and the ML model (3D-CNN) for various volume fractions (

**Left**) and phase contrasts (

**Right**).

Input Symbol | Input Description | CMG Algorithm |
---|---|---|

${L}_{x}$,${L}_{y}$,${L}_{z}$ | Micro/mesostructure size in mm | Assembly algorithm |

${\mathbf{v}}_{{f}_{max}}$ | Maximum volume fraction of inclusions in micro/mesostructure | |

{${\mathbf{l}}_{part}$} | Aggregate size distribution | |

{${\mathbf{v}}_{{f}_{part}}$} | Volume fraction list | |

{$\xi $} | Aspect ratio list | |

{$\mathbf{N}$} | Number of faces list | |

{${\mathbf{S}}_{con}$} | Concave provision list (Yes/No) | |

{${\mathbf{S}}_{ITZ}$} | Coating provision list (Yes/No) | |

{$\mathbf{d}$},{$\mathbf{w}$} | Width and depth parameter list for the concave depression | |

{$\mathbf{t}$} | Coating thickness list | |

${K}_{max}$ | Maximum number of failed assembly attempts for each particle | |

T | Threshold to switch algorithm from RSA to SRSA | |

${D}_{max}$ | Maximum size of aggregate | Aggregate generator—Polyhedron |

N | Number of faces of polyhedron | |

$\xi $ | Aspect ratio of the aggregate | |

${S}_{con}$ | Concave depression boolean (Yes/No) | |

${S}_{ITZ}$ | ITZ provision boolean (Yes/No) | |

M | Number of concave depressions | Aggregate generator—Concave surface |

d | Depth parameter | |

w | Width parameter | |

${\sigma}^{2}$ | Variance parameter | |

t | ITZ thickness | Aggregate generator—ITZ |

Cement Matrix | Fine Aggregates | Coarse Aggregates | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Size [mm] | - | 0.063 | 0.125 | 0.25 | 0.5 | 1 | 2 | 2.8 | 4 | 5.6 | 8 | 11.2 | 16 |

Volume fraction [%] | 29.259 | 1.504 | 1.619 | 1.758 | 1.758 | 3.634 | 12.174 | 5.0626 | 5.146 | 6.743 | 16.606 | 2.904 | 11.832 |

Total [%] | 29.259 | 22.448 | 48.292 | ||||||||||

Total [%] | 29.259 | 70.741 |

Density [kg/m${}^{3}$] | Young ’s Modulus [GPa] | Poisson’s Ratio | |
---|---|---|---|

Cement paste ${}^{1}$ | 1898 | 18.7 | 0.24 |

Quartzitic aggregate | 2560 | 84.6 | 0.12 |

Concrete | 2378 | 48.03 | 0.15 |

**Table 4.**Comparison of model predictions from two homogenisation methods (LS-FFT and FCH) and measured data (Lab.) for the elastic properties of concrete.

Vol. Frac. [%] | Youngs Modulus [GPa] | Poisson’s Ratio | |||||
---|---|---|---|---|---|---|---|

LS-FFT | FCH | Lab. | LS-FFT | FCH | Lab. | ||

Sample 1 | 49.67 | 49.468 | 51.817 | 48.03 | 0.1679 | 0.16 | 0.15 |

Sample 2 | 48.73 | 48.983 | 51.331 | 0.1687 | 0.1607 | ||

Sample 3 | 47.64 | 48.407 | 50.566 | 0.1697 | 0.1621 | ||

Average | 48.68 | 48.952 | 51.238 | 48.03 | 0.16876 | 0.16093 | 0.15 |

Parameters | Values |
---|---|

Standard | AB8, AB16 |

Sample count | 20 |

Volume fraction ${v}_{f}$ | 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.40, 0.45 |

Phase-contrast p | 2, 3, 4, 5 |

total 2 × 20 × 9 × 4 = 1440 mesostructures |

Layer No. | Layer Details | Input Size | Output Size |
---|---|---|---|

1 | Conv3D, 10 filters (10${}^{3}$), strides (3${}^{3}$), ‘ReLU’ | 100 × 100 × 100 × 1 | 31 × 31 × 31 × 10 |

2 | Maxpooling 3D (2${}^{3}$) | 31 × 31 × 31 × 10 | 15 × 15 × 15 × 10 |

3 | Conv3D, 20 filters (5${}^{3}$), strides (2${}^{3}$), ‘ReLU’ | 15 × 15 × 15 × 1 | 6 × 6 × 6 × 20 |

4 | Maxpooling 3D (2${}^{3}$) | 6 × 6 × 6 × 20 | 3 × 3 × 3 × 20 |

5 | Flattening | 3 × 3 × 3 × 20 | 540 |

6 | Dense layer, ‘ReLU’ | 540 | 20 |

7 | Dense layer | 20 | 2 |

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## Share and Cite

**MDPI and ACS Style**

Holla, V.; Vu, G.; Timothy, J.J.; Diewald, F.; Gehlen, C.; Meschke, G.
Computational Generation of Virtual Concrete Mesostructures. *Materials* **2021**, *14*, 3782.
https://doi.org/10.3390/ma14143782

**AMA Style**

Holla V, Vu G, Timothy JJ, Diewald F, Gehlen C, Meschke G.
Computational Generation of Virtual Concrete Mesostructures. *Materials*. 2021; 14(14):3782.
https://doi.org/10.3390/ma14143782

**Chicago/Turabian Style**

Holla, Vijaya, Giao Vu, Jithender J. Timothy, Fabian Diewald, Christoph Gehlen, and Günther Meschke.
2021. "Computational Generation of Virtual Concrete Mesostructures" *Materials* 14, no. 14: 3782.
https://doi.org/10.3390/ma14143782