# Bounding the Multi-Scale Domain in Numerical Modelling and Meta-Heuristics Optimization: Application to Poroelastic Media with Damageable Cracks

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Micro-Scale Problem

#### 2.1. Constitutive Equations of the Homogenized Problem

#### Damage

#### 2.2. Numerical Implementation

#### 2.3. Initial Constriction of Crack Stiffness

## 3. Failure Envelopes

#### 3.1. Maximum Stress Failure Criterion

#### 3.2. Von Misses Failure Criterion

#### 3.3. Stress Ratio to Peak Failure Criterion

## 4. Particle Swarm Optimization

#### 4.1. Objective Function

#### 4.2. Increasing Swarm Size Results

#### 4.3. Increasing Objective Function Resolution Results

#### 4.4. Summary Particle Swarm Optimization Results

#### 4.5. Monte-Carlo Analysis

## 5. Discussion

## 6. Conclusions

- A ratio to peak stress has shown to be a good criteria to characterize the failure of the present micro-structure.
- For the initially given $\lambda $ and $\mu $ elastic coefficients, the multi-scale rich micro-structure behaviour happens in the crack stiffness range $G=[1.0\times {10}^{13}$∼$2.7\times {10}^{13}]$ Pa.
- PSO overcomes the non continuity and non differentiability of the constitutive law for a representative range of function resolutions.
- The metric best $f\left(x\right)$ alone is able to discriminate between local and global optima.

#### Contribution to Science

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PDE | Partial Differential Equation |

AI | Artificial Intelligence |

ML | Machine Learning |

FEM | Finite Element Model |

PSO | Particle Swarm Optimization |

## Appendix A. Convergence, Fitness and Swarm Scatter Results

**Figure A1.**Blue plot is the convergence of the PSO algorithm (minimum value of $f\left(x\right)$), green plot is the fitness of the population (mean value of $f\left(x\right)$) both plots horizontal axes are the PSO iterations. The two scatter plots represent different planes in the optimization variable space, $\mu -{\Delta}_{n}$ and $\mu -G$. The axes are in the units of the corresponding variable, $Pa$ or adimentional for ${\Delta}_{n}$. Swarm Population from top to bottom: $8\u201316\u201324\u201332$. Objective function $f\left(x\right)$ resolution: 8 points.

**Figure A2.**Blue plot is the convergence of the PSO algorithm (minimum value of $f\left(x\right))$, green plot is the fitness of the population (mean value of $f\left(x\right))$, both plots horizontal axes are the PSO iterations. The two scatter plots represent different planes in the optimization variable space, $\mu -{\Delta}_{n}$ and $\mu -G$. The axes are in the units of the corresponding variable, $Pa$ or adimentional for ${\Delta}_{n}$. Swarm Population: 40. Objective function $f\left(x\right)$ resolution from top to bottom: $8\u201316\u201332\u201364$ points.

## Appendix B. Convergence of Δn, μ and G Swarms

**Figure A3.**The three columns represent the evolution of each optimization variable ranges, respectively: ${\Delta}_{n}$, $\mu $ and G. The horizontal axes are the PSO iterations and the vertical axes are in the units of the corresponding variable, $Pa$ or adimensional for ${\delta}_{n}$. Swarm Population from top to bottom: $8\u201316\u201324\u201332$. Objective function $f\left(x\right)$ resolution: 8 points.

**Figure A4.**The three columns represent the evolution of each optimization variable ranges, respectively: ${\Delta}_{n}$, $\mu $ and G The horizontal axes are the PSO iterations and the vertical axes are in the units of the corresponding variable, $Pa$ or adimentional for ${\delta}_{n}$. Swarm Population: 40. Objective function $f\left(x\right)$ resolution from top to bottom: $8\u201316\u201324\u201332$ points.

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**Figure 1.**Separation Ratio Graph based on [34]. Graphical representation of the phase transition between chaos and order. Systems in the target zone are said to possess both a good separation property and ideal dynamic behaviour to produce optimal reservoirs.

**Figure 2.**Geometry of the micro-scale configuration Finite Element Mesh, matrix elements with unit cell adimensional axes (

**left**), crack network links (

**center**) and crack network elements (

**right**). Cracks are totally closed in the initial configuration, figure shows cracks with opening for convenience of representation.

**Figure 3.**Homogenized stiffness function of crack stiffness. Below the red area crack stiffness is dominant and above porous matrix is dominant. The red area corresponds to a material with both crack and porous matrix contribution to the response.

**Figure 4.**Configurations for a macro deformation $\u03f5=0.005$ in all DoF, including crack pore pressure. Colormap represents von Misses strain in Pascals. Crack stiffness: $G={10}^{13}$ $\mathrm{GPa}$ (

**top**) and $G={10}^{14}$ $\mathrm{GPa}$ (

**bottom**). Deformation magnification: 50×.

**Figure 5.**Failure envelopes for different crack stiffnesses, maximum stress failure criterion. Respectively from

**left**to

**right**$G={10}^{12}$ Pa–${10}^{13}$ Pa–${10}^{14}$ Pa–${10}^{15}$ Pa. The concentric envelopes correspond to different applied macro shear. The macro shear spans from zero, for the outmost envelope, until $0.200$ for the innermost, with an increment of $0.040$.

**Figure 6.**Failure envelopes with crack stifness $G={10}^{14}$ Pa. von Misses energy criterion: strains (

**left**), stresses (

**center**). Coordinates in the strain plane to extract configurations of failure region $1a$ and $2a$. (

**right**).

**Figure 7.**Failure according to von Misses energy criterion. From

**left**to

**right**: region $1a$ before failure, one step after failure and stress reduction after damage law expansion. Micro-scale deformation magnification 10×. $G={10}^{14}$ Pa.

**Figure 8.**Failure according to von Misses energy criterion. From

**left**to

**right**: region $2a$ before failure, one step after failure and stress reduction after damage law expansion. Micro-scale deformation magnification 10×. $G={10}^{14}$ Pa.

**Figure 10.**Failure envelope with crack stiffness $G=2.00\times {10}^{13}$ $\mathrm{Pa}$. Stress ratio to peak failure criterion. Coordinates to extract configurations of failure regions $1b\u20132b\u20133b$.

**Figure 11.**Stress ratio to peak failure criterion results. From top to bottom failure region $1b$, $2b$ and $3b$. From left to right: damage parameter $f\left(z\right)$ (Equation (8)), constitutive equation affecting Gauss Points $(1-{d}^{\left(0\right)})$ (Equation (7)), iterations for convergence of the nonlinear problem at each loading step and normalized failure criteria (von Misses and Stress ratio).

**Figure 12.**Failure surfaces for different function resolutions. From left to right: $8\u201316\u201332\u201364$ points.

**Figure 13.**3D scatter plot in the optimization variable space, each circle is a PSO optimization, the colour represents the decimal logarithm of best f(x). The small dark blue cluster is in close proximity of the known target optimization value of x and contains 15 Monte-Carlo realizations.

**Figure 15.**Standard deviation of the growing population N from 1 to 15 divided by the root square of the sampled population N (Blue circles). Power trend line (red line). Trend line equation and ${R}^{2}$ are presented with y representing the vertical axis $(STD{V}_{N}(var.)/{N}^{0.5})$ and x the horizontal axis $\left(N\right)$.

PSO Parameter | Value |
---|---|

Function Tolerance | $1.0\times {10}^{-6}$ |

Inertia Range | [$0.1000$ $1.1000$] |

Min. Neighbors Fraction | $0.25$ |

Objective Limit | $0.0$ |

Self Adjustment Weight | $1.4900$ |

Social Adjustment Weight | $1.4900$ |

**Table 2.**Particle Swarm Optimization simulation summary and results. Run in a computer with Intel Core $i7$-10700 processor @2.90 GHz and 16.0 GB RAM @2933 MHz, all simulations using parallelization with 8 workers.

Sim. | Swarm Size | It. | f-Count | f(x) Res. | Best f(x) | Mean f(x) | Optimal x | Time |
---|---|---|---|---|---|---|---|---|

1 | 8 | 100 | 808 | 8 | 0.01245 | 0.03517 | [0.0056 9.4330 $\times {10}^{8}$ 1.6802 $\times {10}^{13}$] | 42 min |

2 | 16 | 100 | 1616 | 8 | $2.608\times {10}^{-6}$ | 0.003639 | [0.0050 9.5287 $\times {10}^{8}$ 1.9847 $\times {10}^{13}$] | 90 min |

3 | 24 | 100 | 2424 | 8 | 0.0002951 | 0.06567 | [0.0050 9.6105 $\times {10}^{8}$ 2.0029 $\times {10}^{13}$] | 135 min |

4 | 32 | 100 | 3232 | 8 | 0.0379 | 0.1722 | [0.0064 6.6052 $\times {10}^{8}$ 1.0255 $\times {10}^{13}$] | 170 min |

5 | 40 | 75 | 3040 | 8 | 0.02985 | 0.03623 | [0.0044 1.1140 $\times {10}^{9}$ 2.6967 $\times {10}^{13}$] | 3 h |

6 | 40 | 100 | 3640 | 16 | $6.136\times {10}^{-10}$ | 0.00424 | [0.0050 9.5219 $\times {10}^{8}$ 1.9876 $\times {10}^{13}$] | 6 h |

7 | 40 | 100 | 4040 | 32 | $1.635\times {10}^{-8}$ | 0.1146 | [0.0050 9.6970 $\times {10}^{8}$ 2.0132 $\times {10}^{13}$] | 14 h |

8 | 40 | 59 | 2400 | 64 | 0.09868 | 0.1690 | [0.0049 1.2485 $\times {10}^{9}$ 2.5521 $\times {10}^{13}$] | 17 h |

Ref. x | - | - | - | - | - | - | [0.0050 9.6100 $\times {10}^{8}$ 2.0000 $\times {10}^{13}$] | - |

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**MDPI and ACS Style**

Argilaga, A.; Papachristos, E.
Bounding the Multi-Scale Domain in Numerical Modelling and Meta-Heuristics Optimization: Application to Poroelastic Media with Damageable Cracks. *Materials* **2021**, *14*, 3974.
https://doi.org/10.3390/ma14143974

**AMA Style**

Argilaga A, Papachristos E.
Bounding the Multi-Scale Domain in Numerical Modelling and Meta-Heuristics Optimization: Application to Poroelastic Media with Damageable Cracks. *Materials*. 2021; 14(14):3974.
https://doi.org/10.3390/ma14143974

**Chicago/Turabian Style**

Argilaga, Albert, and Efthymios Papachristos.
2021. "Bounding the Multi-Scale Domain in Numerical Modelling and Meta-Heuristics Optimization: Application to Poroelastic Media with Damageable Cracks" *Materials* 14, no. 14: 3974.
https://doi.org/10.3390/ma14143974