# Data Pruning of Tomographic Data for the Calibration of Strain Localization Models

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## Abstract

**:**

## 1. Introduction

## 2. Notations

## 3. Data Pruning by Following an Hyper-Reduction Scheme

#### 3.1. Digital Volume Correlation

#### 3.2. Dimensionality Reduction

#### 3.3. Reduced Experimental Domain

Algorithm 1:k-SWIM Selection of Variables with EmpIrical Modes |

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 3.4. Experimental Data Restricted to the RED

- The pruned reduced basis ${\overline{\mathbf{V}}}_{u}$, and the consecutive reduced coordinates ${({\mathbf{b}}_{u}({t}_{j}))}_{j=1,\dots ,{N}_{t}}$.
- The full mesh of $\mathsf{\Omega}$ and the mesh of ${\mathsf{\Omega}}_{R}$ ($\mathcal{F}$ and $\mathcal{I}$).
- The load history applied to the specimen on the subdomain ${\mathsf{\Omega}}_{user}$.
- Usual metadata related to the experiment (temperature, material parameters, etc.).

#### 3.5. Reduced Mechanical Equations Set Up on the Reduced Experimental Domain

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Illustrating Example: Polyurethane Bonded Sand Studied with X-ray CT

#### 4.1. Material and Test Description

#### 4.2. DVC and Error Estimation

#### 4.3. Building the Reduced Experimental Basis

#### 4.4. RED after DVC on the Specimen

## 5. Assessing the Relevance of the Pruned Data via Finite Element Model Updating-H${}^{2}$ROM

#### 5.1. Constitutive Model MC-CASM

#### 5.1.1. Presentation

- Addition of a damage law whose equation is phenomenological (based on cycled compressive tests).
- The hardening law of the bonding parameter b is different: A first hardening precedes the softening. It is supposed here that the polyurethane resin goes through a first hardening before breaking.

#### 5.1.2. Yield Function and Plastic Flow

#### 5.1.3. Hardening and Damage Laws

#### 5.2. Calibration Protocol by Using the Hybrid Hyper-Reduction Method

- One initial calculation where $\mathit{\mu}={\mathit{\mu}}^{0}$, which gives ${\mathbf{Q}}_{u}^{FE}({\mathit{\mu}}^{0})$;
- m parameters sensibility calculations where $\mathit{\mu}={\mathit{\mu}}^{i}=\{{\mu}_{1}^{0},\dots ,{\mu}_{i}^{0}+\delta {\mu}_{i}^{0},\dots ,{\mu}_{m}^{0}\}$, which give ${\mathbf{Q}}_{u}^{FE}({\mathit{\mu}}^{i})$ for $i=1,\dots ,m$

#### 5.3. Discussion on Dirichlet Boundary Conditions

**Theorem**

**5.**

**Proof.**

#### 5.4. Parameters Updating

#### 5.5. Model Calibration and FEM Validation

- Perform again the whole parameters sensibility study with ${\mathit{\mu}}^{0}={\mathit{\mu}}^{\ast}$.
- Concatenate the previously determined matrix $\overline{\mathbf{X}}$ from Equation (42) with ${\overline{\mathbf{Q}}}_{u}({\mathit{\mu}}^{\ast})$ and perform a new truncated SVD to determine ultimately an enriched reduced basis ${\overline{\mathbf{V}}}^{H}$. No new FEM calculations are needed.

## 6. Discussion

#### 6.1. Limitations of the Pruning Procedure

#### 6.2. About the Reconstruction of Data outside the RED

#### 6.3. Shear Strain Distributions in the RED

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Pruning of experimental data related to DVC, via hyper-reduction. Calibration capabilities of constitutive equations are preserved after data pruning. The experimental data related to the domain occupied by a specimen, denoted by $\mathsf{\Omega}$, are restricted to a reduced experimental domain denoted by ${\mathsf{\Omega}}_{R}$. The way the model calibration is done, depends on the nature of the data available in the storage system.

**Figure 2.**Schematic view of the reduced experimental domain, with linear triangular elements. In both figures, ${\mathsf{\Omega}}_{R}$ is red. On the left, there is the mesh of $\mathsf{\Omega}$. On the right, there is the reduced mesh (i.e., the mesh of ${\mathsf{\Omega}}_{R}$ only). In (

**a**), the nodes having their degrees of freedom in $\mathcal{I}$ are on the green line, the nodes having their degrees of freedom in $\mathcal{F}$ are in blue squares, and the grey nodes have their degrees of freedom in $\mathcal{R}$. In (

**b**), the green line, the blue squares and the grey nodes are related to ${\mathcal{I}}^{\star}$, ${\mathcal{F}}^{\star}$ and ${\mathcal{R}}^{\star}$, respectively.

**Figure 10.**Magnitude of the displacement ($\sqrt{{u}_{1}^{2}+{u}_{2}^{2}+{u}_{3}^{2}}$) for each DEPOD mode depending on $\alpha $.

**Figure 11.**Singular values of $\overline{\mathbf{X}}$ verifying $\lambda (\overline{\mathbf{X}})>{\u03f5}_{POD}{\lambda}_{max}(\overline{\mathbf{X}})$.

**Figure 13.**Probability distribution of shear strain at the last pre-peak step in the whole domain $\mathsf{\Omega}$, comparing FEM calculation (verification step) and experimental data.

Experimental Data | Empirical Modes | Pruned Data | |||
---|---|---|---|---|---|

${\mathbf{Q}}_{u}$ | 474,405 × 7 | ${\mathbf{V}}_{u}$ | 474,405 × 6 | ${\overline{\mathbf{V}}}_{u}$ | 73,911 × 6 |

${\mathbf{b}}_{u}$ | 6 × 7 | ${\mathbf{b}}_{u}$ | 6 × 7 | ||

Memory Saved | 15% | 85% |

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**MDPI and ACS Style**

Hilth, W.; Ryckelynck, D.; Menet, C.
Data Pruning of Tomographic Data for the Calibration of Strain Localization Models. *Math. Comput. Appl.* **2019**, *24*, 18.
https://doi.org/10.3390/mca24010018

**AMA Style**

Hilth W, Ryckelynck D, Menet C.
Data Pruning of Tomographic Data for the Calibration of Strain Localization Models. *Mathematical and Computational Applications*. 2019; 24(1):18.
https://doi.org/10.3390/mca24010018

**Chicago/Turabian Style**

Hilth, William, David Ryckelynck, and Claire Menet.
2019. "Data Pruning of Tomographic Data for the Calibration of Strain Localization Models" *Mathematical and Computational Applications* 24, no. 1: 18.
https://doi.org/10.3390/mca24010018