Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements
Abstract
:1. Introduction
1.1. Overview of Inverse Methods for the Mechanical Characterization of Micro/Meso-Structural Properties
1.2. Multiscale Statistical Identification Method
1.3. Drawbacks and Limitations of the Multiscale Identification Method
1.4. Improvements of the Multiscale Identification Method and Novelty of the Paper
1.5. Outline of the Paper
2. Assumptions for Solving the Multiscale Statistical Inverse Problem
- there exists a scale separation between macroscale and mesoscale, so that a mesoscopic subdomain can be defined and for which the dimensions are sufficiently large with respect to the size of the heterogeneities and sufficiently small with respect to the size of the macroscopic domain. Such a mesoscopic subdomain can then be considered as a representative volume element;
- the random apparent elasticity tensor field at mesoscale is the restriction to one or more bounded mesoscopic subdomain(s) of a second-order stationary random field indexed by , and consequently the mean function of the random elasticity field at mesoscale is independent of the spatial coordinates;
- the random apparent elasticity tensor field at mesoscale is ergodic in average in the mean-square sense, so that the homogenized elasticity tensor at macroscale calculated by stochastic homogenization of the random apparent elasticity field in a mesoscopic subdomain corresponding to a representative volume element can be considered as almost deterministic, in the sense that (i) its spatial average reaches an asymptotic convergence with a very high level of probability for a sufficiently large mesoscopic subdomain size, and therefore (ii) its level of statistical fluctuations around its mean function at macroscale can be considered as negligible, thus yielding a deterministic homogenized elasticity tensor at macroscale.
3. Multiscale Experimental Test Configuration
4. Prior Multiscale Stochastic Model and Its Hyperparameters
5. Objectives and Strategy for Solving the Multiscale Statistical Inverse Problem
5.1. Objectives of the Multiscale Statistical Inverse Problem
5.2. Strategy for Solving the Multiscale Statistical Inverse Problem
- A macroscopic numerical indicator , dedicated to the identification of parameter , that allows for quantifying the distance between the experimental strain field associated to the experimental displacement field measured at macroscale in the macroscopic domain and the strain field associated to the displacement field computed from a deterministic homogeneous linear elasticity boundary value problem (with both Dirichlet and Neumann boundary conditions) that models the experimental test configuration at macroscale and involves the unknown deterministic elasticity tensor ;
- A mesoscopic numerical indicator , dedicated to the identification of hyperparameter , that allows for quantifying the distance between a pseudo-dispersion coefficient modeling the level of spatial fluctuations of the experimental strain field associated to the experimental displacement field measured at mesoscale in a mesoscopic domain of observation , and a random pseudo-dispersion coefficient representing the level of statistical fluctuations of the random strain field associated to the random displacement field computed from a stochastic heterogeneous linear elasticity boundary value problem (with only Dirichlet boundary conditions) that models the experimental test configuration at mesoscale and involves the random elasticity tensor field with an unknown level of statistical fluctuations that must be identified;
- Another mesoscopic numerical indicator , dedicated to the identification of hyperparameter , that allows for quantifying the distance between the 3 different pseudo-spatial correlation lengths of the experimental strain field in each spatial direction, measured at mesoscale in a mesoscopic domain of observation , and the 3 pseudo-spatial correlation lengths of the random strain field in each spatial direction, computed from the same mesoscopic stochastic boundary value problem as for for which the random elasticity tensor field has a spatial correlation structure induced and characterized by an unknown vector of spatial correlation lengths that must be identified;
- A multiscale numerical indicator , dedicated to the identification of hyperparameter , that allows for quantifying the distance between the homogeneous deterministic elasticity tensor at macroscale and the effective elasticity tensor resulting from a computational stochastic homogenization in a representative volume element at mesoscale of the random elasticity tensor field whose mean function is unknown and must be identify.
6. Construction of the Numerical Indicators for Solving the Multiscale Statistical Inverse Problem
6.1. Deterministic Macroscopic Boundary Value Problem for the Macroscopic Indicator
6.2. Stochastic Mesoscopic Boundary Value Problem for the Mesoscopic Indicators
6.3. Macroscopic Numerical Indicator
6.4. Mesoscopic and Multiscale Numerical Indicators
6.4.1. Mesoscopic Numerical Indicator Associated to the Dispersion Parameter
6.4.2. Mesoscopic Numerical Indicator Associated to the Spatial Correlation Lengths
6.4.3. Multiscale Numerical Indicator Associated to Computational Stochastic Homogenization
6.5. Comments
7. Multiscale Statistical Inverse Problem Formulated as a Multi-Objective Optimization Problem
- a macroscale inverse problem formulated as a single-objective optimization problem that consists in calculating the optimal value of parameter in that minimizes the macroscopic numerical indicator , that is
- a mesoscale statistical inverse problem formulated as a multi-objective optimization problem that consists in calculating the optimal value of hyperparameter in that minimizes the two mesoscopic numerical indicators and as well as the multiscale numerical indicator simultaneously, that is
8. Numerical Methods for Solving the Multi-Objective Optimization Problem
9. Probabilistic Model for a Robust Identification of the Hyperparameters
10. Numerical Validation of the Multiscale Identification Method on In Silico Materials in 2D Plane Stress and 3D Linear Elasticity
10.1. Validation on an In Silico Specimen in Compression Test in 2D Plane Stress Linear Elasticity
10.1.1. Parameterization of the Macroscopic and Mesoscopic Models
10.1.2. Resolution of the Single-Objective Optimization Problem at Macroscale
10.1.3. Resolution of the Multi-Objective Optimization Problem at Mesoscale
10.2. Validation on an In Silico Specimen in Compression Test in 3D Linear Elasticity
10.2.1. Parameterization of the Macroscopic and Mesoscopic Models
10.2.2. Resolution of the Single-Objective Optimization Problem at Macroscale
10.2.3. Resolution of the Multi-Objective Optimization Problem at Mesoscale
11. Numerical Application of the Multiscale Identification Method to Real Beef Cortical Bone in Plane Stress Linear Elasticity
11.1. Parameterization of the Macroscopic and Mesoscopic Models
11.2. Numerical Results of the Multiscale Statistical Inverse Identification
11.2.1. Resolution of the Single-Objective Optimization Problem at Macroscale
11.2.2. Resolution of the Multi-Objective Optimization Problem at Mesoscale
12. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
a.s. | almost surely |
RVE | Representative Volume Element |
CCD | Charge-Coupled Device |
CMOS | Complementary Metal-Oxide-Semiconductor |
DIC | Digital Image Correlation |
DVC | Digital Volume Correlation |
CT | micro-Computed Tomography |
MRI | Magnetic Resonance Imaging |
OCT | Optical Coherence Tomography |
LS | Least Squares |
MLE | Maximum Likelihood Estimation |
MaxEnt | Maximum Entropy |
KL | Karhunen-Loève |
PC | Polynomial Chaos |
FP | Fixed-Point |
GA | Genetic Algorithm |
FEM | Finite Element Method |
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[GPa] | [GPa] | |
---|---|---|
Relative error [%] |
ℓ [m] | [GPa] | [GPa] | |||
---|---|---|---|---|---|
3 | |||||
4 | |||||
3 | |||||
3 | |||||
3 | |||||
4 | |||||
4 | |||||
4 | |||||
3 | |||||
4 | |||||
4 | |||||
3 | |||||
4 | |||||
3 | |||||
4 | |||||
4 | |||||
- | |||||
- | |||||
Relative error [%] | - | ||||
855,000 |
ℓ [m] | [GPa] | [GPa] | |||
---|---|---|---|---|---|
- | |||||
() | 855,000 | ||||
Relative error [%] | - | ||||
() | 87,000 | ||||
Relative error [%] | - | ||||
() | 9000 | ||||
Relative error [%] | - |
ℓ [m] | [GPa] | [GPa] | |||
---|---|---|---|---|---|
193 | |||||
202 | |||||
189 | |||||
197 | |||||
207 | |||||
201 | |||||
192 | |||||
199 | |||||
210 | |||||
205 | |||||
203 | |||||
198 | |||||
194 | |||||
208 | |||||
190 | |||||
208 | |||||
- | |||||
- | |||||
Relative error [%] | - | ||||
19,176,000 |
[GPa] | [GPa] | |
---|---|---|
Relative error [%] |
ℓ [m] | [GPa] | [GPa] | |||
---|---|---|---|---|---|
3 | |||||
4 | |||||
3 | |||||
- | |||||
- | |||||
Relative error [%] | - | ||||
150,000 |
[GPa] | [GPa] | |
---|---|---|
ℓ [m] | [GPa] | [GPa] | |||
---|---|---|---|---|---|
5 | |||||
7500 |
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Zhang, T.; Pled, F.; Desceliers, C. Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements. Materials 2020, 13, 2826. https://doi.org/10.3390/ma13122826
Zhang T, Pled F, Desceliers C. Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements. Materials. 2020; 13(12):2826. https://doi.org/10.3390/ma13122826
Chicago/Turabian StyleZhang, Tianyu, Florent Pled, and Christophe Desceliers. 2020. "Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements" Materials 13, no. 12: 2826. https://doi.org/10.3390/ma13122826