Efficient Approaches for Calibrating and Discovering Material Models in Computational Solid Mechanics

Core Concepts

This paper presents a unified framework for traditional parameter estimation methods and novel approaches that can infer the state variables or the model structure itself from experimental data in computational solid mechanics.

Abstract

The paper discusses various approaches for parameter identification and model discovery in computational solid mechanics. It starts by providing an overview of the fundamental equations in solid mechanics, the experimental possibilities for obtaining stress and strain data, and the parameter identification challenges for different classes of constitutive models (e.g., elasticity, hyperelasticity, viscoelasticity, elastoplasticity, viscoplasticity).
The paper then presents computational approaches for parameter identification, including:

The finite element method for discretizing the governing equations
Non-linear least-squares methods using the finite element method
The equilibrium gap method and the virtual fields method
Surrogate models and physics-informed neural networks
Model discovery approaches that infer the model structure from data
Bayesian approaches for quantifying parameter uncertainty
The authors propose a unified framework based on the "all-at-once" approach from the inverse problems community, which can cover both traditional parameter estimation and novel model discovery methods. This framework allows the authors to structure a large portion of the literature on parameter estimation in computational mechanics and identify combinations of methods that have not yet been addressed.
The paper also discusses statistical approaches to quantify the uncertainty in the estimated parameters, including identifiability analysis and a novel two-step procedure for identifying complex material models using both frequentist and Bayesian principles.
Finally, the authors illustrate and compare several of the discussed methods using mechanical benchmarks with synthetic and real data.

Stats

The paper does not contain any specific numerical data or metrics. It is a review and unification of various parameter identification and model discovery approaches in computational solid mechanics.

Quotes

"These developments call for a new unified perspective that is able to cover both traditional and novel parameter estimation and model discovery approaches."
"Adopting concepts discussed in the inverse problems community, we distinguish between all-at-once and reduced approaches."
"With this general framework, we are able to structure a large portion of the literature on parameter estimation in computational mechanics, and we can identify combinations and settings that have not yet been addressed."

Key Insights Distilled From

Reduced and All-at-Once Approaches for Model Calibration and Discovery in Computational Solid Mechanics

by Ulri... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.16980.pdf

Deeper Inquiries

How can the proposed unified framework be extended to handle more complex material behaviors, such as anisotropic or multi-scale models

To extend the proposed unified framework to handle more complex material behaviors, such as anisotropic or multi-scale models, several adjustments and considerations need to be made.

Anisotropic Models:

For anisotropic materials, the constitutive equations need to be modified to account for the directional dependence of material properties. This would involve incorporating tensors to represent the anisotropic behavior accurately.
The identification process would need to consider the additional parameters associated with anisotropy, which could increase the dimensionality of the parameter space.
The framework should be adapted to handle the unique stress-strain relationships and deformation behaviors characteristic of anisotropic materials.

Multi-Scale Models:

Multi-scale models involve interactions between different length scales, requiring a hierarchical approach to modeling and parameter identification.
The framework would need to incorporate coupling between different scales and possibly different constitutive equations at each scale.
Techniques such as homogenization or concurrent multi-scale modeling may be necessary to capture the overall material response accurately.

Parameterization:

Parameterization of anisotropic or multi-scale models may involve a larger number of parameters, necessitating efficient optimization algorithms to handle the increased complexity.
The framework should allow for the inclusion of a diverse range of material parameters that govern the behavior at different scales and orientations.

By adapting the unified framework to accommodate these complexities, researchers can gain a more comprehensive understanding of material behavior across different scales and orientations.

What are the potential challenges in applying the all-at-once approach to real-world experimental data with significant noise and uncertainties

Applying the all-at-once approach to real-world experimental data with significant noise and uncertainties poses several challenges that need to be addressed:

Robust Optimization Algorithms:

Noisy data can lead to inaccuracies in parameter estimation. Robust optimization algorithms, such as Bayesian optimization or evolutionary strategies, should be employed to handle uncertainties effectively.

Uncertainty Quantification:

Incorporating uncertainty quantification techniques, such as Bayesian inference or Monte Carlo simulations, can provide insights into the reliability of the estimated parameters in the presence of noise.

Regularization Techniques:

Regularization methods, like Tikhonov regularization or Lasso regression, can help mitigate the effects of noise by imposing constraints on the parameter space during optimization.

Data Preprocessing:

Preprocessing techniques, such as filtering or denoising algorithms, can be applied to clean the experimental data before parameter identification to improve the accuracy of the results.

Sensitivity Analysis:

Conducting sensitivity analysis to assess the impact of noise on the identified parameters can help in understanding the robustness of the model to uncertainties.

By addressing these challenges, researchers can enhance the reliability and accuracy of parameter identification when dealing with noisy experimental data.

How can the model discovery process be further automated and scaled to handle high-dimensional parameter spaces and complex model structures

Automating and scaling the model discovery process to handle high-dimensional parameter spaces and complex model structures can be achieved through the following strategies:

Machine Learning Techniques:

Utilize machine learning algorithms, such as neural networks or genetic algorithms, to automate the model discovery process and explore high-dimensional parameter spaces efficiently.

Parallel Computing:

Implement parallel computing techniques to expedite the exploration of complex model structures and reduce the computational time required for parameter identification.

Dimensionality Reduction:

Apply dimensionality reduction methods, like principal component analysis or autoencoders, to simplify high-dimensional parameter spaces and facilitate the discovery of essential model features.

Metaheuristic Optimization:

Employ metaheuristic optimization algorithms, such as particle swarm optimization or simulated annealing, to efficiently search the parameter space and discover complex material models.

Integration of Domain Knowledge:

Incorporate domain knowledge and expert insights into the automated model discovery process to guide the exploration of parameter spaces and ensure the relevance of the identified models.

By integrating these approaches, researchers can streamline the model discovery process, handle high-dimensional parameter spaces effectively, and uncover intricate relationships within complex material models.