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Optimal Experimental Design for Reliable Learning of History-Dependent Constitutive Laws

Kaushik Bhattacharya, Lianghao Cao, Andrew Stuart

Abstract

History-dependent constitutive models serve as macroscopic closures for the aggregated effects of micromechanics. Their parameters are typically learned from experimental data. With a limited experimental budget, eliciting the full range of responses needed to characterize the constitutive relation can be difficult. As a result, the data can be well explained by a range of parameter choices, leading to parameter estimates that are uncertain or unreliable. To address this issue, we propose a Bayesian optimal experimental design framework to quantify, interpret, and maximize the utility of experimental designs for reliable learning of history-dependent constitutive models. In this framework, the design utility is defined as the expected reduction in parametric uncertainty or the expected information gain. This enables in silico design optimization using simulated data and reduces the cost of physical experiments for reliable parameter identification. We introduce two approximations that make this framework practical for advanced material testing with expensive forward models and high-dimensional data: (i) a Gaussian approximation of the expected information gain, and (ii) a surrogate approximation of the Fisher information matrix. The former enables efficient design optimization and interpretation, while the latter extends this approach to batched design optimization by amortizing the cost of repeated utility evaluations. Our numerical studies of uniaxial tests for viscoelastic solids show that optimized specimen geometries and loading paths yield image and force data that significantly improve parameter identifiability relative to random designs, especially for parameters associated with memory effects.

Optimal Experimental Design for Reliable Learning of History-Dependent Constitutive Laws

Abstract

History-dependent constitutive models serve as macroscopic closures for the aggregated effects of micromechanics. Their parameters are typically learned from experimental data. With a limited experimental budget, eliciting the full range of responses needed to characterize the constitutive relation can be difficult. As a result, the data can be well explained by a range of parameter choices, leading to parameter estimates that are uncertain or unreliable. To address this issue, we propose a Bayesian optimal experimental design framework to quantify, interpret, and maximize the utility of experimental designs for reliable learning of history-dependent constitutive models. In this framework, the design utility is defined as the expected reduction in parametric uncertainty or the expected information gain. This enables in silico design optimization using simulated data and reduces the cost of physical experiments for reliable parameter identification. We introduce two approximations that make this framework practical for advanced material testing with expensive forward models and high-dimensional data: (i) a Gaussian approximation of the expected information gain, and (ii) a surrogate approximation of the Fisher information matrix. The former enables efficient design optimization and interpretation, while the latter extends this approach to batched design optimization by amortizing the cost of repeated utility evaluations. Our numerical studies of uniaxial tests for viscoelastic solids show that optimized specimen geometries and loading paths yield image and force data that significantly improve parameter identifiability relative to random designs, especially for parameters associated with memory effects.
Paper Structure (87 sections, 1 theorem, 83 equations, 17 figures, 2 tables)

This paper contains 87 sections, 1 theorem, 83 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Main Contributions and Outline of Paper
  4. Related Work
  5. Bayesian Optimal Experimental Design for Material Testing
  6. Details of \ref{['CI']}
  7. Expected Information Gain
  8. Stress State Entropy
  9. Bayesian D-Optimal Design Based on Gaussian Posterior Approximation
  10. Details of \ref{['CII']}
  11. Sensitivity-Based Design
  12. Gaussian Posterior Approximation
  13. Bayesian D-optimal design
  14. Surrogate Modeling For Optimal Experimental Design
  15. Details of \ref{['CIII']}
  16. ...and 72 more sections

Key Result

Lemma 4.1

The EIG, defined as the expected reduction in information entropy from the prior to the posterior in eq:eig_posterior, is equivalent to the mutual information, denoted by $\mathrm{MI}$, between the parameters and the data: Here, the mutual information has two equivalent definitions: where $D_{\mathrm{KL}}$ denotes the Kullback--Leibler divergence.

Figures (17)

  • Figure 1: A visualization of a model for the described uniaxial test with loading path and specimen shape design. The dots on the loading path, i.e., the prescribed strain, are the control points. The color on the deformed material represents the magnitude of the displacement.
  • Figure 2: A visualization of the observation operator for a single snapshot of image data at time $t$ for the uniaxial testing described in \ref{['subsec:material_response']}.
  • Figure 3: An example of the uniaxial testing setup and the design variables. (Left) The setup and the specimen geometry design. (Right) The loading path design.
  • Figure 4: An example of simulated force and image data for learning linear viscoelasticity. This simulated dataset is created via the experimental setup in \ref{['subsec:viscoelastic_setup', 'fig:viscoelastic_setup']}, and the constitutive parameters are drawn from the prior. Note that the resolution of the image data visualized here is much lower than that of the actual data used in the numerical study; see an image snapshot with high resolution in \ref{['fig:high_res_images']}. The material deformation is discernible yet small due to the narrow range of the imposed strain ($<5\%$) suitable for linear viscoelasticity.
  • Figure 5: The surrogate FIM testing accuracy as a function of training sample size for designing uniaxial tests of linear and nonlinear viscoelastic materials in \ref{['sec:results_linear', 'sec:results_nonlinear']}, respectively. The solid and dashed lines indicate the mean relative errors for predicting the log. FIM (measured in the Frobenius norm) and the approximate information gain, respectively. The surrogate prediction of the approximate information gain is obtained from the surrogate FIM predictions using \ref{['eq:approximate_ig']}.
  • ...and 12 more figures

Theorems & Definitions (3)

  • Lemma 4.1: The Expected Information Gain as Mutual Information
  • Remark 1: Gaussian Prior as a Special Case
  • Remark 2: Gaussian Approximation Breaks Form Equivalence