Inference of Heterogeneous Material Properties via Infinite-Dimensional Integrated DIC

TL;DR

This work addresses nondestructive identification of spatially varying material properties from full-field deformation images. It introduces $\infty$--IDIC, a PDE-constrained inverse problem formulated in function spaces to infer high-dimensional modulus fields while enforcing physical consistency. Regularization strategies $L^2$, $H^1$, and primal-dual TV, together with an inexact Newton-CG solver and adjoint/Hessian calculi, yield dimension-independent convergence and robust handling of ill-posedness. Numerical tests on linear elasticity and neo-Hookean hyperelasticity demonstrate accurate recovery of sharp and smooth features, mesh-independent performance, and informative stress post-processing, with an information-gain framework guiding experimental design and data quality impacts.

Abstract

We present a scalable and efficient framework for the inference of spatially-varying parameters of continuum materials from image observations of their deformations. Our goal is the nondestructive identification of arbitrary damage, defects, anomalies and inclusions without knowledge of their morphology or strength. Since these effects cannot be directly observed, we pose their identification as an inverse problem. Our approach builds on integrated digital image correlation (IDIC, Besnard Hild, Roux, 2006), which poses the image registration and material inference as a monolithic inverse problem, thereby enforcing physical consistency of the image registration using the governing PDE. Existing work on IDIC has focused on low-dimensional parameterizations of materials. In order to accommodate the inference of heterogeneous material propertes that are formally infinite dimensional, we present $\infty$-IDIC, a general formulation of the PDE-constrained coupled image registration and inversion posed directly in the function space setting. This leads to several mathematical and algorithmic challenges arising from the ill-posedness and high dimensionality of the inverse problem. To address ill-posedness, we consider various regularization schemes, namely $H^1$ and total variation for the inference of smooth and sharp features, respectively. To address the computational costs associated with the discretized problem, we use an efficient inexact-Newton CG framework for solving the regularized inverse problem. In numerical experiments, we demonstrate the ability of $\infty$-IDIC to characterize complex, spatially varying Lamé parameter fields of linear elastic and hyperelastic materials. Our method exhibits (i) the ability to recover fine-scale and sharp material features, (ii) mesh-independent convergence performance and hyperparameter selection, (iii) robustness to observational noise.

Figures (17)

• Figure 1: Our contributions include the development of an infinite-dimensional IDIC framework that enables the inversion of spatially varying parameter fields engineered for heterogeneous materials. Novel contributions are highlighted: handling heterogeneity, infinite dimensional framework, inference for high dimensional modulus fields, and stress fields.
• Figure 2: Schematic of the numerical setup for generating the reference and deformed states. The domain, $\Omega$, is speckled. A traction condition is applied to the right boundary, $\Gamma_R$ and a clamped condition is applied to the left boundary, $\Gamma_L$.
• Figure 3: Inverse results for Linear Elasticity for varying features. Each inversion is based on a simple tensile experiment with $\sim$0.2% strain. 10% and 5% noise were applied to the image brightness and force measurements, respectively. The inverse problem is solved using the primal-dual total variation formulation ($\gamma_{L^2} = 5\times10^{-8}$, $\gamma_{TV} = 1\times10^{-6}$) with a mesh size of $100 \times 100$, and the speckle correlation length is 0.01. The initial guess was uniformly $m(x)=2$. The synthetic images are shown on the right.
• Figure 4: Inverse results for Hyperelasticity for varying features. Each inversion is based on a simple tensile experiment with $\sim$5% strain. 10% and 5% noise were applied to the image brightness and force measurements, respectively. The inverse problem is solved using the primal-dual total variation formulation ($\gamma_{L^2} = 5\times10^{-6}$, $\gamma_{TV} = 7.5 \times10^{-4}$) with a mesh size of $100 \times 100$, and the speckle correlation length is 0.01.
• Figure 5: Inverse results for Hyperelasticity for varying regularization. Each inversion is based on a simple tensile experiment with $\sim$5% strain. 10% and 5% noise were applied to the image brightness and force measurements, respectively. The inverse problem is solved using a mesh size of $100 \times 100$ and a speckle correlation length of 0.01.

Theorems & Definitions (2)

• Remark 2.1
• Remark 3.1