material parameter estimation

Conventional inverse hyperelastic modeling for heterogeneous materials relies heavily on pre-specified constitutive forms, limiting generalizability and introducing bias. Method: This paper proposes a model-free, physics-embedded end-to-end framework that reconstructs spatially varying constitutive mappings directly from full-field displacement data (e.g., DIC measurements). It integrates Fourier feature networks, neural ordinary differential equations (NODEs), and hypernetworks to inherently satisfy continuum mechanics constraints—namely, kinematic compatibility and local force balance. A physics-informed loss function is formulated using the strong-form equilibrium equations and Neumann boundary conditions, jointly optimized with multi-objective regularization to balance physical consistency and data fidelity. Results: The method demonstrates robustness across scenarios involving parameter heterogeneity, anisotropic transitions, synthetic noise, and real experimental data—significantly reducing reliance on prior constitutive assumptions while achieving superior accuracy compared to traditional inversion approaches.

In PDE-based inverse problems, unreliable constitutive relations induce systematic bias in material parameter estimation. To address this, we propose Weak Neural Variational Inference (WNVI), the first framework to explicitly decouple reliable conservation laws from uncertain constitutive relations. WNVI introduces a weighted-residual virtual likelihood to interpretably localize model error sources and enable probabilistic quantification and robust correction of systematic modeling errors. Integrating weak-form neural networks, variational inference, and latent-variable probabilistic modeling, the method operates without an exact forward solver, ensuring computational efficiency and structural clarity. Evaluated on elastography, WNVI significantly improves both accuracy in material parameter estimation and reliability in uncertainty calibration. It establishes an interpretable, generalizable uncertainty modeling paradigm for PDE-driven inverse problems.

This study addresses the challenge of uncertainty quantification in constitutive model discovery—specifically, when no prior assumptions about model parameters are available. We propose a four-step semi-Bayesian framework: (1) Gaussian process regression to denoise and augment stress–strain data; (2) nonparametric approximation of the joint posterior parameter distribution via normalizing flows; (3) model structure identification and distillation through functional-space distribution matching; and (4) enhanced interpretability using Sobol’ sensitivity analysis. Crucially, the method requires no prespecified parameter priors, handles both linear and nonlinear model libraries uniformly, and automatically discovers interpretable constitutive forms. It significantly improves accuracy in uncertainty propagation. The framework is validated on both isotropic and anisotropic experimental datasets, demonstrating robust performance across diverse material behaviors.

This study addresses the challenge of unreliable parameter identification in history-dependent constitutive models under limited experimental budgets, where insufficient data often compromises inference accuracy—particularly for parameters governing memory effects. To enhance identifiability, the authors propose an efficient framework based on Bayesian optimal experimental design that optimizes experimental protocols by maximizing expected information gain. To mitigate computational costs, a surrogate model is constructed using Gaussian approximations and the Fisher information matrix, enabling rapid optimization of batch experimental designs for complex material testing. Demonstrated on uniaxial tests of viscoelastic solids, the optimized specimen geometries and loading paths significantly outperform random designs, markedly reducing the number of required physical experiments while improving the precision of parameter estimates.

This work addresses the challenge of efficiently inferring constitutive response functions from experimental data, a task traditionally hindered by time-consuming parameter optimization in classical material models. The authors propose two novel frameworks—Physics-Augmented Neural Operator (PANO) and Constitutive Artificial Neural Operator (CANO)—which, for the first time, apply neural operators to solve the inverse problem of constitutive modeling in infinite-dimensional input–output spaces. By encoding inputs via Laplacian eigenfunctions, the approach achieves discretization independence and robustness to noise, while embedding physical constraints in the output layer ensures thermodynamic consistency of the predicted strain energy density function. Requiring only a single forward pass, the model enables near real-time inference of hyperelastic constitutive laws and demonstrates exceptional generalization across unseen geometries, noisy or incomplete data, varying meshes, and different scales.

This study addresses the challenge of identifying nonlinear, spatially heterogeneous constitutive parameters of solid materials when only surface displacements and contact forces are available. To this end, an efficient inverse method based on isogeometric finite element model updating (FEMU) is proposed. By employing low-order Lagrangian interpolation—decoupled from the analysis mesh—to represent spatially varying material fields, and integrating analytical gradient computation, a material-parameter continuation strategy, and a trust-region reflective optimization algorithm, the approach enables non-destructive, high-fidelity parameter identification. Numerical experiments successfully reconstruct heterogeneous material distributions in three-dimensional hyperelastic solids and thin shells, demonstrating the method’s effectiveness and computational efficiency for applications in soft tissue biomechanics and advanced material characterization.

This study addresses the challenge of efficiently and accurately inverting finite-strain anisotropic elastoplastic constitutive models in high-dimensional parameter spaces. To this end, we propose a JAX-based differentiable, GPU-accelerated finite element framework that tightly integrates automatic differentiation with finite element computations, eliminating the need for manual gradient derivation and enabling PDE-constrained inverse parameter identification. By leveraging heterogeneous specimen geometries inspired by topology optimization and full-field displacement measurements, the approach substantially reduces experimental dependency. Implemented on a single H100 GPU, the framework achieves up to 9.4× speedup over an Abaqus implementation running on a 24-core CPU and successfully recovers both homogeneous and spatially varying anisotropic yield and hardening parameters, demonstrating its efficiency and feasibility for high-dimensional inverse problems.

This study investigates the feasibility and geometric dependence of inferring the relative magnitudes of tensile, bending, and bearing loads from stress intensity factor (SIF) distributions along a crack front. Leveraging finite element data from SIFBench, the authors propose a unified crack-front operator that couples a structured forward surrogate model with a differentiable inverse mapping, yielding a set-valued estimator augmented with calibrated uncertainty quantification. Theoretical analysis reveals that load identifiability hinges on the functional linear independence of three canonical load profiles, and introduces an intrinsic stability margin to quantify the ill-posedness of the inverse problem. Experimental validation demonstrates that most corner-crack configurations are well-posed, yielding reliable point estimates, whereas a few inherently ill-conditioned cases produce uninformative estimates—consistent with theoretical predictions and numerical observations.

研究提出了一种数据驱动的多材料本构模型框架，通过神经常微分方程和人工神经网络来预测超弹性和粘弹性材料的组成依赖性行为。

This study addresses the challenge of accurately calibrating strongly coupled thermomechanical materials under finite strains using only surface experimental data. To this end, the authors propose a full-field calibration framework that leverages boundary displacements, reaction forces, and surface temperature measurements. The forward problem is formulated as a nearly incompressible thermo-hyperelastic system based on a Helmholtz free energy constitutive model, and the inverse problem is solved via PDE-constrained optimization. Innovatively relying solely on surface observables—without requiring volumetric measurements—the method integrates weighted multi-source observational terms and exploits automatic differentiation for efficient computation of adjoint gradients. Validation on both synthetic and real experimental data demonstrates the framework’s ability to accurately identify key coupling parameters, such as thermal expansion and directional contraction coefficients.