cs.CEMay 31, 2026

Machine Learning Surrogate Modeling for Homogenization of Hyperelastic Materials with Boolean Microstructures

Authors: Matthias Brändel, Oliver Rheinbach

Abstract

Data-driven surrogate models are an alternative to numerical homogenization of heterogeneous materials. In this contribution, a supervised learning approach is presented for predicting effective Lamé parameters of hyperelastic composites from low-dimensional microstructural descriptors. The data set is based on previously published numerical homogenization results for ensembles of two-phase stochastic microstructures generated by planar Boolean models, covering variations of inclusion shape, phase contrast, and area fraction; see Brändel, Brands, Maike, Rheinbach, Schröder, Schwarz and Stoyan (2022). A neural network is trained on combinations of scalar and curve-valued statistical descriptors, including the area fraction, a derived scalar shape descriptor ττ, the two-point correlation function S2(r)S_2(r), and the lineal-path function ℓ(z)\ell(z). Additional data representing limiting cases of the parameter space are incorporated to stabilize training and improve extrapolation behavior. The surrogate is evaluated by leave-one-grain-type-out cross-validation in order to assess generalization to unseen grain geometries. Numerical results demonstrate that additional descriptors can reduce relative errors. A predictor trained with ττ and S2(r)S_2(r) provides a compact representation with good quantitative accuracy and regular dense response behavior. Adding the lineal-path function ℓ(z)\ell(z) further reduces the error at the available data points, indicating that it is a promising additional descriptor; however, dense post-training response evaluations show that improved pointwise accuracy does not automatically guarantee physically admissible behavior between sampled parameter values. This motivates future work on physically constrained surrogate models, loss formulations, bounded output parametrizations, and a more systematic representation of curve-valued geometric descriptors.

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Emerging sustainable materials increasingly rely on engineered hierarchy and microstructure to achieve control of their properties and mechanical behavior. Optimizing these materials with controllable microstructures requires efficient multiscale simulations. Data-driven surrogate models for the microscale can accelerate multiscale simulations, but require large amounts of data even for a fixed microstructure. When a range of microstructures is considered, as is the case in multiscale optimization, even more data is needed to train a surrogate. To overcome this challenge, we condition a hybrid physics-data surrogate on microstructural variables using a hypernetwork. This approach enables accurate predictions of multiscale mechanical behavior for a mycelium-woodchip composite material, even when trained on small datasets. The conditioned surrogate makes multiscale simulations of functionally graded structures tractable, and we validate it against a full FE^2 simulation. We optimize a graded multiscale disk, and reduce the peak stress by 42% compared to one with a random microstructure. Then, we go one step further, conditioning the network directly on manufacturing variables that can have a complex influence on the microstructure. This is a practical route to engineer the microscale for desired macroscale behavior. This contribution highlights the benefits of microarchitectured structures and demonstrates how conditioned surrogate models enable their multiscale optimization, which will accelerate the development and design of future sustainable materials and structures.
J. Storm, I. B. C. M. Rocha, S. Schyck +2
Porous MediaContinuum DynamicsSurrogate Modeling
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May 4, 2026physics.comp-ph

Multi-fidelity surrogates for mechanics of composites: from co-kriging to multi-fidelity neural networks

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Haizhou Wen, Elham Kiyani, Gang Li +3
Multi-FidelitySurrogate ModelingAnisotropy
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