# On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning

arXiv: https://arxiv.org/abs/2007.07749 · HTML: https://mlsanity.com/paper/2007.07749/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2020.

## Built on by (most cited first, 14 of 14)

- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
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- [Modular machine learning-based elastoplasticity: generalization in the context of limited data](https://mlsanity.com/paper/2210.08343/) (2022)
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- [Model-data-driven constitutive responses: application to a multiscale computational framework](https://mlsanity.com/paper/2104.02650/) (2021)
- [Intelligent Computing: The Latest Advances, Challenges and Future](https://mlsanity.com/paper/2211.11281/) (2022)
- [Cooperative data-driven modeling](https://mlsanity.com/paper/2211.12971/) (2022)
- [Multi-Scale Simulation of Complex Systems: A Perspective of Integrating Knowledge and Data](https://mlsanity.com/paper/2306.10275/) (2023)
- [Physically recurrent neural network for rate and path-dependent heterogeneous materials in a finite strain framework](https://mlsanity.com/paper/2404.17583/) (2024)
- [Recurrent neural networks and transfer learning for elasto-plasticity in woven composites](https://mlsanity.com/paper/2311.13434/) (2023)
- [Integration of Active Learning and MCMC Sampling for Efficient Bayesian Calibration of Mechanical Properties](https://mlsanity.com/paper/2411.13361/) (2024)
- [A Comparison of Surrogate Constitutive Models for Viscoplastic Creep Simulation of HT-9 Steel](https://mlsanity.com/paper/2509.22667/) (2025)

## Abstract (arXiv)

Concurrent multiscale finite element analysis (FE2) is a powerful approach for high-fidelity modeling of materials for which a suitable macroscopic constitutive model is not available. However, the extreme computational effort associated with computing a nested micromodel at every macroscopic integration point makes FE2 prohibitive for most practical applications. Constructing surrogate models able to efficiently compute the microscopic constitutive response is therefore a promising approach in enabling concurrent multiscale modeling. This work presents a reduction framework for adaptively constructing surrogate models based on statistical learning. The nested micromodels are replaced by a machine learning surrogate model based on Gaussian Processes (GP). The need for offline data collection is bypassed by training the GP models online based on data coming from a small set of fully-solved anchor micromodels that undergo the same strain history as their associated macro integration points. The Bayesian formalism inherent to GP models provides a natural tool for uncertainty estimation through which new observations or inclusion of new anchors are triggered. The surrogate constitutive manifold is constructed with as few micromechanical evaluations as possible by enhancing the GP models with gradient information and the solution scheme is made robust through a greedy data selection approach embedded within the conventional finite element solution loop for nonlinear analysis. The sensitivity to model parameters is studied with a tapered bar example with plasticity, while the applicability of the model to more complex cases is demonstrated with the elastoplastic analysis of a plate with multiple cutouts and a crack growth example for mixed-mode bending. Significant efficiency gains are obtained without resorting to offline training.