[论文解读] On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning

I.B.C.M. Rocha, P. Kerfriden+1 more|arXiv (Cornell University)|Jul 11, 2020

Model Reduction and Neural Networks参考文献 56被引用 51

一句话总结

本文提出一种在线贝叶斯代理框架，使用高斯过程来替代FE2中的嵌套微模型，实现并发多尺度分析中自适应、基于不确定性的实时本构代理构建。

ABSTRACT

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.

研究动机与目标

• 阐明对高保真度并发多尺度分析的需求并解决 FE2 计算成本高昂的问题。
• 提出一种在线代理建模方法，用高斯过程代理替代微模型（micromodels）。
• 利用锚点微模型（anchor micromodels）和在线学习来构建具有不确定性量化的本构代理。
• 将 GP 代理整合到标准有限元分析中，实现稳健、数据高效的在线细化。

提出的方法

• 在宏观点上用从少量锚点微模型在线训练的高斯过程代理替代嵌套微模型。
• 使用贝叶斯回归来量化不确定性，并在需要时触发新数据获取。
• 通过梯度信息增强GP以改进预测，并使本构响应的导数预测成为可能。
• 纳入导数观测和互协方差，以在训练点附近提高代理的精度。
• 将代理嵌入到常规有限元求解循环中，以保持数值鲁棒性。

实验结果

研究问题

• RQ1 在线、数据高效的概率代理模型是否能够在不进行离线训练的情况下重现 micromodel 响应？
• RQ2 基于GP的代理如何提供可靠的不确定性估计，以在加载路径期间触发在线细化？
• RQ3 纳入梯度/导数信息对代理准确性和鲁棒性的影响是什么？
• RQ4 在体积和粘结多尺度均质化场景中，在线自适应框架的表现如何？
• RQ5 该方法在保持保真度的同时在塑性、弹塑性和裂纹扩展问题中可在多大程度上降低计算成本？

主要发现

• GP代理使在不进行离线训练的情况下，基于不确定性细化的本构流形在线构建成为可能。
• 纳入梯度和导数观测提高了在训练点附近的预测精度并降低不确定性。
• 该框架通过与FE求解循环耦合、利用在线数据触发细化，展示了鲁棒性。
• 应用包括带有塑性的锥形杆、一块带多个开口的板以及混合模态弯曲中的裂纹扩展示例，以说明其多功能性。
• 通过在线学习和有针对性的微模型评估实现了显著的效率提升。