Deep learning in the finite element method

• Tiefes Lernen in der Finite-Elemente-Methode

Koeppe, Arnd; Markert, Bernd (Thesis advisor); Herty, Michael (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Book, Dissertation / PhD Thesis

In: Report. IAM, Institute of General Mechanics No. IAM-11
Page(s)/Article-Nr.: 1 Online-Ressource : Illustrationen, Diagramme

Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, 2021

Abstract

In mechanics and engineering, the Finite Element Method (FEM) represents the predominant numerical simulation method. It is extraordinarily modular and flexible since it can simulate complex structures assembled from generic elements and utilizing various constitutive models. However, nonlinear problems, such as elastoplasticity, demand many Degrees of Freedom (DOF) and numerous iterations, which make the FEM numerically expensive. To increase numerical efficiency, data-driven algorithms and Artificial Intelligence (AI) offer an attractive approach to infer accurate nonlinear solutions from reduced-order inputs, thereby accelerating simulations. Inspired by the human brain, deep learning algorithms, i.e., (artificial) neural networks, organize and connect numerous neurons in layers and cells to train universal function approximations. Neural networks have demonstrated excellent performance and efficiency through parallelization in various applications. Because of the myriads of neurons and possible ways to connect them, neural networks often elude human understanding. Therefore, simpler models have been favored, even if they exhibit inferior performance. This thesis aims to integrate deep learning algorithms into the FEM, accelerate computations, and interpret neural networks in mechanics. Towards those objectives, a data-driven methodology is developed that deducts strategies to design neural networks for mechanics. Moreover, inductive approaches search optimal neural network configurations and explain neural network learning. Leveraging the fundamental data structure in mechanical balance equations, the data-driven methodology yields strategies and methods to interface neural networks with the FEM at three integration levels. At the highest level, intelligent surrogate models substitute entire finite element models and achieve efficient computations. At the lowest level, intelligent constitutive models offer flexibility, modularity, and straightforward integration. Combining the advantages of both approaches, intelligent meta elements yield considerable speed-ups and flexibility using substructuring. Additionally, strategies for data generation, preprocessing, and postprocessing translate and augmented mechanical data to train new neural network architectures with convolutions and recursions. Finally, a novel explainable AI approach interprets the black box of Recurrent Neural Networks (RNNs). Focusing on elastoplasticity, numerical demonstrators establish the performance of the deducted methods and strategies. Achieving considerable speed-ups by several orders of magnitude, mechanical field quantities are inferred accurately. Lastly, the new explainable AI approach investigates RNNs trained for constitutive behavior.