Machine Learning Models for Computational Structural Mechanics
Creators: name: Jokar, Mehdi
Publisher: name: Purdue University Graduate School
Year: 2024
Subjects: classificationCode: 401707
subject: Solid mechanics
classificationCode: 461103
subject: Deep learning
classificationCode: 400510
subject: Structural engineering
classificationCode: 490302
subject: Numerical analysis

description: The numerical simulation of physical systems plays a key role in different fields of science and engineering. The popularity of numerical methods stems from their ability to simulate complex physical phenomena for which analytical solutions are only possible for limited combinations of geometry, boundary, and initial conditions. Despite their flexibility, the computational demand of classical numerical methods quickly escalates as the size and complexity of the model increase. To address this limitation, and motivated by the unprecedented success of Deep Learning (DL) in computer vision, researchers started exploring the possibility of developing computationally efficient DL-based algorithms to simulate the response of complex systems. To date, DL techniques have been shown to be effective in simulating certain physical systems. However, their practical application faces an important common constraint: trained DL models are limited to a predefined set of configurations. Any change to the system configuration (e.g., changes to the domain size or boundary conditions) entails updating the underlying architecture and retraining the model. It follows that existing DL-based simulation approaches lack the flexibility offered by classical numerical methods. An important constraint that severely hinders the widespread application of these approaches to the simulation of physical systems.
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In an effort to address this limitation, this dissertation explores DL models capable of combining the conceptual flexibility typical of a numerical approach for structural analysis, the finite element method, with the remarkable computational efficiency of trained neural networks. Specifically, this dissertation introduces the novel concept of <em>“Finite Element Network Analysis”</em> (FENA), a physics-informed, DL-based computational framework for the simulation of physical systems. FENA leverages the unique transfer knowledge property of bidirectional recurrent neural networks to provide a uniquely powerful and flexible computing platform. In FENA, each class of physical systems (for example, structural elements such as beams and plates) is represented by a set of surrogate DL-based models. All classes of surrogate models are pre-trained and available in a library, analogous to the finite element method, alleviating the need for repeated retraining. Another remarkable characteristic of FENA is the ability to simulate assemblies built by combining pre-trained networks that serve as surrogate models of different components of physical systems, a functionality that is key to modeling multicomponent physical systems. The ability to assemble pre-trained network models, dubbed <em>network concatenation</em>, places FENA in a new category of DL-based computational platforms because, unlike existing DL-based techniques, FENA does not require <em>ad hoc</em> training for problem-specific conditions.
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While FENA is highly general in nature, this work focuses primarily on the development of linear and nonlinear static simulation capabilities of a variety of fundamental structural elements as a benchmark to demonstrate FENA's capabilities. Specifically, FENA is applied to linear elastic rods, slender beams, and thin plates. Then, the concept of concatenation is utilized to simulate multicomponent structures composed of beams and plate assemblies (stiffened panels). The capacity of FENA to model nonlinear systems is also shown by further applying it to nonlinear problems consisting in the simulation of geometrically nonlinear elastic beams and plastic deformation of aluminum beams, an extension that became possible thanks to the flexibility of FENA and the intrinsic nonlinearity of neural networks. The application of FENA to time-transient simulations is also presented, providing the foundation for linear time-transient simulations of homogeneous and inhomogeneous systems. Specifically, the concepts of Super Finite Network Element (SFNE) and network concatenation in time are introduced. The proposed concepts enable training SFNEs based on data available in a limited time frame and then using the trained SFNEs to simulate the system evolution beyond the initial time window characteristic of the training dataset. To showcase the effectiveness and versatility of the introduced concepts, they are applied to the transient simulation of homogeneous rods and inhomogeneous beams. In each case, the framework is validated by direct comparison against the solutions available from analytical methods or traditional finite element analysis. Results indicate that FENA can provide highly accurate solutions, with relative errors below 2 % for the cases presented in this work and a clear computational advantage over traditional numerical solution methods. 
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The consistency of the performance across diverse problem settings substantiates the adaptability and versatility of FENA. It is expected that, although the framework is illustrated and numerically validated only for selected classes of structures, the framework could potentially be extended to a broad spectrum of structural and multiphysics applications relevant to computational science.