The finite element method has proven itself to be an effective method of performing both linear and non-linear structural analyses. This numerical method, however, has several limitations, of one which is that the discretization of complex geometry requires human effort. The rise of the application of machine learning models has opened new possibilities of approaching challenges within scientific fields. The neural network, and more specifically, the physics informed neural network, is a promising method and allows for performing analyses without any discretization that is necessary. This type of neural network, which is less of a black box than the standard neural network, utilizes the concept of partial differential equations to accurately predict solutions.

The objective of this thesis is to perform a structural analysis of a geometrically non-linear Timoshenko beam using a physics informed neural network. The network is built using a variational principle (the principle of virtual work) and a force residual. Furthermore, two optimization algorithms, the adaptive weight loss algorithm and the adaptive activation function are separately used in conjunction with the model to examine the potential improvements on the convergence rate.

It is found that the geometrically non-linear Timoshenko beam can be accurately modeled (relative error of below 2% with respect to the finite element output) with a physics informed neural network. This accuracy can be achieved with a model possessing a relatively shallow size of four hidden layers containing eight neurons each. The adaptive weight loss algorithm and the adaptive activation algorithm both improve the convergence rate of the model, though they are not necessary to maintain the practicality of the model, as the convergence rate is adequate without these. It is recommended that the hyperbolic tangent function is utilized in conjunction with the Adam optimizer. The adaptive activation function can be incorporated into the model to improve the convergence rate significantly without substantially increasing the computational cost of the model.

Founded in 1910, Boskalis, a leader in offshore operations, contends with limitations in ABB's Ability Marine Advisory System 'OCTOPUS' in predicting maximum vessel motions for heavy-transport vessels (HTVs). Accurate prediction of these motions, especially roll and pitch, is vital for transporting large, heavy structures, as exceeding predefined limits can jeopardize both vessel and cargo integrity. OCTOPUS's challenges, stemming not only from its reliance on linear theory but also potentially from the quality of its environmental data, underline the need for exploring alternatives, such as Machine Learning (ML) approaches, adept at handling complex, nonlinear phenomena, to ensure operational safety and efficiency.

This thesis presents the development and comparison of three new approaches to predict maximum roll and pitch motions. The approaches are compared and evaluated against OCTOPUS. Two validation strategies are used to test their performance under known and unknown loading conditions (LCs). Known LCs in this context refer to the evaluation of data that incorporate LCs that are included in the training dataset for ML-based approaches. On the other hand, unknown LCs refer to the evaluation of data that incorporate LCs that are not included in the training dataset for ML-based approaches. The approaches are trained and validated using sensor data, LC data, and environmental data from 24 different voyages for a specific HTV. They differ in their design and the type of environmental data they use.

The superior performance of ML-based approaches over OCTOPUS in known LCs is mainly due to two factors. First, ML approaches inherently incorporate nonlinear phenomena, which is particularly effective in accurately predicting maximum roll motion. Second, they are better equipped to handle flaws in environmental data. Although these advantages contribute to a significantly lower mean absolute percentage error (MAPE) compared to OCTOPUS, ML-based approaches face challenges in unknown LCs and extreme motion response scenarios. However, it is noteworthy that ML approaches quickly adapt to unknown LCs when small portions of these LCs are included in the training dataset.

ML shows potential in vessel motion prediction, and this thesis underscores the importance of diverse training data to enhance its reliability in unknown LCs and extreme motion response scenarios. For Boskalis, addressing these challenges with strategies such as adjusting the custom loss function, data augmentation, and implementing ensemble methods could improve the accuracy of these approaches. This progress is significant for Boskalis and the wider maritime industry, paving the way for adaptive and efficient prediction systems. Collaborative efforts between industry and academia, using rich data and expertise, are essential to drive these innovations.

Beams are the fundamental structural engineering element, supporting and stabilizing various structures ranging from suspension bridges to buildings and railways. Modeling and analyzing these structures necessitates a comprehensive understanding of the underlying beam dynamics constituting the structures. Simulating and predicting the beam dynamics is pivotal in ensuring structural integrity, optimizing structure design, and selecting appropriate materials. For instance, in railways, tracks and catenary contact wires are conceptualized as beams, allowing for the application of renowned beam theories like Euler-Bernoulli and Timoshenko. These theories provide a foundation for formulating partial differential equations (PDEs) that govern the dynamic behaviors of these beam systems.

These PDEs could be leveraged to simulate the underlying scenarios. The dissertation introduces physics-informed machine learning (PIML) based approaches tailored to simulate the dynamics of beam structures. The aim is to incorporate the physical laws in the neural networks training for more accurate and realistic simulations, handle noisy data effectively, and improve prediction accuracy while mitigating challenges such as multiscale problems and generalization. Chapter 1 outlines the primary challenges tackled in the dissertation. Chapters 2 through 5 detail the methodologies developed to address each challenge.

Chapter 2 presents a physics-informed neural network (PINN) based methodology to simulate complex beam systems with real-world mate- rial properties. In addition, inverse problems are solved in the presence of noisy data to predict unknown parameters, including force acting on the beam systems. It is essential to consider the real-world material parameters to simulate the dynamics of the modeled system and ensure the digital model represents the ground truth. However, incorporating material characteristics leads to multiscale PDE coefficients in the physical model, posing difficulty in training for PINNs. Subsequently, a frame- work is proposed to incorporate nondimensional PDEs into the PINN loss function. This approach facilitates efficient forward and inverse simulations while robust to noise and uncertainty in measurement data. The efficacy of this approach is demonstrated through simulations of Euler- Bernoulli and Timoshenko beam systems, contributing to the challenge of simulating large-scale systems with multiple interconnected components.

Chapter 3 investigates beam dynamic simulations on Winkler foundations for large spatiotemporal domains using PIML. Predictions on expansive spatiotemporal domains are vital for structural integrity, design optimization, and control mechanisms. A causality-respecting PINN frame- work is introduced, enhancing prediction accuracy. Furthermore, integrating transfer learning addresses the need to re-train the network for different initial conditions and computational domains. Numerical experiments based on Euler-Bernoulli and Timoshenko theories validate the methodology for respecting the causality and generalizing the beam dynamics across similar problems. The approach efficiently predicts beam dynamics under diverse engineering scenarios, reducing computational costs and improving convergence.

Chapter 4 explores the generalization abilities of PIML, essential for practical applications requiring accurate predictions in unexplored regions. The proposed framework exploits the inherent causality in the PDE solutions by merging PIML models with recurrent neural architectures, namely neural oscillators. The neural ordinary differential equations in the form of neural oscillators effectively handle long-time dependencies and address gradient-related issues, fostering improved generalization in PIML tasks. Benchmark equations like viscous Burgers, Allen-Cahn, Schrödinger, and biharmonic Euler-Bernoulli beam equations are used to demonstrate the effectiveness of the proposed approach. Through ex- tensive experimentation with time-dependent nonlinear PDEs, the study showcases superior performance compared to existing state-of-the-art methods. The proposed method provides accurate solutions for extrapolation and prediction beyond the training data by enhancing the generalization capabilities of PIML, promising advancements in complex system simulations.

Chapter 5 follows up on generalization of beam dynamics beyond PIML- based approaches. Computer-aided simulations are crucial for advancing engineering industries, but existing simulators often struggle to generalize beyond their training domain. The chapter proposes a two-stage methodology to tackle this challenge. Firstly, it utilizes specialized simulators tailored to the application, such as causal PINNs and black-box finite element simulations. Secondly, it integrates predictions from the first stage into a recurrent neural architecture, incorporating ordinary differential equations to capture intrinsic dynamics and enhance generalization. The approach efficiently captures causality and generalizes dynamics across various data sources. Numerical experiments cover fundamental structural engineering scenarios, including real-world catenary contact wire uplift predictions, and demonstrate superior performance compared to conventional methods, and promise for diverse industrial applications. This dissertation concludes with Chapter 6.

In particular, this dissertation introduces PIML methodologies for simulating complex beam structures, addressing key challenges such as incorporating real material properties, handling noisy data, and improving prediction accuracy. Chapter 2 introduces a PINN-based methodology that efficiently simulates beam systems and predicts unknown parameters, mitigating the difficulties posed by multiscale PDE coefficients. Chapter 3 tackles the challenge of large-domain beam dynamics predictions on the Winkler foundations by using causality-respecting PINNs and integrating transfer learning to reduce computational costs. Chapter 4 addresses the challenge of out-of-domain predictions in PIML by introducing neural oscillators. Chapter 5 proposes a two-stage methodology to generalize beam dynamics simulations, integrating beam dynamics solvers and recurrent neural-based architectures, showcasing its efficacy in real-world applications such as catenary contact wire uplift predictions.

Regular maintenance of civil engineering structures is essential for their safety. Current maintenance regimes involve periodic inspections at regular time intervals. In the time between inspections, there can be a critical development in the structural integrity of a structure, which can be expensive to repair or could even lead to structural failure. A more robust maintenance approach would involve continuous monitoring of the structure. This is the research area of Structural Health Monitoring (SHM). Vibration-based monitoring, which is a subset of SHM, aims to provide new cost-effective maintenance solutions that provide long-term life-safety benefits. Vibration-based monitoring uses vibration measurements from sensors to assess the "Health" of a structure. Damage in a structure will alter the structure's stiffness, mass and damping characteristics, which in turn will change the dynamic properties of the system. This change can then be discovered in the vibration data. The growth of the field is partly due to the significant advances in data-driven science and engineering in recent decades.

One of the issues in vibration-based monitoring is the presence of operational and environmental variability in the vibration data. With this variability present, it is challenging to determine from vibration data the characteristics of the underlying dynamic system. The aim of this thesis is to use Robust Principal Component Analysis (rPCA) to reduce or eliminate the operational variability from the traffic to allow for environmental and damage detection. rPCA is a matrix factorisation method that decomposes a data matrix into a low-rank matrix L and a sparse matrix S. The reconstructed low-rank matrix L contains the main correlations in the data that are robust to outliers and corrupt data that are contained in the sparse matrix S. By applying rPCA to the frequency representation of the vibration data, it is hoped that the underlying coherent structure corresponding to the dynamic system can be recovered.


The vibration data used for this thesis is from two measurement campaigns conducted on the Haringvlietbrug. The Haringvlietbrug is a steel box girder bridge in the Netherlands, and there are several fatigue cracks present in the bridge. This presented an opportunity for damage detection. The goal of the first measurement campaign is to conduct damage detection and discover if there is a difference between vibration data from a damaged area with fatigue cracks and a "healthy" reference area. In the second measurement campaign, the goal was to extract the underlying dynamics of the structure at different temperatures and see if it was possible to distinguish between the different structural states at different temperatures. After applying the rPCA on the vibration data, (regular) principal component analysis (PCA) is used to embed the data into the low-rank subspace of the PCs to distinguish between the different structural states.

The rPCA was successful in extracting the coherent structures in the vibration data corresponding to the underlying dynamic properties of the system. In the subsequent PCA, vibration data with underlying different structural states had different scores in the first three PCs. In other words, it was possible to distinguish between the different structural states based on the first three PCs, which correspond to the main correlation within the data. For the first measurement campaign, this meant it was possible to distinguish between vibration data in the damaged area and the "healthy" reference area and detect "damage". However, there was a difference in the structural configuration between the two areas, so it was not possible to conclude that the differences in vibration data were due to damage caused by the fatigue cracks. In the second measurement campaign, the dynamic system properties at different temperatures were recovered. With the low-rank vibration data from the rPCA, it was possible to distinguish between vibration data with a 1°C difference in the first three PCs from the (regular) PCA. This was not possible without lowering the regularisation parameter in the rPCA. Another method, the Sparse Sensor Placement for Optimisation (SSPOC), was used to determine the locations in the frequency spectrum that contained the largest differences between structural states. For both the first and second measurement campaigns, these locations were at specific natural frequencies of the system.