Fluid-structure interaction (FSI), occurrent in many areas of engineering and in the natural world, has been the subject of much research using a wide range of modelling strategies. However, problems with high levels of structural deformation are difficult to resolve and this is particularly the case for biomedical flows. A Lagrangian flow model coupled with a robust model for nonlinear structural mechanics seems a natural candidate since large distortion of the computational geometry is expected. Smoothed particle Hydrodynamics (SPH) has been widely applied for nonlinear interface modelling and this approach is investigated here. Biomedical applications often involve thin flexible structures and a consistent approach for modelling the interaction of fluids with such structures is also required. The Lagrangian weakly compressible SPH method is investigated in its recent delta-SPH form utilising inter-particle density fluxes to improve stability. Particle shifting is also used to maintain particle distributions sufficiently close to uniform to enable stable computation. The use of artificial viscosity is avoided since it introduces unphysical dissipation. First, solid boundary conditions are studied using a channel flow test. Results show that when the particle distribution is allowed to evolve naturally instabilities are observed and deviations are noted from the expected order of accuracy. A parallel development in the SPH group at Manchester has considered SPH in Eulerian form (for different applications). The Eulerian form is applied to the channel flow test resulting in improved accuracy and stability due to the maintenance of a uniform particle distribution. A higher-order accurate boundary model is developed and applied for the Eulerian SPH tests and third-order convergence is achieved. The well documented case of flow past a thin plate is then considered. The immersed boundary method (IBM) is now a natural candidate for the solid boundary. Again, it quickly becomes apparent that the Lagrangian SPH form has limitations in terms of numerical noise arising from anisotropic particle distributions. This corrupts the predicted flow structures for moderate Reynolds numbers (O(102)). Eulerian weakly compressible SPH is applied to the problem with the IBM and is found to give accurate and convergent results without any numerical stability problems (given the time step limitation defined by the Courant condition). Modelling highly flexible structures using the discrete element model is investigated where granular structures are represented as bonded particles. A novel vector-based form (the V-Model) is identified as an attractive approach and developed further for application to solid structures. This is shown to give accurate results for quasi-static and dynamic structural deformation tests. The V-model is applied to the decay of structural vibration in a still fluid modelled using Eulerian SPH with no artificial stabilising techniques. Again, results are in good agreement with predictions of other numerical models. A more demanding case representative of pulsatile flow through a deep leg vein valve is also modelled using the same form of Eulerian SPH. The results are free of numerical noise and complex FSI features are captured such as vortex shedding and non-linear structural deflection. Reasonable agreement is achieved with direct in-vivo observations despite the simplified two-dimensional numerical geometry. A robust, accurate and convergent method has thus been developed, at present for laminar two-dimensional low Reynolds number flows but this may be generalised. In summary a novel robust and convergent FSI model has been established based on Eulerian SPH coupled to the V-Model for large structural deformation. While these developments are in two dimensions the method is readily extendible to three-dimensional, laminar and turbulent flows for a wide range of applications in engineering and the natural world.

<p>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.</p> <p><br></p> <p>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.</p> <p><br></p> <p>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. </p> <p><br></p> <p>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.</p>

<p dir="ltr">The integration of scientific machine learning with computational structural mechanics offers a range of opportunities to address some of the most significant challenges currently experienced by multiphysical simulations, design optimization, and inverse sensing problems. While traditional mesh-based numerical methods, such as the Finite Element Method (FEM), have proven to be very powerful when applied to complex and geometrically inhomogeneous domains, their performance deteriorates very rapidly when faced with simulation scenarios involving high-dimensional systems, high-frequency inputs and outputs, and highly irregular domains. All these elements contribute to increase in the overall computational cost, the mesh dependence, and the number of costly matrix operations that can rapidly render FEM inapplicable. In a similar way, traditional inverse solvers, including global optimization methods, also face important limitations when handling high-dimensional, dynamic design spaces, and multiphysics systems. Recent advances in machine learning (ML) and deep learning have opened new ways to develop alternative techniques for the simulation of complex engineering systems. However, most of the existing deep learning methods are data greedy, a property that strides with the typically limited availability of physical observations and data in scientific applications. This sharp contrast between needed and available data can lead to poor approximations and physically inconsistent solutions. An opportunity to overcome this problem is offered by the class of so-called physics-informed or scientific machine learning methods that leverage the knowledge of problem-specific governing physics to alleviate, or even completely eliminate, the dependence on data. As a result, this class of methods can leverage the advantages of ML algorithms without inheriting their data greediness. This dissertation aims to develop scientific ML methods for application to forward and inverse problems in acoustics and structural mechanics while simultaneously overcoming some of the most significant limitations of traditional computational mechanics methods. </p><p dir="ltr">This work develops fully physics-driven deep learning frameworks specifically conceived to perform forward <i>simulations</i> of mechanical systems that provide approximate, yet physically consistent, solutions without requiring labeled data. The proposed set of approaches is characterized by low discretization dependence and is conceived to support parallel computations in future developments. These characteristics make these methods efficient to handle high degrees of freedom systems, high-frequency simulations, and systems with irregular geometries. The proposed deep learning frameworks enforce the governing equations within the deep learning algorithm, therefore removing the need for costly training data generation while preserving the physical accuracy of the simulation results. Another noteworthy contribution consists in the development of a fully physics-driven deep learning framework capable of improving the computational time for simulating domains with irregular geometries by orders of magnitude in comparison to the traditional mesh-based methods. This novel framework is both geometry-aware and maintains physical consistency throughout the simulation process. The proposed framework displays the remarkable ability to simulate systems with different domain geometries without the need for a new model assembly or a training phase. This capability is in stark contrast with current numerical mesh-based methods, that require new model assembly, and with conventional ML models, that require new training.</p><p dir="ltr">In the second part of this dissertation, the work focuses on the development of ML-based approaches to solve inverse problems. A new deep reinforcement learning framework tailored for dynamic <i>design optimization</i> tasks in coupled-physics problems is presented. The framework effectively addresses key limitations of traditional methods by enabling the exploration of high-dimensional design spaces and supporting sequential decision-making in complex multiphysics systems. Maintaining the focus on the class of inverse problems, ML-based algorithms for <i>remote sensing</i> are also explored with particular reference to structural health monitoring applications. A modular neural network framework is formulated by integrating three essential modules: physics-based regularization, geometry-based regularization, and reduced-order representation. The concurrent use of these modules has shown remarkable performance when addressing the challenges associated with nonlinear, high-dimensional, and often ill-posed remote sensing problems. Finally, this dissertation illustrates the efficacy of deep learning approaches for experimental remote sensing. Results show the significant ability of these techniques when applied to learning inverse mappings based on high-dimensional and noisy experimental data. The proposed framework incorporates data augmentation and denoising techniques to handle limited and noisy experimental datasets, hence establishing a robust approach for training on experimental data.</p>