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Multi-material physics arise in an innumerable amount of engineering problems. A broadly scoped numerical model is developed and described in this thesis to simulate the dynamic interaction of multi-fluid and solid systems. It is particularly aimed at modelling the interaction of two immiscible fluids with solid structures in a coastal engineering context; however it can be extended to other similar areas of research. The Navier Stokes equations governing the fluids are solved using a combination of finite element (FEM) and control volume finite element (CVFE) discretisations. The sharp interface between the fluids is obtained through the compressive transport of material properties (e.g. material concentration). This behaviour is achieved through the CVFE method and a conveniently limited flux calculation scheme based on the Hyper-C method by Leonard (1991). Analytical and validation test cases are provided, consisting of steady and unsteady flows. To further enhance the method, improve accuracy, and exploit Lagrangian benefits, a novel moving mesh method is also introduced and tested. It is essentially an Arbitrary Lagrangian Eulerian method in which the grid velocity is defined by semi-explicitly solving an iterative functional minimisation problem. A multi-phase approach is used to introduce solid structure modelling. In this approach, solution of the velocity field for the fluid phase is obtained using Model B as explained by Gidaspow (1994, page 151). Interaction between the fluid phase and the solids is achieved through the means of a source term included in the fluid momentum equations. The interacting force is calculated through integration of this source term and adding a buoyancy contribution. The resulting force is passed to an external solid-dynamics model such as the Discrete Element Method (DEM), or the combined Finite Discrete Element Method (FEMDEM). The versatility and novelty of this combined modelling approach stems from its ability to capture the fluid interaction with particles of random size and shape. Each of the three main components of this thesis: the advection scheme, the moving mesh method, and the solid interaction are individually validated, and examples of randomly shaped and sized particles are shown. To conclude the work, the methods are combined together in the context of coastal engineering applications, where the complex coupled problem of waves impacting on breakwater amour units is chosen to demonstrate the simulation possibilities. The three components developed in this thesis significantly extend the application range of already powerful tools, such as Fluidity, for fluids-modelling and finite discrete element solids-modelling tools by bringing them together for the first time.

The scientific problems related to granular matter are ubiquitous. It is currently an active area of research for physicists and earth scientists, with a wide range of applications within the industrial community. Simple analogue experiments exhibit behaviour that is neither predicted nor described by any current theory. The work presented here consists of modelling granular media using a two-dimensional combined Finite-Discrete Element Method (FEM-DEM). While computationally expensive, as well as modelling accurately the dynamic interactions between independent and arbitrarily shaped grains, this method allows for a complete description of the stress state within individual grains during their transient motion. After a detailed description of FEM-DEM principles, this computational approach is used to investigate the packing of elliptical particles. The work is aimed at understanding the influence of the particle shape (the ellipse aspect ratio) on the emergent properties of the granular matrix such as the particle coordination number and the packing density. The diff erences in microstructure of the resultant packing are analysed using pair correlation functions, particle orientations and pore size distributions. A comparison between frictional and frictionless systems is carried out. It shows great diff erences not only in the calculated porosity and coordination number, but also in terms of structural arrangement and stress distribution. The results suggest that the particle's shape a ffects the structural order of the particle assemblage, which itself controls the stress distribution between the pseudo-static grains. The study then focuses on describing the stress patterns or \force chains" naturally generated in a frictional system. An algorithm based on the analysis of the contact force network is proposed and applied to various packs in order to identify the force chains. A statistical analysis of the force chains looking at their orientation, length and proportion of the particles that support the loads is then performed. It is observed that force chains propagate less efficiently and more heterogeneously through granular systems made of elliptical particles than through systems of discs and it is proposed that structural diff erences due to the particle shape lead to a signifi cant reduction in the length of the stress path that propagates across connected particles. Finally, the e ffect of compression on the granular packing, the emergent properties and the contact force distribution is examined. Results show that the force network evolves towards a more randomly distributed system (from an exponential to a Gaussian distribution), and it confi rms the observations made from simulations using discs. To conclude, the combined finite-discrete element method applied to the study of granular systems provides an attractive modelling strategy to improve the knowledge of granular matter. This is due to the wide range of static and dynamic problems that can be treated with a rigorous physical basis. The applicability of the method was demonstrated through to a variety of problems that involve di fferent physical processes modelled with the FEM-DEM (internal deformations, fracture, and complex geometry). With the rapid extension of the practical limits of computational models, this work emphasizes the opportunity to move towards a modern generation of computer software to understand the complexity of the phenomena associated with discontinua.