Numerical modelling has the potential to probe the complexity of the interacting physics of rubble mound armour systems. Through forward modelling of armour unit packs, stochastic variables such as unit displacement and maximum contact force per unit during an external oscillatory disturbance can be predicted. The combined finite-discrete element method (FEMDEM) is a multi-body method ideally suited to model the behaviour of the armour layer system and the stresses generated within complex shape units. In this paper we highlight the latest developments made with the application of FEMDEM technology to breakwater modelling including realistic rock underlayer and concrete unit layer topologies, maximum contact force distributions, internal unit stresses, fracture and unit breakages. Finally, fully coupled wave and multi-body armour unit motion with internal dynamic stress generation is illustrated.

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The aim of this study is to model the interactions between fluids and solids using fully nonlinear models. Non-linearity is important in the context of floating wind turbines, for example, to model breaking waves impacting on the structure and the effect of the solids elasticity. In this work, the fluid- and solid-dynamics equations are solved using separate finite-element models, which are coupled at every time step. This enables the mutual interactions between fluids and moving solids to be modelled. Importantly, the coupling algorithm ensures that the action-reaction principle is satisfied at a discrete level, independently of the order of representation of the discrete fields in each model. To the authors knowledge, the present algorithm is novel in that it can simultaneously handle (i) non-matching fluid and solid meshes, (ii) different polynomial orders of the basis functions on each mesh, and (iii) different fluid and solid time steps. Results are shown for: (i) a bottom-mounted pile subjected to small-amplitude waves in a numerical wave tank, and (ii) a truncated pile floating at an interface between air and water.@en

FEMDEM modelling which combines the multi-body particle interaction and motion modelling (i.e. Discrete Element Model, DEM) with the ability to model internal deformation of arbitrary shape (Finite Element Model, FEM) has been applied to breakwater models. There are two versions of a FEMDEM solver developed; Y3D-D is for deformable materials and is required for dynamic and static stress analysis and Y3D-R is the rigid version often used to numerically construct the armour unit packs. This paper also reports the placement protocols: POSITIT. FEMDEM modelling deals with solids interactions and is one modelling component that is to be coupled to other modelling technologies e.g. CFD, interface tracking, wave models, porous media etc. so that the key fluid-solid interactions can be modelled in a full scale virtual breakwater alongside work on scaled hydraulic laboratory models and prototype structures. The latest developments of two-way coupled interactions of waves with coastal structures are also described in this paper.

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Discrete-fracture and rock matrix (DFM) modelling necessitates a physically realistic discretisation of the large aspect ratio fractures and the dissected material domains. Using unstructured spatially adaptively refined finite-element meshes, we find that the fastest flow often occurs in the smallest elements. Flow velocity and element size vary over many orders of magnitude, disqualifying global Courant number (CFL)-dependent transport schemes because too many time steps would be necessary to investigate displacements of interest. Here, we present a higher-order accurate implicit pressure-(semi)-implicit transport scheme for the advection-diffusion equation that overcomes this CFL limitation for DFM models. Using operator splitting, we solve the pressure and the transport equations on finite-element, node-centred finite-volume meshes, respectively, using algebraic multigrid methods. We apply this approach to field data-based DFM models where the fracture flow velocity and mesh refinement is 2-4 orders of magnitude greater than that of the matrix. For a global CFL of ≤10,000, this implies sub-CFL, second-order accurate behaviour in the matrix, and super-CFL, at least first-order accurate, transports in fast-flowing fractures. Their greater refinement, however, largely offsets this numerical dispersion, promoting a highly accurate overall solution. Numerical and fracture-related mechanical dispersions are compared in the realistic DFM models using second-order accurate runs as reference cases. With a CFL histogram, we establish target error criteria for CFL overstepping. This analysis indicates that for extreme fracture heterogeneity, only a few transport steps can be sufficient to analyse macro-dispersion. This makes our implicit method attractive for quick analysis of transport properties on multiple realisations of DFM models.

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