Boundary formulations for discrete lattices: to describe the non-smooth dynamic response of solid media in the time domain

This thesis presents a modelling framework to describe the non-smooth dynamic response of solid media in the time domain under dynamic loading conditions, such as those typically found in soil-structure and ice-structure interaction. The approach divides the medium into two domains: a near-field domain capable of capturing nonlinear phenomena, and a surrounding linear far-field domain. The near field is modelled as a discrete lattice incorporating rheological elements, such as springs, dashpots, and dry-friction elements, for example enabling stick-slip behaviour.
The far-field domain is incorporated through a boundary integral formulation that accounts for the domain's properties solely at the interface with the near field, allowing for accurate wave transmission and minimal reflections at the boundary. Boundary integral equations (BIEs) are derived for both continuous and discrete representations of the far field, including one of the first derivations of BIEs for finite or semi-infinite discrete particle systems.
To address the computational challenges of time-domain simulations involving nonlinearities, a novel mixed time-frequency domain (MTFD) method is introduced. This non-iterative hybrid approach combines the efficiency of frequency-domain methods during periods of linear behaviour while accounting for the changing properties of the lattice over time whenever a nonlinear event occurs.

Results demonstrate the effectiveness of lattice models and discrete-based BIEs in capturing non-smooth dynamics, while highlighting the importance of robust numerical implementation. The proposed framework offers a promising tool for simulating wave propagation in nonlinear media and supports improved analysis and design in civil, geotechnical, and offshore engineering applications.