Dynamic adaptive concurrent multi-scale simulation of wave propagation in 3D media

Over the last decades the use of numerical simulations to characterize the response of real structures has proven to be a valid tool that can accelerate the design process. However, the correct interpretation of the mechanical behaviour including stress localisations and geometrical features, requires the adoption of fine discretisations that drastically increase the computational cost. Even when using one numerical description for the mechanichal representation of the structure (e.g. continuum mechanics), the use of concurrent adaptive multi-scale frameworks in which finer temporal and/or spatial discretisation scales are generated during computation within the regions of interest and coupled both spatially and temporally to the original coarse scale, can lead to drastic reductions of the computation times. In this context, the challenge is the formulation of a stable coupling between the discretisation scales, which avoids the generation and propagation of numerical artefacts such as spurious wave reflection. Moreover, the detection of the error in dynamic simulations based on the the popular super-convergent patch recovery technique, even if formally correct, requires a substantial computational effort, that can result in a bottleneck for the whole simulation.
Based on these current limitations, the conducted research was to develop a novel numerically efficient concurrent dynamic finite element framework which automatically detects the regions of interest and simulates them in a finer time and length scales. In particular, this thesis investigates the concurrent coupling between domains with the same numerical description (i.e. Finite Elements for fine and coarse scales), where a scale is defined as a computational domain which presents a finer/coarser temporal and spatial discretisation. The first main contribution of this work is the formulation of an error estimator based on an hermitian interpolation of the kinematic variables that is local to each element. The proposed procedure, avoids the need for neighbour searches and resolution of complex systems of equations. Another key feature of this methodology is that it can be used to directly transfer the variables from the coarse to the fine scale, resulting in smooth strain and stress field without the use of a balance step. Another key contribution of this thesis is represented by the coupling methodology, formulated in terms of nodal forces over an evolving coupling volume. This novel formulation enforces the kinematic link between the scales over a volume defined as the set composed by the first neighbours of the elements highlighted for refinement. The resulting coupling can be evaluated explicitly at every node of the coupled domain, resulting in a procedure numerically efficient and easy to implement.
The properties of the framework are demonstrated both analytically and through a set of numerical simulations, using the one dimensional wave propagation in elastic rods for validation. Firstly, the proposed error estimator is compared against the analytical error showing similar rate of convergence. The above hermitian strategy is used to transfer the variables to the finer scales, where the equilibrium among internal, external and inertial forces is respected without the use of an intermediate balance step. Subsequently, parametric studies have demonstrated the paramount importance of suitable coupling lengths and weighting parameters to avoid the formation of spurious wave reflections.
The novel mathematical findings, expanded in the three dimensional space, result in the fundamental building blocks of a novel dynamic adaptive recursive concurrent multi-scale framework. The proposed implementation idealises the relationship among scales as parent-child in which one coarse scale can generate several child scales based on the implemented refinement criteria. The validation of the framework has been proven in reversible and irreversible settings. Finally, after the rigorous demoniii stration of the validity of the methodology future research lines are suggested, based on the versatility of the framework.

Advisor: Petrinic, N
Barbieri, E