Mathematics

Every transient problem in continuum mechanics is characterized by three variables: space, time and parameters. The space defines the physical domain, enabling the definition of diverse systems. The time captures dynamic processes, allowing for transient behavior analysis. The parameters control system and modeling characteristics. Together, these elements drive the accuracy and relevance of computational science, making them essential for understanding and predicting real-world phenomena.The complexity arising from managing space, time, and parameters in numerical simulations can be particularly challenging when dealing with thin structures, small time steps combined with long time intervals, and a high number of parameters over large domains.The numerical simulation of three-dimensional models in thin geometries presents important challenges since maintaining the mesh granularity proportional to the thickness dimension requires an impractical number of elements for the entire structure. This issue is currently encountered in automotive industry when considering vehicle crash simulations, where most of the components are thin structures.When time multiscale behaviours occur, standard discretization techniques are constraint to mesh up to the finest scale to predict accurately the response of the system. This results in a prohibitive computational when the phenomena must be observed over a long duration. This occurs, for instance, in material science when dealing with fatigue damage assessments and cyclic visco-elasto-plastic fatigue problems.A large number of parameters increases the dimensionality of the parameter space exponentially, making its exploration computationally intensive. The data generated from numerous simulations can be difficult to manage and advanced meta-modeling techniques are required. This typically happens in optimal design problems of multi-component parametric structures.To address these challenges, it is essential to strike a balance between accuracy and computational efficiency, requiring ad-hoc advanced developments. In this thesis the three challenges are separately addressed via novel separation-based techniques.