At the heart of understanding material behavior lies the examination of phenomena occurring at diverse length and time scales. Historically, individual disciplines—ranging from quantum mechanics to continuum mechanics—have specialized in exploring specific scales, from atomic and molecular up to macroscopic levels.

Multiscale Materials Modelling endeavors to bridge these scales, employing a hierarchical approach to couple models from the finest atomic scales to broader macroscopic scales. At the atomic level, electronic structures dictate material properties, and understanding the positions of atoms and their underlying potentials is crucial. As we transition to larger scales, atomistic models are often too computationally expensive, necessitating the use of alternate representations which can capture the essence of the material behavior without delving into atomic detail.

This upscaling is a meticulous process, requiring the development of transition models that maintain the continuity of local physical variables, such as displacement or temperature. Such continuity ensures that the physics at one scale transition seamlessly into the next.

Additionally, a central challenge in multiscale modelling is uncertainty quantification. As we derive coarse-scale properties from finer-scale models, there's inherent uncertainty involved. Quantifying this uncertainty is pivotal, as the fidelity of the macroscopic model depends on the accuracy and reliability of its foundational microscopic models.