Discrete to continuum limits of fully atomistic and quasicontinuum systems with potentials of Lennard-Jones type – Anja Schloemerkemper


Sala Seminari (Dip. Matematica).


In recent years several engineering models, e.g., in the context of elasticity theory, were justified by a discrete to continuum analysis, i.e., by a passage from discrete/atomistic systems to continuum problems. I will present results with M. Schäffner in a one-dimensional setting with interaction potentials that are of Lennard-Jones type and thus have a convex-concave shape. This allows the formation of cracks. The focus will be on finite range interactions. In particular I will show a method that allows to trace the proof back to earlier proofs in the case of nearest and next-to-nearest neighbor interactions. We will discuss fully atomistic systems as well as systems which mimic some computational quasicontinuum method. This provides a rigorous analytical understanding of the quasicontinuum method and yields a condition on the choice of the finite element mesh that ensures a reasonable approximation.

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