Bridging scales - from Quantum Mechanics to Continuum Mechanics. A Finite Element approach.

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The concurrently coupled Quantum Mechanics (QM) - Continuum Mechanics (CM) approach for electro-elastic problems is considered in this proposal. Despite the fact that efforts have been made to bridge different description of matter, many questions are yet to be answered. First, an efficient Finite Element (FE)-based solution approach to the Kohn-Sham (KS) equations of Density Functional Theory (DFT) will be further developed. The h-adaptivity in the FE-based solution with non-local pseudo-potentials, as well as the mesh transformation during the structural optimization and formulation of the deformation map are the main topics to be studied. It should be noted that until now there exists no open-source implementation of the DFT approach which uses a FE basis and provides hp-refinement capabilities. A FE basis is very attractive in the context of the DFT theory because of its completeness, refinement possibility as well as good polarization properties based on domain decomposition. Second, QM quantities will be related to their CM counterparts (e.g. displacements, deformation gradient, the Piola stress, polarization, etc). This will be achieved using averaging in the Lagrangian configuration. To that end the full control over a FE-based solution of the KS equations is required. The procedure is then to be tested on a representative numerical example - bending of a single wall carbon nanotube. On the CM side, the surface-enhanced continuum theory will be utilized to properly capture surface effects. It should be noted that although several theoretical works exist on this matter, no numerical attempts have been made to check their validity on test examples. Lastly, based on the correspondence between different formulations, a concurrently coupled QM-CM method will be proposed. Coupling will be achieved in a staggered way, i.e. QM and CM problems will be solved iteratively with a proper exchange of information between them. A test-problem of crack propagation in a graphene sheet will be considered. As a long term goal of the project, coupling strategies for electro-elastic problems will be developed. To the best of my knowledge, non of the QM-CM coupling method is capable to handle electro-elastic problems.