A FEM-based formulation for nanostructures: from allotropes of carbon to proteins
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One of the great successes in the field of engineering, whether aeronautical, civil or mechanical, was the development of the Finite Element Method, thanks to which problems of solid mechanics, fluids and even fluid-structure interaction can be solved with great precision. In its origins, the method was applied mainly to macroscale problems, while at the atomic scale the use of Molecular Dynamics with Verlet-type integration schemes has been the more usual way. Currently, most molecular dynamics software uses the same approach to solve structural dynamics problems under the Born-Oppenheimer hypothesis. Whereas, in the case of the Finite Element Method, the method also allows the analysis of structures in different analysis regimes, such as first order, linear buckling or geometric nonlinear. This makes FEM an extremely versatile method and leads to considerable savings in computational time. For this reason, we have developed a formulation that allows, directly from the force field, to perform a structural analysis at molecular level in the different regimes mentioned above [1, 2]. In the case of first-order analysis, we applied this formulation to the vibrational analysis of different molecules [1], proposing a method that allows the parameterization of the potentials known the experimental vibrational frequencies of the structure [2]. Their use was not only limited to small molecular structures, but also the vibrations of the spikes of coronaviruses in their two fundamental states were studied. In the case of linear buckling, it has been applied to the buckling of nanotubes and graphene sheets, in combination with equivalent continuous models. Finally, nonlinear effects such as kinks in nanotubes and graphene sheets and the behaviour of the fullerene family under different loading cases were also studied. The possibility of predicting mechanical properties of different nanostructures solely based on atomic geometry and force field parameterization is demonstrated. REFERENCES [1] A. Fernández-San Miguel, L. Ramírez, I. Couceiro & F. Navarrina. A first order FEM-based formulation for the analysis of molecular structures with bonded interactions. Engineering with Computers, 2024. [2] A. Fernández-San Miguel, L. Ramírez, I. Couceiro & F. Navarrina. A Comparative Review of FEM Like Techniques Applied to the Linear Analysis of Molecular Structures. Archives of Computational Methods in Engineering, 2025.
