The Molecular Element Method (MEM): A FEM-based Formulation for Linear and Non-Linear Molecular Elasticity

• Autores: A. Fernández-San Miguel
• Directores de la Tesis: Iván Couceiro Aguiar (dir. tes.) , Luís Ramírez (dir. tes.)
• Lectura: En la Universidade da Coruña ( España ) en 2026
• Idioma: inglés
• Número de páginas: 252
• Tribunal Calificador de la Tesis: Eugenio Oñate Ibáñez de Navarra (presid.) , José París (secret.) , Juan Carlos Verduzco (voc.)
• Enlaces
  • Tesis en acceso abierto en: RUC
• Resumen

  • Since its beginnings, humanity has always been characterized by building. From shelters to protect us from wind and rain, to pyramids erected to honor the pharaohs, to stone cathedrals and today’s slender cable-stayed bridges, we need knowledge about materials to make these structures possible. Until the end of the 20th century, the design and analysis of complex structures was particularly limited, but the advent of the Finite Element Method (FEM) has made it possible today to analyze them in different elasticity regimes, incorporating nonlinear effects and to design new and larger structures. On another level, the nanoscale has not been left without its advances and, despite the size, has claimed its relevance too. The discovery of materials with extraordinary and promising properties such as graphene and carbon nanotubes, or the devastating effects of the SARS-CoV-2 pandemic, has highlighted the impact that, even the smallest things, might have in our lives. However, unlike what happens at the macroscale with the FEM, only small systems of the order of hundreds of atoms can be analyzed at the nanoescale with techniques such as Density Functional Theory (DFT). This thesis proposes a new formulation coined the Molecular Element Method (MEM) that enables elastic analysis of molecular structures, from small molecules to proteins with thousands of atoms. The method relies on force fields developed in recent years by combining experimental and ab initio techniques, which have been exploited in Molecular Dynamics (MD). However, unlike MD, the proposed formulation is not restricted to dynamic analysis using explicit integration schemes. The proposed method allows to obtain linearized responses in first-order theory rigorously, to perform instability analyses such as buckling, and include nonlinear effects, exploiting typical FEM algorithms superior in order of convergence. The formulation permits working directly on geometries and potential energy at the atomic level, making it particularly versatile and competitive against laborious processes such as the generation of approximated continous models for specific problems. To perform first-order theory analysis, it has been proven that stiffness matrices only require the calculation of geometrical descriptor gradients. So, in an innovative way, the different gradients have been calculated exactly, treating possible singularities as in the case of aligned molecules or graphene sheets. The application of the formulation in this regime has made it possible to analyze different geometries and degrees of localized phenomena: from the precise determination of molecular vibration frequencies to the prediction of the mechanical properties of graphene and carbon nanotubes. The accurate determination of stiffness matrices has facilitated, unlike Molecular Dynamics where a correct selection of temporal discretization is essential to ensure simulation stability, the use of the Newmark method in its unconditionally stable version. As an example, the dispersion of mechanical waves in a nanotube was successfully analyzed with results equivalent to Molecular Dynamics. This work also analyzed the vibrations of protein systems, such as the spikes of SARS-CoV-2. The precise assembly of the stiffness matrices makes it feasible to determine the vibration frequency required to disable the proteins by resonance, and suggests the possibility of building an ultrasonic device for this purpose. Additionally, based on the results established in first-order theory, an extension to the nonlinear regime has been carried out. Consistently, the tangent stiffness matrices are determined exactly, which creates the possibility of using Newton algorithms or the arc length method. These algorithms are used to simulate elastic instabilities with experimental evidence such as single-walled and multi-walled nanotubes kink formation and the indentation of fullerenes. In turn, given that the extension makes it feasible to determine exactly the geometric stiffness matrices of nanostructures such as nanotubes, their critical loads have been calculated under different boundary conditions. This has enabled the calibration of a consistent continuous model using only geometry and force field parameterization as data, yielding mechanical properties that are in agreement with experimental results. Thefore, the Molecular Element Method allows to obtain, directly from the force fields the structural response, either static, dynamic or even non-linear of any molecular structure. In this sense, the method developed is a bridge between energy and structural response.