Novel multiscale models in a multicontinuum approach to divide and conquer strategies

• Published in: Computational & applied mathematics 2023-06, Vol.42 (4), Article 143
• Main Authors: Feijóo, Raúl A., Blanco, Pablo J., de Souza Neto, Eduardo A., Sánchez, Pablo J.
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Applied physics; Boundary conditions; Computational mathematics; Computational Mathematics and Numerical Analysis; Exact solutions; Iterative methods; Kinematics; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Microstructure; Scale models
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-023-02288-9

This contribution presents a comprehensive, in-depth analysis of the solution of the mechanical equilibrium problem for a generic solid with microstructure. The exact solution to this problem, referred to here as the reference solution , corresponds to the full-scale model of the problem that takes into account the kinematics and constitutive behavior of its entire microstructure. The analysis is carried out based on the Principle of Multiscale Virtual Power (PMVP) previously proposed by the authors. The PMVP provides a robust theoretical setting whereby the strong links between the reference solution and solutions of the mechanical equilibrium obtained using coarser scale models are brought to light. In this context, some fundamental properties of coarser scale solutions are identified by means of variational arguments. These findings unveil a new homogenization landscape for Representative Volume Element (RVE) multiscale theories, leading to the construction of new Minimal Kinematical Restriction (MKR)-based models where either displacements or tractions may be prescribed on the RVE boundary. A careful observation of the aforementioned landscape leads naturally to the proposal of a new, multicontinuum strategy (a generalized continuum counterpart of multigrid strategies) to approximate the reference solution at low computational cost. In the proposed strategy, the mechanical interactions among neighboring microcells are accounted for in an iterative fashion by means of suitably chosen boundary conditions enforced alternately on the new MKR-based models. The proposed developments are presented assuming a classical continuum at all scales, but the results are equally valid when different kinematical and constitutive assumptions are made at different scales.