Alternative multiscale material and structures modeling by the finite-element method

In this study, we present an alternative finite element to multiscale analysis. In this strategy, strain energy comes only from semi-discrete or lattice elements immersed in a continuum without stiffness, enabling mechanical analysis from molecular scales to macroscopic scales. Some characteristics of the proposed element are: (1) geometrically non-linear exact description that allows the presence of large displacements and large strain, (2) general mapping that allows curved and distorted elements generation with automatic immersions, (3) total compatibility with standard finite elements, and (4) huge degrees of freedom reduction with a small loss of continuum mobility. Throughout the text, the proposed strategy is presented in detail and applied in the determination of suitable meshes for any scale of analysis, which is an important information for future applications. To be direct, a Lennard–Jones-like (LJL) potential is chosen to build different crystalline-like structures that, when immersed in finite elements without stiffness, results in the desired continuous behavior. In this sense, some space of the paper is used to determine the energy constant of the LJL potential for these different "crystalline" structures at any scale. Taking advantage of the total compatibility of the proposed element with continuum elements, the multiscale strategy is straightforward applied. Selected examples are used to demonstrate the good behavior of the proposed element and its applicability. Future developments to enhance applications are commented at the conclusion section.