A multiscale meshfree method for the mechanical analysis of low dimensional nanostructures

A multiscale meshfree approach for the mechanical analysis of low dimensional nanostructures is proposed, with the objective that both computational efficiency and accuracy can be achieved for systems involving large number of atoms or molecules. This method is implemented through a so-called "bridging scale" term, which is constructed by projecting the molecular dynamics (MD) solution onto its coarse scale approximation. The multiscale decomposition is obtained by subtracting the bridging scale component from the MD solutions. With this decomposition and a physics-based atomic potential, the coupled nonlinear governing equations between the coarse and fine scales are derived and solved using Newton's method. The advantages of the method can be briefly summarized as follows: First of all, the multiscale projection effectively decouples the incremental form of the multiscale equations. In addition, the coarse scale and fine scale treatments are unified in one formulation, in particular, the classical hyperelastic stress (sometimes involves higher order stress) update scheme is replaced with a simpler treatment at the atomic level in the coarse scale part of the method. Meshfree approximations are introduced for interpolating the low dimensional nano-structures. Particular advantage of this approach is that the approximation automatically satisfies the high order continuity requirement, which is essential for properly representing curved surfaces and lines at nanoscale. Finally, Comparisons with MD on bending of carbon nanotubes are made. Good agreement on the buckling pattern and energy shows the robustness of the method.