Multiscale methods for mechanical science of complex materials: Bridging from quantum to stochastic multiresolution continuum

Recent development in the multiscale method based on the bridging scale concept is presented with an emphasis on complex material systems. The bridging scale method (BSM) was originally proposed by Wagner and Liu (J. Comput. Phys. 2003; 190:249-274) as an effective way of treating the interface in coupled atomistic/continuum simulation. Since its publication, the BSM has become a very useful paradigm that has been applied to solve a host of problems in mechanical sciences of complex material systems. In this paper, we present a review on the recent developments. We first describe the application of BSM to the coupled atomistic/continuum simulation of dynamic fracture. The important extensions within the framework of space-time method and multiscale non-equilibrium molecular dynamics are then presented. We then focus on the multiresolution continuum theory that inherits the BSM concept in the analysis of heterogeneous material structures. Recent work of incorporating statistical factors into this model based on the concurrent nested homogenization of randomness of the material structures is highlighted. Finally, we present the use of the bridging scale concept in resolving the electron-mechanical coupling mechanism. The robustness of the BSM is demonstrated through many benchmark problems and application examples.