Multiscale adjoint design sensitivity analysis of atomistic-continuum dynamic systems using bridging scale decomposition

To obtain design sensitivity in molecular dynamics (MD), finite differencing is impractical from the viewpoint of efficiency and accuracy since MD problems could include highly nonlinear design parameters and generally require a lot of computation time. In this paper, using a bridging scale decomposition method, we develop a multiscale adjoint design sensitivity analysis (DSA) method for the coarse-scale performance of atomistic-continuum dynamic systems considering fine scale effects. Due to the decomposition of total solution into fine and coarse scales using a mass-weighted projection operator that possesses the orthogonal property of complimentary projector to mass matrix, each scale can be independently considered in both response and sensitivity analyses. To reduce computing costs, using a generalized Langevin equation and a lattice mechanics, the fine scale MD region is confined locally while the coarse-scale finite element analysis is utilized in the whole domain. The multiscale adjoint sensitivity includes only explicitly design-dependent terms together with the original and adjoint responses so that one additional time integration process is sufficient to evaluate the design sensitivity. Numerical examples demonstrate the accuracy of the developed DSA method for various design variables in coarse and fine scales.