Continuum‐based sensitivity analysis for coupled atomistic and continuum simulations for 2‐D applications using bridging scale decomposition

The concept of linking simulations at multiple length and time scales is found useful for studying local physical phenomena such as crack propagation. Many multi‐scale methods, which couple molecular dynamics models with continuum models, have been proposed over the last decade. One of the most advanced methods developed recently is the bridging scale method, in which the total displacement is decomposed into orthogonal coarse and fine scales. This paper presents the continuum‐based sensitivity analysis for two‐dimensional coupled atomistic and continuum problems using the bridging scale method. A variational formulation for the bridging scale decomposition is developed based on the Hamilton’s principle. The continuum‐based variational formulation provides a uniform and generalized system of equations from which the differential equations can be obtained naturally. The sensitivity expressions for both direct differentiation method (DDM) and adjoint variable method (AVM) are derived in a continuum setting. Due to its efficiency for crack problems, the direct differentiation method is chosen to be implemented numerically and applied to two examples, including a crack propagation problem. Both material and sizing design variables are included to reveal the impact of design changes at the macroscopic level to the responses at the atomistic level. Also demonstrated is the feasibility of achieving the variation of the time history kernel computed using numerical procedures. The sensitivity coefficients calculated are shown to be accurate compared with overall finite difference method. The physical implications of the sensitivity results are also discussed, which accurately predict the behavior of the structural responses.