Adjoint design sensitivity analysis of crack propagation using molecular-continuum multiscale approach

We have developed a multi-scale design sensitivity analysis method for transient dynamics using a bridging scale method by a projection operator for scale decomposition. Employing a mass-weighted projection operator, we can fully decouple the equations of motion into fine and coarse scales using the orthogonal property of complimentary projector to the mass matrix. Therefore, independent solvers in response analysis can be utilized for the fine scale analysis of molecular dynamics and the coarse scale analysis of finite element analysis. To reduce the size of problems and to improve the computational efficiency, a Generalized Langevin Equation (GLE) is used for a localized Molecular Dynamics (MD) analysis. Through demonstrative numerical examples, it turns out that the derived sensitivity analysis method is accurate and efficient compared with finite difference sensitivity.