A new formulation is proposed for shape design sensitivity analysis of nonlinear solid mechanics problems. The formulation is based on the Lagrangian description commonly used in continuum mechanics problems. A new kinematic framework for the sensitivity formulation is established by describing design changes in Lagrangian viewpoint. Shape design sensitivity expressions are derived by the present approach for the total Lagrangian and the updated Lagrangian formulations of response analysis. For the total Lagrangian formulation, the same sensitivity results are obtained as those from other approaches in literature and some observations are pointed out. For the updated Lagrangian formulation, which requires objective stress rate-based constitutive relation, the consistency between response and sensitivity analysis is established and the results seem to be promising. The use of the present formulation is illustrated clearly through several analytical and numerical examples.
As an application of the sensitivity analysis of nonlinear structural systems, the sensitivity analysis on buckling load is studied in this paper. A new technique is presented to approximate design sensitivity the buckling load at prebuckling points. Attention is focused on how to get the stable estimation of the design sensitivity convergent to its actual value near the buckling point. The estimation is easily done by using information already known from the eigenvalue buckling analysis. The approximations based on one- and two-point approaches are numerically compared with each other for truss and beam structures.
