Design sensitivity analysis and optimization of non-linear transient dynamics. Part I - sizing design

A continuum-based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non-linear structural systems with elastic}plastic material and large deformation. The methodology is aimed for applications in non-linear dynamic problems, such as crashworthiness design. The "rst-order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct di!erentiation method and updated Lagrangian formulation are used since they are more appropriate for the path-dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central di!erence method and the "nite element method are used to discretize the temporal and spatial domains, respectively. The Hughes}Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate-based incrementally objective stress integration scheme are used to handle the "nite strain and rotation. An elastic}plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the "nite di!erence results. Design optimization is carried out using the design sensitivity information