We extend the hierarchical multiscale design framework of Nakshatrala et al. (2013) to nonlinear elastodynamics wherein we use topology optimization to design material micro-structures to achieve desired energy propagation in nonlinear elastic material systems subjected to impact loading. As in Part I, a well-posed topology optimization problem is obtained via (a) relaxation to design the macroscale which requires homogenization theory to relate the macroscopic homogenized response to its micro-structure and (b) via restriction to design the microscale to obtain a well-defined micro-structural length scale. It is assumed that the primary wavelengths of interest are much longer than the micro-structural length scale and hence the effective properties are computed using the static homogenization theory. An adjoint sensitivity analysis is performed to compute the derivatives of the objective function with respect to the micro-structural design parameters and a gradient-based optimization algorithm is used to update the design. The numerical implementation of the computationally challenging terminal-value adjoint problems is discussed and a structural design example for tailored energy propagation is provided. 
Nonlinear structural design using multiscale topology optimization. Part II: Transient formulation