State-Space Reduced Newton's Algorithm for Topology Optimization

In the topology optimization community it is customary to formulate the optimization problem in terms of the design variables only—this is a basis for a so-called nested approach. Such formulations foster numerical algorithms operating in the design space only, which has a significantly smaller dimensionality than the full (design times state times adjoint state) space of the problem. On the flip side, the function/derivative evaluations in this framework are very expensive, as they require solving the governing/adjoint partial differential equations. Even more importantly, it is impractical to use the Newton’s iteration in this space. Indeed, the dimensionality of the design space is still very large for the topology optimization problems, yet the reduced Hessian matrix is fully populated and is exceedingly expensive to compute.
An alternative approach is to formulate and solve the problem in the full space. By significantly increasing the dimensionality of the space one ends up with inexpensive function/derivative evaluations and preserves the sparsity of the discretizations of the governing equations, which act as equality constraints in the resulting optimization problem. We propose a third alternative, which is based on reducing the optimal design problem onto the state space. Indeed, in certain topology optimization problems encountered in practice, the design space has a very simple structure. This allows us to eliminate the design variables from the problem with tiny computational effort and without destroying the sparsity of the problem. Therefore the resulting optimization algorithm operates in somewhat smaller space (when compared to the full space) while maintaining the inexpensiveness of the function evaluations enjoyed by the full space methods and the possibility of utilizing the Newton’s algorithm due to the preserved sparsity.