  The present work proposes an extension of the third medium contact method for
solving structural topology optimization problems that involve and exploit
self-contact. A new regularization of the void region, which acts as the
contact medium, makes the method suitable for cases with very large
deformations. The proposed contact method is implemented in a second order
topology optimization framework, which employs a coupled simultaneous solution
of the mechanical, design update, and adjoint problems. All three problems are
derived and presented in weak form, and discretized with finite elements of
suitable order. The capabilities and accuracy of the developed method are
demonstrated in a topology optimization problem for achieving a desired
non-linear force-displacement path.
