In the current element density-based topology optimization, element stiffness is penalized to yield either solid or void (or very weak) materials, but low-density elements appear during and even after optimization iterations. Especially when nonlinear problems involving large deformation are considered, low-density finite elements cause serious numerical problems as their tangent stiffness matrices lose positive definiteness. To completely eliminate the problem caused by low-density elements, we propose a new method: all finite elements are kept solid throughout the optimization process; zero-length elastic links are introduced to parameterize inter-element connectivity; the link stiffness is penalized. In this approach, the design variables are defined on the links, and vary from 0 and 1 corresponding to the unconnected and rigidly-connected states. Since the finite elements used to discretize the analysis domain always remain solid, typical numerical problems encountered by the standard element density-based formulation disappear. To implement the present method, several issues such as the handling of the mass constraint and raster imaging are also investigated.