Simultaneous discrete and continuum multiresolution topology optimization

In the field of continuum structures, the density-based methods for topology optimization are well known and broadly studied. Likewise, in the area of (quasi-) optimal discrete element structures, there is significant work which can even consider nonlinear constitutive models. This work seeks to set a precedent by combining these strategies, in other words, to topology optimize continuum and discrete elements simultaneously with the possibility of including nonlinear constitutive models for the discrete elements. Reinforced concrete, reinforced masonry, fiber-reinforced materials, rib-reinforced shell structures, and others are problems which are conveniently modeled using a combination of both, discrete and continuum elements. Thus, a combined and simultaneous framework to topologically optimize these type of hybrid structures breaks down the barrier often separating both fields. The simultaneous optimization of continuum and discrete poses several mathematical and numerical challenges, some of which have been previously documented. The present work addresses a large number of these challenges and presents a robust and stable computational implementation as a proof-of-concept.