  We introduce a lightweight structure optimization approach for problems in
which there is uncertainty in the force locations. Such uncertainty may arise
due to force contact locations that change during use or are simply unknown a
priori. Given an input 3D model, regions on its boundary where arbitrary normal
forces may make contact, and a total force-magnitude budget, our algorithm
generates a minimum weight 3D structure that withstands any force configuration
capped by the budget. Our approach works by repeatedly finding the most
critical force configuration and altering the internal structure accordingly. A
key issue, however, is that the critical force configuration changes as the
structure evolves, resulting in a significant computational challenge. To
address this, we propose an efficient critical instant analysis approach.
Combined with a reduced order formulation, our method provides a practical
solution to the structural optimization problem. We demonstrate our method on a
variety of models and validate it with mechanical tests.

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