Fig. 1. We present a method for lightweight structure design for scenarios where external forces may contact an object at a multitude of locations that are unknown a priori. For a given surface mesh (grey), we design the interior material distribution such that the final object can withstand all external force combinations capped by a budget. The red volume represents the carved out material, while the remaining solid is shown in clear. Notice the dark material concentration on the fragile regions of the optimum result in the backlit image. The cut-out shows the corresponding interior structure of the 3D printed optimum.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 con guration capped by the budget. Our approach works by repeatedly nding the most critical force con guration and altering the internal structure accordingly. A key issue, however, is that the critical force con guration changes as the structure evolves, resulting in a signi cant computational challenge. To address this, we propose an e cient 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.