We present a method for designing truss structures, a common and complex category of buildings, using non-linear optimization. Truss structures are ubiquitous in the industrialized world, appearing as bridges, towers, roof supports and building exoskeletons, yet are complex enough that modeling them by hand is time consuming and tedious. We represent trusses as a set of rigid bars connected by pin joints, which may change location during optimization. By including the location of the joints as well as the strength of individual beams in our design variables, we can simultaneously optimize the geometry and the mass of structures. We present the details of our technique together with examples illustrating its use, including comparisons with real structures.

We present a method for the rapid and controllable simulation of the shattering of brittle objects under impact.An object to be broken is represented as a set of point masses connected by distance-preserving linear constraints.This use of constraints, rather than stiff springs, gains us a significant advantage in speed while still retaining fine control over the fracturing behavior.The forces exerted by these constraints during impact are computed using Lagrange multipliers.These constraint forces are then used to determine when and where the object will break, and to calculate the velocities of the newly created fragments.We present the details of our technique together with examples illustrating its use.