Truss structures are an important element of engineering constructions. It is a main goal of engineers to make such constructions as lightweight as possible, subject to the condition that they be able to sustain a prescribed load.

Taggart and Dewhurst describe a numerical scheme for the optimization of such structures. The scheme is based on a combination of finite element techniques with probabilistic ideas. It works in an iterative way. In the beginning, the desired final mass of the structure is distributed uniformly throughout the design domain. All nodes of the finite element mesh are assigned equal initial relative densities. This distribution is then modified in an iteration loop. The modification strategy that eventually leads to an optimized structure is based on a probabilistic interpretation of the distribution, that is, on constructing a probability distribution function that is equivalent, in a certain sense, to the mass distribution in the truss structure. The condition that the structure must be able to sustain a certain load can also be interpreted in this probabilistic language. The authors construct a class of such probability distributions that satisfy the load criterion. Then, they iteratively improve the initial distribution in a smooth way within this class, via extensive use of Euler’s beta functions. The final probability distribution is then translated back into mechanics language; it turns out that it represents a material distribution with two strictly separated regions: in the first region, the material is fully dense (this is the region in which one is actually interested), while the density of the second region vanishes.

In mechanics language, the optimization is based on modifying the nodal densities. Specifically, the nodal strain energies of the present state are sorted and the new densities are computed on the basis of the old densities and the position of the corresponding entry in this sorted list of energies. To be precise, if the strain energy at a node is relatively small, then its nodal density is decreased; if the strain energy is relatively high, the density is increased. Of course, it must be ensured that the densities of all material points are always between zero and one.

A number of examples indicate that the approach works well in practice.