Imperfect symmetry: A new approach to structural optima via group representation theory

While imperfect symmetry (i.e., small deviation from symmetry) is abundant in Nature, it is not common among engineering structures. Here we investigate whether, and under which conditions the response of engineering structures with given symmetry group Γ may be improved by adding small perturbations x_i to the geometry. We will prove that a naturally emerging representation of Γ in the space of the variables {x_i} plays a key role and, based on this representation, we formulate exact conditions of improvability, utilizing classical representation theory of finite groups. We also present various examples among which optimal structures with imperfect symmetry emerge, somewhat counter to the engineer’s intuition.