The evolutionary structural optimization method: theoretical aspects

In 1993, Y.M. Xie and G.P. Steven introduced an approach called evolutionary structural optimization (ESO). ESO is based on the simple idea that the optimal structure (maximum stiffness, minimum weight) can be produced by gradually removing the ineffectively used material from the design domain. The design domain is constructed by the FE method, and furthermore, external loads and support conditions are applied to the element model. Considering the engineering aspects, ESO seems to have some attractive features: the ESO method is very simple to program via the FEA packages and requires a relatively small amount of FEA time. Additionally, the ESO topologies have been compared with analytical ones, e.g. Michell trusses, and so far the results are quite promising. On the other hand, ESO does not have a solid theoretical basis, and consequently, the ESO minimization problem is still unresolved. Since the good agreement between the results cannot be just a coincidence, in this paper, we will study whether the gradual removal of material can be explained mathematically and whether the theoretical basis of ESO can be outlined. First, a minimization problem solved by ESO is examined. Based on the results of earlier publications, it was assumed that the ESO method minimizes the compliance-volume product of a structure or a finite element model. It was noted that the sequential linear programming (SLP)-based approximate optimization method followed by the Simplex algorithm is equivalent to ESO if the strain energy rejection criterion is utilized. However, ESO should be applied so that the elements corresponding to the design domain are equally sized. If this requirement is not met, the rejection criterion, which also considers the varying sizes of the elements, should be used. Additionally, the element stiffness matrices and element volumes should be linearly dependent on the design variables. Also linearly elastic material is assumed. At each iteration the rejected elements should be removed completely. Most often only element removal is allowed in ESO. If the design variables are initially assigned values other than the maximum value, however, the elements should be allowed to reenter the design domain. This subject, obviously, needs further study. Typically, ESO is applied to problems having a planar design domain with in-plane forces only. In these cases, ESO produces truss-like, equally stressed and maximum-stiffness topologies. It is often recommended that, based on the topology optimization, a new finite element discretization should be employed. After that, the sizing optimization procedure can be performed. Since ESO seems to be producing truss-like topologies, ESO should be applied to structural problems having pin-jointed connections. For other types of structures ESO should be studied further.