On an optimal property of compliant topologies

An optimal structural property for compliant topologies is presented in this paper for general multi-criteria formulations that comprise the conflicting flexibility and stiffness requirements. The property deduced from the first-order necessary conditions for optimality implies that the ratio of the mutual potential energy density to the strain energy density is uniform throughout the continuum, but for portions otherwise bounded by gage constraints. This property is used to develop an optimality criteria method for synthesizing compliant topologies. It is also noted that the multicriteria formulations considered here are nonconvex and can result in nonunique solutions. However, by incorporating a one-variable search along the direction determined by: the above optimal property, it is ensured that the converged solution is a minimum. Several synthesis examples are included with linear frame finite elements which are easy for implementation and are capable of appropriately accounting for the bending behaviour in the continuum. Examples with previously reported density based design parameterization using bilinear plane-stress elements are also included to illustrate the synthesis procedure.