A formulation for optimal continuum structures with a decomposition of material properties into specified and designable parts

A formulation is presented for optimal design in the setting of linear elastostatics for continuum structures, where the modulus tensor field is expressed as a decomposition into a set of independent tensors. For the linear case at hand, net material properties simply equal the sum of the constituent tensor fields. The design variables are comprised of one or more of the these fields, while the remaining ones in the "mix" are taken to be specified and fixed. The optimal continuum structure is designated to be such combination of designed and specified constituent fields that minimizes structural compliance. A set of unit energies is introduced to serve as a basis for the general expression of both unit cost in the isoperimetric (cost) constraint, and unit strain energy. The design problem is characterized in the form of a max-min problem, where the max applies to design and the (inner) min is w.r.t. the state variables. Partitioning of energies among constituents in the optimal mixture is identified directly from the govering equations.