Optimality criteria for the components of anisotropic constitutive matrix

In a free material formulation, the problem of minimizing a weighted sum of compliance’s from multiple load cases, subject to an active constraint on material volume, is solved in a formulation with two optimality criteria. The first optimality criterion for the distribution of material volume densities is equal value for the weighted elastic energy densities, as a natural extension of the optimality criterion for a single load case. The second optimality criterion for the components of a constitutive matrix (of unit norm) is proportionality to corresponding weighted strain components with the same proportionality factor \(\widehat \lambda \) for all the components, as shortly specified by \(C_{i j k l} = \widehat \lambda \sum _{n} \eta _{n} (\epsilon _{i j})_{n} (\epsilon _{k l})_{n}\), in traditional notation (n indicate load case). These simple analytical results should be communicated, in spite of the practical objection against design for weighted sum of compliance’s, as compared to worst case design and design considering strength. The application of the approach of the two optimality criteria is illustrated by a 2D example with 8 load cases. Stable and fast convergence is shown.