On Optimizing the Properties of Hierarchical Laminates Using Pontryagin's Maximum Principle

The problem of optimizing the structure of a hierarchical laminate, constructed from a given finite set of materials, in order to achieve extremal properties such as its overall heat or electrical conduction, dielectric, magnetic, elastic, thermoelastic, and piezoelectric properties, or combinations thereof, possibly in conjunction with minimizing the overall cost (when each phase is assigned a given cost), is considered. The controls in constructing a hierarchical laminate are the directions of lamination and the volume fractions chosen at each stage in the lamination process. It is shown how this optimization problem can be formulated in such a way that the discrete version of Pontryagin's maximum principle can be applied. The state variables at a given stage of construction consist of the set of effective tensors of the hierarchical laminates which are being built and which will ultimately be combined to form the final hierarchical laminate. The adjoint variables turn out to have a simple physical inte...

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