An energy model for the optimal design of linear continuum structures

Abstract

An analytical model is presented for the optimal design of linearly elastic continuum structures. To facilitate the expression of the combined analysis and design problem in general form, a basis is introduced covering a general set of energy invariants. Both internal (strain) energy and the expression of generalized cost are represented conveniently in terms of this basis, and as a result the optimality conditions for the design problem have a particularly simple form. Present developments comprise a reinterpretation and an extension of existing models where the design variable is the material modulus tensor, and where “cost” is represented in a general form. The conventional potential energy statement for linear continuum elastostatics is restated in the form of an isoperimetric problem, as a preliminary step. This interpretation of the mechanics is then incorporated in a max-min formulation applicable for the general design of linear continuum structures. To exemplify its application, the model is interpreted as it would apply for certain materials with particular geometric structure, e.g. crystalline forms. Also problems treated earlier where optimal material properties are predicted for the case where unit cost is proportional to the trace of the modulus tensor are identified as examples within the generalized formulation. The application of a recently developed technique to predict optimal black-white structures, i.e. designs having sharp topological features, is considered in the setting of the present generalized model.