Optimal Shape Remodeling of Elastic Bodies By the Finite Element Method.

Abstract

Variational principles of structural analysis are used in conjunction with the design problem to derive optimality conditions for optimal shape remodeling of elastic bodies. The term remodeling is used to identify a procedure for the prediction of the optimal modification to a given shape, where the extent of modification is constrained to be within a prescribed global measure of change. Design for optimal reinforcement, design for optimal reduction in weight, and optimal combined reinforcement and removal of material comprise separate categories of optimal remodel design problems. The optimality conditions are derived for various types of boundaries; a boundary where a state variable is free to take on any compatible value, a boundary where the value of a state variable is specified (including specified constant as a special case), and a boundary where the external loading, independent of shape change, is specified. The formulation and the derived optimality conditions are broad; they apply to any two-dimensional design problem that can be identified with the general problem statement made in this work. Sufficiency of the derived optimality conditions is established through a uniqueness proof of the design which satisfies the optimality conditions for a case of optimal shape remodeling of elastic bars under torsion for maximum torsional rigidity. Based upon the derived optimality conditions a computational scheme is proposed. Experience indicates that the method provides a considerably simple way of obtaining the optimal design. Convergence properties of the solution method are discussed: this is done with use of a study based on the fixed-point theorem, and through a different interpretation of the computational scheme. Numerous example problems are solved for optimal shape remodeling of elastic bars in torsion for maximum torsional rigidity, and of two-dimensional elastic bodies for minimum mean structural compliance.