Pareto Optimal Design of Non‐Homogeneous Isotropic Material Properties for the Multiple Loading Conditions

Abstract

The present paper concerns two methods: IMD (isotropic material design) and YMD (Young modulus material design) concerning Pareto optimal distributions of the material and its isotropic properties. The merit function is the weighted sum of compliances corresponding to n independent loading conditions. The unit cost of the design is assumed as equal to the trace of the elastic moduli tensor C. In the IMD method, the design variables are the bulk and shear moduli. In the YMD method, the Poisson ratio is assumed as given, possibly non‐uniformly distributed in the design domain, the Young modulus being the only design variable. Both the methods reduce to the auxiliary minimization problem involving statically admissible stress fields corresponding to the subsequent loading conditions. The integrand is expressed by a norm of the collection of these stresses, hence has linear growth. The numerical method proposed refers to the primal formulation: it is based on piece‐wise polynomial approximation of the set of statically admissible stresses. The exemplary Pareto optimal solutions show that admitting negative values of Poisson's ratio may contribute to a decrease of the total compliances.