Constraints on integral measures of stress state in topology optimization problems

Abstract

Topology optimization (TO) is a computational method of determining material distribution in a given design area to create the optimal shape of a part under given boundary conditions. The increased interest in the development of effective methods of designing parts of the optimal topology testifies to the relevance of these theoretical studies and the important applied value of the obtained results. In the classic formulation of maintenance, the minimization of the flexibility of the part under restrictions on the volume (mass) of the optimization result is chosen as a criterion for finding the specified distribution. Closer to practical application is the formulation of the maintenance problem, which involves minimizing the volume of the part, taking into account the condition of its strength. The inclusion of aggregate functions for the calculation of integral measures of the stress state has a number of advantages over the traditional check of the maximum value of mechanical stress: significant saving of time for solving the maintenance problem, reduction of computational costs and ensuring the stability of the computational process. This work presents and analyzes the specialization of the applied application of aggregate functions, which have been most widely used in modern research on maintenance issues, taking into account the strength of the optimized part. In particular, the P-norm and P-mean functions, the Kreiselmeier-Steinhauser functions, the smoothed Heaviside function, the measure of exceeded stresses, and the measure of uneven distribution of the stress state are described. The large number of options available in the literature for the mathematical formulation of limitations for integral measures of the stress state of designed parts indicates that the issue of developing a universal and effective method of designing parts, taking into account the criterion of its strength, remains open.