Multi-criteria parametric optimization of the displacement and weight of a shell of minimal surface on a circular contour consisting of two inclined ellipses under thermal and power loading with consideration of geometric nonlinearity

Abstract

Thin shells are well suited to optimal design problems, and the finite element method and gradient descent method make it possible to solve inverse problems in structural and applied mechanics. The calculation process takes into account geometric nonlinearity. In modern numerical studies, a formulation taking into account geometric nonlinearity in the finite element method is used, namely, a stepwise loading procedure. It becomes necessary to take into account the relations between displacement vectors and their derivatives and strain increments. These relations help to determine the stiffness matrix of the finite element at each loading step, which leads to a qualitative study of real displacements in the structure. The geometrically nonlinear calculation of a shell of minimum surface on a circular contour consisting of two inclined ellipses in a multi-criteria parametric optimization is performed by the finite element method. The finite element method is a universal variational method that is focused on solving the most complex problems of elasticity theory and applied and structural mechanics using separate calculation complexes. Stiffness matrix - for the whole body is formed on the basis of the finite element stiffness matrix. The system of solving equations of the finite element method is formed using the Lagrange's variational principle, according to which the total potential energy P of a finite element model of a body in a state of stability and equilibrium has a minimum value. As part of the sensitivity analysis, gradients of the design variables of the structure, displacements in the form of partial derivatives of these characteristics along the design variables, and shell thickness are calculated. The sensitivity information serves as the basis for building an optimal design algorithm using the gradient descent method of the objective function. Standard multi-criteria parametric optimization allows for an average of 10% steel savings, with geometric nonlinearity increasing to 20%. At this study site, 24.4% of sheet steel was saved, which is a significant relative saving. This methodology of the authors shows high results for innovative design and production of steel thin shells in Ukraine and around the world, and also makes it possible to apply several types of optimization to one research object.