# Topology and Reinforcement Layout Optimization of Disk, Plate, and Shell Structures

• L. A. Krog
• N. Olhoff
Chapter
Part of the International Centre for Mechanical Sciences book series (CISM, volume 374)

## Abstract

This chapter deals with topology optimization problems for disks, plates and shells, and with problems of layout optimization of different types of reinforcement of plates and shells. Special emphasis is devoted to the solution of multiple load case stiffness maximization problems and to the solution of eigenfrequency maximization problems.

Two design parametrizations based on the application of layered microstructures of different rank are applied. In both formulations, a parametrization of the material/reinforcement distribution for a disk, plate or shell structure is obtained by modelling the material within each of the elements of a finite element discretized structure as a layered microstructure with a continuously variable density of material/reinforcement. The design variables of the optimization problem are the variables which control the composition of the layered microstructure, and hereby the density of material/reinforcement within each finite element. Furthermore, by allowing for four different configurations of each of the layered microstructures, we obtain formulations for solution of both topology optimization problems for disk, plate and shell structures, and for solution of reinforcement layout optimization problems for rib-stiffened plates and shells and internally stiffened honeycomb and sandwich plates and shells.

Stiffness maximization problems are treated as minimization problems for total elastic energy, and in the case of several independent load cases we either minimize a weighted sum of the total elastic energies, or the maximum total elastic energy from among all the load cases. The optimization problems are solved by means of mathematical programming based on analytical design sensitivity analysis, and examples of solution of maximum stiffness layout problems are presented.

Eigenfrequency maximization problems are considered as maximization problems for the lower bound on a given set of eigenvalues of vibration. The main difficulty associated with solution of such problems is that multiple eigenvalues may exist and that these are non-differentiable with respect to the design parameters. However, it is shown that, despite the lack of usual differentiability properties, such problems may be treated like differentiable optimization problems, if some restrictions are imposed on the vector of design changes at each iteration. In this way, we have developed a new general method for solution of eigenfrequency optimization problems which can handle problems with simple as well as multiple eigenvalues of vibration. This method is particularly attractive since it only requires ordinary methods for design sensitivity analysis and mathematical programming. Several numerical examples pertaining to solution of layout optimization problems with multiple eigenvalues are presented.

## Keywords

Design Sensitivity Stiffness Tensor Topology Optimization Problem Mindlin Plate Multiple Eigenvalue
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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