Topology Optimization of Geometrically Nonlinear Structures

Abstract

So far, structural topology optimization has been mainly focused on linear problems. Much less attention has been paid to geometrically nonlinear problems, although it has a lot of interesting applications. The objective of this work is to implement a method to solve topology optimization problems under finite displacements and rotations. The main focus is on designing stiff structures, but also an outlook is presented on the design of structures that follow a prescribed equilibrium path.

The Newton-Raphson incremental-iterative procedure is implemented to solve nonlinear problems. Arc-length control is used to be able to overcome limit points and to obtain faster convergence. It is shown by means of numerical experiments that this method leads to correct results.

Since nonlinear analysis, especially in combination with topology optimization, is computationally expensive, an attempt is made to develop a new reduction method. This method uses load dependent Ritz vectors as a reduced basis and was originally used in linear dynamic problems. However, it is demonstrated that this method will in general not lead to accurate results in nonlinear static problems. This is due to the fact that deformation modes that are not excited by the external force, cannot be described by the basis.

The topology optimization process is implemented without reduction method. An adjoint formulation is derived to obtain sensitivities in a computationally efficient way. By means of several examples of end-compliance minimization, it is demonstrated that in most cases, especially for shell structures, this method leads to better performing designs than topology optimization based on linear analysis.