Stability constraints for geometrically nonlinear topology optimization

Abstract

This paper compares four methods for formulating stability constraints in topology optimization with geometric nonlinearity. The methods are: a direct approach to compute the critical load factor, an approximation using an eigenvalue analysis at a load factor of 1, a new method based on an eigenvalue analysis at the constraint limit load factor, and an implicit method based on stiffness reduction, which has not previously been investigated for stability constraint formulation. These four methods are described in detail and then compared qualitatively and quantitatively (including optimization examples) in terms of accuracy, robustness, and computational efficiency. The results show that formulating the constraint using an eigenvalue analysis at a load factor of 1 is the most robust approach, as it is least likely to experience mode switching or mode skipping during optimization, which leads to poor convergence for the other three methods. It is also the most efficient, as it only requires a single eigenvalue solve, whereas other methods require additional linear solves to compute the constraint value. However, an eigenvalue analysis at a load factor of 1 only approximates the critical load factor, which may be over, or under-estimated. Therefore, none of the methods fully satisfy the criteria of accuracy, robustness, and efficiency, highlighting the need for further research, e.g., by improving the accuracy of the method based on an eigenvalue analysis at a load factor of 1, or by improving the robustness and efficiency of the direct approach.