# Topology optimization for geometrically nonlinear structures

### Description

The majority of studies in topology optimization is carried out under the assumptions of linear elastic material behavior and geometric linearity. Thus, the material is idealized and equilibrium is enforced with respect to the reference (undeformed) configuration of the structure. Indeed, these simplifications are suitable for a large class of problems. However, linearity assumptions are too restrictive for advanced engineering problems. By definition it is necessary to account for geometric nonlinearity in the analysis and design of flexible thin-walled structures, compliant mechanisms, and multi-stable structures with snap-through behavior. In particular if these structures have to perform at or beyond the limits of conventional design envelopes. Consequently, seeking efficient and effective methods to carry out topology optimization for structures exhibiting geometrical nonlinearity is of great practical relevance. The main challenges in the field of geometrically nonlinear topology optimization are:

• Efficiency
• Low-density areas
• Convergence problems
• Spurious modes

### Objective

In this research, we will focus on structures exhibiting geometrical nonlinearity. Consistent with the challenges presented in the above, the objectives are followed:

• Enhance the efficiency of Newton iteration and sensitivity analysis thus accelerate the topology optimization progress;
• Deal with the difficulties introduced by low-density elements;
• Design flexible structures exhibiting snap-through behaviors.

### Methods

SIMP method is adopted and MMA is used to solve the topology optimization formulations. Reduced order modeling is introduced to accelerate the optimization process. Various techniques are presented to construct, update, and maintain the ROM basis vectors.

• Augmentation
• Bootstrapping
• Rejection
• Reordering
• Orthogonalization

Path derivatives are included in ROM basis to assist to simulate inextensional structures. A resetting displacement method and a ROM-based eigenvector elimination technique are presented to deal with the spurious modes in ROM basis.

### Results

Figure 1. Topology optimization results: (a) linear (b) nonlinear.  (a)(b)
Figure 2. Efficiency test: The horizontal axis represents the optimization steps and the vertical axis represent the number of solves. Pure bending problem with path derivatives