A maximum filter for the ground structure method: An optimization tool to harness multiple structural designs

The ground structure method seeks to approximate Michell optimal solutions for real-world design problems requiring truss solutions. The single solution extracted from the ground structure is typically too complex to realize directly in practice and is instead used to inform designer intuition about how the structure behaves. Additionally, a post-processing step required to filter out unnecessary truss members in the final design often leads to structures that no longer satisfy global equilibrium. Here, a maximum filter is proposed that, in addition to guaranteeing structures that satisfy global equilibrium, leads to several design perspectives for a single problem and allows for increased user control over the complexity of the final design. Rather than applying a static filter in each optimization iteration, the maximum filter employs an interval reducing method (e.g., bisection)to find the maximum allowable filter value that can be imposed in a given optimization iteration such that the design space is reduced while preserving global equilibrium and limiting local increases in the objective function. Minimization of potential energy with Tikhonov regularization is adopted to solve the singular system of equilibrium equations resulting from the filtered designs. In addition to reducing the order of the state problem, the maximum filter reduces the order of the optimization problem to increase computational efficiency. Numerical examples are presented to demonstrate the capabilities of the maximum filter, including a problem with multiple load cases, and its use as an end-filter in the traditional plastic and nested elastic approaches of the ground structure method.