Topology optimization for programmable materials and structures with nonlinear behaviors

As a physics-based free-form design methodology, topology optimization has long served as an inverse design tool for creating materials and structures with targeted linear mechanical properties. In the past decade, its extension to nonlinear and potentially multi-physics mechanical behaviors has grown rapidly, enabling the development of wide-ranging new materials and structures with highly complex mechanical responses. This review paper discusses recent advances in this dynamic and fast-evolving field, highlighting key achievements, existing challenges, and potential future directions. The focus is on nonlinear mechanical behaviors, complex responses, and various functions enabled by topology optimization. We explore several categories of nonlinear mechanical phenomena arising from various sources, including material nonlinearity, large deformations, contact mechanics, and multi-physics interactions. As detailed in the paper, the field still faces substantial and multifaceted challenges ranging from accurate modeling and efficient computation to advanced fabrication techniques and experimental validation. Meanwhile, there is considerable room for innovation—both in advancing technical methods and in exploring new nonlinear mechanical responses and functions, particularly through inverse design. This paper aims to serve as a guide and reference for topology optimization of nonlinear mechanics in materials and structures.