Topology Optimization of Hyperelastic Continua
Authors: Pham Trung , Hoyle Christopher , Zhang Yue , Nguyen Tam 
Published: December 2016

Topology optimization (TO) aims to find a material distribution within a reference domain, which optimizes objective function(s) and satisfies certain constraints. Topology optimization has various potential applications in early phases of structural design, e.g., reducing structural weight or maximizing structural stiffness. However, most research on TO has focused on linear elastic materials, which has severely restricted applications of TO to hyperelastic structures made of, e.g., rubber or elastomer. While there is some work in literature on TO of nonlinear continua, to the best knowledge of the authors there is no work which investigates the different models of hyperelastic material. Furthermore, topology optimized designs often possess complex geometries and intermediate densities making it difficult to manufacture such designs using conventional methods. Additive Manufacturing (AM) is capable of handling such complexities. Continuing advances in AM will allow for usage of rubber-like materials, which are modeled by hyperelastic constitutive laws, in producing complex structures designed by TO. The contribution of this paper is an investigation of different models of hyperelastic materials taking account of both geometrical and material nonlinearities, and their influences on the resulting topologies.

Topology optimization of nonlinear continua is the main topic of few papers. This paper considers different isotropic hyperelastic models including the Ogden, Arruda–Boyce and Yeoh model under finite deformations, which have not yet been implemented in the context of topology optimization of continua. This paper proposes to start with a reference domain having known boundary and loading conditions. Material parameters of different models that fill the domain are also known. Maximizing the stiffness of the structure subject to a volume constraint is used as the design objective. The domain is then meshed into a large number of finite elements, and each element is assigned a density between 0 and 1, which becomes design variable of the optimization problem. These densities are further penalized to make intermediate densities (i.e., not 0 or 1) less favorable. Optimized material distribution will be constructed from optimized values of design variables. Because of the penalization factors that make the problem nonlinear, the Method of Moving Asymptotes (MMA) is utilized to update it iteratively. At each iteration the nonlinear finite element problem is solved using the Finite Element Analysis Program (FEAP), which has been modified to accept penalized densities on element stiffness matrices and internal nodal forces, and a filtering scheme is applied on the sensitivities of objective function to guarantee the existence of solution. The proposed method is tested on several numerical examples. The first two examples are common benchmark models, which are a simply supported beam , and a beam fixed at two ends. Both models are subjected to a concentrated force at midpoints of their edges. The effects of linear and nonlinear material behaviors are compared with regards to resulting designs. The third example is a foremost attempt to reflect on TO in design of airless tire through a simple model, which demonstrates capability of the method in solving real-world design problems.