Topology Optimization Design of Functionally Graded Structures

1. Abstract Functionally Graded Materials (FGMs) possess continuously graded material properties and are characterized by spatially varying microstructures. Such materials are studied in conjunction with the concept of topology optimization design which determines holes and connectivities of the structure by adding and removing material in the extended fixed design domain. The objective is to design FGM structures by using the concept of continuum topology optimization which considers a continuum distribution of the design variable inside the finite element domain. The traditional formulation for stiffness design problem is considered where the objective is to find the material distribution that minimizes the mean compliance. Two distinct, but related, applications are considered. The first application considers the objective of designing a structure in an FGM domain where the properties change in a certain direction according to a specified law, which leads to a structure with asymmetric stiffness properties. A new material model, called FGM-SIMP (Functionally Graded Material – Solid Isotropic Material with Penalization), is defined based on the traditional SIMP model. The second application consists of finding the optimal material gradation law inside the design domain. In this case we expect to obtain intermediate material properties inside the design domain. A material model obeying the Hashin-Strikman bounds is applied. Because current FGM manufacturing techniques emphasize layered systems, a layered material constraint is adopted. Different from the traditional topology optimization problem which focuses on a 0-1 design, we seek intermediate properties at the end of the optimization process. The optimality criteria method is applied to solve the optimization problem. The algorithm was implemented using C language. A few numerical examples are given to illustrate the proposed methods. It is shown that the resulting topology can change significantly as a function of the material gradient.