Topology optimization of repetitive near-regular shell structures using preconditioned conjugate gradients method

Abstract

During the topology optimization process, the stiffness matrix of the structure changes continuously. Suppose we have a structure whose stiffness matrix is in a special form such as block tridiagonal matrix at the beginning of topology optimization process. The question now arises: How this special property can be used in the topology optimization process in an efficient manner? To find the answer, we examined the use of an appropriate form of the stiffness matrix at the beginning of the optimization process by constructing preconditioners for conjugate gradient method, where this approximate method is utilized to solve the nested analysis equations in topology optimization. As we already know, using a suitable nodal ordering of the repetitive near-regular shell structure, the stiffness matrix will have a suitable block form. In this study, this block form is utilized to construct some of the well-known block preconditioners. Constructing preconditioners only once at the beginning and employing in the entire design process of optimization is previously proposed as an effective approach for the examples they studied. Here, we examine the effectiveness of this method on shell structures.