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.

摘要

在拓扑优化过程中，结构的刚度矩阵不断变化。假设我们有一个结构，其刚度矩阵在拓扑优化过程开始时采用特殊形式，例如块状三对角矩阵。现在出现的问题是：如何在拓扑优化过程中以有效的方式使用这种特殊属性？为了找到答案，我们在优化过程开始时通过构建共轭梯度法的预条件子来检查刚度矩阵的适当形式的使用，其中这种近似方法用于求解拓扑优化中的嵌套分析方程。正如我们已经知道的，使用重复近规则壳结构的合适节点排序，刚度矩阵将具有合适的块形式。在这项研究中，这种块形式用于构建一些众所周知的块预处理器。在开始时只构建一次预条件子并在整个优化设计过程中使用之前被提议作为他们研究的示例的有效方法。在这里，我们检查了这种方法在壳结构上的有效性。