中文摘要：

大量工程问题要求结构的局部区域在不同承载工况下保持位移响应的几何稳定性．在结构的特定区域引入可以随承载i况调节的补偿位移是实现这一目标的有效手段．在线弹性小变形范围内，通过最小的变形控制成本，研究了多工况下使结构特定位置的位移响应保持几何稳定性的拓扑优化问题．设计目标为在保持结构位移响应几何稳定性的同时实现结构的最大刚度；优化模型包含两类设计变量：结构拓扑变量及补偿位移变量，两类变量采用分层寻优技术进行耦合．采用伴随法分别推导了目标函数对两类设计变量的敏度求解格式．结果表明，该优化模型能够在兼顾成本的同时较好地实现结构的变形控制目标．

英文摘要：

The special parts of structures are always required to be geometric stable on output displacements under different loadings in many engineering cases. Introduction of the adjustable compensation displacements in the special region of structures is the effective technique. In the scope of the linearly elasticity and small deformation, the purpose of this paper is to study the topology optimization of structures with the geometric stability in the given region through the lowest compensation cost. The objective of the present optimization model is to obtain the maximum stiffness of structures with the stable ideal displacement responses of structural given region. There are two types of design variables in the model which are the structural topology variables and the compensation displacement variables. These two types of variables are coupled through the two-level search method. The adjoint method is performed to compute the sensitivities of the objective function with respect to the two types of variables respectively. The results show that the present optimization formulation can realize the objective of shape control of structures through the low controlling cost.