特定方向"零膨胀"的最小柔顺性结构优化设计

工程中很多承载结构必须面对苛刻的温度变化工作环境,如卫星天线、太空照相机和电子器件等。剧烈的温度变化引起较大的热变形,造成仪器信号失真,精度下降;同时温度应力也会造成结构破坏甚至失效,因此零膨胀材料的研制备受关注。近年来国内外很多学者对此进行了研究,设计出具有特定等效膨胀系数的微结构,但考虑到制备工艺的限制,这类具有复杂微结构的材料制备起来比较困难,成本较高;同时这类材料一般不具备足够的刚度,难以满足承载性能的要求。本文基于结构优化设计技术,采用拓扑优化方法直接设计出具备较高的承载性能和特定方向变形较少受热载荷影响的结构。本文提出采用多目标优化的方法设计圆环结构,使其具有较高的刚度和在热载荷下圆环内表面具有较好的热几何稳定性。由于用单相材料无法同时满足高刚度和低热膨胀的要求,因此假设结构由两种不同的材料构成,用连续体拓扑优化的方法设计三相材料(两种实体材料MAT-I、MAT-II和空材料)在设计域上的最优分布,使结构满足设计要求。由对称性,设计域取为圆环的一个扇面,将设计域离散成有限元网格,每个单元具有两个设计变量:实体材料的体分比和MAT-I在实体材料中所占的体分比,采用伴随法进行灵敏度分析,用GCMMA方法求解此问题,采用体积守恒的Heaviside密度过滤函数保证获得清晰的最优拓扑构型以及避免棋盘格式的出现。通过两个数值算例,表明使用本文提出的多目标优化模型能够得到特定方向"零膨胀"同时具有一定刚度的结构设计,且这种宏观结构尺度上的两种材料组成的拓扑构型相对易于制造。

Multi-objective optimization of low structural compliance and thermal directional expansion

Many load carrying engineering structures experience large temperature changes which lead to an extreme thermal stress and activate structural failure. Structural design with high stiffness and low thermal expansion in the specified direction is interesting and challenging. Design of materials with low or zero isotropic thermal expansion coefficients were investigated intensively in the field of structural topology optimization and solid mechanics. The obtained materials have low isotropic thermal expansion, however, mostly they have very low stiffness and are not adequate for constructing load carrying structures. Therefore, the authors attempt to use topology optimization method to design the structure, which combines high stiffness with low thermal expansion in a predefined direction.In this paper, we develop a new multi-objective optimization formulation for circular cylinder structure design. In its circumferential direction, the cylinder is composed of a number of sectors with the same topology and size. The structural topology of the sector, the material distribution within the sector is to be designed. We aim at optimal design of low thermal expansion in circumferential direction and high stiffness in the radial direction. Since the combination of the two attributes cannot be found with any single constituent material, three-phase material including two different material phases and one void phase is utilized to compose the sector structure. Topology optimization technique is adopted here to optimize the structure with high stiffness and low thermal directional expansion. The density of solid materials and fraction of phase 1 material are chosen as the design variables and penalty technique is implemented. The sensitivity analysis is accomplished with the adjoint method. The GCMMA method is adopted and the volume preserving nonlinear density filtering based on Heaviside step function is used to prevent checkerboard patterns and to obtain a clear design. The numerical examples are illustrated and the resulting optimum designs are discussed in the scope of multi-objective optimization.