Abstract This study presents a general approach for the multi-scale design of variable stiffness composites (VSCs). The first-level problem of the multi-scale two-level optimisation strategy (MS2LOS) is solved to determine the optimal distribution of the VSC stiffness properties at the macroscopic scale satisfying the requirements of the problem at hand. In this phase, the VSC laminate is modelled as an equivalent homogeneous anisotropic plate whose behaviour is described in terms of polar parameters (PPs), which vary locally over the structure. The First-order Shear Deformation Theory is used to take into account the influence of the transverse shear stiffness on the mechanical response of the VSC and Basis Spline (B-Spline) surfaces are employed to represent the PPs fields. In this background, the expression of the gradient of the buckling factor is determined analytically by exploiting the properties of the polar formalism and of the B-Spline surfaces. Moreover, the effect of the discrete variables, involved in the definition of the B-Spline surfaces, on the performances of the optimised solution is investigated. The effectiveness of the approach is proven on two benchmark problems dealing with the maximisation of the first buckling load of a VSC laminate, subject to feasibility and geometric requirements, taken from the literature. The results obtained by means of the MS2LOS based on the polar formalism outperform those reported in the literature, which are obtained through an optimisation strategy based on lamination parameters.

中文翻译：

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用于优化可变刚度复合材料的通用等几何极坐标方法：在特征值屈曲问题中的应用

摘要 本研究为可变刚度复合材料 (VSC) 的多尺度设计提供了一种通用方法。解决了多尺度两级优化策略（MS2LOS）的一级问题，以确定满足手头问题要求的宏观尺度上VSC刚度特性的最优分布。在此阶段，VSC 层压板被建模为等效的均质各向异性板，其行为根据极性参数 (PP) 进行描述，该参数在整个结构上局部变化。一阶剪切变形理论用于考虑横向剪切刚度对 VSC 机械响应的影响，并采用基本样条 (B-样条) 表面来表示 PPs 场。在这样的背景下，屈曲因子梯度的表达式是通过利用极坐标形式和 B 样条曲面的特性来分析确定的。此外，还研究了 B 样条曲面定义中涉及的离散变量对优化解的性能的影响。该方法的有效性在两个基准问题上得到证明，这些问题涉及 VSC 层压板的第一次屈曲载荷的最大化，受可行性和几何要求的限制，取自文献。通过基于极性形式的 MS2LOS 获得的结果优于文献中报道的结果，这些结果是通过基于层压参数的优化策略获得的。