Traditionally, the main structural system of tall buildings is designed iteratively to resist extreme wind loads, which provides safe, but typically suboptimal building systems. Topology optimization provides a general approach to obtain optimal material layout to carry the required load within specified design constraints. The wind loading on a structure is typically modeled as an equivalent static load or as a spatio-temporal stochastic field. While a few models for stochastic wind excitation are available, these approaches are focused on obtaining samples for time history analyses. In this study the stochastic wind excitation is modeled as a filtered vector white noise; the state space representation of the filter is obtained by solving a regularized optimization problem from known stationary wind power spectral densities. An augmented state space representation is formed by combining the equation of motion for the structure with the excitation filter. The stationary covariances of the structural responses of interest are then obtained by solving the associated Lyapunov equation. Dynamic condensation of the equations of motion is employed to increase the efficiency of the proposed approach. Objective functions for the optimization scheme are defined in terms of the stationary covariance of the response; various objectives are considered corresponding to typical design considerations. An equivalent smooth formulation is employed to solve the non-smooth min–max problem. A gradient-based method is used to update the design variables, while the sensitivities are computed from the solution of an adjoint Lyapunov equation. The proposed topology optimization scheme is illustrated for a tall building subjected to along-wind and across-wind loads. The results presented herein demonstrate the efficacy of the proposed approach for efficient topology optimization of buildings subjected to stochastic wind excitation.

传统上，高层建筑的主要结构系统经过迭代设计以抵抗极端的风荷载，这提供了安全的但通常不是最佳的建筑系统。拓扑优化提供了一种通用方法，可获取最佳的材料布局，以在指定的设计约束内承受所需的载荷。通常将结构上的风荷载建模为等效静荷载或时空随机场。尽管有一些随机风激励模型可用，但是这些方法集中于获取用于时程分析的样本。在这项研究中，随机风激励被建模为滤波后的矢量白噪声。滤波器的状态空间表示是通过从已知的固定风能谱密度中解决正则化优化问题而获得的。通过将结构的运动方程与激励滤波器组合在一起，形成增强的状态空间表示。然后，通过求解相关的Lyapunov方程，可以获得感兴趣的结构响应的平稳协方差。运动方程的动态压缩被用来提高所提出方法的效率。优化方案的目标函数是根据响应的平稳协方差定义的；各种目标都被认为与典型的设计考虑相对应。采用等效的光滑公式来解决非光滑的最小-最大问题。使用基于梯度的方法来更新设计变量，而灵敏度是通过伴随的Lyapunov方程的解来计算的。针对经受顺风和横风荷载的高层建筑，说明了所建议的拓扑优化方案。本文呈现的结果证明了所提出的方法对于经受随机风激励的建筑物进行有效拓扑优化的功效。