Topology optimization of buildings subjected to stochastic wind loads

Abstract

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.