Topology optimization of structures subjected to stochastic dynamic excitation
The field of topology optimization has progressed substantially in recent years, with applications varying in terms of the type of structures, boundary conditions, loadings, and materials. Nevertheless, topology optimization of stochastically excited structural systems has received relatively little attention. Due to the vast number of degrees of freedom typically required for topology optimization, conventional approaches for solving the random vibration problem are time prohibitive; moreover, the application of non-gradient-based optimization algorithms is not feasible due to the extensive number of design variables. Consequently, new techniques are required to obtain the response due to stochastic excitation and perform the sensitivity analysis of these quantities efficiently for large systems. In this research, a direct approach to this problem is proposed, modeling the excitation as a filtered white noise. The excitation model is combined with the structural model to form an augmented representation, and the covariance of the structural response is obtained by solving a Lyapunov equation. The objective function is defined in terms of the response covariance. For the stationary problem, a fast large-scale solver of the Lyapunov equation is implemented for sparse matrices; and an efficient adjoint method is proposed to obtain the sensitivities of the objective function. Model reduction techniques are also considered to improve the efficiency of the approach for buildings. The proposed formulation and numerical solutions are extended to consider other representative problems such as non-stationary excitations or minimizing the maximum response. Furthermore, the proposed method is extended to perform simultaneous optimization of topology and supplemental damping distribution of buildings subjected to stochastic dynamic excitation. The proposed topology optimization framework and its components are illustrated through several numerical examples: application to idealized structures, application to buildings subjected to ground motions, application to tall buildings subjected to dynamic wind loading, application to buildings with braces and dampers. The results presented herein demonstrate the efficacy of the proposed approach for efficient topology optimization of stochastically excited structures.
Topology optimization, stochastic dynamics, Lyapunov equation, seismic design, wind design, damping devices, finite element
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign