Data-driven performance-based topology optimization of uncertain wind-excited tall buildings

Abstract

Topology optimization is traditionally framed in a static and deterministic setting notwithstanding the uncertain dynamic nature of many problems. This paper presents a new data-driven simulation-based framework for the effective topology optimization of uncertain and dynamic wind-excited tall buildings. The performance of the system is described through probabilistic performance integrals that encapsulate state-of-the-art performance-based design driven by climatological, aerodynamic and fragility data sets for describing the site-specific hazard, aerodynamic response and damage susceptibility of the system. To solve the resulting probabilistic topology optimization problem, a sequential optimization strategy is developed that is based on solving a series of high quality approximate sub-problems. A suite of case studies demonstrate the effectiveness of the approach.

Appendices

Appendix A: Reduced system

The stiffness of the complete system K can be easily reduced to the degrees of freedom (DOFs) of the master nodes (see Section 3.1) through static condensation, therefore defining a reduced stiffness matrix $\tilde {\textbf {K}}$. By observing that the mass of the complete system is zero outside the DOFs of the master nodes, a reduced mass matrix, $\tilde {\textbf {M}}$, can be defined by simply eliminating the rows and columns of the complete mass matrix, M, associated with DOFs of non master nodes. The first m modes of the reduced system can be calculated by solving the following standard eigenvalue problem in terms of $\tilde {\textbf {K}}$ and $\tilde {\textbf {M}}$:

$$ \left( \tilde{\mathbf{K}} - \tilde{\omega}^{2}_{i} \tilde{\mathbf{M}} \right) \tilde{\boldsymbol{\phi}}_{i} = 0 $$

(36)

In particular, because the mass of the complete system is located only at the DOFs of the master nodes, the natural frequencies found by solving (36) will be identical to those found by solving (12) while, for the same reason, the mode shapes $\tilde {\boldsymbol {\phi }}_{i}$ will be identical in the DOFs of the master nodes ($\tilde {\boldsymbol {\phi }}_{i}$ is not defined outside these DOFs) to ϕ _i . By now observing that the wind loads also enter the complete system exclusively at the DOFs of the master nodes (see Section 3.1), the elimination of the zero terms from f will define a reduced forcing function vector $\tilde {\textbf {f}}$ that allows the following modal equation to be defined:

$$ \ddot{q}_{i}(t) + 2 s_{2_{i}} \xi_{i} s_{3_{i}} \tilde{\omega}_{i} \dot{q}_{i}(t) + (s_{3_{i}}\tilde{\omega}_{i})^{2} q_{i}(t) = \tilde{\boldsymbol{\phi}}_{i}^{T} \tilde{\mathbf{f}}(t;\bar{v}_{H}) $$

(37)

Because the original forcing functions are zero in all non master DOFs, while the natural frequencies $\tilde {\omega }_{i}$ are identical between the two systems as are the mode shapes in the master DOFs, (37) will yield the exact same modal responses as (14). Because the background modal response of (15) is also identical between the two systems for the same reasons stated above, the resonant modal response vector $\tilde {\textbf {q}}_{r,m}$ will be identical for the two systems. This implies that, in the DOFs of the master nodes, the quasi-static loads calculated from the following reduced equations will be identical to those calculated in the complete system:

$$ \tilde{\pmb{\mathcal{F}}}(t) = s_{1}\tilde{\mathbf{f}}(t;\bar{v}_{H})+ s_{1}\tilde{\mathbf{K}} \tilde{\boldsymbol{\Phi}}_{m} \tilde{\mathbf{q}}_{r,m}(t) $$

(38)

If it is now observed that the entries of quasi-static loads $\pmb {\mathcal {F}}$ of the complete system ((11)) must be identically zero in all non master DOFs (a consequence of the fact that the mass matrix, M, and the forcing functions, f, of the complete system have zero terms in all non master DOFs), it follows that for the response process calculated in the reduced system as:

$$ \tilde{R}(t) = \tilde{\boldsymbol{\Gamma}}_{R}^{T} \tilde{\pmb{\mathcal{F}}}(t) $$

(39)

where $\tilde {\boldsymbol {\Gamma }}_{R}$ is the vector of influence function of R calculated from Γ _R by simply eliminating the entries of Γ _R corresponding to non master DOFs, the following must hold:

$$ R(t) \equiv \tilde{R}(t) $$

(40)

In other words the response processes calculated in the reduced system are identical to those calculated in the complete system.

Appendix B: The equivalent load profile and the reduced system

To transfer the vector $\tilde {\boldsymbol {\Psi }}$ to the complete system the following transformation can be used:

$$ {\Psi}_{k}(\boldsymbol{\rho},\lambda_{0}) = \left\{\begin{array}{ll} \tilde{\Psi}_{l}(\boldsymbol{\rho},\lambda_{0}), & k = l \in {\Xi} \\ 0, & k \neq l \in {\Xi} \end{array}\right. $$

(41)

where Ξ is the set of degrees of freedom associated with the master nodes while k=1,…,D with D indicating the total number of degrees of freedom of the complete system.