# Adaptive Harmony Search for Tuned Liquid Damper Optimization under Seismic Excitation
## 3 Numerical Examples
### Single-Story Structures  
By selecting a mass of 100t for all single-story structures, a three-degrees-of-freedom (3 DOF) model was created with the sloshing liquid, the structure, and the TLD plus non-sloshing liquid moving together. The stiffness values obtained were 15.8 MN/m, 3.95 MN/m, and 1.75 MN/m, respectively, and the damping coefficients were 0.13 MN s/m, 0.06 MN s/m, and 0.04 MN s/m, respectively; the structure period values were between 0.5 s and 1.5 s. The TLD was optimized to withstand structure period values between 0.5 and 1.5. After the height and radius values of the cylindrical TLD tank were added to the single-story structure model, the TLD period and damping ratio were optimized. The results obtained from optimization with the classical harmony search algorithm (HS), adaptive harmony search algorithms (AHS), and two improved harmony search algorithms (IHS [24] and a modified variant of HS (LHS) [41]) are shown in Table 3, for the structure with a 1 s period. Maximum displacement and total acceleration values were calculated as a result of the analysis performed under FEMA earthquake warnings, with TLD parameters optimized for the 1 s period of the structure. Table 4 shows the values obtained from the critical earthquake analysis. A critical earthquake record for a single-story structure was recorded in 1999 in Duzce (BOL090) for the 1 s period. Based on the results, AHS is the best method for minimization of the objective function, taken as displacement of the top story. After the verification of AHS as the best modified algorithm, the other optimization cases were performed for only AHS and classical HS. HS and AHS optimized values are shown in Table 5 for single-story models with structure periods of 0.5 s and 1.5 s. For the 0.5 s and 1.5 s structure period, the critical earthquake records were 1994 Northridge (MUL279) and 1999 Kocaeli (DZC270) earthquakes, respectively. Tables 6 and 7 show the maximum displacement and total acceleration values obtained from the critical earthquake analysis at the 0.5 s and 1.5 s structural periods. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Definition** & **Value** \\ \hline \(HMCR\) & \begin{tabular}{c} Harmony Memory \\ Considering Rate \\ Initial Harmony Memory \\ Considering Rate \\ Fre Width \\ Initial Fre Width \\ Maximum iteration number \\ Iteration number \\ \(pn\) \\ Population number \\ \end{tabular}  & 0\(-\)0.5 \\ \(-\)0. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{
\begin{tabular}{c} **Structure** \\ **Without TLD** \\ \end{tabular} } & \multicolumn{2}{c}{**Structure with TLD**} \\ \cline{2-4}  & & **HS** & **HS** & **AHS** \\ \hline Displacement (m) & 0.1774455 & 0.1273578 & 0.1268410 \\ \hline Total Acceleration (m/s\({}^{2}\)) & 28.0477691 & 20.4824088 & 20.3932509 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: TLD plus single-story building model; critical earthquake analysis results (T = 1.5s). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{**Without TLD**} & \multicolumn{3}{c}{**Structure with TLD**} \\ \cline{2-4}  & & **HS** & **HS** & **LHS** & **AHS** \\ \hline Displacement (m) & 0.2873419 & 0.2473086 & 0.2460654 & 0.2450165 & 0.2446071 \\ \hline Total Acceleration (m/s\({}^{2}\)) & 11.4077630 & 9.6358656 & 9.5542167 & 9.4776471 & 9.5648807 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: TLD plus single-story building model; critical earthquake analysis results (T = 1.0 s). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{
\begin{tabular}{c} **Structure** \\ **Without TLD** \\ \end{tabular} } & \multicolumn{2}{c}{**Structure with TLD**} \\ \cline{2-4}  & & **HS** & **HS** & **LHS** \\ \hline Displacement (m) & 0.2927541 & 0.2574074 & 0.2567679 \\ \hline Total Acceleration (m/s\({}^{2}\)) & 5.2223281 & 4.5565000 & 4.5520700 \\ \hline \hline \end{tabular}
\end{table}