# Bridge Model Updating Based on Wavelet Neural Network and Wind-Driven Optimization
## 3 Wind-Driven Optimization Algorithm
### Derivation of Update Equations  
To derive the velocity update equation, Equation (29) is simplified; let the air particle volume be \(\delta V=1\), \(\Delta t=1\). Both sides of the equation are divided by \(\rho\), and then it is converted to \(\rho=\mathbf{Q}/RT\) according to the ideal gas law, which leads to the following:  
\[\Delta\mathbf{u}=\frac{-\Delta\mathbf{Q}\times RT}{\mathbf{Q}}-\gamma\mathbf{u}+\mathbf{g}-\frac{ 2\mathbf{\Omega}\times\mathbf{u}\times RT}{\mathbf{Q}} \tag{30}\]  
In Equation (30), the gravity vector \(\mathbf{g}\) in the three-dimensional coordinate system centered at the earth will change with the distance between the air particle and the center of the earth, and the WDO algorithm replaces the earth-centered three-dimensional coordinate system with a \([-1,\,1]\)-type one-dimensional coordinate system with \(0\) as the origin and assumes that the relationship between the vector \(\mathbf{g}\) and the current position \(\mathbf{x}_{cur}\) of the air particle is \(\mathbf{g}=|\mathbf{g}|\cdot(0-\mathbf{x}_{cur})\). Therefore, the position of the air particle is also limited to the range of \([-1,\,1]\) so that the optimization range of the algorithm is \([-1,\,1]\). Meanwhile, in the one-dimensional coordinate system, the direction of the pressure gradient force \(F_{p}\) is from the high-pressure region to the low-pressure region, and the pressure value at the current position \(\mathbf{x}_{cur}\) of the air particle is defined as \(\mathbf{Q}_{cur}\), and the current minimum pressure value in the air particle swarm is \(\mathbf{Q}_{opt}\), which is the optimal pressure value, corresponding to the current optimal position \(\mathbf{x}_{opt}\) in the air particle swarm. Then, it can be assumed that the relationship between the pressure gradient \(-\Delta\mathbf{Q}\) and the position of air particle is \(-\Delta\mathbf{Q}\ =\Big{|}\mathbf{Q}_{opt}-\mathbf{Q}_{cur}\Big{|}\cdot(\mathbf{x}_{opt}-\mathbf{x}_{cur})\). The gravity force \(F_{g}\) and the pressure gradient force \(F_{p}\) are shown in Figure 3.  
Figure 3: The friction force (\(F_{g}\)) and pressure gradient force (\(F_{p}\)) in the one-dimensional coordinate system. In Equation (30), the velocity variable \(\Delta{\mathbf{u}}\) can be expressed as \(\Delta{\mathbf{u}}={\mathbf{u}}_{new}-{\mathbf{u}}_{cur}\), where \({\mathbf{u}}_{cur}\) is the current velocity of air particle, and \({\mathbf{u}}_{new}\) is the updated velocity of air particle. The friction force \(F_{u}\) is affected by the current velocity; therefore, \({\mathbf{u}}={\mathbf{u}}_{cur}\) is the friction force term, but the Coriolis force \(F_{c}\) describes the influence of inertia on current velocity, and thus, the velocity \({\mathbf{u}}_{cur}^{other~{}dim}\) is used to describe the velocity of the current dimension that is affected by other dimensions; that is, \({\mathbf{u}}={\mathbf{u}}_{cur}^{other~{}dim}\) is the Coriolis force term, and this velocity ensures the balance between the algorithm's exploration ability and development ability. Thereafter, Equation (30) can be further derived:  
\[ \begin{split}{\mathbf{u}}_{new}=(1-\gamma){\mathbf{u}}_{cur}-RT\big{(}{ \mathbf{x}}_{opt}-{\mathbf{x}}_{cur}\big{)}\,\frac{|Q_{opt}-Q_{cur}|}{Q_{cur}}\\ +g{\mathbf{x}}_{cur}-\frac{2\Omega RT{\mathbf{x}}_{cur}^{other~{}dim}}{Q_{cur }}\end{split}  \tag{31}\]  
To further improve Equation (31), let \(=-2|{\mathbf{\Omega}}|RT\). In addition, to make the WDO algorithm operational, the algorithm uses the ratio of air particle sorting numbers to replace the actual air pressure ratio; that is, \(\frac{Q_{opt}}{Q_{cur}}=\frac{1}{k}\), where \(k\) is the sorting number obtained by sorting the air particles in ascending order according to the distance between the current position \({\mathbf{x}}_{cur}\) and the current optimal position \({\mathbf{x}}_{opt}\), and the sorting number for \({\mathbf{x}}_{opt}\) is 1.