# Bridge Model Updating Based on Wavelet Neural Network and Wind-Driven Optimization
## 3 Wind-Driven Optimization Algorithm
### Fundamentals of the Algorithm  
The motion of the atmosphere inspires WDO; that is, the atmosphere is considered to be composed of many air particles with a certain mass and volume, and the air particles satisfy the ideal gas law, so the motion state of each air particle can be described by [PERSON]'s second law [45]. The fundamentals of the WDO algorithm are derivations of velocity and position update equations during the movement of air particles. [PERSON]'s second law and ideal gas law are as Equations (23) and (24):  
\[\rho\mathbf{a}=\sum_{i}\mathbf{F}_{i} \tag{23}\]  
\[\mathbf{Q}=\rho RT \tag{24}\]  
where \(\mathbf{a}\) is the acceleration of air particles; \(\rho\) is the density of air particles in the atmosphere; \(\mathbf{F}_{i}\) is the different forces acting on the air particle; \(Q\) is the atmospheric pressure; \(R\) is the ideal gas constant; \(T\) is the atmospheric temperature.  
The atmosphere moves on the earth's surface and has a certain speed relative to the earth, usually called wind speed. When air particles move in a straight line with the wind speed vector in a rotating system, their inertia will produce a linear motion offset relative to the rotating system. Since [PERSON]'s law is established in the inertial coordinate system, when dealing with the problem of the non-inertial rotating coordinate system, an imaginary force, the Coriolis force, needs to be introduced to describe the linear offset phenomenon in the rotating system. Combined with the physics of atmospheric motion, it is assumed that air particles are in a state of hydrostatic equilibrium, and the atmospheric motion is simulated by the four main forces acting on air particles, which are as follows:  
(1) The pressure gradient force \(F_{p}\) that makes the gas move from the high-pressure region to the low-pressure region is calculated by Equation (25):  
\[F_{p}=-\Delta\mathbf{Q}\times\delta V=-\Delta\rho RT\times\delta V; \tag{25}\]  
Figure 2: Combination of WDO and WNN.  
(2) The friction force \(F_{u}\) that prevents the gas from flowing due to the pressure gradient force is calculated by Equation (26):  
\[F_{u}=-\rho\gamma\mathbf{u}; \tag{26}\]  
(3) The gravity force \(F_{g}\) which is perpendicular to the center of the earth is calculated by Equation (27):  
\[F_{g}=\rho\times\mathbf{g}\times\delta V; \tag{27}\]  
(4) The Coriolis force \(F_{c}\) describing the linear offset of air particles is calculated by Equation (28):  
\[F_{c}=-2\mathbf{\Omega}\times\mathbf{u} \tag{28}\]  
where \(-\Delta\mathbf{Q}\) is the pressure gradient, and the negative sign indicates the direction of descent along the gradient; \(\delta V\) is the finite volume of air particles; \(\gamma\) is the friction coefficient; \(\mathbf{u}\) is the wind velocity vector; \(\mathbf{g}\) is the gravitational acceleration vector; \(\mathbf{\Omega}\) is the earth's rotation angular velocity vector. The acceleration of air particles can be expressed as \(\mathbf{a}=\Delta\mathbf{u}/\Delta t\), which can be obtained by substituting the four main forces into [PERSON]'s second law as Equation (29):  
\[\begin{split}\rho\frac{\Delta\mathbf{u}}{\Delta t}=F_{p}+F_{u}+F_{g} +F_{c}\\ =(-\Delta\mathbf{Q}\times\delta V)+(-\rho\gamma\mathbf{u})+(\rho\times \mathbf{g}\times\delta V)+(-2\mathbf{\Omega}\times\mathbf{u})\end{split} \tag{29}\]