## 2 Methodology
### ALE-LBFS for Fluid Solver  
#### 2.2.1 Macroscopic Conservation Laws  
As one of the core components of flutter analysis, the lattice Boltzmann flux solver (LBFS) acts as a fluid solver in the ALE formulation. LBFS is a novel numerical solver developed from lattice Boltzmann equations. It involves the evaluation of flux at the interface and obeys laws of conservation for mass, momentum, and energy. In a Cartesian coordinate system of unstructured meshes, Equation (1) is rewritten, and the unsteady ALE-LBFS for a finite control volume is  
\[\frac{\partial\textbf{W}}{\partial t}+\
abla\cdot\textbf{F}_{c}-\
abla\cdot \textbf{F}_{g}-\
abla\cdot\textbf{F}_{v}=0 \tag{8}\]  
\[\textbf{W}= \begin{bmatrix}\rho\\ \rho\textbf{u}\\ \rho E\end{bmatrix} , \tag{9}\]  
\[\textbf{F}_{c}= \begin{bmatrix}\rho u_{n}\\ \rho\textbf{u}u_{n}+p\textbf{n}\\ (\rho E+p)u_{n}\end{bmatrix} ,\textbf{F}_{g}= \begin{bmatrix}\rho u_{g,n}\\ \rho\textbf{u}u_{g,n}\\ \rho\textbf{u}u_{g,n}\end{bmatrix} \text{ and }\textbf{F}_{v}= \begin{bmatrix}0\\ \boldsymbol{\tau}\cdot\textbf{n}\\ \boldsymbol{\Theta}\cdot\textbf{n}\end{bmatrix}  \tag{10}\]  
where the vector **W** is the conservative variable, \(\textbf{F}_{c}\) denotes the convective flux, \(\textbf{F}_{g}\) indicates the additional flux contributed by moving mesh, and \(\textbf{F}_{v}\) is the viscous flux. For a detailed description, it is necessary to state that \(\rho\) is density, \(E\) is the total energy in unit mass. **u** and \(\textbf{u}_{g}\) are the fluid velocity vector and the mesh velocity vector. The subscript n in the \(u_{n}\) and \(u_{g,n}\) indicates the normal velocity component. **n** is the unit normal vector. Aside from static pressure \(p\), unit tensor **I**, and time t, tensor \(\boldsymbol{\tau}\) is the viscous stress as common and \(\boldsymbol{\Theta}\) describes the contribution made by other physical effects, such as heat conduction. Adapting spatial discretization with a vertex-centered finite-volume scheme, Equation (8) is semi-discrete as  
\[\frac{d\textbf{W}_{i}}{dt}=-\frac{1}{d\Omega}\sum_{k=1}^{N_{f}}\Big{(}\textbf {F}_{ck}-\textbf{F}_{gk}-\textbf{F}_{vk}\Big{)}S_{k} \tag{11}\]  
where \(\textbf{W}_{i}\) is the vector of aforementioned conservative variables specified for the i-th control volume, and \(\textbf{F}_{ck}\) and \(\textbf{F}_{vk}\) subscripted with k are the vectors of convective and viscous flux across the kth interface of control volume for index i, respectively. \(\textbf{F}_{gk}\) is the additional flux contributed by mesh motion on the kth interface. It is common to say that \(S_{k}\) denotes the area of the kth interface, let alone cell volume \(\Omega_{i}\) and the number of the interface \(N_{f}\). Figure 2 presents a schematic diagram for the control volume of a median-dual cell-vertex scheme. The interfaces of a control volume are made up of the lines for 2D or faces for 3D linked by the midpoint of the edge and element center around the considered grid node. Contrary to the cell-centered approach, **W** is stored at the grid node together with mesh velocity \(\textbf{u}_{g}\). As a consequence, the solutions of governing equations and mesh velocity are directly computed and obtained at the nodes at the same time. Unlike the conventional LBFS built on a cell-centered scheme treating dynamic meshes where it has to integrate and average grid velocity from grid nodes to the cell center, it is novel and simple to establish LBFS with a cell-vertex scheme. To solve Equation (11), it needs to separately evaluate the convective, viscous, and additional ALE flux according to Equation (10). First, the non-free parameter D1Q4 model [42; 47] is responsible for building connections between local fluid properties and lattice velocity and density distribution. Figure 3 shows the distribution of discrete lattice particles. In the assumption of the 1D model, the lattice particles transport in lines.