## 2 Panel/Viscous Vortex Particle Aerodynamic Model
### Fast Summation Method  
As shown in Equation (12), calculation of the convection velocity requires, for each particle, summation over all the particles within the computation domain as well as the velocity gradient, which renders the vortex particle method a typical \(N\)-body problem with a direct summation computation complexity of \(O(N^{2})\) at each time step, where \(N\) is the number of particles. Two effective acceleration approaches are the TreeCode [22] and the Fast Multipole Method (FMM) [21; 26], which reduce the computational complexity from \(O(N^{2})\) to \(O(N\log(N))\) and \(O(N)\), respectively. Both approaches utilize an Octree data structure to hierarchically subdivide the whole computation space occupied by particles into three dimensional cells and to exactly calculate the near-filed interactions by particle-to-particle (P2P) interactions while approximating the far-field interactions of the particles that are well separated. The separation threshold is defined by the Multipole Acceptance Criteria (MAC) \(\frac{r_{1}+r_{2}}{R}<\theta\), where \(r_{1}\) and \(r_{2}\) are the radius of the source and target cell, respectively, and \(R\) is the distance between the two cells. However, FMM can approximate far-field interactions to a specified tolerance. TreeCode approximates the interaction between well-separated particles using the interactions between particle and cells by multipole expansion, which is a multipole-to-particle (M2P) interaction. In FMM, for two well-seperated cells, as depicted in Figure 2, the influence of the source cell on the center of the target cell is first calculated by a multipole-to-local (M2L) transformation, the influence can be propagated to offspring cells of the target cell by local-to-local (L2L) transformation down to the finest level, and finally the velocity of each particle in the target cell can be evaluated by accumulating the local-to-particle (L2P) interactions at the finest level. In the present work, an adaptive Octree is built, the dual tree traversal algorithm is implemented to identify the well-separated interaction list, and the classical FMM in spherical coordinate is adopted to speed up the calculations of the convection velocity and stretching term for the particle-particle interactions. Meanwhile, as in [17] and due to the rapid decay of the kernel function, only the particles close to the target particle contribute significantly to the PSE for \(\mathrm{d}\mathbf{\alpha}_{i}/\mathrm{d}t\); thus, the PSE can be considered only in the process of the particle-to-particle (P2P) interaction of FMM. In the numerical experiments, as the rotor wake develops, the number of vortex particles increases rapidly and the vortex panel-particle interaction becomes a time-consuming computation step. A fast panel-particle evaluation is thus required as well. The velocity potential at \(\mathbf{x}_{j}\) generated by a constant dipole quadrilateral and source quadrilateral can be expressed as  
\[\phi_{\sigma} = \int_{S}\sigma\frac{1}{|\mathbf{x}_{i}-\mathbf{x}_{j}|}\mathrm{d}S,\] \[\phi_{\mu} = \int_{S}\mu\mathbf{n}_{j}\cdot\
abla(\frac{1}{|\mathbf{x}_{i}-\mathbf{x}_{j}| })\mathrm{d}S. \tag{18}\]  
Figure 2: Flow of FMM calculation. Compared with the potential generated by point sources and point dipoles, one needs to compute an integral over the panel. When the target point is outside the panel, the integral is regular and can be computed accurately by a numerical method, such as a Gauss quadrature rule. When the target point is inside the panel, the integral is singular. Thus, specific approaches are needed depending on the location of the target point. In the present work, a semi-analytical method is adopted, where the inside integral is computed by resorting to an analytical method and the outside integral is evaluated by Gauss quadrature. For the inside integral, analytical expressions are available; see, e.g., [27]. When the target point is located outside the panel, as depicted in Figure 3, the integral over the quadrilateral panel is evaluated numerically.