## 2 Panel/Viscous Vortex Particle Aerodynamic Model
### Fast Summation Method
By applying the Gauss quadrature on the panel surface, Equation (18) yields  
\[\phi_{\sigma} = \sum_{k=1}^{K}\sigma w_{k}\frac{1}{|\mathbf{x}_{i}-\mathbf{x}_{k}|},\] \[\phi_{\mu} = \sum_{k=1}^{K}\mu w_{k}\mathbf{n}\cdot\
abla(\frac{1}{|\mathbf{x}_{i}-\mathbf{ x}_{k}|}), \tag{19}\]  
where \(\mathbf{x}_{k}\) is the Gauss quadrature point on the element; \(\mathbf{n}\) is the element normal vector; \(\sigma\) and \(\mu\) are panel source and dipole strength, respectively; and \(w_{k}\) is a suitable integration weight. In order to compute the integration weight, the coordinates on the panel are interpolated by linear Lagrangian polynomials \(N_{j}(\xi,\eta)\) by \(\mathbf{x}(\xi,\eta)=\sum_{j}N_{j}(\xi,\eta)\mathbf{x}_{j}\), where \(\xi\) and \(\eta\) are the local coordinates in the local panel coordinate system and span the range \([-1,1]\). An integral \(\int_{A}f(\mathbf{x})\mathrm{d}A\) over a quadrilateral element can thus be approximated, as customary in finite element modeling, as  
\[\int_{A}f(\mathbf{x})\mathrm{d}A=\int_{-1}^{1}\int_{-1}^{1}f(\mathbf{x}(\xi,\eta)) \left\|\frac{\partial\mathbf{x}}{\partial\xi}\times\frac{\partial\mathbf{x}}{\partial \eta}\right\|\mathrm{d}\xi\mathrm{d}\eta\approx\sum_{k}w_{k}^{0}f(\mathbf{x}(\xi_ {i},\eta_{i}))\left\|\frac{\partial\mathbf{x}}{\partial\xi}\times\frac{\partial\bm {x}}{\partial\eta}\right\|_{|\xi_{i},\eta_{i}}, \tag{20}\]  
where \(w_{k}^{0}\) is the Gauss quadrature weight. Thus, \(w_{k}=w_{k}^{0}\left\|\frac{\partial\mathbf{x}}{\partial\xi}\times\frac{\partial \mathbf{x}}{\partial\eta}\right\|_{|\xi_{k},\eta_{k}}\). From Equation (19), it can be seen that the quadrilateral element can be represented by several particles located at the Gauss point and with a weighted strength. This makes it possible to implement the panel FMM into the classical particle-based fast multipole method. The multipole expansion for source and dipole quadrilateral elements can be obtained:  
\[\phi=\sum_{n=0}^{P}\sum_{m=n}^{n}M_{n}^{m}r^{-n-1}Y_{n}^{m}(\theta,\phi) \tag{21}\]  
\[\bar{M}_{n}^{m} = \sum_{k=1}^{K}w_{k}\sigma\rho_{k}^{n}Y_{n}^{-m}(\alpha_{k},\beta_ {k})\] \[\bar{M}_{n}^{m} = \sum_{k=1}^{K}w_{k}\mu\cdot\
abla(\rho_{k}^{n}Y_{n}^{the multipole and local expansion coefficients have 5 components: 3 for the vector-valued vorticity, 1 for the weighted dipole strength, and 1 for the weighted source strength. A case with randomly distributed particles within a unit cubic was studied to demonstrate the convergence and acceleration of the FMM. The numerical experiments were performed on an Intel(r) Xeon(r) CPU E5-2640 0 @ 2.50 GHz desktop. Figure 3(a) shows the \(L^{2}\)-norm of the relative difference between the induced velocity computed by the FMM and the result computed by the analytical method. It can be seen that, when the truncation order is larger than 5, a relative error of \(10^{-4}\) can be obtained. Figure 3(b) shows the time consumption of the two methods; the FMM shows its advantage for an increasing numer of particles, as expected, and when the particle number is \(10^{6}\), an acceleration of over 100 times is obtained. It is clear that the computational complexity is proportional to the number of particles, \(O(N)\), as it should be. In order to verify the validity of the proposed approach for the panels, the interaction of \(n\times n\times n\) panels that are uniformly distributed in the unit cube was studied and the velocity measured at the center of each panel was compared with the analytical results computed with the formulae from [27]. Figure 4(a) shows the relative errors obtained with \(30\times 30\times 30\) panels. Figure 4(b) reports the computational time cost and shows that a nearly linear computation complexity was obtained. Figure 4: Fast multipole method (FMM) for vortex particles. Figure 3: Gauss quadrature diagram on the vortex panel (the red circles stand for the Gauss quadrature points). The overall speedup allowed by applying the FMM also to panels is significant for real-word simulations.