# Iterative MLFMA-MADBT Technique for Analysis of Antenna Mounted on Large Platforms
## 3 Formulation  
### Point Source Modeling and Size Estimation of the MLFMA Region  
The first problem to be solved is how to build the mutual relationship between the MLFMA region and BT region. To analyze the radiation field from the MLFMA region to the BT region, a general way is calculating the integral from all RWG basis functions in the MLFMA region. When the platform is large, this solution requires large memory and long computational time. Thus, a novel method is proposed in this paper to reduce the computational complexity. In this paper, considering the mechanism of the MADBT method, the radiation field from the MLFMA region to the BT region is regarded as transmitted from a point source at the geometrical center of the MLFMA region. In addition, the electric field radiated by the MLFMA region can be calculated by  
\[\mathbf{E}(\mathbf{r}_{i})=-jw\mu\int_{S}G(\mathbf{r}_{i},\mathbf{r}_{j})[ \mathbf{J}(\mathbf{r}_{j})+\frac{1}{k^{2}}\
abla^{\prime}\
abla^{\prime} \cdot\mathbf{J}(\mathbf{r}_{j})]d\mathbf{r}_{j} \tag{1}\]  
where \(w\), \(\mu\) and \(k\) are the angular velocity, permeability and wave number in free space, \(\mathbf{r}_{i}\) is the observation point located in the BT region, \(\mathbf{r}_{j}\) is the source point in the MLFMA region, \(G(\mathbf{r}_{i},\mathbf{r}_{j})\) is the Green's function in free space, \(\mathbf{J}(\mathbf{r}_{j})\) is the electric current at \(\mathbf{r}_{j}\). When the source point and observation point is relatively far, the fast far-field approximation technique combined with MLFMA (FAFFA-MLFMA) can be used to alleviate the burden of field calculations [18; 19]. In this paper, the oct-tree boxes in the MLFMA region define the source groups, and each triangle in the BT region becomes a single far-field observation group. Suppose the \(\mathbf{r}_{m}\) and \(\mathbf{r}_{n}\) are the centers of the observation and source groups. For the far-field BT region, the spherical-wave function can be approximated as  
\[\frac{e^{-j|k|\mathbf{r}_{i}-\mathbf{r}_{j}|}}{|\mathbf{r}_{i}-\mathbf{r}_{j}| }\approx\frac{-jk}{4\pi}e^{-j|\mathbf{k}_{0}\cdot(\mathbf{r}_{mn}+\mathbf{r} _{nj})}\alpha_{mn}^{far} \tag{2}\]  
where \(\mathbf{k}_{0}=k\hat{k}_{0}\), \(\hat{k}_{0}\) is unit spatial vector from \(\mathbf{r}_{n}\) to \(\mathbf{r}_{m}\), \(\mathbf{r}_{im}=\mathbf{r}_{i}-\mathbf{r}_{m}\), \(\mathbf{r}_{nj}=\mathbf{r}_{n}-\mathbf{r}_{j}\) and \(\alpha_{mn}\) is a translator which is defined as  
\[\alpha_{mn}^{far}=4\pi\frac{e^{-j\
u_{mn}}}{-jkr_{mn}} \tag{3}\]  
where \(r_{mn}\) is the distance between \(\mathbf{r}_{m}\) and \(\mathbf{r}_{n}\). Therefore, the MLFMA region can be simply treated as a point source with the newly defined coupling formula to reduce the number of radiation sources dramatically, while the original is the number of RWG basis functions in the MLFMA region. The size of the MLFMA region must be determined before calculating the radiated field from the MLFMA region. As mentioned in [18], the quantitative criterion for the far-field approximation in the BT region can be presented as  
\[r_{mn}\geq 3\gamma D \tag{4}\]  
where \(\gamma\geq 1\) is a threshold set as 1.5 in this paper and \(D\) is the maximum value of \(|\mathbf{r}_{im}+\mathbf{r}_{inj}|\). In the BT region, \(\mathbf{r}_{im}\) is 0 because of the observation point in each triangle (group) is exactly the center of the group. Then we have  
\[D=max|\mathbf{r}_{nj}|=\frac{1}{2}\sqrt{D_{x}^{2}+D_{y}^{2}+D_{z}^{2}} \tag{5}\]  
where \(D_{x}^{2}\), \(D_{y}^{2}\) and \(D_{z}^{2}\) are the side lengths of the box (group) in the oct-tree of the MLFMA. Suppose there is no transition region and only an antenna is included in the MLFMA region. According to (4), for a large platform, only the triangles whose distances from the center point to any part the antenna is bigger than \(3\gamma D\) can be identified as the far-field BT region. The remaining triangles in the platform must be treated as the transition regionand calculated as a part of the MLFMA region.