# Iterative MLFMA-MADBT Technique for Analysis of Antenna Mounted on Large Platforms
## 3 Formulation
### MADBT Method  
In this paper, a modified adaptive beam division algorithm is introduced to analyze radiation problem where the initial beams are pyramids with different corner ray directions. Beams in this method are dynamically split when they are divergent, and more details are described as follows. After the generation of initial beams, beams are launched toward the BT region and the recursive beam tracing is performed using the MADBT. During the tracing process, several beam-triangle intersections are performed. Please note that the kd-tree algorithm can be used to accelerate the time-consuming beam-triangle intersections when the platform is complex. Figure 3: The generation of initial beams. P is the point source, M and \(\hat{n}\) are the center and normal of the triangle. As shown in Figure 4, there are three basic cases of the beam-triangle intersections, i.e., non-intersection, partial intersection and full intersection. Different cases can be determined by checking the relative position between the beam and the projected mesh triangle on the beam base plane. Among the three intersection cases, the non-intersection and full intersection are easy to handle. For the non-intersection case, the beam-triangle intersection will come to an end and other beam-triangle intersections will be done for the beam until it exits the platform or intersects with one of the mesh triangles. For the full intersection case, the reflected beam will be generated, and the information of the mesh triangle will be stored for further field calculation. The generation of reflected beam is shown in Figure 5a. When it comes to the partial-intersection case, the situation is quite complicated. First, a beam may have partial intersections with several adjacent mesh triangles, if all the normal of the intersected triangles are the same, the beam can be treated as the full intersection without partition. Otherwise, as shown in Figure 5b, the original beam should be dynamically split into several new hit beams and miss beams according to the projection of the mesh triangle on the beam base plane using the partition method introduced in [7]. The number of corner ray directions of the beam are limited to three or four because of the  
Figure 4: Three cases of intersection. Project the mesh triangle on the beam base plane, we can get the relative position between beam and mesh triangle. Figure 5: (**a**) The generation of reflected beams. (**b**) The generation of hit beams, e.g., IDE, IEF, IFGH, and miss beams, e.g., CIH, DAE, and GFB. consistency, and both are supported by the partition method and the beam tracing used in the MADBT method. And for the hit beams, reflected beams will be generated. After that, all current beams, including reflected beams and miss beams will continue to be traced and adaptively divided until they exit the platform, or the order of reflection is larger than the threshold. At the same time, the beam tree that records the tracing procedure for each initial beam is generated. Generally, the number of beams will increase for considering the multi-bounce effects within the BT region. In actual applications, this problem will not bring an unacceptable negative impact to the efficiency of the proposed method for several reasons: First, not all beams with the partial intersection need to be split since the first condition of the partial intersection. Second, the large platforms discussed in this paper are not the enclosing objects with antennas inside, and most beams will be reflected into the free space within a finite order of reflection. Third, considering diminishing returns in multiple bounces, a proper threshold of the maximum order of the reflection can control the number of the beams while keeping a reasonable accuracy. Throughout the entire BT process, each beam is adaptively and recursively subdivided according to the intersections with the mesh triangles of the BT region. In addition, the tracing process of the MADBT method is only related to the geometry of the BT region.