The use of Huygens' equivalenceprinciple for solving 3D volume integral equation of scattering

A three dimensional (3D) version of the nested equivalent principle algorithm (NEPAL) is presented. In 3D, spherical wave functions are used to represent the scattered field of the subscatterers. Subscatterers are divided into different levels of groups in a nested manner. In other words, each group consists of two subgroups, and each subgroup contains two sub-subgroups, and so on. For each subgroup, the scattering solution is first solved and the number of subscatterers of the subgroup is then reduced by replacing the interior subscatterers with boundary subscatterers using Huygens' equivalence principle. As a result, when the subgroups are combined to form a higher level group, the group will have a smaller number of subscatterers. This process is repeated for each level, and in the last level, the number of subscatterers is proportional to that of boundary size of the subscatterers. This algorithm has a computational complexity of $O(N^2)$ in three dimensions and has the advantage of solving large scattering problems for multiple excitations. This is in contrast to Gaussian elimination which has a computational complexity of $O(N^3)$.