Solutions of electromagnetics problems involving hundreds of millions of unknowns with parallel multilevel fast multipole algorithm

We present the solution of extremely large electromagnetics problems formulated with surface integral equations (SIEs) and discretized with hundreds of millions of unknowns. Scattering and radiation problems involving three-dimensional closed metallic objects are formulated rigorously by using the combined-field integral equation (CFIE). Surfaces are discretized with small triangles, on which the Rao-Wilton-Glisson (RWG) functions are defined to expand the induced electric current and to test the boundary conditions for the tangential electric and magnetic fields. Discretizations of large objects with dimensions of hundreds of wavelengths lead to dense matrix equations with hundreds of millions of unknowns. Solutions are performed iteratively, where the matrix-vector multiplications are performed efficiently by using the multilevel fast multipole algorithm (MLFMA). Solutions are also parallelized on a cluster of computers using a hierarchical partitioning strategy, which is well suited for the multilevel structure of MLFMA. Accuracy and efficiency of the implementation are demonstrated on electromagnetic problems involving as many as 205 million unknowns, which are the largest integral-equation problems ever solved in the literature.