Multiscale electromagnetic computations using a hierarchical multilevel fast multipole algorithm

Abstract

A hierarchical multilevel fast multipole method (H-MLFMM) is proposed herein to accelerate the solutions of surface integral equation methods. The proposed algorithm is particularly suitable for solutions of wideband and multiscale electromagnetic problems. As documented in Zhao and Chew (2000) that the multilevel fast multipole method (MLFMM) achieves O(N log N) computational complexity in the fixed mesh size scenario, hk = cst, where h is the mesh size and k is the corresponding wave number, for problems discretized under conventional mesh density. However, its performance deteriorates drastically for overly dense meshes where the couplings between different groups are dominated by evanescent waves or circuit physics. In the H-MLFMM algorithm, two different types of basis functions are proposed to address these two different natures of physics corresponding to the electrical size of the elements. Specifically, for the propagating wave couplings, the plane wave basis function adopted by MLFMM are effective and they are inherited by H-MLFMM. Whereas in the circuit physics and for the evanescent waves, H-MLFMM employs the so-called skeleton basis. Moreover, the proposed H-MLFMM unifies the procedures to account for the couplings using these two distinct types of basis functions. O(N) complexity is observed for both memory and CPU time from a set of numerical examples with fixed mesh sizes. Numerical results are included to demonstrate that H-MLFMM is error controllable and robust for a wide range of applications.

A Novel Broadband Multilevel Fast Multipole Algorithm With Incomplete-Leaf Tree Structures for Multiscale Electromagnetic Problems

“…Apart from the EPA, some other methods based on the hierarchical MLFMA [16], multiresolution basis functions [17], and accelerated Cartesian expansion [18] have also been used to attack multiscale problems. In various advanced computing [19], chemistry [20], biology [21], and physics [22] applications, especially in N-body problems, a modified version of the FMM, namely, the adaptive FMM, has been used.…”

ADAPTIVE AND PARALLEL SURFACE INTEGRAL EQUATION SOLVERS FOR VERY LARGE-SCALE ELECTROMAGNETIC MODELING AND SIMULATION (Invited Paper)

“…It requires that the surface currents in each independent subdomain be radiated to all other subdomains. The computation is accomplished with two mathematical ingredients: (i) a hierarchical multi-level fast multipole method (H-MLFMM) [49,50], which leads to the seamless integration of multi-level skeletonization technique [51] into the FMM framework and results in an effective matrix compression for non-uniform DG discretizations; (ii) a primal-dual octree partitioning algorithm for separable subdomain coupling [52]. Namely, instead of partitioning the entire computational domain into a single octree as in the traditional FMM, we have created independent octrees for all subdomains.…”

Solution box: SOLBOX-11

“…When the variety in the element size is large, conventional implementations often suff er from inaccuracy, instability, and/or ineffi ciency issues due to numerical breakdowns in the context of discretization, expansion, and/or matrix solution. As a natural consequence, there is an enormous collective eff ort in the literature [1][2][3][4][5][6][7][8][9] to develop accurate, stable, and effi cient numerical solvers for multi-scale problems involving nonuniform discretizations.…”