Efficient Implementation of Inner-Outer Flexible GMRES for the Method of Moments Based on a Volume-Surface Integral Equation |
|
Hidetoshi Chiba ・ Toru Fukasawa ・ Hiroaki Miyashita ・ Yoshihiko Konishi |
|
|
The spectacular progress in the field of scientific computing in recent times has enabled researchers to perform large-scale, complex, and practical computations that were previously impossible. Nevertheless, further improvement of the computation speed and accuracy is still desired. Hence, increase in productivity using computer-aided engineering (CAE), especially in the front line of practical design scenarios, can be realized only when faster and more efficient numerical techniques are established.
In this study, acceleration of the method of moments incorporated with the fast multipole method (FMM) is investigated. Since the coefficient matrix cannot be explicitly generated using the FMM, only limited information based on near-field interactions, which correspond to direct computation part of the coefficient matrix, can be used when applying a conventional static preconditioner. However, this conventional approach becomes less effective because information to be used for the preconditioner becomes scarce as the problem size increases. On the other hand, a flexible version of the iterative methods, in which variable preconditioners are acceptable, can utilize far-field interactions, which are evaluated using the multipole expansion. Thus, the efficiency of the FMM can be maximized. However, there is no definite rule for determining the accuracy of the FMM in the variable preconditioner. This study focuses on the following feature of multipole techniques: the accuracy and computational cost of the FMM can be controlled by appropriate choice of the truncation number for multipole expansion. Further, in this study, a method that constructs two FMM operators with different levels of accuracy-a highly accurate operator for the outer solver and a less accurate and relatively inexpensive operator for the inner solver-is proposed. Numerical experiments with practical and complex geometries reveal that once the truncation number in the inner solver is appropriately chosen, the proposed method significantly accelerates the convergence with sufficiently accurate solutions than does the conventional method.
As described above, a very fast and efficient numerical method is developed and it is conclusively validated via practical examples. The accomplishments would contribute to not only the facilitation of the utilization of numerical methods for electromagnetic wave problems but also the further development of the CAE technology.
