The Crout version of ILU preconditioning applied to the fast‐multipole method for solving large‐scale electromagnetic open PEC problems

Abstract

PurposeThe purpose of this paper is to investigate and analyze the efficiency and stability of the implementation of the Crout version of ILU (ILUC) preconditioning on fast‐multipole method (FMM) for solving large‐scale dense complex linear systems arising from electromagnetic open perfect electrical conductor (PEC).Design/methodology/approachThe FMM is employed to reduce the computational complexity of the matrix‐vector product and the memory requirement of the impedance matrix. The numerical examples are initially solved by the quasi‐minimal residual (QMR) method with ILUC preconditioning. In order to fully investigate the performance of ILUC in connection with other iterative solvers, a case is also solved by bi‐conjugate gradient solver and conjugate gradient squared solver with ILUC preconditioning.FindingsThe solutions show that the ILUC preconditioner is stable and significantly improves the performance of the QMR solver on large ill‐conditioned open PEC problems compared to using ILU(0) and threshold‐based ILU (ILUT) preconditioners. It dramatically decreases the number of iterations required for convergence and consequently reduces the total CPU solving time with a reasonable overhead in memory.Practical implicationsThe preconditioning scheme can be applied to large ill‐conditioned open PEC problems to effectively speed up the overall electromagnetic simulation progress while maintaining the computational complexity of FMM. More complex structures including wire‐PEC junctions and microstrip arrays may be addressed in future work.Originality/valueThe performance of ILUC has been previously reported only on preconditioning sparse linear systems, in which the ILU preconditioner is constructed by the ILUC of the coefficient matrix (e.g. matrix arised from two‐dimensional finite element convection‐diffusion problem) and subsequently applied to the same sparse linear systems; so it is important to report its performance on the dense complex linear systems that arised from open PEC electromagnetic problems. In contrast, the preconditioner is constructed upon the near‐field matrix of the FMM and subsequently applied to the whole dense linear system. The comparison of its performance against the diagonal, ILU(0) and ILUT precoditioners is also presented.