A fast singular boundary method for 3D Helmholtz equation

Abstract This paper presents a fast singular boundary method (SBM) for three-dimensional (3D) Helmholtz equation. The SBM is a boundary-type meshless method which incorporates the advantages of the boundary element method (BEM) and the method of fundamental solutions (MFS). It is easy-to-program, and attractive to the problems with complex geometries. However, the SBM is usually limited to small-scale problems, because of the operation count of O ( N 3 ) with direct solvers or O ( N 2 ) with iterative solvers, as well as the memory requirement of O ( N 2 ) . To overcome this drawback, this study makes the first attempt to employ the precorrected-FFT (PFFT) to accelerate the SBM matrix–vector multiplication at each iteration step of the GMRES for 3D Helmholtz equation. Consequently, the computational complexity can be reduced from O ( N 2 ) to O ( N l o g N ) or O ( N ) . Three numerical examples are successfully tested on a desktop computer. The results clearly demonstrate the accuracy and efficiency of the developed fast PFFT-SBM strategy.

[1]  D. L. Young,et al.  Regularized meshless method for multiply-connected-domain Laplace problems , 2006 .

[2]  Jun Zhang,et al.  Precorrected FFT accelerated BEM for large‐scale transient elastodynamic analysis using frequency‐domain approach , 2012 .

[3]  Jiheng Zhang,et al.  Rapid solution of 3-D oscillatory elastodynamics using the pFFT accelerated BEM , 2010 .

[4]  Chuanzeng Zhang,et al.  Singular boundary method for wave propagation analysis in periodic structures , 2018 .

[5]  Ji Lin,et al.  Numerical treatment of acoustic problems with boundary singularities by the singular boundary method , 2014 .

[6]  Jun Lu,et al.  Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions , 2016, Comput. Math. Appl..

[7]  Yan Gu,et al.  Investigation on near-boundary solutions by singular boundary method , 2012 .

[8]  Wen Chen,et al.  A novel numerical method for infinite domain potential problems , 2010 .

[9]  Chuanzeng Zhang,et al.  Singular boundary method for solving plane strain elastostatic problems , 2011 .

[10]  Wen Chen,et al.  The singular boundary method: Mathematical background and application in orthotropic elastic problems , 2014 .

[11]  Simulation of acoustic scattering by the fast BEM approach , 2010 .

[12]  Weiwei Li,et al.  Band gap calculations of photonic crystals by singular boundary method , 2017, J. Comput. Appl. Math..

[13]  Xing Wei,et al.  Potential Problems by Singular Boundary Method Satisfying Moment Condition , 2009 .

[14]  Changjun Zheng,et al.  Diagonal form fast multipole singular boundary method applied to the solution of high‐frequency acoustic radiation and scattering , 2017 .

[15]  Ji Lin,et al.  Method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems , 2017 .

[16]  Zhuo-Jia Fu,et al.  Precorrected-FFT Accelerated Singular Boundary Method for Large-Scale Three-Dimensional Potential Problems , 2017 .

[18]  Yan Gu,et al.  Error bounds of singular boundary method for potential problems , 2017 .

[19]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[20]  Wenzhen Qu,et al.  Singular Boundary Method: Three RegularizationApproaches and ExteriorWave Applications , 2014 .

[21]  Wenjing Ye,et al.  Fast BEM solution for coupled 3D electrostatic and linear elastic problems , 2004 .

[22]  B. Teng,et al.  A precorrected-FFT higher-order boundary element method for wave–body problems , 2012 .

[23]  Wen Chen,et al.  Numerical Investigation on Convergence Rate of Singular Boundary Method , 2016 .

[24]  Ji Lin,et al.  Simulation of Seismic Wave Scattering by Embedded Cavities in an Elastic Half-Plane Using the Novel Singular Boundary Method , 2018, Advances in Applied Mathematics and Mechanics.

[25]  Zhuo-Jia Fu,et al.  Explicit empirical formula evaluating original intensity factors of singular boundary method for potential and Helmholtz problems , 2016 .

[26]  Xiao-Wei Gao,et al.  The development of the pFFT accelerated BEM for 3-D acoustic scattering problems based on the Burton and Miller's integral formulation , 2013 .

[27]  Yan Gu,et al.  Fast multipole accelerated singular boundary method for the 3D Helmholtz equation in low frequency regime , 2015, Comput. Math. Appl..

[28]  Yan Gu,et al.  Burton–Miller-type singular boundary method for acoustic radiation and scattering , 2014 .

[29]  Wen Chen,et al.  A modified singular boundary method for three-dimensional high frequency acoustic wave problems , 2018 .

[30]  Guofei Pang,et al.  Singular boundary method for acoustic eigenanalysis , 2016, Comput. Math. Appl..

[31]  C. W. Lee,et al.  Singular meshless method using double layer potentials for exterior acoustics. , 2006, The Journal of the Acoustical Society of America.

[32]  Chein-Shan Liu,et al.  Numerical simulation of 3D nonlinear Schrödinger equations by using the localized method of approximate particular solutions , 2017 .

[33]  Chein-Shan Liu,et al.  Optimal sources in the MFS by minimizing a new merit function: Energy gap functional , 2018, Appl. Math. Lett..

[35]  An Improved Formulation of Singular Boundary Method , 2012 .

[36]  N. Nishimura,et al.  Large-scale modeling of carbon-nanotube composites by a fast multipole boundary element method , 2005 .

[37]  Jacob K. White,et al.  A precorrected-FFT method for electrostatic analysis of complicated 3-D structures , 1997, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..