A regularized approach evaluating the near-boundary and boundary solutions for three-dimensional Helmholtz equation with wideband wavenumbers

Abstract Efficient evaluation of near-boundary and boundary solutions for the Helmholtz equation with wideband wavenumbers by the boundary collocation method has been a difficult task for a long time. This study provides a regularized approach to bypass this limitation. The singular boundary method avoids the near singularity by using the nearly singular factors to replace the corresponding nearly singular terms. The core idea of the regularized approach is to substitute an artificially constructed general solution of the Helmholtz equation into the boundary integral equation or hyper boundary integral equation to determine the nearly singular factors. The core difficulty is the construction of the appropriate general solutions. The proposed regularized approach is free of integrations, easy-to-use and independent with particular wavenumbers. Numerical experiments show that accuracy of the near-boundary and boundary solutions of the singular boundary method improved by several orders of magnitude through application of the proposed regularized approach.

[1]  Weiwei Li,et al.  A fast singular boundary method for 3D Helmholtz equation , 2019, Comput. Math. Appl..

[2]  Qing Hua Qin,et al.  Application of hybrid-Trefftz element approach to transient heat conduction analysis , 1996 .

[3]  Jun Lu,et al.  A novel meshless method for fully nonlinear advection-diffusion-reaction problems to model transfer in anisotropic media , 2018, Appl. Math. Comput..

[4]  Wen Chen,et al.  A modified dual-level fast multipole boundary element method for large-scale three-dimensional potential problems , 2018, Comput. Phys. Commun..

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

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

[7]  Wen Chen,et al.  A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations , 2018, Comput. Math. Appl..

[8]  Wen Chen,et al.  A modified dual-level fast multipole boundary element method based on the Burton–Miller formulation for large-scale three-dimensional sound field analysis , 2018, Computer Methods in Applied Mechanics and Engineering.

[9]  Qing Hua Qin,et al.  A modified multilevel algorithm for large-scale scientific and engineering computing , 2019, Comput. Math. Appl..

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

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

[12]  Junpu Li,et al.  A Dual-Level Method of Fundamental Solutions in Conjunction with Kernel-Independent Fast Multipole Method for Large-Scale Isotropic Heat Conduction Problems , 2019, Advances in Applied Mathematics and Mechanics.

[13]  Q. Qin,et al.  A meshless method for generalized linear or nonlinear Poisson-type problems , 2006 .

[14]  Zhuojia Fu,et al.  A dual-level method of fundamental solutions for three-dimensional exterior high frequency acoustic problems , 2018, Applied Mathematical Modelling.

[15]  Xiaolin Li,et al.  The element-free Galerkin method for the nonlinear p-Laplacian equation , 2018, Comput. Math. Appl..

[16]  Wen Chen,et al.  Numerical investigation on the obliquely incident water wave passing through the submerged breakwater by singular boundary method , 2016, Comput. Math. Appl..

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

[18]  Qingsong Hua,et al.  Boundary function method for inverse geometry problem in two-dimensional anisotropic heat conduction equation , 2018, Appl. Math. Lett..

[19]  Wen Chen,et al.  Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method , 2017, Comput. Math. Appl..

[20]  Xing Wei,et al.  A frequency domain formulation of the singular boundary method for dynamic analysis of thin elastic plate , 2019, Engineering Analysis with Boundary Elements.