Solution of the 3D-Helmholtz equation in exterior domains using spherical harmonic decomposition

“…Gerdes (1998) studied the convergence and performance of finite-infinite element which was applied to arbitrary geometries to obtain the discretization method of Helmholtz equation in an outer region of arbitrary shape. Garate et al (2012) developed the finite element formula with an effective semi-analytical method for the direct calculation of a global finite element matrix which is obtained by using separate variables to solve the exterior domains Helmholtz equation. At the same time, there has been some developments on using the boundary element method to solve the 3D Helmholtz equation (Sun et al, 2015(Sun et al, , 2017(Sun et al, , 2020aKlaseboer et al, 2017Klaseboer et al, , 2019a, which mainly focused on the method of boundary regularized integral equation formulation.…”

Section: Introductionmentioning

confidence: 99%

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

Yang

2021

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PurposeThis meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.Design/methodology/approachIn this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.FindingsThe numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.Research limitations/implicationsCompared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.Practical implicationsThe Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.Originality/valueThis meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

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“…Then the expansion coefficients for purely outgoing waves can be obtained by solving a linear system resulted by matching boundary data on the scatterers. The proposed algorithm actually provides an accurate way to compute the far-field scattering waves from local element based approximations [12,31]. Besides, they can be combined with some appropriate solvers [2] to tackle practical 3D problems involving scatterers with complex shapes.…”

Section: Introductionmentioning

confidence: 99%

Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids

Wang

Xie

2017

Adv Comput Math

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We present in this paper a spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map.2000 Mathematics Subject Classification. 65Z05, 74J20, 78A40, 33E10, 35J05, 65M70, 65N35.

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Solution of the 3D-Helmholtz equation in exterior domains using spherical harmonic decomposition

Cited by 3 publications

References 25 publications

ADITU: A mesh-free formulation for the solution of Helmholtz equation in bounded and unbounded domains

ADITU: A mesh-free formulation for the solution of Helmholtz equation in bounded and unbounded domains

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids

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