Localized singular boundary method for solving Laplace and Helmholtz equations in arbitrary 2D domains

Wang, Fajie; Chen, Zengtao; Li, Po-Wei; Fan, Chia‐Ming

doi:10.1016/j.enganabound.2021.04.020

Cited by 18 publications

(11 citation statements)

References 45 publications

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“…Xi and Fu et al [37] presented a localized collocation Trefftz method for heat conduction analysis in two kinds of heterogeneous materials (functionally graded materials and multi-medium materials) under temperature loading. Wang et al proposed a localized SBM and a localized Chebyshev collocation method [38,39]. Fan and Chen et al [40] proposed the localized MFS (LMFS) for solving boundary value problems governed by Laplace equation and biharmonic equation.…”

Section: Introductionmentioning

confidence: 99%

Localized Method of Fundamental Solutions for Acoustic Analysis Inside a Car Cavity with Sound-Absorbing Material

null¹,

Wang²

2023

AAMM

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This paper documents the first attempt to apply a localized method of fundamental solutions (LMFS) to the acoustic analysis of car cavity containing soundabsorbing materials. The LMFS is a recently developed meshless approach with the merits of being mathematically simple, numerically accurate, and requiring less computer time and storage. Compared with the traditional method of fundamental solutions (MFS) with a full interpolation matrix, the LMFS can obtain a sparse banded linear algebraic system, and can circumvent the perplexing issue of fictitious boundary encountered in the MFS for complex solution domains. In the LMFS, only circular or spherical fictitious boundary is involved. Based on these advantages, the method can be regarded as a competitive alternative to the standard method, especially for high-dimensional and large-scale problems. Three benchmark numerical examples are provided to verify the effectiveness and performance of the present method for the solution of car cavity acoustic problems with impedance conditions.

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Section: Introductionmentioning

confidence: 99%

Localized Method of Fundamental Solutions for Acoustic Analysis Inside a Car Cavity with Sound-Absorbing Material

null¹,

Wang²

2023

AAMM

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“…For the above reasons, a new class of meshless collocation techniques, called the localized semianalytical meshless methods, has been proposed to solve various mathematical and mechanical problems. Such methods mainly include the localized method of fundamental solution (LMFS) [30], the localized singular boundary method (LSBM) [31], the localized Trefftz method (LTM) [32], and the localized boundary knot method (LBKM) [33]. The LMFS and the LSBM use the singular fundamental solutions as the basis functions, the LTM employs the T-complete functions, and the LBKM uses the non-singular general solutions.…”

Section: Introductionmentioning

confidence: 99%

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

Zhang¹,

Wang²,

Chen³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method (LKM) with the dual reciprocity method (DRM). Firstly, the temporal derivative is discretized by a finite difference scheme, and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation. Secondly, the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution. And then, the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation, respectively. The LKM is a recently proposed local radial basis function collocation method with the merits of being simple, accurate, and free of mesh and integration. Compared with the traditional domain-type and boundary-type schemes, the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains. Numerical experiments, including two-and three-dimensional heat transfer models, demonstrated the effectiveness and accuracy of the new methodology.

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“…In recent years, the local semi-analytical meshless collocation methods, such as the localized method of fundamental solutions (LMFS), 24,25 localized singular boundary method (LSBM) 26 and LKM, have attracted more and more attention due to their high-accuracy and meshless. This kind of method adopts semi-analytical basis functions satisfied governing equations to approximate the solution of partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

Local knot method for solving inverse Cauchy problems of Helmholtz equations on complicated two‐ and three‐dimensional domains

Wang

Chen

Gong

2022

Numerical Meth Engineering

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This article presents a local knot method (LKM) to solve inverse Cauchy problems of Helmholtz equations in arbitrary 2D and 3D domains. The Moore-Penrose pseudoinverse using the truncated singular value decomposition is employed in the local approximation of supporting domain instead of the moving least squares method. The developed approach is a semi-analytical and local radial basis function collocation method using the non-singular general solution as the basis function. Like the traditional boundary knot method, the LKM is simple, accurate and easy-to-program in solving inverse Cauchy problems associated with Helmholtz equations. Unlike the boundary knot method, the new scheme can directly reconstruct the unknowns both inside the physical domain and along its boundary by solving a sparse linear system, and can achieve a satisfactory solution. Numerical experiments, involving the complicated geometry and the high noise level, confirm the accuracy and reliability of the proposed method for solving inverse Cauchy problems of Helmholtz equations.

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Localized singular boundary method for solving Laplace and Helmholtz equations in arbitrary 2D domains

Cited by 18 publications

References 45 publications

Localized Method of Fundamental Solutions for Acoustic Analysis Inside a Car Cavity with Sound-Absorbing Material

Localized Method of Fundamental Solutions for Acoustic Analysis Inside a Car Cavity with Sound-Absorbing Material

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

Local knot method for solving inverse Cauchy problems of Helmholtz equations on complicated two‐ and three‐dimensional domains

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