Element-Free Galerkin solutions for Helmholtz problems: fomulation and numerical assessment of the pollution effect

Bouillard, Ph.; Suleaub, S.

doi:10.1016/s0045-7825(97)00350-2

Cited by 124 publications

(51 citation statements)

References 13 publications

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“…The robustness of the numerical scheme plays a more important role whilst dealing with high frequencies since the numerical pollution and dispersion affect the accuracy in a significant way [1]. Meshless methods have received attention in the last a few decades due to their several advantages over usual boundary element methods hence some approaches, using Galerkin method [2] and hybrid boundary node method [3], are present for solving the Helmholtz equation. In this report, we present a truly meshless Radial Basis Integral Equation Method (RBIEM) in order to solve the governing Helmholtz equation in a 2-D setting.…”

Section: Abstract: Meshless Method 2d Helmholtz Equation Circular Smentioning

confidence: 99%

The radial basis integral equation method for 2D Helmholtz problems

Doğan¹,

Popov²

2011

Boundary Elements and Other Mesh Reduction Methods XXXIII

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A meshless method for the solution of 2D Helmholtz equation has been developed by using the Boundary Integral Equation (BIE) combined with Radial Basis Function (RBF) interpolations. BIE is applied by using the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source point always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green's identities and the remaining equations are the derivatives of the first equation in respect to space coordinates. RBF interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing in this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs are used, namely 1 ) ln( ) ( 2 1

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Section: Abstract: Meshless Method 2d Helmholtz Equation Circular Smentioning

confidence: 99%

The radial basis integral equation method for 2D Helmholtz problems

Doğan¹,

Popov²

2011

Boundary Elements and Other Mesh Reduction Methods XXXIII

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“…In particular, meshfree methods are widely applied to solve acoustics problems, because field points used in this method are arbitrarily distributed and the approximation smoothness order is chosen flexibility. The method of fundamental solutions (MFS) [6], the multiple-scale reproducing kernel particle method (RKPM) [7], the element-free Galerkin method (EFGM) [8], and other meshfree methods [9][10][11][12] are used to address certain acoustic problems.…”

Section: Introductionmentioning

confidence: 99%

Modeling Sound Propagation Using the Corrective Smoothed Particle Method with an Acoustic Boundary Treatment Technique

Zhang

2017

Preprint

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Abstract:The development of computational acoustics allows simulation of sound generation and propagation in complex environment. In particular, meshfree methods are widely used to solve acoustics problems through arbitrarily distributed field points and approximation smoothness flexibility. As a Lagrangian meshfree method, smoothed particle hydrodynamics (SPH) method reduce the difficulty in solving problems with deformable boundaries, complex topologies, or multiphase medium. The traditional SPH method has been applied in acoustic simulation. This study presents the corrective smoothed particle method (CSPM), which is a combination of SPH kernel estimate and Taylor series expansion. The CSPM is introduced as a Lagrangian approach to improve accuracy in solving acoustic wave equations in the time domain. Moreover, a boundary treatment technique based on the hybrid meshfree and finite difference time domain (FDTD) method is proposed to represent different acoustic boundaries with particles. To model sound propagation in pipes with different boundaries, soft, rigid, and absorbing boundary conditions are built with this technique. Numerical results show that the CSPM algorithm is consistent and demonstrates convergence with exact solutions. Main computational parameters are discussed, and different boundary conditions are validated to be effective for benchmark problems in computational acoustics.

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“…Meshfree methods has been used to solve the acoustic wave equation in quiescent media in the time domain or the Helmholtz equation in the frequency domain, as in the multiplescale reproducing kernel particle method (RKPM) [29], the element-free Galerkin method (EFGM) [30], the method of fundamental solutions (MFS) [31], etc… [32,33]. When using a meshfree method, the Lagrangian approach proposed in this paper not only has the advantages of the Lagrangian property mentioned before, but also has the benefits of the meshfree method itself, such as avoiding mesh generation.…”

Section: Introductionmentioning

confidence: 99%

A Lagrangian Approach for Computational Acoustics with Meshfree Method

Zhang

Smith

Zhang

et al. 2017

Preprint

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Abstract:Although Eulerian approaches are standard in computational acoustics, they are less effective for certain classes of problems like bubble acoustics and combustion noise. A different approach for solving acoustic problems is to compute with individual particles following particle motion. In this paper, a Lagrangian approach to model sound propagation in moving fluid is presented and implemented numerically, using three meshfree methods to solve the Lagrangian acoustic perturbation equations (LAPE) in the time domain. The LAPE split the fluid dynamic equations into a set of hydrodynamic equations for the motion of fluid particles and perturbation equations for the acoustic quantities corresponding to each fluid particle. Then, three meshfree methods, the smoothed particle hydrodynamics (SPH) method, the corrective smoothed particle (CSP) method, and the generalized finite difference (GFD) method, are introduced to solve the LAPE and the linearized LAPE (LLAPE). The SPH and CSP methods are widely used meshfree methods, while the GFD method based on the Taylor series expansion can be easily extended to higher orders. Applications to modeling sound propagation in steady or unsteady fluids in motion are outlined, treating a number of different cases in one and two space dimensions. A comparison of the LAPE and the LLAPE using the three meshfree methods is also presented. The Lagrangian approach shows good agreement with exact solutions. The comparison indicates that the CSP and GFD method exhibit convergence in cases with different background flow. The GFD method is more accurate, while the CSP method can handle higher Courant numbers.

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Element-Free Galerkin solutions for Helmholtz problems: fomulation and numerical assessment of the pollution effect

Cited by 124 publications

References 13 publications

The radial basis integral equation method for 2D Helmholtz problems

The radial basis integral equation method for 2D Helmholtz problems

Modeling Sound Propagation Using the Corrective Smoothed Particle Method with an Acoustic Boundary Treatment Technique

A Lagrangian Approach for Computational Acoustics with Meshfree Method

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